First Course in Bulk Queues

o our parents

Copyright 0 1983 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United Statescopyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, lnc. Library o f Congress Cataloging in Publiention Data

Chaudhry, M. L. A first course in b u k queues.

"A Wiley-Interscience publication." Bibliography: p. Includes index. 1. Queuing theory. l . Templeton, James 6 .C. 11. Title. T57.9.C46 1983 519.8'2 82-21764 ISBN 0-471-86260-6 Printed in the United States of America

The theory of queues had its origin early in the present century with the work Markov and Erlang on stochastic systems. Markov chains and processes ren among the principal analytical tools of the theory of queues, while the teleph systems studied by Erlang constitute one of the principal ,areas of applicatior queuing models. Queuing theory and its applications have expanded greatly since the days of pioneers. The theory has attracted the interest of some very capable app mathematicians, some of whom (Neuts, Pollaczek, and Takics, for example) known mainly for their work on queues and closely related topics, while o t (such as Kendall, Khintchine, and Lindley) have done important work on que but are probably more widely known for work in other fields. Published applications of queuing models have expanded beyond telephone include ambulance dispatching, aircraft and automotive traffic, computers, mili operations of various kinds, medical appointment scheduling, machine rep inventory studies, water reservoirs, processing of steel ingots, and many ot. Other applications, such as the fairly recent replacement of parallel single-se queues of bank customers by a single multiserver queue, require no new theory are therefore unlikely to be described in research journals. The introduction public acceptance of the single multiserver queue, in banks and elsewhere, see] depend on a fairly widespread appreciation of some of the simpler facts a1 queues, and on a much more widespread demand for "equal treatment for all." Queuing theory has sometimes been dismissed by mathematicians as trivial by engineers as inapplicable t o practical problems. Nevertheless, some of the n ematics done in queuing theory has been and continues t o be of very good quz most of the mathematics is competent, and many applications of old and theories are made. In this book we discuss bulk queuing models in which customers arrive in gr or are served in groups. Both the theory of bulk queues and its application t o tical problems can be said t o have begun in 1954 with a paper in which I\; Bailey obtained a steady-state distribution of queue size for a particular bulk-se queue, and applied the results t o a practical problem in scheduling medical app ments. Since Bailey's pioneer work, hundreds of papers have been publishe bulk queues. Some of these papers are of substantial practical value or mathemi interest or both: some are not. Many techniques have been developed or exte t o handle the additional analytical complexities which result from the introdu of bulk arrival, bulk service, or both. Bulk queues exist in the real world,

viii

PREFACE

have an extensive mathematical theory, and they often cannot be satisfactorily approximated by simpler queues. It has been difficult to learn more about bulk queues than is available in the short sections devoted t o this topic in most introductory textbooks in queuing theory. A student or practitioner, seeking either a general knowledge of bulk queues or a model appropriate to a particular problem, has been faced with a search through a large and scattered literature, consisting of research papers together with a few sets of lecture notes on special topics. This literature lacks consistency of notation, sometimes uses methods not well adapted t o the problem at hand, has some surprising gaps, and contains a substantial number of uncorrected errors. We therefore conclude that the theory of bulk queues is a large and useful body of knowledge which needs and deserves the extended unified account which we have tried t o provide here. This book is neither a formal treatise for the mathematician nor a collection of case histories for the practitioner. We try to follow a middle path by discussing a variety of models and techniques, thus giving the reader the means of constructing other models best suited t o a particular problem. In so doing, we have filled in some gaps in the theory and have corrected some errors in the literature. Results are not always presented in the most general known form, but in some places we have generalized the published work on which we have drawn. We do not give a systematic discussion of the art of modeling or of specific applications, but have chosen models and techniques of solution with a view t o application. We have tried to make the work accessible by keeping the mathematical prerequisites t o a reasonable minimum. We expect that most readers will have made a study of the theory of queues as set forth in one of the well-known introductory textbooks, such as those by Conolly, Cooper, Giffin, Cross and Harris, Kleinrock, Kosten, and Page. Study of these books implies some knowledge of elementary probability theory and linear algebra, Markov chains, differential equations, Laplace transforms, and some rudiments of complex analysis, in particular RouchC's theorem. Except in Section 5.3 and Chapter 6, the reader who is prepared t o have faith in the stated properties of complex power series will require very little knowledge of complex analysis. Since we could not give complete coverage to the existing literature in the field, we have tried to give fairly detailed coverage t o the methods and models we thought most important and useful, and t o provide a fairly full list of references to other material not covered. While we have tried t o give proper credit for all the material we have presented or adapted from the work of other authors, we apologize t o any authors whose work we may have slighted inadvertently, either by omission or by inadequate citation. We hope that readers will notify use of any remaining errors or omissions that they may find. At our chosen mathematical level, we were unable to include important work by Bhat, Borovkov,Dagsvik, DeSmit, Keilson,Kingrnan, Loris-Teghem, Neuts, Pollaczek, Takics, and other authors. We have also excluded work that seemed to be of purely mathematical interest, some work included in existing monographs, and most work on time-dependent solutions. In Appendix B we give references to some publications on queues which combine bulk arrival and

PREFACE

bulk service, but have not given a detailed discussion of this work because of complexity. One of the most important features of the book is the provision of a large c lection of problems and complements. Few of these are routine drill problems, L: many are fairly easy with the aid of the text. Some problems invite the reader obtain alternative derivations for results already derived by other methods in t text. Some of the problems represent substantial extensions t o the content of t book, and are likely t o be found challenging by most readers. This is particula true of the Miscellaneous Problems and Complements. Almost every problc includes a statement of the required answer, and many problem statemel include hints or outline solutions to be completed by the reader. Chapters 1 and 2 are introductory, and do not discuss bulk queues in any det: Chapter 1 gives results in analysis and probability which are Bsed in later chapte Chapter 2 is a self-contained introduction to (nonbulk) queuing theory, in particu t o the M/G/1 and GIIM/l queues, with emphasis on the techniques that may used t o investigate them. This chapter could be used as a textbook for a sha introductory course on queues, but its main purpose is t o introduce methods wh are used repeatedly in later chapters. We expect that most readers will have se some of the material in Chapters 1 and 2, and many readers will have seen most it. Even well-prepared readers, however, may find it useful t o have this mate] readily available, in a notation consistent with that used in the rest of the bol when studying later chapters. In Chapters 3, 4, 5, and 6 we discuss bulk queues: single-server bulk-arri queues in Chapter 3 , single-server bulk-service queues in Chapter 4, and mu channel queues with bulk arrival or bulk service in Chapter 5 . Chapters 3 , 4 , an1 may be read in any order. Chapter 6 discusses relations among different queu systems, and expected values for busy periods and idle periods. Some import: mathematical results have been collected in Appendix A, and a guide t o the notat of the book is given in Appendix C. References t o publications cited in the tl are listed at the end of each chapter. There is also a list of additional publicatil cited only in the Overview section at the end of each of Chapters 3 to 6. This book is addressed to several (overlapping) classes of readers. Students us the book as a text should be well versed in advanced calculus and element probability, at a level to be found in a good undergraduate program in engineeri computer science, or statistics, preferably supplemented by an introduction queues. Chapter 2 could be used by itself as a text or reference for an unorthoc short introductory course on queues at the senior undergraduate or first-y graduate level. Chapters 3 t o 6 could be used, together with the first two chap. or separately, as a graduate textbook on queues, emphasizing techniques. Fc student who has studied an introductory book on queues such as Kleinrock Gross and Harris, Chapters 1 and 2 could be used for reference and Chapters 3 t would be almost entirely new material. For practitioners who construct queL models of real systems in the course of their work (management consulta members of industrial operational research groups, and so on), we offer a 1; collection of queuing models, most of which are amenable to computation. We .

x

PREFACE

offer some guidance in constructing other models. For professionals who read the journal literature on queues, we offer a guide t o that literature and a unified discussion of a large and significant part of it. Finally, we wish our readers a pleasant journey through the collection of queuing models presented here, and success in applying or improving them. M.L. CHAUDHRY J.G.C. TEMPLETON Kingston, Canada Toronto, Canada March 1983

We take great pleasure in acknowledging the generous support provided by many our professional colleagues who contributed t o this book in one way or anoth Our special thanks go t o Professor J. Gani, University of Kentucky, who read pa of the first draft and encouraged us t o approach a publisher. Others who wc contacted during the process of writing this book, or who r
Chapter 1 1.I 1.2 1.3 1.4 1.5 1.6

Lattice Random Variables Addition of Random Variables: Convolutions Other Useful Distributions Laplace Transform On Certain Functions Used in Queuing Theory Partial Differential Equations Problems and Complements References

Chapter 2 2.1 2.2 2.3 2.4

Some Techniques of Queuing Theory

Basic Material in Queues Techniques and History of Analysis Basic Renewal Theory Overview Problems and Complements Miscellaneous Problems and Complements References

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Preliminaries

Bulk-Anival Queues

The System M ~ / E ~ / I The System G I r / M / l The System M r / M / l The System MX/G,/1 The System E?/E,/I with Generalized Erlang Input The system M ~ ZM?, /GI, G 2 / 1 Overview Problems and Complements Miscellaneous Problems and Complements References

xii

xiv

CONTENTS

Chapter 4 4.1 4.2

4.3 4.4 4.5 4.6

Bulk-Service Queues

The System M/GB/1 The System M / G B / 1 / ~with Customers Served in Batches of Variable Capacity The System M / G ~ / Iwith Customers Served in Batches of Exactly Size k The System G E O M / G ~ / I The System M/MB/l/MB Overview Problems and Complements Miscellaneous Problems and Complements References

CONTENTS

Generalized Argument Principle Rouch6's Theorem Lagrange's Theorem The Riemann-Stieltjes Integral and the Laplace-Stieltjes Transform Some Theorems on Laplace-Stieltjes Transform and Generating Functions Linear Difference Equations with Constant andvariable Coefficients Inversion of a Discrete Transform References Bulk-Arrival, Bulk-Service Queues References Glossary of Symbols AUTHOR INDEX

Chapter 5

Multichannel Bulk Queues SUBJECT INDEX

5.1 5.2 5.3 5.4 5.5 5.6 5.7

The System M x ( t ) / ~ / - with the Input Rate A ( t ) The system MX/M/c The System M X / ~ / c The System DS/M/c The System M/MB/c The System GI/M~/c Overview Problems and Complements References

Chapter 6

Relations Among Queuing Systems

The System GIr/E/ 1 The System GI/E,/l Relationship between GI/E,/l and GIr/E/l The System E / G ~ / ~ The System Ek/G/l Relationship between Ek/G/l and E / G ~ / I Expected Busy and Idle Periods Overview Problems and Complements Miscellaneous Problems and Complements References Appendices A.l A.2 A.3

Differentiation of a Definite Integral with Respect to a Parameter Fundamental Lemma of Branching Process Theory Transform Inversion and Residue Calculus

Queues, or waiting lines, in which customers arrive, wait for service, are serl and then leave the system are a familiar feature of daily life. While the prese of waiting lines may be predictable, their detailed behavior usually is not. In ( structing mathematical models applicable to real queuing syskems, therefore, we stochastic models characterized by random variation over time. Mathematically, queuing theory is the study of certain stochastic processes tc discussed later. The study of these processes deals with the probabilistic deve ment of the states of the process with changes in the value of the parameter, usu time. Thus to study queuing theory, we need to know something about the thc of stochastic processes as a part of probability theory. It will be assumed, for purpose of this section, that the reader has a preliminary knowledge of probab theory. In particular, at least familiarity with the following is assumed:

1 2

3 4

Random variable (r.v.), discrete, continuous, or mixed. The cumulative distribution function (c.d.f.), the probability mass func (p.m.f.1, or the probability density function (p.d.f.)of an r.v. X. Where t seems to be little danger of confusion, we may refer to these function: the single word 'distribution'. Moments of X, and moments of functions of X. The probability laws dealing with the calculation of probabilities of un of events, intersections of events, and so on.

Though some knowledge of the above is assumed, it is still felt that some sl should be devoted to rudimentary ideas concerning the various tools. This will help us in establishing the notation used for the rest of the book. The first t with which we start is the probability generating function (p.g.f.). Smce one ol problems in queuing theory is to solve a set of difference equat~onswhich ; from the consideration of the limiting behavior of the queuing system, this ele and powerful technique is often helpful in expressing solutions to these problen closed forms. Means, variances, and higher-order moments can easily be compi from these functions. We do not claim that we are giving a full account of t functions in the next section. For a fuller treatment of thls, we refer the readt an introductory textbook on probability theory, such as Blake (1979), or a sc what more advanced treatise (without using measure theory), such as Feller (19 The discussion of probability theory from a measure theoretic point of view ma

PRELIMINARIES

2

found in the books by Loeve (1977,1978), Neveu (1965), and Kingman and Taylor (1966), among others. It is also assumed that the reader has some basic knowledge of the theory of complex variables. However, some of the theorems of complex variables that are needed in certain sections are given in Appendix A. For a more complete treatment of these and some other related topics, the reader is referred to books mentioned there.

Among the discrete r.v.'s those assuming only values of the form nd, where n is a nonnegative integer and d is a real constant >0, play an important role. In other words, if X is such an r.v., then P(X = x ) = 0 if x # nd, d > 0, and n = 0, 1 , 2 , . . . . The set of integer multiples of a real constant is called a lattice (in a geometric sense, not to be confused with the algebraic system called a lattice, nor with the statistical design called a lattice square). R.v.'s whose values are confined (with probability 1) to such a lattice are called lattice r.v.'s. In view of this, binomial, Poisson, hypergeometric, and other nonnegative integer-valued r.v.'s may be considered to fall under the broader class of lattice r.v.'s. R.v.'s that do not satisfy the above property may be called nonlattice and the corresponding distributions, nonlattice distribution~.Examples are absolutely continuous r.v.'s and mixed r.v.'s.

1.1

LATTICE RANDOM VARIABLES

Thus P(z) is absolutely convergent at least for Iz I < 1. Polya (1954) compares a p.g.f. to a bag. Instead of carrying several little separately, it may be convenient to carry a single bag which contains all th pieces. Quite similarly, instead of handling each term of the probability sec px(0), px(l), px(2), . . . , px(n), . . . individually, we put them all in : C= ;, px(n)zn = P(z), and then we have only one mathematical piece to f the p.g.f. It should, however, be made clear that taking the pieces out of the bag m be as difficult a problem as getting the probabilities out of the p.g.f. In simpl, one can invert the p.g.f. and get the explicit probabilities. However, when tht becomes more complicated, the problem does not rem& that simple, as below. The p.g.f. Px(z) = Z px(n)zn is a special case of the z-transform discus Jury (1964) and other authors, and used in the solution of difference equ In the z-transform Z a n z n or E anz-n the coefficients {a,) are not nece: probabilities. Among the useful quantities obtained from p.g.f.'s, and from other I functions introduced later, are the mean and the variance of the probability bution {px(n)}. They are easy to derive and are recorded here in terms of thc of the p.g.f. of X and its derivatives, evaluated at z = 1. The mean is

ability Generating Function

Let X be a nonnegative lattice r.v. assuming integral values 0,1,2,. . . ,such that the probability that X equals n is, say, P(X = n) r px(n). The sequence of probabilities {px(n)),"=, or {px(n), n > 0 ) (henceforth called the probability distribution of X) is said to be proper if px(n) >, 0 and E;,, px(n) = 1. In this book we shall be concerned only with proper distributions, and hence the adjective proper will not be repeated. Thus if {px(n)) is the distribution of X, then

where

m

E(X(X- I)) = n

C=

n(n

- l)px(n)

=~ ( ~ ) ( 1 ) ,

0

the variance o; is given by is called the p.g.f. of the sequence jpx(n)},"=, or more frequently, the p.g.f. of the r.v. X. Since P&) is an analytic function of the complex variable z , we may use the theory of analytic functions to obtain results concerning Px(z) and hence {px(n)). The subscript X of the function Px(z) will not normally be used as long as there is no confusion regarding the r.v. involved. Also px(n) is very often written as p(n), p,, or Pn. We shall be using all these notations in the subsequent chapters when the context of the r.v.'s involved is clear. As P(1) = 1 , it is easy to show that

Higher order moments about the origin, E(Xr), or about the mean, E((X - I;

r = 1,2, 3, . . . , may be obtained similarly.

It is a simple matter to consider the p.g.f.'s of some of the standard discrc tributions such as Poisson, geometric, and binomial. However, problems in q theory frequently involve finding the p.g.f.'s of sequences such as (px(n {px(n + 2)), {npx(n)). For a short table of z-transforms we refer the rea Beightler et al. (1961), and for further details to Jury (1964) or Cadzow (197

1.2

1.2

ADDITION OF RANDOM VARIABLES: CO

In queuing theory we are frequently concerned with sums of r.v.'s. In particular, we often have to consider the distribution of an r.v. which is the sum of N independently and identically distributed r.v.'s (henceforth frequently abbreviated as i.i.d. r.v.'s). When the r.v.'s in the sum are i.i.d., much simpler formulas and techniques are available than in the general non4.i.d. case. In this section we shall consider separately two cases of sums of r.v.'s: ( 1 ) the sum of discrete i.i.d. r.v.'s; and ( 2 ) the sum of continuous i.i.d. r.v.'s.

ADDITION OF RANDOM VARIABLES: CONVOLUTIONS

{ p w j ( l ) ) = (PX,(OI*( P X , Y ) J * - * { p x j ( z ) ) (J In practice there are many problems in which the r.v.'s involved are not independent but identically distributed. Let now the r.v.'s Xi be i.i.d. wit1 common distribution {px(n)] and the p.g.f. px(z). Then equation (1.2.3) c: replaced by the much simpler formula In the i.i.d. case we write the convolution of X , , X 2 , . . . ,Xi as

Case 1. Sum of Discrete i.i.d. r.v.'s. Consider the partial sum f---

j factors

>

in case there is no confusion between the r.v.'s involved, wit& where the X's are nonnegative, independently distributed discrete r.v.'s and N is an r.v. independent of the Xi's. We discuss first the particular case of eq. (1.2.1) when N is fixed and equal to 2 and X , and X , are integer valued. Thus if px,(n) = P(Xl = n ) , px2 (m) = P(X2 = m), n, m > 0 , then considering various mutually exclusive cases for the event W, = 1 and using the condition of independence, it can easily be shown that

Since by definition the sequence {px(n))O*is { 1,0,0, . . . ), its p.g.f. is Px(z) The notation ~ ( ~ ) (isnused ) for convenience. It may, however, be remarked tl the subscripti in pU)(n) is without the parentheses (1, the meaning is different If N instead of being a constant is an r.v. independent of Xi, then W, is the of a random number of r.v.'s. If P(Xi = n ) = px(n), for every i,P(N=j ) = p, P(WN = I ) = pwN(Z), and the corresponding generating functions are Px(z),PI and PWN(z),it follows that

When X I and X , are not independent, then

where P ~ , , ~ , ( Xx,), , is the joint probability mass function defined by p x l , ~ (,n , m ) = P(X1 = n , X2 = m). The sequence { p w 2 ( l ) j ~ =iso called the convolution of the sequences { p x l ( l ) ) , { p x 2 ( l ) j and is denoted by { P X , ( ~ ) ){ *P X , ( ~ )or) b x 2 ( l ) } *f P x , ( l ) } . In terms of the p.g.f.'s, this can be written as

The distribution ] p w N ( l ) ) whose p.g.f. is given by equation (1.2.4) is callec compound distriljution. If j = 0, the value of W N , by convention, is taken t zero. The concept of univariate p.g.f. can easily be extended to bivariate and n variate p.g.f.3. In the case of two discrete nonnegative r.v.'s XI and X,, we havc bivariate p.g.f. px, ,x,(Y, Z ) = E

(

Y

~

~

~

~

)

if X, and X 2 are independent. This is easy to prove by considering E ( Z ~ I + ~ ' ) . If N = j > 1, the whole concept can be generalized, and we have in the case of independent r.v.'s i

pwi(z) = The sequence in this case is denoted by

n

m = l pxm

(z).

where p ~ , , ~ ~m() nis ,a joint probability and stands for P(X1 = n , X 2 = m). It be remarked that sometimes one r.v. is discrete and the other is continuous. : cases are dealt with through the Laplace transform (L.T.) and the p.g.f. ) the marginal p.g.f.'s of the d The p.g.f.3 P x , , x , ( y , 1) and P x , , X 2 ( 1 , ~are

6

PRELIMINARIES

butions { p x , ( n ) ] and { p x 2 ( m ) ) ,respkctively. This concept may be generalized to more than two r.v.'s, but the mathematics involved, in general, will become cumbersome. However, certain results connected with the sum (1.2.1) are easy to derive, whether N is a fixed constant or an r.v. For example, if XI,X2, . . . ,Xn (n fixed) are independently Poisson distributed with parameters A,, . . . ,An, the p.g.f. of W , is Pwn(z) = exp [u(z - I)], where u = Cr=lh i . Since the p.g.f. of a Poisson r.v. X with parameter h is exp[h(z - I)], one can see that the sum of n independent Poisson distributed r.v.3 is itself a Poisson r.v. Case 2. Sum of Continuous i.i.d. r.v.'s. Let N be fixed and equal to 2 , and let XI and X2 be two nonnegative continuous independent r.v.'s. Then the distribution function (d.f.) Fx, + x 2 ( t ) [with p.d.f. f x l + x 2 ( t ) ]of the sum W 2= X , 4- X2 is given by Fw2(t) = P(W2 f t )

1.2

ADDlTlON OF RANDOM VARIABLES: CONVOLUTIONS

where fx, ,x2(x,y) is the joint p.d.f. of the r.v.'s X, and X 2 . Convolutio~lsof the above type are handled easily with the help of L.T. are discussed in Section 1.4. We have restricted the r.v.'s to nonnegative value limits of integration (or summation) are to be adjusted appropriately when t h ~ take values over the whole real line from --w to w. As in the discrete case, the concept of convolutions in the continuous cas be extended to N r.v.'s, where N may be either a constant or an r.v., as following example, which occurs in queuing theory. Consider the sum (1.2.1). Let first N be fixed and equal to n. For the case N is an r.v., see the Problems and Complements. Let Xi's be i.i.d. exponentia with the common p.d.f. f&)

where the last result has been obtained by using the independence of X, and X 2 . We use the more common definition F x ( x ) = P(X < x ) of the d.f. of an r.v. X , rather than the alternative definition Fx(x) =P(X<x). Since limh+, , Fx(x +h) = limb ,,,P(X<x 12) = P(X f x ) = Fx(x), the d.f. Fx(x) is continuous from the right (but not in general from the left). By conditioning on the value of X,,we could have obtained

+

= pe-'lt,

p>O,

t>O.

(

The p.d.f. (1.2.5) is called the (negative) exponential p.d.f. However, the tive 'negative' will not be repeated since we shall be concerned with only thi: of exponential distribution. Now if we let fwn(t) be the p.d.f. of W,,

It is easy to show by induction that

W , has mean n / p and standard qviation +/p. This p.d.f. is used in various by changing the parameters. For example, if p is replaced by kz,the p.d.f. becl

The corresponding p.d.f. is

and consequently the mean now is llh. The c.d.f. corresponding to the (1.2.6) is We call the p.d.f. f w 2 ( t ) the convolution of fx,(t) and f x z ( t ) and write it as

fw,(0 = fx,( t )* f x , (0= fx,W * fx,@>. If X, and X, are also identically distributed as X, then fw,W = ( f x W 2 " or f(,)(t) if there is no confusion with the r.v.'s involved. Furthermore, if X1 and X2 are not necessarily either independent or identically distributed, then we have

The distribution (1.2.6), (1.2.6a), or (1.2.7) is called the Erlangian distrib (or n-Erlang) and will be denoted by En. In particular, it may be noted that an exponential distribution, one of whose properties is that its mean is equal standard deviation. The letter E in En is the first letter of the name of A. K. Erlang, who wc for the Copenhagen Telephone Company during 1909-1920 and did pione

8

PRELIMINARIES

work in the development of the theory of queues and its applications to telephony. For a study of Erlang's work, one may see Brockrneyer et al. (1948). In queuing theory the subscript n in E n is called the number of phases. The phases may have a physical meaning, or they may be just an artifice to allow us to obtain a wide class of distributions. We have seen that the exponential distribution is a particular case of the Erlangian distribution. Another important distribution, which can be derived from the Erlangian by letting n -+ 00, is known as the constant or deterministic distribution in which P(X = c ) = 1 for some constant c . This is denoted by D and is easy to derive with the help of an L.T. or a characteristic function (c.f.). For the procedure, see Problems and Complements. The Erlangian distribution may be generalized to the gamma distribution with p.d.f.

where oc instead of being a positive integer is a positive real number and r ( a ) is the gamma function defined in Section 1.5. The p.d.f. of the n-Erlang distribution is identical (except for a scale factor) with that of the X2 distribution with 2n degrees of freedom, since

Since the x2 distribution is important in statistics and its c.d.f. is extensively tabulated, it is fairly easy to obtain values of n-Erlang c.d.f.'s. Limiting Results

Two of the most important limiting results connected with the sums of r.v.3 are stated below without proof: 1

2

The strong law of large numbers. If W, = 2Ll Xi, where the Xi's are i.i.d. r.v.'s with the common mean p < =, then

The central limit theorem. If W, = ZinmlXi, where the Xi's are i.i.d. r.v.'s with the common mean p and common variance u2,p < = and 0 < a2 < =, respectively, then dx . n-+-

This is one of the simplest (though it will be sufficient for our needs) of a large number of different but related theorems which have been proved concerning the

11.3

OTHER USEFUL DlSTRlBlUTlONS

convergence of sums of r.v.'s lo a limiting Gaussian r.v. Any of these theorems I in fact, be called a central limit theorem. These theorems are discussed in n books on probability, but one particularly good reference is Cnedenko and mogorov (1954).

Just as the Erlang distribution (1.2.6) is obtained by considering the sum of pendent exponential r.v.3, so we may obtain another useful family of r.v.' considering mixtures of exponential r.v.'s. Consider an r.v. Swith p.d.f.

Then S is said to have a hyperexponential or HE, distribution. The HE, c butions are useful in queuing theory and are complementary to the En distribu of Section 1.2, as discussed in Problem 10(b). The HE, distribution may be generated as follows. Consider a queue u service channel consists of n branches in parallel, working in such a way that customer entering service is independently assigned to one of the branche~ probability of assignment to branch r being or, r = 1 , 2, . . . , n. The service distribution in the rth branch is exponential with rate p,, and a customer con ing service in any b r a n c h M e s the system at once. While the customer occ any branch, the service channel is considered completely occupied and no bra1 available to serve another customer. Then the customer's total service time ,! have an HEn distribution with the p.d.f. given above. If all p, = p , then the distribution reduces to the exponential El . Truncated Distributions

In several important applications in the theories of queues, reliability, life-te and so on, one is interested in the truncated distributions of r.v.'s, truncated ( on the left of a given point or on the right of another given point, or boll illustrate this concept by means of two simple cases, (I) when the r.v. is cc uous, and (2) when it is discrete. The concept, however, may be extended obvious way to other distributions. Case 1. Truncated Exponential r.v. Let X be an exponential r.v. with fx x 2 0 . Then for such an r.v. truncated on the left at X = T, T > 0 , we \ have the p.d.f.

PRELIMINARIES

10

1.4

LAPLACE TRANSFORM

The constant A is determined from ' the normalizing condition J , fx,(x) ~ dx = 1, which gives A = eTP,and fx,(x)

= pe-r(X-T),

x

> T.

Once A is known, one can easily determine the moments of the new r.v. X,. In particular, one may note that E(X,) = 7 + 1/p.

and zero elsewhere. As an application of this, let us consider the case of particles which are er from a radioactive source following the Poisson distribution with parame A counting mechanism recording these emissions functions only if not more two particles arrive. If more than two particles arrive during a specified time val, the counting mechanism gets locked. Thus if X is the number of pal recorded during a specified time interval, then proceeding as above it can eas seen that X has the distribution

It may also be noted that when T + 0, the distribution of X, becomes the same as that of X, as it should. If we are interested in the r.v. X , truncated on the right at X = 8 , B > 0, then we should have

and

For further applications, see Problems and Complements.

Note again that as B + m, X0 -+X (in distribution). Furthermore, if we are interested in an r.v. X, truncated both to the left of X = T, 7 > 0 and also to the right of X = T + 6 , 8 > O , t h e n wewouldhave

1.4 LAPLAGE TRANSFORM

\ 0,

otherwise

with

Note that when B discussed earlier.

-+

-

with T a finite quantity, or when 7

-+

0, we get the two cases

Case 2. Truncated Poisson r.v. Truncated discrete r.v.'s are treated in a manner similar to the one used for truncated continuous r.v.'s. For instance, if a Poissondistributed r.v. X having parameter h is truncated on the right at X = i, then X has the following distribution:

Using the normalizing condition 2; =, Pj= 1, we find C and consequently

.

The Laplace transform (L.T.), in the form of the moment generating functioi long been used in elementary probability theory. In the terminology of Sectioi the L.T. provides a "bag" in which to carry an r.v. in much the same way s p.g.f. provides a "bag" in&ich to carry an integer-valued r.v. As happens also the p.g.f., taking the r.v. out of the bag (inverting the L.T. to obtain the formed distribution) may be difficult. For this reason, the use of L.T.'s in qu theory, and especially the reporting of final results in L.T. form, has been I criticized. In some cases, results using L.T.'s have been superseded by more ex results not requiring inversion. Nevertheless, the L.T. still provides useful i mation in a fairly simple form, and it is constantly used in many parts of qui theory. We will most frequently use the L.T. in the slightly more general form o Lap!ace-Stieltjes transform (L.-S.T.), which is applicable to a wider class of than the simple L.T. Let X (continuous, discrete, or _mixed) be a nonnegative r.v. with the c bution function Fx(x). The L.-S.T. fx(or) of Fx(x) is given by

The integral on the right-hand side of equation (1.4.1) is a Stieltjes integral. ! TX(a) is the L.-S.T. of FX(x), we call Fx(x) the inverse L.-S.T. of &(a). t brief sketch of this integral (transform), see Appendix A.7, but for-more det discussion, see the references quoted therein. A transform such as fx(a) not

PRELIMINARIES

12

affords a substantial simplification, but permits us to discuss the discrete and continuous cases simultaneously. However, it may be said that a deep knowledge of such transforms is not required to study this book. A look at Appendix A.7 would enable the reader not already familiar with such transforms to handle them without much difficulty. It may be mentioned here that when a is complex, say a = g if, g and f real, the integral converges for Re a g > 0. Furthermore, if X is continuous and its p.d.f. fx(x) = dFx(x)/dx exists, we have

+

1.4

LAPLACE TRANSFORM

Characteristic Function

Another important transform, which avoids the convergence problems of the and the L.T., is the characteristic function (c.f.) Let Fx(x) be the distrib~ , o is a real variable defined ovel function of an r.v. X. The c.f. ~ ( a ) where is defined by whole real line and i = f i , w

which is the usual L.T. When X is discrete, the integral needs to be replaced by a summation. Some of the properties of the L.T. that we shall be needing are discussed below. Similar properties for the L.-S.T. can be derived, but we shall not discuss them here. One of the important properties is the convolution property. The convolution of two independent continuous nonnegative r.v.'s XI and X 2 , which we discussed in Section 1.2, isfW2(t)=fXl(t) *fX2(t). It can be shown that the L.T. of fw2(t) is

,yx(w) = ~ ( e ' =~ ~ )eiw'dFx(x). If Fx(x) has a p.d.f.fx(x), the c.f. becomes

When Fx(x) has a p.m.f. {px(k));=,,

(1

X

equation (1.4.2) becomes a series express

Further, if the r.v.'s are identically distributed wtih common p.d.f.fx(x), The moments about the origin may be calculated from

c /.4iw)Tlr! ca

These important results, which may be easily generalized t o the case of n r.v.'s, n > 2, are used repeatedly in later chapters. When the indicated limits exist,

-

lim a&(&) a

+

O+

-

= lim fx(t), t

-+

a -+ 0 (also written as a $ 0 ) means cu -+ 0 through values greater than zero. The moments about the origin of the r.v. X having the p.d.f. fx(x), if they exist, can be obtained directly from its L.T. For example, E(X) = -72'(0);

E(X2) = +@)(o).

Since

higber order moments about the origin are given by p: = E(Xr) = coefficient of (-ay/r! in the power series expansion offx(a). Similarly, the rth moment h a b o u t the mean may be obtained from the formula pr =-E(X -E(X))') = coefficient of (-a)r/r! in the power series expansion of eaE(X)fx(cu). Sometimes we are interested in the L.T. of the derivative of a function. If h(t) = dnfx(t)/dtn = f g ) ( t ) , for n 0, it can easily be shown that

xx(w> = // .. . The rth moment about the origm is

r=O

1.1; = E ( X r ) = coefficient of (iw)'/r ! .

The relation between d.f.'s and c.f.'s is one to one. There are several books deal with the properties of the c.f.'s and their applications. For the propertit the c.f.'s one may see Lukacs (1960); for applications of c.f.'s to probal: theory see Neuts (1973), to stochastic processes see Parzen (1962), to statj (and probability) see Lukacs and Laha (1964), among others. The transform equation (1.4.2) is called the Fourier-Stieltjes transform, e tion (1.4.3) is called the Fourier transform, and equation (1.4.4) is called Dirichlet series. Moment Generating Function and Curnutant Generating Function

Sometimes the moment generating function (m.g.f.) and the cumulant generz function are also used. The m.g.f. of a d.f. Fx(x) is given by Mx(o) =l?(eWX) = J e W X d F x ( x ) ,

areal.

The two cases when X is absolutely continuous or discrete can be separated

i

1.4

the case of the c.f. However, one disadvantage of the m.g.f. is that it may not exist, whereas the c.f. always exists. The cumulant generating function (c.g.f.) is the logarithm to the base e of the m.g.f. or the c.f., and in certain situations it is easier to calculate moments through the cumulant generating function. For relations between moments and cumulants, one may refer to Kendall and Stuart (1 969). The p.g.f. Px(z), the L.T. &(a), the c.f. xx(w), the m.g.f.&(a), and the c.g.f. In xx(w) or lnMx(oc) are related one to another, as may be seen easily from their xx(w) ) = Mx(iw) if X i s a discrete definitions. For example, Px(el") = ~ ( e ~ "= ~ r.v. If a = iw, where w is a real number, then fx(a) = xx(-w).

of inversion of Transforms Moments of low order can conveniently be obtained from the p.g.f. or the L.T. of the r.v. under study. We often find, however, that we need to obtain the distribution of our r.v. from its transform, that is, to invert the transform. Transform inversion is often very difficult, but sometimes it is simple, possibly with the aid of the relations between different types of transforms, as discussed above. Since we are mostly concerned with the p.g.f., we discuss briefly the inversion problem connected with the p.g.f. While an inverse transform can always be expressed as an integral over a contour in the complex plane, it may be difficult or impossible to obtain the inverse in closed form. For cases in which the inverse is known, the reader is referred to some standard text book in L.T. or tables of L.T. Extensive tables of L.T.'s and their inverses are available; see for example, Erdilyi et al. (1954), Roberts and Kaufman (1966), or Oberhettinger and Badii (1973). A nice but elementary discussion relating to some queuing problems may be found in Kleinrock (1975). In the case of the p.g.f., one method of recovering the distribution is

Sometimes Px(z) may be expressed as the ratio of two power series in z. Abramowitz and Stegun (1964) show how to express the ratio of two power series as a power series, namely, ce

1

+ x uizi

where px(n) can be obtained recursively from

LAPLACE TRANSFORM

15

This recursive relation is good for numerical evaluation. In practice, when the function Px(z) is complicated, calculations may become cumbersome or even impossible. In that case some sort of approximation is desirable, such as the following method based on partial fractions. Let us suppose that Px(z) is a rational function (in application, many of the transform functions do take this form) of the form

where Q(z) and R(z) are polynomials without common factors. Without going into the algebra of decomposing the function into partial fractions, we give below several results which hold under different conditions. For details, see Feller (1968). 1 If R(z) is a polynomial of degree m and Q(z) is a poly~omialof degree n<m,andR(z)=Ohasmdistinctrootszi,i= 1 , 2 , . . . m , then,for IzI<minlzil, l
(The case of repeated roots is discussed later.) Equation (1.4.5) is exact. It may be used with great accuracy when all the roots can be found exactly, as they can when m < 4. For m > 4, the zeros, in general, can be found only by algori&ic techniques. Several efficient algorithms, including Bairstow's and Muller's, are available for finding zeros (real or complex) of polynomials of fairly high degree. Although some of the methods fail to give repeated roots, Muller's method is successful in finding even the repeated roots. On the limitations of solving polynomials for zeros, the reader is referred to books on numerical analysis such as those by Conte and de Boor (1972) and by Henrici (1964). Bairstow's method has been successfully used by several authors, including Russell and Taylor (1977), in computing results that occur in the theory of bulk queues. When the calculation of m roots (for m large) becomes infeasible, an approximate expression for px(n) is desirable and is given in equation (1.4.6). 2 If one loot, say z l , is smaller in absolute value than all other roots, the first term of equation (1.4.5) is dominant as n increases, and thus we may write

Once again it may be said that algorithms are available for finding the root of smallest magnitude of a polynomial; for example, see Bernoulli's method in Henrici (1964). Equation (1.4.6) often gives surprisingly good results even for relatively small values of n. The sign -- between two sides of a relation indicates that the two sides are asymptotically of the same order, that is, their ratio tends to 1 asn-+-. In the derivation of the result (1.4.6) we supposed that the degree of Q ( z ) is less

16

PRELIMINARIES

than that of R(z). If we now relax this restriction and say that Q(z) is of degree n = m + r, division by R(z) leads to the form already discussed, added to an rthdegree polynomial which affects only the first r + 1 terms of the p.g.f. under consideration, thus making equation (1.4.6) still true. If zi, i > 1, is a double zero of R(z), then the partial fraction expansion involves one more term, which will add an extra term to the exact form, equation (1.4.5). However, this will not make any difference to the limiting case presented in equation (1.4.6), provided the smallest root z 1 is a simple root. Thus under the relaxed conditions equation (1.4.6) still holds. 3 When z1 is repeated, if R ( z ) = 0 has a root z , that is smaller in absolute value than all other roots, and if z , is of multiplicity r, then

where

-

and means that the ratio of the two sides tends to 1 as n + -. 4 Douglas (1963) gives still another approximation. If Px(z) = Zr,, the method of steepest descent gives an expansion with leading term

ON CERTAIN FUNCTIONS USED IN QUEUING THEORY

17

stant. The gamma function has numerous properties, but we discuss only those that we shall use; other details and properties may be found in Abramowitz and Stegun (1964). The gamma function, denoted by P(a), is defined by the integral

The integral converges for all a with real part Re or > 0, but does not converge for Re or < 0 . However, it is possible to define r ( a ) by analytic continuation for complex values of a, such that Re a + -n, n = 0 , 1 , 2, . . . . We do not go into further detail for the case Re a < 0, as we shall be concerned with r ( a ) for real and positive values of a only. The value of the gamma function is obtained by series expansion or some type of numerical integration, unless a is a positive integer. Some of its properties are:

-(

1

r@)

= a

2

r(4

=

3

r(4)

=J;;

if a > 0

a- )

(a-I)!,

if a is a positive integer

px(n)zn,

Properties 1, 2, and 3 are given in text books on calculus, and property 4 is proved below. By definition

where

and z , satisfies gC1)(zn) = 0, provided that g ( 2 ) ( z n > ) 0. Douglas gives a number of examples which show that the accuracy of approximation can vary widely. However, used with some care, approximation (1.4.7) can give useful information when it is difficult to get general explicit results for px(n). The approximation of Feller (1968) has been used by Cromie et al. (1979) to queuing system, and the results were good. We have not had analyze the M ~ / M / C enough opportunity to use the approximation of Douglas (1963), nor did we find his approximation used in the literature on bulk queues. It is planned, however, to test the accuracy of the approximation of Douglas at some later date.

1.5

1.5

O N CERTAIN FUNCTIONS USED I N QUEUl

Integration of the right-hand side of equation (1 S . 2 ) by parts leads to

since

tff lim -e-f a

t+-

tr

- = 0.

r=o

r!

The vanishing of the limit may be seen by applying L'Hapital's rule to each term of the sum or otherwise. Equation (1 S.1) could also have been obtained more simply by applying induction on k and using only property 1. For more similar properties see Problems and Complements.

The Gamma Function

The Bessel Functions and Their Modifications

In Section 1.2, Case 2, while discussing the Erlangian distribution, we talked about the gamma distribution which involves the gamma function as a normalizing con-

Bessel functions, which occur in solutions to several queuing problems, are particular solution of the Bessel equation

18

PRELIMINARIES

1.5

ON CERTAIN FUNCTIONS USED I N QUEUING THEORY

19

The number n is called the order of the equation, and in our applications to queues n is a nonnegative integer. It is easy to verify by tern-by-term differentiation that y = Jn(f) is a solution of equation (1.5.3), where

where ( L Y , ) ~ = ~ , ( ~ ~ , ) ~ = a ~ ( f, .f. ~. ,+( al, +) k - l ) , e t c . , is called the generalized hypergeometric function. The ah's &e the numerator parameters, the ,oh's are the denominator parameters, and z is the variable. One notation prevalent for the function is pFq(ai,a2,...,ffp; The function Jn(t) is called the Bessel function of the first kind of order n . The function In(!) is called the modified Bessel function of the first kind of order n , and is related to the function Jn(t) through the equation &(t) =

.

where i =

P ~ , P z , . . - , P Q ;Z)

or simply pFq when there is no confusion about the arguments of the function. As an example note that

\/=i

In the theory of queues, the following modified form of I,(t) has been used:

This is called an ordinary hypergeometric function. Most of the very extensive literature on hypergeomehiZ'functions considers the cases when p and q take values between 0 and 2, both inclusive, such as F o , F, , and so on. The following examples may further clarify the notation. The exponential function:

,

It may be noted that I;(t) is the same as In(t) defined above and that I,h(t) is not a solution of the Bessel equation (1 S.3) when k > 1. For the properties of I,h(t) see Luchak(1956),and for the properties of I,(t) and Jn(t) see Abramowitz and Stegun (1964). It may be noted for future reference that the L.T. of In(t) is given by

ez =

-

1

k=0

,

zk

lzl

= i F l ( a l ; a l ; z ) = ,Fo(z),

< rn.

(1.5.6)

and the negative binomial function:

Iow

e -a'In(t) dt =

e Generalized Hypergeometric Functions

= 2F1(~,~l;~l;-z)

= lFo(~;-z),

Quite a number of important functions can be expressed by means of hypergeometric functions. The series

lzl

<

1.

(1.5.7)

It may be observed from equations (1.5.6) and (1.5.7) that when the parameter value occurs in both the numerator and the denominator, both parameters can be

20

PRELIMINARIES

omitted from the argument of the function, and the subscripts of F are each reduced by 1. Under certain conditions it can be proved by using the ratio test, that the series (1.5.5) (or its particular cases) converges in a certain circle lzl < r in the general case of p and q. For example, note that the series (1.5.6) and (1.5.7) are convergent for lzl< w and lzl< I , respectively. For a more detailed study of these and several other functions see Luke (1969), Copson (1948), or Abramowitz and Stegun (1964).

1.6

PROBLEMS AND COMPLEMENTS

Section 1.I

1

Prove the following results, which are useful in many queuing problems:

(b)

PARTIAL DIFFERENTIAL EQUATIONS

For some problems of multichannel queues in Chapter 5 we need to consider the solutions of linear partial differential equations in two independent variables, which may be solved by a general procedure known as Lagrange's method, which we now discuss briefly. Consider an equation of the form

-

C npx(n)zn n=O

= z~(l)(z).

(d) Ifqx(n) = P(X

-

> n), then C qx(n)zn

-

< n), then C hx(n)zn n=O m

(f) Ifgx(n) = P(X

< n),then

2

=

P(z> .

1-z

-. gX(n)zn = zP(z) 1-2

Hint:

Since fx(x) = 0 forx

E (X) =

jOmxdFx (x)

<0

< 0, =

Bm yFx(y) s+-

The general solution, which is a function of u and v may be written in the form F fu (x, y, z), v(x, y, z)) = 0

or

u = f (v)

or

v = g(u).

The form of the function may be determined by the initial conditions. One may gain experience with Lagrange's method by attempting the problems given at the end of this chapter.

.

In many problems v g theory we deal with an r.v. X which is continuous and whose density funct~onvanlshes for negative values. The mean of such r.v.'s may be obtained heuristically as follows:

if fx(x) exists and fx(x) = 0 for x

v(x, y, z) = constant = C2

1-2

n=O

Let two independent solutions of the first and second equations be u(x, y, z) = constant = C,

1 - P(z) .--------

n=O

(e) If hx(n) = P(X where P, Q and R are functions of x, y , and z. We give here only an outline of a procedure for solving this standard-type equation. For the general theory of such equations, one may refer to standard text books, such as Forsyth (1929) and Piaggio (1958). For an application of these equations to generalized birth and death processes, see Bailey (1964). The associated Lagrange (also called subsidiary or characteristic) equations are

=

m

=

[l -Fx(x)]

dx.

PROBLEMS AND COMPLEMENTS

This problem shows that [ I - ~ ~ /E(x) ( i )may]be treated as a p.d.f. For a rigorous proof of this, see Chapter 2.In case of a positive discrete integer-valued r.v., the analogous result is

5

It may be seen that if P(N = n) = 1, then we recover the Erlangiai distribution (1.2.6). If N in problem 4 is distributed according to the geometric law (a)

then show that

Section 1.2

3

2

Let Xi be independently distributed r.v.'s with Wn = Z i t l Xi. (a)

(b)

(c)

If {Xi) are identically distributed as X, such that P(X = x) = pxql -", x = 0 , 1, with p + q = 1, then the r.v. W, is binomial with parameters n and p. Hint: The p.g.f. of X is (q + pz). If (Xi) are binomially distributed with parameters ni andp (p remaining the same), then W, is binomially distributed with parameters C.Zl ni and p. If {Xi) are geometrically i.i.d. as X, such that P ( X = x) = q x p , p > 0, q > 0 , x = O,1,2, . . . , with p q = 1 , then the r.v. W, has the negative binomial distribution. Hint: The p.g.f. of X is p/(l - qz).

+

(b)

If N has the zero-truncated Poisson distribution

then show that

]

p (t a) .1,(2Jallt). fw,(t> = ~ e x p [ -1 -aexp

6

(a)

Continuation of Problem 4. If N follows a Poisson distribution wit1 mean X a n d P ( X = l ) = p , P ( X = O ) = q , with O < p < 1 andp t q = 1 then using equation (1.2.4), show that

N

If N in WN = .Z Xi, instead of being a constant, as assumed in the d e r i v a t S z=1

of the ~ r l a n ~ idistribution,is in an r .v., we have a generalization of the Erlangian distribution, which may be called the modified Erlangian distribution and may be denoted by EN. Consider the sum (1.2.1), where N is now an r.v. and the Xi's are independently exponentially distributed with a parameter K . Prove that the density function of WN is given by

whereOGc, = P ( N = n ) < l and C.:==, cn = 1. The EN distribution, when considered for service (or interarrival) times, consists of an indefinite number of phases. In practice, however, the values of c, and ~1 are to be determined by equating a few moments of the experimentally determined service times to the theoretical moments which can be determined from this distribution. A large class of distributions that are of practical interest can be handled by this distribution.

(b)

which is the p.g.f. of a Poisson distribution with mean Xp. This show that the Poisson distribution is preserved under random selection. Thi property of the Poisson distribution is very useful in many applications some of which may be found in Parzen (1962) or Feller (1968). We giv' two simple applications. In transportation and traffic problems N could be the number of car passing an intersection on a certain highway during a period of obser vation and p the probability of a car turning at the intersection. The] WN is the total number of cars that turned at the intersection. In biologj cal sciences, N may represent the number of female insects in a certai~ region and X the number of eggs laid by an insect. Then clearly W3 represents the total number of eggs laid by all the insects. In general, irrespective of the distribution of N and X, WN is the sum o f . random number N of i.i.d. r.v.'s {Xi). From equation (1.2.4) or otherwise, show that: (i) E(WN)=E(N)E(X). (ii) E(W&) = E(N) Var (X) 3.E(N2)E2(X). If N is a geometric r.v. with P(N = n) = qpn-', n = 1 , 2 , 3 , . . . show that

PRELIMINARIES

26

PROBLEMS A N D COMPLEMENTS

case perhaps occurs when Y is assumed to be independent of X and exponentially distributed with parameter, say h. Show that in this case, the L.T. of (X - Y ) + is given by

27

By using FZ(0) = 1, it can be shown that

and hence

(c)

The function y ( x ) is also called the intensity function or conditional rate of failure function, and is widely used in renewal theory, reliability, actuarial science, and so on. In particular, it may be seen that if y ( x ) = A, a c b s t a n t , then X is exponential with parameter X,namely, Fx(x) = 1 - e-'%. Conversely, if X is an exponential r.v., then

where&(a) = ~ ( e - ' ~ ) : ) '=: x ) = 1;;e-"(x-Y)),e-XY dy + e - h x , and Hint: E ( ~ - ' ( ~ - ~IX SO on. Continuation of (b). Consider the problem in (b) and show that if Y has an Ek distribution with p.d.f.

then

which shows that, given that the service did not end until time t , the remaining service time has the original exponential distribution. It may be remarked that ~ ( x is) not a p.d.f., conditional or unconditional. The requirements that it must satisfy are

P(X)>O,

J:y(x)dx=I0,

the first of which is easy to see and the second can be seen to hold since Fx(m) = 1. As an example of the use of the hazard function p(x) let XI and X2 be i.i.d. exponential r.v.'s with parameter A. Then W.2 = XI X , will have an E2 distribution with mean 2/X. In this case

+

where 72)(or)= fx ( a ) . int: Proceed as in (b) and use the identity discussed in Problem 17 of Chapter 2. Another distribution that has been used in queuing theory may be discussed in relation to the following example. Let X be an r.v. representing the service time of a customer with p.d.f. f x ( x ) It is sometimes useful to express the p.d.f. of a continuous nonnegative r.v. such as X in terms of the hazard function p(x), where

Fw2(t) = 1 - e - A ' and consequently

Hint:

See Problem 7.

Section 1.3

10 (a) and

Fg(x) = 1-Fx(x).

Discuss the mathematical derivation of the hyperexponential distribution f o r n = 2. Hint: If X I and X2 are the r.v.'s representing the times a customer spends in channels 1 and 2, respectively, then

PRELIMINARIES

28

P(S

(b)

> t)

> tlcustomei selects first channel) u, + P(X2 > t 1 customer selects second channel) a2

= P(XI

PROBLEMS A N D COMPLEMENTS

29

(c)

Suppose that the arrivals to a givensingle-serverqueuing system follow the Poisson process with rate h, and the service times are exponential with mean I/p. Suppose further that it is known that the distribution of waiting time (in queue) consists of two parts, the probability mass concentration at the origin given by (i)P(Vq = O ) = 1 - p together with the p.d.f. t>O (ii) fvq(t) = h(1 - p ) e - p ( l - p ) f , where V, is the r.v. representing the time a customer will spend in the queue before getting served, and p = X/p. Show that the truncated p.d.f. of waiting time (in queue) for those who have to wait is the exponential distribution with parameter (p -A).

(d)

In the queuing problem discussed above, suppose that itis known that the distribution of the r.v. N , which represents the number in the system (queue t service), is given by

The derivative of this gives the required p .d.f. The case of a general value of n may be discussed similarly. Show that the mean and the variance of the hyperexponential are

The variability of the distribution of a nonnegative r.v. X can be measured through the coefficient of variation, defined by

Show that the truncated distribution of N truncated to the right of N=Mis

\

Find C.V. (X) for the following distributions (i) Exponential with parameter p. (ii) Erlangian given by equation (1.2.6). (iii) Deterministic. (iv) Hyperexponential with n = 2. Show further that for n > 1 , the Erlangian distribution E, has less variability (as measured by the coefficient of variation) than the exponential, whereas the hyperexponential HE, has greater variability than the exponential. These distributions are found to adequately represent many real-world problems in the application of queuing theory. In any application the values of n and for the Erlangian distribution, and n , or, and p, for the hyperexponential must be determined empirically. The hyperexponential distribution is very useful in approximating statistical distributions for which the coefficient of variation lies between 1 and m, whereas the Erlangian distributions are useful when the coefficient of variation lies between 0 and 1. In practice the coefficient of variation can be used gainfully in identifying the distribution of r.v.'s involved in queuing theory. For example,if the sample standard deviation is nearly equal to the sample mean, then one might suspect that the underlying distribution is exponential. If the sample standard deviation is significantly less than the mean, then the underlying distribution may well be Erlangian. Finally, if the sample standard deviation is greater than the sample mean, then it may be appropriate to test whether the hyperexponential fits the observed data.

(e)

In the theory of queues, the truncated distribution of N is useful where the queuing system has a limited waiting room, say of size M. Find the p.d.f. of the truncated hyperexponential r.v. T (truncated on the right at T = 8 ) discussed in Section 1.3, (when n = 2).

Section 1.4

11 Show that the L.T. of E n , namely,

Prove from this that

+ ?

c:

5 5 g

3

0 , "

;$3 r.

N

n

Pi;

PRELIMINARIES

32

Further show that if ~ ( y = ) y , then Q = (q + p ~ exp ) ~

An interpretation of n(y) = y is given in Chapter 5 on multichannel queues. Let h,instead of being a constant, be a function of t, say X(t). The partial differential equation of Problem 16 then becomes

17 Continuation of Problem 16.

with the initial condition Q = x i when t = 0. Show that the solution in this case is

A slight variation of this with different initial conditions is discussed in Chapter 5 on multichannel queues. 18 The following equations arose in a study of the growth of animal herds. If

with the initial conditions Q = x i when t = 0,then show that X

= ( p + k1n-l

-

[ p - p e x p [-(h+p)t]

[p+hexp[-(X+p)t]

+ x ( h + p e x p [--(~+p)f]}]'-~

+x{h-hexp[-(h+p)t])ln-'.

For further details, see Bunday (1970).

REFERENCES Abramowitz, M., and I. A. Stegun (Eds.) (1964). See Chapter 2. Bailey, N. T. J. (1964). The elements of stochastic processes. Wiey, New York. Beightler, C. S., L. G. Mitten, and 6.L. Nemhauser (1961). A short table of z-transforms and generating functions. Oper. Res. 9,574-578. Blake, I. F. (1979). A n introduction t o applied probability. Wiley, New York. Brockmeyer, E., H. L. Halstrem, and A. Jensen (1948). The life and works o f A. K. Erlang. The Copenhagen Telephone Co., Copenhagen. Bunday, B. D. (1970). The growth of elephant herds. Math. Gaz. 54,38-40. Cadzow, 3. A. (1973). Discrete-time systems. Prentice-Hall, Englewood Cliffs, NJ. Conte, S. D., and C. de Boor (1972). Elementaly Numerical Analysis. McGraw-Hill, New York.

REFERENCES

33

Copson, E. T. (1948). A n introduction to the theory o f functions o f a complex variable. Oxford University Press, London. Cromie, M. V., M. L. Chaudhry, and W. K. Grassmann (1979). Further resuits for the queueing system MX/M/c.J. Oper. Res. Soc. 30,755-763. Douglas, J. B. (1963). Asymptotic expansion for some contagious distributions. Proceedings o f the International Symposium on Discrete Distributions, Montreal, 291 -302. Erdelyi, A., W. Magnus, F. Oberhettinger, and F. 6. Tricomi (1954). Tables of integral transforms. McGraw-Hill, New York. Feller, W. (1968). A n introduction to probability theory and its applications, vol. 1, 3rd ed. Wiley, New York. Forsyth, A. R. (1929).A treatise on differential equations. MacmiUan, London. Gnedenko, B. V., and A. N. Kolmogorov (1954). Limit distributions for sums o f independent random variables. Addison-Wesley ,Cambridge, MA. Henrici, P. (1964). Elements o f numerical analysis. Wiley, New York. Jury, E. I. (1964). Theory andapplication o f the z-transform method. Wiley, New Yb,t. Kendall, M. G., and A. Stuart (1969). The advanced theory o f statistics, Vol. 1,3rd ed. Griffin, London. Kingman, J. F. C., and S. J. Ta~lor(1966).Introductionto measureand probability. Cambridge, London. Kieinrock, L. (1 975). Queueing sysrems, Vol. 1, Wiley ,New York. Loive, M. (1977). Probability theory I, 4th ed. Springer-Verlag,New York. (1978). Probability theory 11, 4th ed. Springer-Verlae. New York - -, - .... Luchak, G. (1956). The solution of the singlechannel queueing equations characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Oper. Res. 4,711-732. Lukacs, E. (1960). Characteristic functions. Hafner, New York. and R. G. Laha (1964). Applications of characteristic functions. Griffin, London. Luke, V. L. (1969). The special functions and their approximations, Vol. 1. Academic Press, New York. Neuts, M. F. (1973). Probability. Allyn and Bacon, Boston. Neveu, J. (1965). Mathematical foundations o f the calculus o f probability. Holden-Day, San Francisco. Oberhettinger, F., and L. Badii (1973). Tables o f Laphce transforms. Springer-Verlag, New York. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Piaggio, H. T. H. (1958). An elementary treatise on differential equations and their applications, rev. ed. G. Bell & Sons, London. Polya, G. (1954). Mathematics and plausible reasoning, Vol. 1 : Induction and analogy in mathematics. Princeton University Press, Princeton, N.J. Roberts, G. E., and H. Kaufman (1966). Table o f Laplace transforms. Saunders, Philadelphia. Russell, A., and R. G. Taylor (1977). Numerical analysis and relationships for the queueing models M / E ~ / IE,; / M / ~ .Unpublished thesis, Royal Military Collegeof Canada, Kingston, Ontario.

2.1

The history of queuing theory goes back to the work of Erlang in the early years of the twentieth century, and important work was done in the 1930s and 1940s by Khintchine, Palm, Pollaczek, and others. About 1950 the theory of queues entered a period of intensive investigation by workers in a variety of fields, particularly in mathematics and operati ns research. Research in the theory of queues has developed new general meth s, including methods which can be used to give explicit solutions to many particular problems. At the same time, research has taken place in the development and application of new and old queuing models for particular applications. Queuing systems which have been studied include such everyday activities as waiting for service by a teller in a bank, as well as a wide range of systems in manufacturing, communications, transportation, and other fields. A systematic study of the theory of queues provides a base of knowledge which can be applied to improve the efficiency of many queuing systems in the real world.

&i

A queue, in the sense in which we use the word, is a waiting line which usually is formed in front of some service facility as a result of an irregularity in the pattern of arrivals or departures, or both. Thus it appears that a waiting line has three elements - a queue, a server who may be active (such as a bank teller) or passive (such as a parking lot), and a service facility. In the mathematical analysis, a queuing system is normally characterized by means of the following terminology due to KendaU (1951). The Arrival Process

This is the process that governs the arrival of some given service-seeking entities, which we shall call customers, be they human or otherwise, to the system. The customers normally receive service from a server or service channel and then leave the system. If a customer arrives when all servers are busy, the customer is usually assumed to wait for service. Here, as elsewhere in queuing theory, our technical

BASIC MATERIAL IN QUEUES

35

terms such as "customer" and "server" are often words taken from nontechnical English, but they are used here in an extended sense. For example, a customer might be a computer program waiting to be computed, and the server might be the computer itself which performs calculations on the program. The simplest hypothesis about the arrival process assumes that arrivals follow a Poisson process (random arrivals). The arrivals may occur singly or in bulk (groups or batches). If all the arriving customers are allowed to join the queue, we have an infinite waiting space queuing or delay system. We also consider systems, called finite waiting space systems, in which customers do not wait, but leave the system immediately if they arrive when a fixed number K of customers are already waiting. A loss system is the extreme case of a finite waiting space system, in which K = 0 so that customers never wait, but leave unless they receive immediate service on arrival. Infmite waiting space queuing systems, which are sippler to handle mathematically, serve as a reasonable approximation to the (more realistic) finite waiting space queumg systems, particularly when the traffic intensity is low ( p < I), that is, when the traffic is light. In case the traffic intensity is large but less than 1, the finite waiting space queuing system converges very slowly to the infinite waiting space queuing system as the waiting space tends to infinity. When p > 1 , it will be essential to consider finite waiting space queuing systems. While queues with infinite waiting space may not exist in the real world, they provide excellent models for real systems in which the (large but finite) waiting space is rarely filled. We may therefore say that in practice both infinite and finite waiting space systems are useful and worthy of study. Approximate (or exact) results obtained from the infinite waiting space systems (or finite waiting space systems) may be good under the conditions stated above. There are also queues in which customers may not join the system if, on arrival, they get some information which deters them from joining the queue. Such customer behavior (not joining the queue) is known as balking. Moreover, it is possible that a customer having waited for some time becomes impatient and thus leaves the system without getting service; this is known as reneging. Connected with the arri
36

SOME TECHNIQUES OF OUEUING THEORY

U's, instead of being independent, may be correlated. For example, if the arrivals are scheduled for predetermined times and if a customer arrives early, there is likely to be a short interval before one arrival, followed by a long interval before the next arrival. However, we shall not consider such correlated interarrival times in this book, unless stated otherwise. The Queue Discipline

This is the rule that determines the formation of the queue and the take-up of customers for service. For example, customers may be picked up for service randomly or on some set basis. In the latter case, the discipline, which is the simplest in concept and also from an analytic point of view, is first-come, first-served (FCFS), that is, corii;n;tencement of service is in the order of arrival. Besides these disciplines, several others are possible, such as last-come, first-served (LCFS). One example, among others, of LCFS is last hired, first fired because of low seniority. Unless otherwise stated explicitly, we shall assume throughout our study that the queue discipline is FCFS. Service, like arrivals, might be in batches of fixed or varying size. Usually it is assumed that intake for service is instantaneous. However, this may not be true in certain situations. For example, if the service facility is manually controlled, the server may be available intermittently, in the sense that after a customer leaves the service facility, the server may not take up the next customer immediately, but may spend some time on a job left over by the departing customer before taking up the next customer. As another example, consider a one-way transportation system such as a ski lift in which the gondolas or chairs are regarded as servers. Then the servers are not instantaneously available for service after customers' departures, since each server must return to the lower station of the ski lift before picking up new passengers. Service Mechanism

The time interval from the instant of initiation of service on a customer t o the instant when this service ends is called the senrice time (or holding time in telephone networks). Let the customers depart at the instants u,, a*, . . . , o k , . . . , k = 1 , 2, 3, . . . . If, however, there is a departure at time t = 0 , we may write uo = 0, and call oo an initial departure instant. If Vk, k = 1, 2, 3 , . . . , is the duration of the kth service interval, we assume that V's are independent of U's and also mutually independent, with the common distribution function

Its p.d.f. will be denoted by b(v). As we did with interarrival time distributions, we consider various service time distributions. For example, consider the E k distribution for service times, in which service takes place in k exponential "phases". In general, phases may have no real physical significance. Recall from Chapter 1 that the Ek distribution may be obtained from a sum of k i.i.d. exponential r.v.'s. This artifice, introduced by Erlang and generalized by later writers, permits us to use

2.1

BASIC MATERIAL 1N QUEUES

37

the "memoryless" property of the exponential in studying a large, though not completely general, class of service time (or interarrival time) distributions. Generally speaking, the service facility can be thought of as being composed of a given combination of servicing stations arranged in series, in parallel, or in some more general configuration. In both applied and theoretical studies the most common configuration of service channels consists of c servers in parallel, where l < c 9., The single-server case c = l has been thoroughly studied and much information is available about it. In the infinite-server case c = m , all customers can be served without waiting, and our interest is in the number of busy servers. The infinite-server queue can be a useful and easily solved approximation for multiserver queues in which c is fairly large but finite, and the system is lightly loaded, so that almost all customers can be served without waiting. Denote E(Ui) and E(Vi), i = l , 2, 3 , . . . , by a and b, respectively, so that h = l/a and p = l l b , and assume that 0 < a < m, 0 < b < m. The traffic intensity p plays a fundamental role in the study of queuing systems and is defined by mean arrival rate of customers = maximum service rate when operating at maximum capacity

'

In many, but not all, cases p is equal to the utilization factor, defined as the proportion of time the system is busy. The cases in which p differs from the utilization factor include finite waiting space systems and single-server bulk-service systems (excluding the case when service is in batches of fixed size k), as well as other systems to be discussed later. For bulk arrival single-server queues with mean arrival group size Z, we have p = hii/p. Similarly, for the bulk-service queue M / G ~ / I p, = X/Bp, where B is the capacity of the server. For multichannel queues with customers served singly and c(< m) identical servers, p = h/cp. In the case when c = or waiting space is finite, p does not play such a significant role as in the other systems. Two types of solutions of queuing systems are of interest - the limiting solution and the transient solution. There are other equivalent terms for both of these. For the former, we have time-independent, intransient, homogeneous in time, or steady-state, whereas for the latter we have time-dependent, nonlimiting, and so on. In the limiting case, the effect of the initial conditions is worn out. Let us explain these concepts by the following example. Suppose that one is interested in the distribution of the number of customers in a given queuing system. In the limiting case, although the probabilities are independent of time, the distribution is not deterministic. The queue fluctuates following a distribution independent of time. This may be explained as follows. If the probability of 20 vehicles waiting at a toll gate on a highway between 2.30P.M. and 2.31 P.M. (on a given day) is 0.15, we will have the same probability between the same times on other days. that is, the probability depends on the length of the time interval and not on the initial conditions. When the limiting solutions are inapplicable or do not exist it is desirable to consider transient solutions. It may be mentioned here that the limiting solution, provided it exists, may be obtained by letting t + m , or by setting the derivative with respect to t equal to zero and then solving the resulting limiting

SOME TECHNIQUES OF QUEUING THEORY

38

equations. A limiting solution will exist if'O < p < 1 . If p > 1, there is, in general, no limiting solution. However, there are exceptions to this. For example, when c = or waiting space is finite, limiting solutions hold for any value of p , though in the finite waiting space case the limiting solution may be reached very slowly when p > 1. The transient solutions, of course, hold for any value of p. Transient solutions, however, are in general difficult to obtain in explicit form. Even if they can be obtained, numerical analysis seems difficult to carry out, and this is especially true when customers arrive or are served in batches. In view of this, we shaU be discussing mostly the limiting solutions. Transient solutions are discussed when they have some other interesting feature. The most important problems connected with the probabilistic study of queuing systems include the determination of probability distributions for the system length, waiting timqactual and virtual), busy period, and idle period. We now discuss these problems. 1 The Distribution of System Length. Let N(t) be the number of customers in the queuing system, including those in service, if any, at any epoch t. We will sometimes refer to N(t) informally as the system length, but the reader should be aware that the term "queue length or size" has been used in the literature in at least two different senses. Some authors use queue length to refer to the number of customers in the system, whether waiting or in service. Other authors use queue length to refer to the number of waiting customers only, excluding those being served. When we need to discuss the number of customers waiting, the number of customers in the system just before the nth arrival, or the number of customers in the system just after the nth departure, appropriate notation will be introduced to distinguish these quantities from one another and from N(t). For convenience, we have used the term queue size in Chapter 6.

2 The Waiting-Time Distribution. The time spent by a customer from the instant of joining the queue to the instant of entering service is called the customer's waiting time in the queue, and is generally denoted by Vq. The customer's time in the system. or sojourn time, is the sum of the customer's waiting time in queue and service time, and is generally denoted by V. We must distinguish between the actual waiting time v?) of the n t h arriving customer and the virtual waiting time Vq ( t ) , which is the waiting time experienced by a (fictitious) customer arriving at epoch t . The actual and virtual waiting times are connected by the relation v?) = Vq (a; - 0), where oh is the arrival epoch of the arriving customer. Similarly v ( ~=) V(5; - 0), where V(t) is the virtual sojourn time at epoch t and v@)is the sojourn time of the n t h arriving customer. It should be clear that in the same queuing system, the two processes, though related, are quite different. In fact, Vq (t) [or V(t)] is considered in continuous time, whereas v?) is considered at a discrete set of points in time. In order to distinguish between the two processes, Vq ( t ) is called the virtual waiting time at epoch t , whereas V?) is called the actual waiting time of the n t h arrival. However, in the sequel, the adjectives virtual or actual will usually be omitted since the meaning should be clear from the processes under consideration. In the case of statistical equilibrium, Vq(t) or will be replaced by Vq in the continuous as well as in the discrete case,

vF)

2.1

BASIC MATERIAL I N QUEUES

39

although the two processes are in general different. This is done for notational convenience. The same argument applies to the time spent in the system. The d.f.'s of the actual waiting (in queue) time and the virtual waiting (in queue) time are denoted by Wi(t) and Wq(t), respectively. If the two coincide, as they do in the steady state for Poisson arrivals, we prefer to use the notation Wq (t). Connected with the two waiting times (virtual and actual) is the occupation time of the server. The occupation time of the server at epoch t is denoted by {(t), and is defined as the residual time at t during which the server will be continuously busy serving customers already in the system at t. Thus if no customers arrive after t, the server will be busy from t to t {(t) and will then become idle. Note that {(t) = 0 if and only if (iff) the server is idle at t. We shall consider occasionally the limiting behavior of the process {{(t), t > 0). The reader will see that in some situations the limiting behavior of the process {{(t), t 2 0 ) coincides with the limiting behavior of {Vq(t), t 2 0 ) . One example in which the limiting . details, see behaviors do not coincide is that of the queuing system M / M ~ / IFor Miscellaneous Problems in Chapter 4. It may be remarked here that some authors refer to the virtual waiting time at time f as the unfinished work (or backlog) in the system at time t [see, for example, Kleinrock (1975)l. The unfinished work at time t is defined as the time interval required to empty the system of all customers present in the system at time t. In view of the definition of unfinished work it is clear that it applies to all queue disciplines, whereas the virtual wait applies to FCFS disciplines. Since in this book we shall not be dealing with queue disciplines other than FCFS, we prefer the use of virtual waiting time to unfinished work.

+

3 The Busy-Period Distribution. For a single-server queue in which arrivals are by singlets, a busy period may be defined as the interval of time from the instant of arrival of a customer at an idle channel to the instant when the channel next becomes free for the first time. For this type of queue, the initial busy period may start with i > 1 customers and hence may be different from other busy periods which start with the arrival of a single customer. Results for busy periods of singleserver, single-arrival, Poisson-input queues can rather easily be generalized to the corresponding bulk-arrival queues. For multi-server queues, any one of several different definitions of busy period may be appropriate. Some of the known results on busy period distributions and related problems, such as idle periods or number served during a busy period, will be discussed in later chapters.

4 The Idle-Period Distribution. For single-server queuing systems the duration of the period of time from the instant when the server becomes idle after serving a customer to the instant when the server restarts service on the arrival of a customer, is called the actual idle period and will be denoted by I. An actual idle period separates two successive distinct busy periods, and it therefore follows from the definition that I > 0. One may also define a virtual idle period, with duration I,(n)> 0, where I,(n) is the duration of the actual idle period (if there is one) immediately following the departure of the n t h customer, and I,(n) = 0 if there is no idle period immediately following the departure of the nth customer. In steady state as n -+m, I,(n) converges in distribution to I,, and

SOME TECHNIQUES OF QUEUING THEORY

40

the conditional distribution of (I,II, > 0) 'will be the same as the distribution of I. For details, see the discussion of the queuing system GI/M/l in Section 2.2 and of GI'/M/~ in later chapters. We will also make some use of the busy cycle, defined as the sum of a busy period and the following idle period.

5 Utility of These Distributions. At this point a comment on the utility of the system-length, waiting-time, busy-period, and idle-period distributions may be worthwhile. The distribution of system length is useful from the design point of view as an aid in estimating the cost of operation if a waiting room must be built. It is also u s e M o m the customer's point of view. The customer on arrival is interested not only in the system length as measured by the number of customers in the system, but also in the interval of time he has to wait before service on him can be started or in the total time he has to spend in the system. The busy-period distribution is important from the server's point of view. In recent years, analysis of busy periods has found increasing use in the design and study of real-time computer systems, as in Kleinrock (1976). It may be mentioned here that the theory of queues is quite similar in its mathematical structure t o seemingly unrelated mathematical theories of inventories, insurance, and so on. See, for example, Prabhu (1965a), Takics (1967), or Seal (1969). The theory of dams and reservoirs has been developed by Moran (1959), Cani (1957, 1969), Kendall (1957), Prabhu (1964, 1965a), and others. The mathematical models of dam theory are, perhaps surprisingly at first sight, closely related to queuing models. For example, the wet period of an infinite' reservoir corresponds to the busy period of a queue. The study of idle periods also helps in the design of queuing systems. An analyst might probably like to strike a balance between idle time of the servers and customers' waiting time according to some cost structure. These distributions are related to each other and can give useful information to the management of an organization where there are congestion problems.

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

In Section 2.1 we gently introduced some of the ideas and terminology of queuing theory, with some mathematical symbols, but without mathematical manipulation of those symbols. Now we introduce some of the mathematical techniques used to solve queuing models, and some simple queuing models to illustrate the use of these techniques. The mathematical theory of queues is a small part of the more general mathematical theory of stochastic processes, which we now consider briefly. The word "stochastic" has a long and interesting history, but has now come to be used in the sense "depending on probability distributions," more or less synonymously with such words as "random," "chance," or "probabilistic." Stochastic Processes

For a formal treatment of stochastic processes, consult Bartlett (1978), Cox and Miller (1965), Parzen (1962), Karlin and Taylor (1975), Cinlar (1975), or more

2.2

TECHNIQUES AND HISTORY OF ANALYSBS

41

advanced texts by Doob (1953), Gvy (1965), and others. We give here a brief informal introduction to a few of the ideas contained in these books. A stochastic process N(t) [or ~ ( t ) [(t), , etc.] has three elements: (1) the state space S, (2) the parameter (or index) set T and (3) the dependence relations among the r.v.'s N(t) when t varies in T. The state space S, which is the set of values which may be taken by N(t) for typical values of t E T , could be discrete or continuous, scalar or vector. The parameter set T could also be discrete or continuous, scalar or vector. While in queuing theory t is normally a time parameter, in other applications it need not be so. [An example in which T is neither one-dimensional nor a time parameter set is that of waves in oceans. If one regards the longitudinal and latitudinal coordinates as the components of a 2-vector t, N(t) then is the height of the wave at the location t . ] The arrival process, the distribution of system length, and so on, are further examples of stochastic processes that arise in the study of queuing systems (see Section 2.1). Among the stochastic processes, the Markov processes occupy a prominent place. A stochastic process is called a Markov process if the future development of the process depends on its past only through its present and is otherwise independent of the past history of the process. More precisely, if for any n > 0 and any set of n points t1 < t z < . . . < t, in T, the conditional distribution of N(t,) given N(tl), . . . ,N(t,-, ) is the same as the conditional distribution of N(tn) given N(tn- ) only, then (N(t)) is a Markov process. In a Markov process, the dependence relations (3) mentioned earlier between the r.v.'s N(t) and N(t s) suffice to determine the probability structure of the process. For a discrete-state Markov process, often called a Markov chain, the dependence relations take the form of transition probabilitiesPij(t, s), defined by

+

We will usually consider the important special case of a homogeneous Markov chain, in which Pij(t, s) does not depend on the initial epoch s E T. The transition probability Pij(t, s) then may be written asPij(t), where Pij(t) = P{N(t

+ s)

= jlN(s) = i},

i, j E S,

s Z 0.

Pii(t) satisfies the following conditions:

Furthermore from (2.2.1) it is easy to establish the continuity of Pii(t> for every t > 0. For let h > 0 ; then as h -+ 0 , Pij(t + h) +Pij(t). Also P,(t - h) -, Pij(t) Thus Pij(t) is continuous at each t > 0. Similarly, using the continuity property of Pij(t), it can be shown that the derivative of Pij(t) exists for every t > O . However, at f = 0, continuity and differentiability hold only from the

SOME TECHNIQUES OF QUEUING THEORY

42

2.2

TECHNIQUES A N D HISTORY OF ANALYSIS

43

and not, for example, on t l . As a consequence one may use the notation N(t2 - t l ) to represent both N ( t z )- - N ( t l ) andN(t2 + h ) - N ( f , h).

right, if we define

+

Orders of Magnitude o ( h )and O ( h )

One states informally that the function g(t) is o ( A t ) as A t

It is supposed that the probability Pij(t) depends on the length of the time interval t and not on the position of the starting epoch s on the time axis, as indicated through the notation that we have adopted. Such processes are called time-homogeneous Markov processes or Markov processes with stationary transition probabilities. An excellent detailed mathematical study of these processes may be found in Dynkin (1965). In discussing time-homogeneous Markov processes, we shall make frequent use of equations (2.2.1), which are known as the Chapman-Kolmogorov equations. To begin with, the use of equations (2.2.1) is illustrated through one simple stochastic process, the bulk-arrival Poisson process, which will be frequently used in this book. The Time-Homogeneous Poisson Process wherein Events May Occur in Bulk

Since this process (also called the compound Poisson process) occurs frequently in the subsequent chapters, it is appropriate to discuss it here briefly. First let us consider informally certain other things needed for its discussion (and also needed later). Let N ( t ) denote the number of customers arriving at a queuing system during the time interval ] 0, t ]. For each fixed t > 0 , N ( t ) is an r.v. but if t varies, N ( t ) represents a single-parameter stochastic process to be denoted by { N ( t ) ,t 2 0 ) ,or { N ( t ) )when the range of t is evident, or simply by N ( t ) for notational ease. The object here is to study the process { N ( t ) ,t & 0 ) under certain assumptions, which are stated in terms of technical statements and notations. We proceed to explain these before taking up the process itself.

if

lim g(t) -= 0 At

At+O

that is,g(t) tends to zero faster than At. Formally one may define o(g(x)) as follows. We write f ( x )= o(g(x))as x + x o whenever

In addition, it may be appropriate here, though it is not used until later, to introduce the big 0 as well. Let f ( x ) and g(x) be two real-valued functions defined on a set S of real numbers and assume that g ( x ) is nonnegative. We now write f ( x ) = O(g(x)) for x in S if there exists a positive constant K such that 1 f (x)l d Kg(x) for every x in S. Now we are ready to discuss the homogeneous Poisson process with bulk arrival. A stochastic process may be called a homogeneous Poisson process with bulk arrival if: 1 2

3

The process has stationary increments. The process has independent increments. Pik@) = Aqk-ih + o(h), k - i 2 0 , where A > 0 is the mean rate of occurrence of the homogeneous Poisson process. If X is an r.v. representing the size of batches arriving at each of the points of the homogeneous Poisson process, then qk = P(X = k )

Independence of Increments

-+ 0

with

- qk 1

= 1

k=1

Here the existence of X and qk is assumed.

Suppose that to < t l < t2 < . . . < t , < m. For every n > 1, if the increments N ( t l )--N(to), . . . , iV(tn)-N(tn-,) are mutually independent r.v.3, the process { N ( t ) ,t 2 0 ) having this property is called a process with independent increments. This expresses the fact that increments of N ( t ) over nonoverlapping time periods are independent r.v.'s.

We may also assume without loss of generality that to = 0 and let N(0) = 0. Define Po, ( t ) to be the probability that n events occur by time t , that is,

Stationarity of Increments

It may be noted that Pon(0)= S o n , where 60n is a Kronecker symbol. Moreover, we shall use the convention that Po, (t)= 0 whenever n is negative. Now from the above definitions,

+

If the distribution of N(t2 4-h ) -N ( t , h), h > 0 , is the same as that of N ( t z ) N ( t l ) , the process is said to have stationary increments, that is, the distribution h ) depends only on the length of the time interval t2 - tl of N ( t , h ) - N ( t ,

+

+

SOME TECHNIQUES OF QUEUING THEORY

44

Thus

which reduces, as h

-+ 0 , to

the differential equation ~ & p ( t )=

- hP,(t)

where

a notation that will be followed henceforth. Equation (2.2.3) has been derived after using conditions (1) and (2). On the same lines, we may write:

- h P o l ( t )+ XqlPoo(t) ) hq,Pm(t) ~ $ i ) ( t=) - XPo,(t) + h q l P o ~ ( t+

P$:'(t) =

2.2

TECHNIQUES AND MISTORV OF ANALYSIS

45

In the subsequent chapters we shall frequently meet with the bulk-arrival as well as the unit-arrival time-homogeneous Poisson process. The above results hold when h , the rate of occurrence, is independent of the parameter t and the state of the process. However, if X is time dependent, for example, X = h ( t ) , say, the process is known as nonhomogeneous in time. We shall meet with this case in multichannel queuing systems. For a detailed discussion of this process and various other processes connected with the basic unit-arrival time-homogeneous Poisson process, the reader is referred to Parzen (19621, Gross and Harris (19741, and Problems and Complements. It may be remarked here that for notational convenience, the first subscript 0 in Pon(t),which represents the initial state of the process, will normally be omitted, except when we are dealing with the transition probabilities of the process, in which case we shall continue to use both subscripts. A process that does not possess the Markov property is termed non-Markovian, In queuing processes discussed in this book, we shall consider both Markovian and non-Markovian processes. In general, the single-server system-length process { N ( t ) ) is non-Markovian and difficult to handle directly. N ( t ) can often be represented by a vector Markov process of the form { N ( t ) ,X ( t ) , Y ( t ) ) ,where N ( t ) = number in system at epoch t

X ( t ) = elapsed service time of a customer under service at epoch t Y ( t ) = time elapsed since last arrival. Defining the generating functions Q(z) and Po(z;t ) by

multiplying the above equations successively by 1, z , z 2 , . . . , and adding, we get (2.2.4 a) P$')(z;t ) = h { Q ( z )- l ) P o ( z ;t). This is an ordinary differential equation whose solution is easy to obtain and is pjven by Po(z; t ) = exp [ h ( Q ( z )- I ) t ] , where we have used the initial condition P0(z;O)= 1 . If q , = 1, qk = 0 , k 1, then we get the usual time-homogeneous Poisson process (with unit arrivals) with the p.g.f. exp [ X (z - I ) t ] ,the inversion of which gives Po,(t)=e-ht(ht)n/n!, h>O, n = 0 , 1 , 2 , . . . . Also, if instead of taking to = 0 , we start at to = s > 0 , then

+

P o ( z ; t - - s ) = exp [ X ( Q ( z ) - l ) ( t - s ) ] . This is due to the assumption of stationary increments of the process.

The vector Markov process is often more tractable than the original non-Markovian process {N(t)). The variables X(t), Y ( t ) are called supplementary variables (for details see below). In principle, it is possible to study multiserver queuing processes by introducing a sufficient number of supplementary variables; in practice the theory becomes more complicated. However, particular cases of both the singleserver and the multi-server queuing processes can be studied more elaborately by several techniques, some of which we proceed to discuss before illustrating their use in the chapters that follow. It should also be mentioned here that for singleserver queues with bulk input, in which the service times or the interarrival times, or both, have Erlangian distributions, some results have been obtained by studying the vector Markov process { N ( t ) ,X(t), Y ( t ) )(see Chapter 3). Phase Technique

This technique is essentially due to Erlang, although modifications have been made by various authors. We assume that service on a customer consists of k imaginary phases, which are mutually independent and exponentiaIly distributed with the common expected sojourn time I/p in any of the phases. A customer on arrival passes in sequence through all the k phases before it is discharged. After a customer leaves the server, a new customer is taken up instantaneously if one is waiting in the queue; otherwise the server remains idle. As pointed out in the first chapter,

46

SOME TECHNIQUES OF QUEUING THEORY

these distributions are known as k-Erlang distributions. For more details on the meaning of phases in particular cases, see Chapter 1. Gaver (1954) modifies this technique. He assumes that instead of a finite number k of phases there are (potentially) an infinite number of them, and further that an arrival demands j phases of service with probability cj. It has been demonstrated by Gaver (1954) and later by Luchak (1956) that it is possible to obtain (or approximate) a wide class of service time distributions of practical interest by varying { c j f . In this connection, see also Wishart (1959). In these papers the state of the system is defined by the number of phases in the system - either waiting or being processed. Jaiswal (1960) assumed the same modified Erlangian service time as Gaver (1954) and Luchak (19561, but in place of their phase process used a vector process ( N ( t ) , R(t)) as a system-length process. In Jaiswal's method, which we call a modified Erlangian method, N(t) is the number of customers (not phases), and R(t) is the number of phases of service that remain to be completed by the customer currently in service at epoch t . Since R(t) is the number of phases of service that remain to be completed, it is convenient to number the phases "backward." A customer, having entered phase j (1 <j < m) with probability c j , then moves after an exponential service time to phase ( j - I), and so on down to phase 1, and thence out of the system. The modified Erlangian method may be divided into two submethods, with N(t) being defined as the number of customers in the system in the first submethod, and as the number of customers in the queue in the second submethod. The difference between the two submethods will be important in some bulk-service queues.

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

47

t is normally replaced by n and N(t) by N,. Such a chain may be explained briefly as follows. Consider a physical system which is observed at a discrete set of points 0 , 1 , 2, . . . . Let the successive observations be No, N , , . . . ,N,, . . . , where N, is an r.v. Further, assume that each of these variables is capable of taking the values 0, 1, 2, . . . . The sequence {N,, n 2 0 ) is said to form a Markov chain if for all n, n = 1, 2 , . . . , the conditional distribution of N,, for given values of No, N, , . . . , N n - , , depends only on Nn-, , the most recent known value. In symbols this becomes

where io, il , i2, . . . , in- ,k take the values 0 , 1 , 2 , . . . . The system is said to be in state k at the n t h time point (or step) if Nn = k. Also if N n - I = i and N, = k, the system is said to have made a transition i + k at the nth time point. The conditional probabilities P(Nn = klN,-, = i j are called one-step transitional probabilities. These probabilities may depend on i, k, n, and n - 1. However, if they are independent of n, the discrete-time variable, then the Markov chain is said to be a homogeneous Markov chain, and in this case the one-step transitional probabilities may be denoted by

In the case of a homogeneous Markov chain, the n-step transitional probabilities are denoted by

rnbedded Markov Chain Technique

Before we explain this technique, it is essential to know about Markov chains which are a subclass of Markov processes in which the state space is discrete (finite or countably infinite), but the time parameter may be discrete or continuous. Essentially a Markov process is a Markov chain iff the state space is discrete. A discrete-state Markov process with discrete time parameter is known as a discretetime Markov chain, or simply a discrete Markov chain. Similarly a discrete-state Markov process with continuous time parameter is called a continuous-time Markov chain, or simply a continuous Markov chain. For example, the bulk-arrival Poisson process discussed earlier is a continuous Markov chain. We give here a condensed account of Markov chains. For further details see Feller (1968), Kemeny and Snell (1960), Bharucha-Reid (1960), or for a deeper study Chung (1967). The choice of state space of a Markov chain or process (that is, the set of values that N(t) may take corresponding to the values that t takes in the set T) is not uniquely specified by a physical phenomenon under consideration. However, usually one particular choice stands out as the most suitable. The same remarks apply to the time parameter. A case of immediate interest to us is the discrete-time Markov chain in which

and are independent of m. It may be noted that P i k ( I )r P i k . The one-step transitional probabilities are sometimes written by means of a matrix:

SOME TECHNIQUES OF QUEUING THEORY

48

We define for later use

Matrices whose elements satisfy the above requirements are known as stochastic matrices. In fact, these are infinite stochastic matrices (the number of rows and columns is m). In general it is more difficult to handle such infinite stochastic matrices than their counterpart finite square stochastic matrices. A systematic study of finite stochastic matrices is given by Romanovsky (1970) whose book deals with, among other things, discrete Markov chains. See also Seneta (1973). Homogeneous Markov Chains

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

49

chain is defined to be the greatest common divisor (GCD) of all integers n > 1 such that Pi&) > O . When the period is unity, that is, the GCD is 1, the state is called aperiodic. Also if Pii(n) = 0, for every n > 1, define the period as 0. Two states of the Markov chain are said to be of the same type if both have the same period or both are aperiodic; if both are transient or if both are positive recurrent, or if both are null recurrent. All states of an irreducible Markov chain are of the same type. As an example consider a Markovian queue in which the process {N(t)) is a Markov chain. In Markovian queuing problems with infinite waiting space, it is possible to show that the classification criterion depends on the traffic intensity p . If p < I , the states are positive recurrent, if p = 1, they are null recurrent, and if p > 1, they are transient. Intuitively, this concept may be explained by considering ansexample M ~ / G / I in which p = xatp

In what follows we consider only homogeneous Markov chains, and the adjective homogeneous will therefore be omitted. The Markov chain IN,, , n >, 0 ) is completely determined once the value of No (or more generally the distribution of No) and the one-step transition probabilities are known. A state k of the Markov chain is said to be accessible from state i, i-+ k, if there exists a number n >, 0 such that Pik (n) > 0. If two states i and k are accessible from each other, then they are said to communicate, i ++ k. A Markov chain is called irreducible if all its pairs of states communicate. Consider an arbitrary, but fixed state i. Let fii(n), n > 1 , be the probability that the system starting from state i returns to i for the first time in n steps. Notationally, fii(n) = P{Nn = i , N , f i,r = 1,2 ,..., n - l l N o

= i)

with fii(0) = 1. Note also that f i i ( l ) = Pii. The probability of the system eventually returning to i is denoted by

Classification of States of Markov Chain

A state i is said to be recurrent iff f z = 1; otherwise it is transient (also called nonrecurrent). The number of steps required for the first return to i is called the recurrence time, its expectation being the mean recurrence time pii, where pii = Z r = l nfii(n). A recurrent state can be further classified either as positive recurrent or null recurrent, according to whether pii < or pii = w. In order to study the limiting distribution of the Markov chain, we need to define periodicity of the states of the chain. The period of a state i of the Markov

where h = mean arrival rate

Z = mean of arrival group size

1/,u = mean service time Then p represents the expected number of customers arriving during a mean service time. Consequently, if p > I , then on the average more customers arrive per mean service time than the server can handle, and hence we could expect the system length to grow without limit. On the other hand, if p < 1, on the average there is less than one customer arriving per service time unit, and hence the server can cope with the load of customers. In this case the process reaches a steady state. The crucial case is when p = 1. In this case the mathematical system breaks down or degenerates, in the sense that all state probabilities become zero, or equivalently the system length becomes infinite. An irreducible aperiodic Markov chain possesses a limiting distribution limn,,. P(N, = k) = Pk , k 2 0, which is independent of the initial distribution P(N, = i ) , i > 0 . Two cases arise: 1

2

All states are null recurrent or transient. In this case Pk = 0 for every -+ rn, Nn w with probability I. All states are positive recurrent. In this case Pk > O for every k > 0 , and {Pk ) is a probability distribution such that Pk = ( p k k ) - ' . Thus to determine Pk we need to determine pkk which may not be easily determined. However, P's may also be determined uniquely as the solution of the following system of linear equations:

k > 0. As n

-+

SOME TECHNIQUES OF QUEUING THEORY

50

Equivalently any nonnull solution of the system of linear equations w

C xiPik = xk

with

i=O

ixkl

<

k=O

when normalized gives (Pk), that is,

Clearly the case of interest is case 2. Since the Markov chain possesses a limiting distribution {Pj), it possesses a stationary distribution. A Markov chain is said to possess a stationary distribution {Pk) if

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

51

Thus to study the process {N(t)) of the queuing system M / G / l , let ul , u, , . . . , o n , . . . be the epochs of departure of successive customers. If there is a departure at time t = 0 , we may put 00 = 0 and call it an initial departure epoch. Let V,, n = 1 , 2, . . . , be the successive service times which are i.i.d. r.v.'s with d.f. B(v) = P(Vn Gv). The sequence { V n ) is independent of the arrival process, which is Poisson with rate A. The service initiation is instantaneous as long as there are customers to be served. After an idle period during which there are no customers remaining to be served, neither in the queue nor with the server, let us suppose that the next count (increase in the index n) on the process occurs at the end of the service interval that begins with the first subsequent arrival. Define N; = N(u, + O), that is, N i is the number in the system immediately after the n t h departure or (if N: > 0) just before the senice on the (n 1)th customer starts. If there is a departure at 00 = 0, write Nof =Q(oo + 0). The stochastic process is said to be imbedded in the continuous-time process {N(t)). The process {Nz) forms a homogeneous Markov chain, as can be seen from the relation

+

{x)

If equation (2.2.5) holds, then where (x)+ = max (0,x) and x,+, is the number of customers arriving during a service period ending at o n + ,. The probability distribution of x,,, is given by Consequently, if the initial unconditional distribution is P(No = i ) = Pi, then, for every n , the unconditional distribution

In other words, a Markov chain that starts in steady state will remain in steady state. In closing it may be mentioned that positive recurrent aperiodic states are called ergodic states. We have stated several important results of the theory of Markov chains, for which proofs are given in some of the references cited earlier. We are now ready to explain the concept of imbedded Markov chain (IMC). This can be done by means of two examples. As a first example, let us consider the non-Markovian system-length process ( N ( t ) ) in the queuing system M / G / l . This process may be studied by considering the Markovian vector process {N(t), X(t)), where N(t) represents the number in the system and X(t) the unexpended (or alternatively the expended) service time of the customer currently undergoing service. In the relatively simple M/G/1 case we can give a full solution in continuous time for the vector process {N(t), X(t)), as we do later (in steady state) by means of the supplementary variable technique. Alternatively, we may obtain useful information about the M / G / l queue with less effort by studying the process N(t) at a suitably chosen discrete set of time points. The discrete-time solution can indeed be used as an approximation to the continuous-time solution. For a more complex model, such as the GI/M/c queue, an exact continuous-time solution may be excessively difficult, and a discrete-time solution may therefore provide the most practical alternative.

kj = ~ ~ P ( X . += , jlservice period = v) dB(v)

Here we have used the independence of the arrival process and the service times, and the fact that in a Poisson process the number of arrivals during an interval of time of length v , the service period, depends only on Xv. Also because of the independent increments of the Poisson process, the x's are i.i.d. r.v.3. Note also that X, is independent of No+,N;, . . . ,N i . The one-step transition probabilities

are given by

As a second example, consider the queuing system G I / M / l , a single-server queuing system in which the interarrival times of successive customers are i.i.d. r.v.'s and the service times are independently distributed exponential r.v.'s

SOME TECHNIQUES OF QUEUING THEORY

52

with mean 1/p. If N(t) is the number 'in the system at any time, then let N i = N(o; - 0), where oh = 0, and a;, a;, a $ , . . . are the epochs of arrivals of successive customers. The notation implies that N i is the number of customers in the system just before the arrival of the n t h customer, and it can be seen that

where D, represents the number of potential departures during the (n interarrival period. If {D,) are i.i.d. r.v.'s, their distribution is given by

+ 1)th

k j = j O m ~ (= ~ jlinterarrival , time = u) dA(u)

where A(u) is the d.f. of interarrival times. Now the process ( N i ) , which is imbedded in the continuous-time process {N(t)), is a homogeneous Markov chain which has the following one-step transition probabilities:

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

53

Integral Equation Technique

This technique has been used for finding the waiting time (in queue) distribution for queues with general input and general service time distributions and may best be explained by first considering the queuing system GI/G/1 and then its modified form. Later this technique will be used for some bulk queuing systems. Suppose that customers arrive at the instants 0 = ah, a;, cr;, . . . so that the interarrival times U, = a;+, - a;, n = 0, 1 , 2 , . . . are i.i.d. T.V.'S with common d.f. A(u) and finite mean. The customers are served individually by a single server on an FCFS basis. If V, is the service time of the n t h customer to be served, then it is assumed that {V,; n = 1, 2, . . .) is a sequence of positive r.v.'s with common d.f. B(v). The two sequences (V,) and {U,)need not be independent, but are often taken to be so for practical applications, and we shall assume that this latter condition holds. * Define v?) = V(o; - 0 ) so that v?) is the waiting time (in queue) of the (n + 1)th arrival (since first arrival is at ah = 0). It can be seen that the following recurrence relation between the r.v.'s and v?) for the queuing system GI/G/l holds:

~2'"

where The queuing systems M/G/I and GI/M/l, with bulk arrival or bulk service, will be studied in Chapters 3 and 4. One might think that in statistical equilibrium the limiting behavior of N(t) when t -+ or of N , (or N:) when n -+ should be the same. However, except in certain special cases, this is not so, as will be seen in the chapters that follow. The imbedded Markov chain technique is due to Kendall (1951, 19531, who gives a more precise definition based on the concept of regeneration points due to Palm (1943). Later this technique was widely accepted and applied to various queuing systems. We may explain intuitively the concept of regeneration process, which is more general than that of a process having an imbedded Markov chain. Points on a time scale at which a stochastic process restarts itself probabilistically are called regeneration points. In other words, if the continuation of a stochastic process { X ( t ) ]beyond a point T I ,which exists with probability 1, is a probabilistic replica of the process starting at 0 , so that {X(t T I ) ) has the same stochastic structure as { X ( t ) ) , then the process is called a regenerative process. The points such as T I , Tz , T 3 ,. . . are called regeneration points (the existence of the points T, , T 3 ,. . . is implied by the definition). As an example, note that a GI/G/l queue is a regenerative process with regeneration points at the start of each busy cycle. A consequence of this is that if we can solve a queuing model between successive regeneration points, we can consider that we have a full solution.

-

+

S n = Vn+l - U n + l . It may be remarked that when v?) + S, < 0, the (n + 21th customer will arrive to find the server idle after service on the (n + 1)th customer has been completed. The above relation may be rewritten in a compact form:

where (x)' = max (0, x),

-w

<x <

m.

We note that the stochastic process {v?)) is a discrete-time Markov process, since the behavior of v?") is dependent on v?) but is otherwise independent of the past history of waiting time. By virtue of the assumptions we have made about the interarrival time and service time distributions, the S,'s are i.i.d. r.v.'s such that E(S,) is finite. Also S, and v?) are independent. Now using probability reasonings, we can see that

+

= P ( V ~ S,

< t).

SOME TECHNIQUES OF QUEUING THEORY

54

y use of the convolution formula.and the independence of

VP)and S ,

we

2.2

TECHNIQUES A N D HlSTORY OF ANALYSIS

5!

equation (2.2.7) so that it has values for all t. Let W,*(t)be defined by the equation'

get where

S,(x) = P(S, G x ) .

Lindley (1952) has shown that limn,, w i ( " ) ( t )= W i ( t ) exists and is a unique proper distribution iff p < 1, as one should expect intuitively. For the proof of existence of the limit, one may alternatively see Prabhu (1965a) or Karlin and Taylor (1 975). Using W i ( t ) as the common limit of ~ i ( ~ ) and ( t )W,("+')(t) as n -+ m, we get

or equivalently

c4

W&t) =

-

0-

W&X)dS(t - x ) ,

t P0

(2.2.8)

where S ( x ) is the common distribution of { S n ( x ) ]and is given by

Note that equation (2.2.10) is defined for the whole real line, whereas equation (2.2.7) is defined only for nonnegative values of t. We discuss equation (2.2.10: further as follows. First (when t < 0 ) we see that

Substituting t - x = u and x 3.z = v, we get

where

= ceAt, t
= a constant

+

S ( x ) = ~ o m ~ (z )xd A (z). It should be noted that equation (2.2.9) can be written in many forms. In practice, the integral is to be evaluated so that the inequalities z P 0 and x z P 0 are satisfied. The integral equation (2.2.7) or (2.2.811 is of the Wiener-Hopf type, which requires sophisticated mathematical tools for a solution in its most general form. For methods of obtaining particular solutions in certain special cases, see for example Feller (1971), Prabhu (1965a), Gross and Harris (1974), and Kleinrock (1975). We give below Lindley's solution for the M/G/l queue. It may, however, be remarked here that this approach does not yield practical solutions to all GI/G/l queuing systems. [In this connection, see also Smith (1953)].

+

e now employ Lindley's approach to obtain the waiting-time distribution of the queuing system M/G/I. Let the d.f.'s of the senice time Y and the interarrival time U be B ( x ) and A ( x ) , respectively. Both B ( x ) and A ( x )vanish for x < 0 . Now using equation (2.2.91, ~ ( x =) h L w ~ ( x z)e+' dz.

>0 ,

and we have used B ( t ) = 0 for t < 0. Second, W,*(tj = Wq(tj,

t > 0.

Finally, we can rewrite equation (2.2.10) as follows:

At t = 0 the limiting value of W z ( t ) is Wq(O+) when approached from the right (through positive values of t ) and C when approached from the left (through negative values of t). It can be shown that C = Wq(O"), as shown by, for example, Kleinrock (1975). The solution is now obtained by taking the two-sided L.-S.T. of equations (2.2.10) and (2.2.1 1). Using equation (2.2.1 I), we have

The two-sided L.-S.T. of the right-hand side of equation (2.2.10) is Gq(cu)P(a), which is the product of the individual transforms of Wq(t) and S(t), namely,

+

Our interest now is t o solve the integral equation (2.2.7) for t P 0. But the integral on the right-hand side of equation (2.2.7) does not have a value for t < 0. In order to get around this difficulty, we redefine the integral on the right-hand side of

*When the arrivals follow a Poisson process, W;(t) is the same as Wq(t)Hence we shall not distinguish between them and instead write Wq(t).For details on this remark, see Section 2.1, Miscellaneous Problem 4 of Chapter 3, and so on.

SOME TECHNIQUES OF QUEUING THEORY

54

By use of the convolution formula and the independence of get

~ ; ( " + ' ) ( t )= where

lo-Witn)(t-x)~s,,(x),

tZ 0

v?) and Sn we

Lindley (1952) has shown that limn,, w i f n ) ( t )= W;(t) exists and is a unique proper distribution iff p < 1, as one should expect intuitively. For the proof of existence of the limit, one may alternatively see Prabhu (1965a) or Karlin and Taylor (1975). Using W,(t) as the common limit of w i C n ) ( tand ) w;("+')(t) as n -+ -, we get

Wi(t) =

-

Io: W;(x)dS(t - x ) ,

t P0

TECHNIQUES A N D HISTORY OF ANALYSIS

55

equation (2.2.7) so that it has values for all t. Let W:(t) be defined by the equation*

(2.2.6)

S n ( x ) = P(Sn <x).

or equivalently

2.2

(2.2.8)

where S ( x ) is the common distribution of { S n ( x ) ]and is given by

It should be noted that equation (2.2.9) can be written in many forms. In practice, the integral is to be evaluated so that the inequalities z Z 0 and x + z 2 0 are satisfied. The integral equation (2.2.7) [or (2.2.8)l is of the Wiener-Hopf type, which requires sophisticated mathematical tools for a solution in its most general form. For methods of obtaining particular solutions in certain special cases, see for example Feller (1971), Prabhu (1965a), Gross and Harris (1974), and Kleinrock (1975). We give below Lindley's solution for the M/G/1 queue. It may, however, be remarked here that this approach does not yield practical solutions to all GI/G/l queuing systems. [ I n this connection, see also Smith (1953)l.

Note that equation (2.2.10) is defined for the whole real line, whereas equation (2.2.7) is defined only for nonnegative values of t. We discuss equation (2.2.10) further as follows. First (when t < 0 ) we see that

W:(i) = h Substituting t - x = u and x

where

I t

x=-_

~z~oWq(t-~)e-h~dB(~+z)dz.

+ z = v, we get

= ceht,

C = h

t
IomJom~ ~ ( u ) e - ' ( ' +dB(v) ~)

= a constant

du

>0 ,

and we have used B ( t ) = O for t < 0. Second, W,*(t) = W,(t),

t 2 0.

Finally, we can rewrite equation (2.2.10) as follows:

At t = 0 the limiting value of W:(t) is W,(O +) when approached from the right (through positive values of t ) and C when approached from the left (through negative values of :). It can be shown that C = W,(O+), as shown by, for example, Weinrock (1975). The solution is now obtained by taking the two-sided L.-S.T. of equations (2.2.10) and (2.2.1 1). Using equation (2.2.1 I), we have

The System M/GA

We now employ Lindley's approach to obtain the waiting-time distribution of the queuing system M/G/l. Let the d.f.'s of the s e ~ c time e V and the interarrival time U be B ( x ) and A ( x ) , respectively. Both B ( x ) and A ( x ) vanish for x < 0. Now using equation (2.2.9), ~ ( x =) A j o w ~ ( x+ z ) e-*' dz. Our interest now is to solve the integral equation (2.2.7) for t 2 0. But the integral on the right-hand side of equation (2.2.7) does not have a value for t <0. In order to get around this difficulty, we redefine the integral on the right-hand side of

The two-sided L.-S.T. of the right-hand side of equation (2.2.10) is G,(a)F(a), which is the product of the individual transforms of Wq(t)and S ( t ) , namely,

* When the arrivals follow a Poisson process, W&) is the same as W q ( t ) Hence we shall not distinguish between them and instead write Wq(t).For details on this remark, see Section 2.1, Miscellaneous Problem 4 of Chapter 3, and so on.

SOME TECHNIQUES OF QUEUING THEORY

56

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

57

Note that this is a generalization of the result (2.2.6). Following Lindley (1952), Finch (1959) has shown that for the queuing system under consideration the limiting waiting-time (in queue) d.f., namely,

Equation (2.2.1 2) then gives

W&t) = n-+lim P ( v ~ < ) t) and thus, since f(a) = b(a)A/(h - a),

exists, is a proper d.f. (if p

< 1) ,and is the unique solution of the integral equation

The constant G is found by using the normalizing condition i3,(0) = 1. Since E(U) = h-' and - E ( V ) = lim,,o [- 1 $(a)] /a, we have

+

where

The result (2.2.13) has been obtained by several methods by many authors, and is now popularly known as the Pollaczek-Klzintchine transform formula. Let us now consider a generalization of Lindley's results to the queuing system GI/G/1 with the following modifications. In GI/G/1 it is assumed that if the nth customer arrives to find the server idle, service begins instantaneously. Now relax this restriction and assume that when the nth customer arrives to find the server idle, his service commences after a random time T,, where the (T,)'s are i.i.d, with mean d and d.f.

In this case the following recurrence relations between the r.v.'s and Tn can be seen to hold:

v?+')and v?)

The sequence (S,) is independent of the sequence (T,). Since v;")> 0, for every n, P(v;") < t ) = 0 for t < 0. For t > 0, we find from the above relation


w ; ( n + y t ) = p(v;n+"

+ S , < 0, T,+% > t ) ,

Since T,+, is independent of S, and V r ) for r = 1 , 2 , . . . , n, we obtain t
t > 0.

Supplementary Variable Technique

This technique was used in 1942 by Kosten [see Kosten (1973)l. The name appears to be due to Cox (1955), who used a supplementary variable to study the queuing system M / G / l . Kendall (1953) considered this technique, which he labeled "augmentation", but preferred the use of an imbedded Markov chain as leading to simpler calculations. In spite of this, supplementary variables have been used by many authors to solve a good number of queuing problems. Jaiswal's (1968) book on priority queues makes heavy use of supplementary variables. While Kendall's imbedded Markov chain technique is very powerful and elegant, it gives only approximate solutions to queuing problems considered in continuous time. However, in the steady state the approximation seems to be pretty good, and the mathematics involved become less cumbersome. Even in the steady-state case, many problems are more readily treated by the supplementary variable technique than by the imbedded Markov chain technique. See, for example, Regis (1973). Waiting-time distributions in queues with Poisson input are easily obtained by the supplementary variable technique. For details, see the discussion of waiting-time distribution for the system M/GB/l in Chapter 4, for the system M ~ / G /in~ Chapter 3 , and so on. In the supplementary variable technique a nonMarkovian process in continuous time is made Markovian by the inclusion of one or more supplementary variables. To illustrate this point, first consider the queuing system M/G/l studied by Cox (1955). In M/G/I, since the service-time distribution is general, let the state of the system be defined by a pair of variables, the number N ( t ) in the system [or Nq(t) in the queue] at epoch t, and the elapsed service time X ( t ) of the customer who is undergoing service. Then we may study the bivariate Markovian process { N ( t ) , X ( t ) ) in order t o obtain results for the non-Markovian process {N(t)}.Thus Cox defines P,(v, t ) to be the joint probability and p.d.f. of n, the number of customers in the system, including the one being served, and v the elapsed service time of the customer in service. The inclusion of a single supplementary variable makes the process Markovian in continuous time. The equations of the process can be written down by considering, in the usual way, the transitions occurring in time At. In this

SOME TECHNlQUES OF QUEUING THEORY

58

case ~ ( vdv ) is taken as the probability that the service is completed in the interval ] v, v dv] ,conditional upon its remaining incomplete up to time V . We illustrate the use of this technique by discussing the limiting behavior of the queuing system M/G/l. In Chapter 3 we use this technique to solve a typical bulk-arrival priority queuing system and in Chapter 4 the system M / G ~ / Iand its generalizations are considered using the same technique. In M/G/1customers arrive following a Poisson process with mean rate X and are served one at a time on an FCFS basis. Service times are i.i.d. r.v.'s whose p.d.f. (see Chapter 1, Problems and Complements) is

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

+

with mean l / p , 0 < 1/p = JTvb(v) dv < m. Also note that we have shown that the corresponding d.f. is given by

An equilibrium solution exists when p = X/p < 1 . In this case we can put the limiting derivatives with respect to t equal to zero; and by letting lim,,, Po(t) = P o , we get Pn(x) and lim,,,

1 Pl(x)q(x)dx

P,(x, t

)

m

0 = -kPO 4-

.o

(2.2.15)

These equations are to be solved under the so-called boundary conditions m

P ~ ( ~ ) = ~ P ~ + ~ ( X ) V (n X > l) ~ X >

(2.2.1 7 )

a result we shall use in the sequel. Let us define:

I

2

P,(x, t ) dx 3. o(dx), n > 0 as the probability that at time t there are n customers in the system with the elapsed service time of the customer undergoing service lying between x and x + dx. Po(t) as the probability that the system is empty at time t.

Since (N(t), X ( t ) ) is Markovian in continuous time, one can write the equations of the process in the usual Erlangian procedure by considering the transitions occurring in At. First we may note that P,(x, t ) = 0 or in the equilibrium case Po(x)= 0. Then considering the empty state, Po(t + At) = Po(i)(l h a t )

+ (1 - h a t )

so that

-- dPO(t) dt

- XPo ( t ) +

I-

J m p , ( x , t ) q ( x ) dxAt

+ o(At),

or

0

+ At) -Pn(x + At, t ) At

Then

and

+ At, t + At) = (1 - XAt)Pn(x, t ) [ l - q ( x ) A t ] + hAtPn-,(x, t ) [ l --q(x)At] + o ( A t )

Pn(x + At, t

The solution now can be obtained in closed form with the help of a p.g.f., Po@;x ) , which we define as

pl ( x , t )Q ( x ) dx.

Similarly, we get Pn(x

and the normalizing condition

+

Pn(x + At, t ) --Pn(x, t ) At

We multiply equation (2.2.16) by z n , add from n = 1 to m, use equations (2.2.20) and (2.2.21), and get ap0(z;x) -----ax

[hz - X

- q ( x ) ] Po(Z;x).

Similarly from equations (2.2.17), (2.2.18), (2.2.221, and (2.2.15) we get

~

O'l "I

"s

t;. A

N

w w

Crr

99.

-

SOME TECHNIQUES OF QUEUING THEORY

~ : = ~ J ~ ~ ~ + ~ ( x ) v ( x )nd> O x , where D is a normalizing constant to be determined later. The p.g.f. of P; is then given by ce (1 - p)(z - 1) 6(h - hz) (2.2.27) P'(z) = C P:zn = n =O 2 -6(hhz) The result (2.2.27) is the same as the result (2.2.25) and establishes the assertion P i = P,. To prove equation (2.2.27), we proceed as follows:

2.2

TECHNlQUES AND HISTORY OF ANALYSIS

63

on such relations when the arrivals or services are in bulk, see Chapters 3, 4, and 6. The Busy-Period and the Idle-Period Distributions for the M/G/1 Queue

The busy-period distribution for the queuing system M/G/l could be derived by continuing the discussion of the systemM/G/l through the supplementary variable technique [see, for example, Jaiswal (1968) or Problems and Complements]. However, the following excellent method due to Takbcs (1962) is more elegant. Since the busy period is independent of the order of service (the waiting-time distribution is not), we may assume LCFS queue discipline. Let T be an r.v. representing the busy period of an M/G/I queuing system, which starts with the arrival of a customer. Let G(t) be the distribution function of T, and let V denote the service time of the arrival with its d.f. B(v). Then sin= each arrival during the initial service time of the busy period will generate its own busy period, we may write T = V + T I + T2 + . . . + T N , where Ti, i = 1, 2 , . . . , N, is the r.v. representing the busy period generated by the ith arrival that arrives during V, and N is a r.v. representing the number of arrivals that arrive during V. Now as Ti's may be supposed to be independent of N and V and also to be independent r.v.'s identically distributed as T, we have by conditioning on V and N

The constant ADPo is obtained by the normalizing condition P+(1-) = 1, giving hDPo = 1 - p. Thus P+(z) is completely known as stated in equation (2.2.27). It may be noted that the constant D is really l/h, the reciprocal of the mean input rate A. Waiting-Time (in Queue) Distribution for the M/G/l Queue

Since we are considering a queue with FCFS queue discipline, the customers left behind by a departing customer are just those that arrive during the time the customer was delayed in the system (queue service). Since arrivals are random, it follows that P+(z) = %,(X - hz) i;(X - Xz)

+

where %,(a!) is the L.T. of the distribution of queuing time. We have used the independence of the service time and the queuing time of the customer, and the argument used in the derivation of equation (2.2.26). Using the transformation h - hz = a! and equation (2.2.27), we finally obtain

which is the L.T. of the distribution of the actual waiting time (in queue). Because of Poisson input, the virtual waiting time distribution (discussed in Miscellaneous Problems and Complements) would be the same. The result P; = Pn was first proved for M/G/l by Khintchine (1932) and later by several other authors. The derivation given here appears to be new. For details

where { ~ ( t ) ) " *is the n-fold convolution of G(t) with itself in which (G(t - v)l0* = 6 (t - v) corresponds to no arrival during the initial senice time of the busy period. If g(a) and g(a) are the L.-S.T.s of G(t) and B(v), respectively, then by taking the transform of the above equation, we find that

where we have interchanged limits. Now making the transformation t - v = y and then using the convolution property, we get

SOME TECHNIQUES OF QUEUING THEORY

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

65

d.f. 1 - e-", x > 0, with mean E ( I ) = 1/A, where lis the length of the idle period. In view of this the mean length of the busy-period distribution for the queuing =

:/

exp [- ( A 4- a -

u] dB (v)

This is a functional equation, inversion of which is difficult except in the simple case 6 2 M . For more details regarding. its inversion, one may refer to TakCcs (1962). However, one can easily obtain moments from it. For example, the mean length of the busy period is

system M/G/I can also be easily obtained without the use of the transform. For details, see Section 2.3. Next consider the system GI/M/I. Instead of service time being general as in M/G/I, we now have the assumption of general independent input. In this case the supplementary variable Y ( t ) , which measures the time since the last arrival, removes the non-Markovian aspect of the system, and the system is solvea by means of the backward Kolmogorov equations. Readers not familiar with renewal theory will understand the discussion below better if they first read the section on renewal theory. The System Gl/M/1

which gives

1 I-(-

h'

i f p = -h- < I I-1

E(T) = O",

ifp2l.

From this one can infer that (1) if p < 1, the busy period terminates with probability P and has a finite mean; ( 2 ) if p = 1, the busy period terminates with probability 1, but has infinite mean; ( 3 ) if p > I , the busy period may not terminate at all. This corresponds to positive recurrent, null recurrent, and transient cases in Markov chain analysis. Although we have considered the busy period starting with the arrival of a singe customer, the busy period may also start with the arrival of a group of size r > 1 (r fixed) of customers. Then one may conceive each customer of the group generating its own busy period in an independent way, and consequently one may write the L.-S.T. of the distribution of the busy period c ( a ) , generated by a group of size r, as ?,(a) = [g(a)]' = [6{a+ h - hg(a))] with & ( a ) r g(a). We mav also interpret - E ( a ) as the transform of the duration of the busy period initiated by the existing r customers in the queue, which is called the initial busy period. If arrivals are single, the other busy periods will have transforms g l ( a ) or g(a), as discussed earlier. It is an easy matter to find the idle-period distribution for the queuing system M/G/1, for when the busy period terminates, an idle period must start, which in turn terminates as soon as the next customer arrives. In the language of renewal theory (see later), an idle period may be thought of as the residual interarrival time. Thus using the characteristic property of the exponential, the idle period has

AU the assumptions in the discussion of the queuing system GI/M/l are the same as those in the system M/G/l, except that interarrival times are now arbitrarily and independently distributed with d.f. A(t), p.d.f. a(t), t 2 0, and 1 / X = :j ta(t) dt, and service times are independently exponentially distributed with p.d.f. ~ e - ~ ~ , p>O,t>O. Since the limiting behavior of the system does not depend on initial conditions, we may formulate the model with any convenient initial conditions before taking limits as t -+ -. We choose the simple initial conditions N(0 -) = 0 and N(0) = 1. Thus t = 0 is an epoch at which a customer joins an empty system and immediately enters service. Throughout our discussion, it will be assumed that p = < 1. Let us now define the following probabilities and probability densities:

= P { N ( t ) = n and last customer arrived in

In words, P,(t, y ) is the joint probability and the probability density for the system state in which there are n customers in the system [one in service and (n - 1) in the queue] ,the last customer having arrived at ( t -y). Related to the above probability is the probability P,(t, 0 ) dt + o ( d t ) , which is associated with the arrival of a customer in the interval ] t - d t , t ]. The joint probability and probability density Pn(t, 0), which is denoted by P:(t), has meaning only for n 2 1, for it refers to the case when there are n in the system after an arrival. Clearly then Pt(t)=O. P:(t) is the rate at which customers join the system in state (n - 1). In words, P,(t) is the probability that there are n in the system, whatever may be the time of arrival of the last customer.

2.2

SOME TECHNIQUES OF QUEUING THEORY

66

i

n=l

wi)y

dy

Equation (2.2.33) is the solution of equation (2.2.31a) which holds for n 2=2, but we have assumed that equation (2.2.33) holds for n = 1 as well. The reason for this assumption will become apparent later. Since by equation (2.2.30) 2% Pz must be a convergent series, only those yi for which lyil < 1 can be included on the right-hand side of equation (2.2.34). Now using Rouchk's theorem (Appendix AS), it can be seen that equation (2.2.32) has only one root inside the unit circle lzl = 1 when p = hli. < 1. Let this root be y,. Consequently we may write equation (2.2.33) as

Note that equation (2.2.29) is simply the ordinary renewal equation given later as equation (2.3.9a) with m(t) = Pa(t). We may therefore appeal to renewal theory for a rigorous proof of the intuitively reasonable result

t-+-

= JOma(y)e-r(l

Equation (2.2.32)is called the characteristic or operator equation. Consequently, if y,,i = 1 , 2, 3 , . . . , are the roots of equation (2.2.32), then the elementary theory of difference equations (see Appendix A.9) gives the solution *

P2(t) then represents the renewal density related to the event of an arrival of a customer in the interval ] t - dt, t] . Pa(t) is related to the interarrival density a(t) through the integral equation

x-

67

interval after the arrival in (I) is of lengthy; and (3) m customers depart during the interval y, with a customer in senice at the end of the interval y . This result summed over all m = 0, 1 , 2 , . . . and any y , 0 < y < t, gives the required probability and probability density PP,(t) of the left-hand side. We now proceed to solve the set of equations (2.2.31a). Replacing P: by z R in equation (2.2.31a), we get

The inclusion of the single supplementary variable Y = Y(t), which is measured backward from t to the epoch of arrival of the last customer, makes the queuing process {N(t), Y(t)] for the GI/M/l queue Markovian in continuous time. Thus one can write the equations of the process by using the backward Kolrnogorov equations rather than the forward ones, which were used while discussing the process M/G/l. First we investigate P:(t), n > 1, and its limiting behavior before discussing the other probabilities. Let

lim Pa(t) = Pa =

TECHNIQUES AND HISTORY OF ANALYSIS

PE = h

where l / h = jzu dA(u). We now proceed to discuss the probabilities PP, = limt+, P:(t) for n > 2. The integro-difference equations for these probabilities are given (for any t) by

Using equation (2.2.30), we finally write P; as

P i given in equation (2.2.34) gives us the limiting probabilities just after an arrival. It may be stated that PP, is an arrival rate and not a true distribution, as C P: is h. From equation (2.2.34) one can, of course, get the true distribution of the number just before an arrival. or in the limit as t -+ -,

Number in System just Before Arrival, P,

CQ

Pi =

P:-l+m J m=o

~

~

~ ( m!

~ dy,)

( n~ > 2). ~ (2.2.31a) ~ - ~

Since the first and the second equations of (2.2.31) may be explained similarly, we choose to explain the second. Its left-hand side is the probability that the state of the system has just increased to n@ 3) by an arrival at epoch t. The term P:-, +m(t-y)a(y)(py)me-pY/m! on the right-hand side is the joint probability and probability density which is obtained by considering three cases: (I) the state of the system increases to (n - 1 m) at (t -y) by an arrival: (2) the interarrival

+

-'

~

We now proceed to find the distribution of the number in the system GI/M/l just before an arrival epoch (imbedded Markov chain). Since

where D is a normalizing constant, we have, using C;P=, P; = 1,

Equation (2.2.35) gives the distribution of the number in the system just before an arrival epoch, and this can easily be used to get the waiting-time distribution.

SOME TECHNIQUES OF QUEUING THEORY

68

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

69

Waiting-Time (in Queue) Distribution fo; the Gl/M/l Queue

Let w;(T) d~ be the probability that the waiting time Vq of a customer lies between ] 7, 7 + d ~ . ]Then using the characteristic property of the exponential distribution and the FCFS queue discipline, the L.T. @,(a) of the p.d.f. is given by

where (I - y l ) is the concentration of probability mass at the origin. Equation (2.2.36) on inversion gives P [ V q = 01 = (1 - y,) and .,? >o. (2.2.37) w,(T) = E I Y I (- ~- Y I )- ~ 1 - 7 7 , Number in System at Random Epoch, P,

Let {P,) represent the limiting distribution of the number in the system GI/M/l at a random epoch, To get the distribution {P,}, we first get the limiting joint distribution and then

We now derive a probability expression for P,(y) in terms of P,". Recollect the definition of P,(y) in equation (2.2.38), which represents the limiting probability and probability density that the number in the system is n and the last customer joined the system y time units before the current epoch. Define A C ( y )= Piinterarrival time > y ) = f; dA(t). Since during the period y only departures can occur, adapting the argument used to get equation (2.2.3 la), we have

Thus For an alternative derivation of Po using Po(y), see Problems and Complements. Equations (2.2.42) and (2.2.43) completely determine the distribution of the number in the system at a random instant of time. Distribution of Virtual ldle Period and Actual ldle Period for the GI/M/1 Queue First we obtain the virtual idle-period distribution for the GI/M/I queue, which can be found by using some of the analysis discussed earlier. Then the actual idle-period distribution can be deduced from the virtual one. For an alternative and elegant denvation of these distributions, see Miscellaneous Problems and Complements. The procedure which is used here can be extended easily to get the idle-period distributions for the bulk-arrival queue GIr/M/l. For details of this, see Miscellaneous Problems in Chapter 6. Recall that for the M/G/1 queue the idle-period distribution was simple and easily obtained, while the waiting-time distribution was more complicated in form and more difficult to derive. It is therefore not surprising to find that for the GI/M/l queue, the dual of the M/G/l, it is the waiting-time distribution which is simple and easily found. The idle-period distribution can be found by a somewhat more complicated argument given below. Recall that I is an r.v. which represents the actual idle period, and that I, is an r.v. which represents the virtual idle period. The distribution of I, consists of two parts given by P(IV = 0 ) = 1 -P (arrival finds system empty)

The distribution of I is related to that of I, by the following relation:

Using equation (2.2.341, we get

p,(y) = M C ( y ) ( l- y,) yy-' ee-pY(l- 7 1 ) )

n

> 1.

(2.2.41)

or eauivalentlv.

Putting equation (2.2.41) into equation (2.2.39) gives, on integration,

~ , = ~ ( ~ - y ~ ) y ; - ' ,n > l where we have used equation (2.2.32) to obtain

The term Po is obtained by the normalizing condition

(2.2.42)

In some situations it is convenient to use equation (2.2.45) rather than the equivalent form (2.2.44). We now derive the distribution of I,,, and thus encounter the same difficulty as in the derivation of the waiting-time distribution for M/G/l. As the waiting-time distribution for M/G/l (see Miscellaneous Problem 4 ) depends on the residual service time of the customer undergoing service, so the idle-period

SOME TECHNIQUES OF QUEUING THEORY

70

distribution for GI/M/l depends on how long an interarrival period has already elapsed since the start of the idle period. Now a(r y)/AC(y) represents the probability density that an interarrival interval lasts for time T y , given that it did not end before or at time y ; Po(y) represents the limiting joint probability and probability density that the system is idle at some instant of time (all customers having departed during y). Hence we have, if we define F&(T) to be the probability that the system is idle at some instant of time and that the idle time thereafter is

+

+

To evaluate the integral in some compact form, we need to know first the value of P,(y). This may be evaluated from

2.2

TECHNlOUES AND HISTORY OF ANALYSIS

71

case and the distribution {P,} given in equations (2.2.42) and (2.2.43) are contained in Conolly (1958). ConoUy (1958) derives Pn by first considering the transient solution and then takes the L.T. and the limit. Our procedure here has been based on several results due to Conolly (1960, 1975). We have attempted to unify many of the results for the queuing system G M 1 . Some of the results discussed here are obtained later in this chapter and also in other chapters using other techniques. In this connection see Cohen (1980), Prabhu (1965a) and the references therein, among others. Wishart (1961) has solved the system M/G/I by defining the supplementary variable as the time to service completion instead of the time since the beginning of the last service. Henderson (1972) extends Wishart's work by giving a transient solution for n/l/G/1 using the time to service completion as supplementary variable, and also discusses GI/M/I using the time to next arrival as supplementary variable. In this connection, see also Cooper (19&1). Discrete Time Analysis

where Pn(y) is given by equation (2.2.41). Note that since E;=, P,(y) represents the p.d.f. of the past life of an arrival (see Section 2.3), it must be X[1 -A(y)]. Using equation (2.2.41) in equation (2.2.47), we get

For an independent proof of equation (2.2.48), see Problems and Complements. Ignoring the norming constant h and using equation (2.2.48) in equation (2.2.46), gives = a ( y T) dy - j m e - ~ ( l * ~ l ) yya ( T) dy F;,(T)

jm + 0

0

+

Considering the complement, we get

which is the d.f. of I,. Note that equation (2.2.49) is true for T = 0. For when T = 0, F,,(O) = P(I, = 0) = y l, as it should be. See the definition of P(I, = 0) given earlier. An alternative derivation of the d.f. of I, is discussed in Problems and Complements. The moments of I, may be obtained by taking the L.T. of I,, as discussed in Problems and Complements. The distribution of I may be obtained by using equation (2.2.45) or equation (2.2.44). In particular, note that for the system

Most studies of queuing systems either consider time as a continuous parameter or define a convenient set of discrete points in time (imbedded Markov chain case). However, in practice, situations do arise when events occur at discrete points in time which may not coincide with the time points of the imbedded Markov chain. Such systems may be found in electronic installations whose operations are governed by internal clocks, or missile bases which fire at oncoming airplanes at somewhat regularly spaced intervals of time. By a discrete-time queuing system we mean a system in which customers arrive or are processed only at fixed time points separated by uniformly spaced fixed intervals of time., Thus arnval of customers or their processing is initiated and completed in an integral number of periods of time. Mathematical formalism required for the study of discrete-time queuing systems is analogous to that required for the continuous-time case. However, from the practical point of view it may be worthwhile to study discrete-time queuing systems in order to show the modifications needed in their study, and the ease with which computer calculations can be handled. For more details of the advantages of discrete-time queuing models, see, for example, Dafermos and Neuts (1971). We shall study some discrete-time queuing systems with bulk service or bulk arrival, which have been discussed in the literature. The study of queuing systems in discrete time was begun by Meisling (1958), who studied the system Geom/G/l and in a way used a hidden Markov chain. We shall consider Meisling's (1958) approach while discussing the system Geom/GB/l. The geometric distribution is used here as the discrete analog of the negative exponential, whose memoryless property it shares. Similarly the binomial distribution is the discrete analog of the Poisson distribution. Conservation Principle

as it should be. The distribution of the number in the queuing system GI/M/1 in the transient

Conservation principles play a fundamental role in several fields, such as physical sciences, engineering, and economics. The ideas of conservation of mass, energy, momentum, and charge often provide us with a system of equations which can, in

72

SOME TECHNlQUES OF QUEUING THEORY

many cases, be easily solved to give the desired information. The principle of conservation states that in equilibrium the flow must be conserved in the sense that the input flow must equal the output flow. This principle can be used to ascertain some limiting characteristics of queuing systems. We use this principle to discuss a few of the simple queuing results in the limiting case. For applications of the principle of conservation to some other queuing models we refer the reader to Kleinrock (1976) or Krakowski (1973, 1974). However, the reader is warned here that the result (S.23) in Krakowski (1974) appears to have been incorrectly reported. The correct version of this may either be obtained by taking correctly the probability of the position of a customer within a group [see, for example, Problem 15(e)] or else it may be seen in Chapter 3, where it has been derived using other techniques. For the bulk-arrival queuing systems M X / ~ / 1or GIX/M/l with mean group size a, the mean arrival rate is hz, and the mean service (or departure) rate when the system is occupied is p. Since the mean departure rate p is conditional upon the server being busy, we have, in the limiting case, using the fact that the average input is equal to the average output,

2.3

BASIC RENEWAL THEORY

which are being used by a certain machine or equipment. Also suppose that as soon as the item under use fails, it is replaced by a similar item from the stock. If the life lengths U,, n = 1, 2, 3, . . . , of the items are i.i.d. r.v.'s, then N(t), which represents the number of items renewed (or replaced) during the time interval ] 0, t ] ,is a renewal process. This section is largely self-contained. It discusses that portion of renewal theory which is needed for the study of bulk queuing theory, and some other basic properties of renewal theory. These properties are best discussed through the L.-S.T. (or the L.T. when the r.v.5 are continuous, as we shall assume here). For the discrete case we refer the reader to Feller (1968), Prabhu (1965b), or Neuts (1973). For a more detailed study of renewal theory in the case of continuous r.v.'s see Cox (1962), Feller (1971), or Karlin and Taylor (1975). It may also be remarked here that a renewal process is called by some authors a recurrent process [see Feller (1968) and Takics (1962)l. Let the renewals occur at instants of time a : , a;, . . . and suppose that Un = I unr - un-, ,n = 2 , 3 , . . . , are i.i.d. r.v.'s with common distribution A(u)=P(Un
that is,

Further, let U1 = u', which is the probability of the server being idle for both i W X / ~ / land GIX/~W/1. In subsequent chapters many of the results of this section will be derived using other techniques.

As we have remarked earlier, queuing theory originated with the probabilistic study of problems connected with telephony. Similarly, renewal theory was first applied to problems connected with the failure and replacement of equipment. Later, however, in renewal theory, as in queuing theory, models developed for one type of application turned out to be useful for other applications. Thus renewal theory can be applied to nuclear particle counters and to queues. Renewal theory deals with the study of renewal processes. A process (N(t), t > 0 ) whose state space belongs to a denumerable set {O, 1 , 2 , . . . 1and for which , n = 1, 2, 3, . . . ;oh = the interarrival (or interoccurrence) times U, = a; 0 between successive arrivals (or occurrences) are positive i.i.d. r.v.'s, is called a renewal process. Clearly, as N(t) counts the number of arrivals during a time period ] 0, t ] , it may also be called a counting process. The interarrival times are called the renewal periods, and the arrival instants (or epochs) the renewal instants (or points in time). As an example, consider airplanes arriving singly at an airport. If interarrival times are i.i.d. and N(t) is the number of arrivals during ] 0, t] , then {N(t)J is a renewal process. As a further example, suppose that we have a stock of items

uL-,

73

n >2.

a = E(U,),

- ob be independent

n>2

of other U's and

< u). Un, n > 1, one may define A (O+) = 0.1

A1 (u) = P(Ul

[To signify the positiveness of the r.v.'s We have taken the distribution of the first renewal period to be possibly different from that of other renewal periods. In practice, however, two possibilities arise : 1 A,@) = A@). In this case the renewal process is said to be an ordinary renewal process or simply a renewal process. This situation arises when the renewal instant occurs at a; = 0 , thus making U1 identical with the other U's. We may, however, caution the reader here that some authors do not count the renewal at 0; = 0 while studying an ordinary renewal process, whereas others do. We have preferred to use the former approach and as such may call the renewal at a; = 0 an initial renewal. The essential effect of counting the initial renewal at ub = 0 is that the total number of renewals in [0, r] is N(t) + 1 instead of N(t) in ] 0, t] . The real distinction between the two versions, though trivial, needs to be made. 2 Practical cases do arise when A (u) f A@). This situation arises when the renewal instant does not occur at ah = 0, thus making Ul not identical with the other U's. To illustrate this point, consider the second example discussed before. Suppose now that we started observing the process when the first item was already in use. Thus in this case it is natural to suppose that U1 need not be distributed as the other U's. However, U1 will be taken to be independent of the remaining U's. In this case the renewal process is said to be a modified (delayed or general) renewal process. One particular type of modified renewal process is extremely important, in which

SOME TECHNlQUES OF OUEUlNG THEORY

74

2.3

BASIC RENEWAL THEORY

Thus

P(N(t) > n) = P(Wn < t )

where

h this case the process is said to be an equilibrium (or stationary) renewal process. As we show later [see equation (2.3.14)], the equilibrium renewal process can be obtained as a limiting case of a renewal process which has been in operation for a sufficiently long time. Since Un is nonnegative for all n , we see that a > 0 , except in the trivial case in which P(U, = 0 ) = 1 . We assume a > 0 henceforth. Consider the partial sum

This gives the distribution of N(t). We could also write P,(t) = Fwn(t) - Fwn+,(t), n > 0 , if we define Fw, ( t ) = 1 for t > 0, or zero otherwise. The Renewal Function

We say that a renewal occurs at time t if W, = t for some n. Clearly, Wn gives the waiting time until the nth renewal. It is common practice to refer to either the counting process { N ( t ) ,t > 0 ) or the partial sum process {W,, n 2 0 ) as the renewal process, for as we shall see later, one is related to the other. Using independence and identity of the r.v.'s, we have from equation (2.3.1),

where * indicates convolution, and Af,-,)(u) is the (n - 1)-fold convolution of A(u) with itself. Note that A(*)(u)= 0 for u < 0, A(,)(u) = 1 for u > 0 , and A ( l ) ( u )= A ( u ) , Introduce the transforms

The mean value of the renewal process N ( t ) is known as the renewal function. Thus the function M ( t ) = E(N(t)) is the mean value or renewal function of the process {N(t), t 2 0). Much of renewal theory is concerned with the properties of M(t). It can be seen easily from equation (2.3.5) that

where we define M(0) = 0. For the ordinary renewal process, since A ( t ) = A(t), we can write equation (2.3.5) as Pn(t) = Acn)(t) -A Q (2.3.7) and consequently for this case w

A simple case of the ordinary renewal process arises when the renewal pe~iodsare positive i.i.d. r.v.'s. If the renewal periods are exponentially distributed with common mean, a = l/h, then using the convolution formula for the exponential distribution, we have from equation (2.3.7),

From equation (2.3.2) we have

Distribution of the Number of Renewals N ( t )during the Time Period 10, tl

In order to obtain the distribution of N(t), we need the basic relation that exists between N ( t ) and the sequence { W,).By definition of N ( t ) and W, and for t > 0, N ( t ) >n

iff

W,

< t.

(2.3.4)

which is a Poisson distribution and has been discussed earlier. The process {N(t)}, when t varies in T , is called a Poisson process, and is a particular case of the ordinary renewal process for whichM(t) = At.

SOME TECHNIQUES OF QUEUlNG THEORY

76

The Renewal Density

The derivative of the renewal function M(t) is called the renewal density rn(t). Thus from equatTon (2.3.6) we have

where al(t) and a(,-,)(t) are the p.d.f.3 corresponding to the d.f.'s A,(t) and A ( , )(t), respectively, and we have assumed that the d.f.'s of the U's are absolutely continuous. Note also that a l (t)*a(, -,)(t) is a notational way of writing the derivative of A,(t)*A(,-,)(t). The probabilistic interpretation of m(t) is that it gives the probability density of a renewal at time t. It can be seen that for the Poisson process, m(t) = A.

-,

The Renewal Equation for rn ( t l

The equation

t

m(t) = a l ( i ) + ~ o r n ( t - u ) a ( u ) d u

2.3

BASIC RENEWAL THEORY

77

which is the required density. Since the inverse transform uniquely determines the function, rn (t) is unique. If a,(t) and a(t) are given, the interest lies in finding the unknown function m(t) as a solution of equation (2.3.9). We must caution the reader here that though rn(t) is called a renewal density, it is not a p.d.f., for its integral diverges j; m(x) dx = even in the typical case of a Poisson process for which lim,,, lim,,, k t - + = . rn(t), in fact, gives the renewal rate at time t. In view of this, equation (2.3.9) may be given a probabilistic interpretation. Roughly speaking, up to first order in At, the left- and right-hand sides of equation (2.3.9) are:

LHS of eq. (2.3.9) = P (a renewal occurs in ] t, t

+ At] )

s RHS of eq. (2.3.9) = P (first renewal in 1t, t + At]) + sum over u of P (last renewal before t in ] t - u, t - u + At11 x P (renewal component introduced at epoch t - u fails in interval ] u, u + Au] of renewal life)

(2.3.9)

where a(t) is the common density of U's and a,(t) the density of U , , is known as the renewal equation for m(t). In the special case ai(t)Ea(t), equation (2.3.9) reduces to the ordinary renewal equation

There are several methods of finding the renewal equation (2.3.9) for the renewal density m(t), and one of them, which we adopt here, is through the transforms. Let m E(a) = e-%?{t) dt.

j0

Then as we got equation (2.3.3) from equation (2.3.2), we have, by using equation (2.3.81,

and the result follows. A similar probabilistic interpretation may be given to the renewal function (2.3.1 2). The renewal equation (2.3.9) for m(t), which is an integral equation, is a particular case of the more general Volterra integral equation of the second kind:

where G(t), H(t) and A(t) are defined for t > 0. In equation (2.3.1 1) as in equation (2.3.9), if H{t) and A(t) are given, then the interest lies in finding the unknown function G(t) as a solution of the integral equation. In renewal theory, equation (2.3.1 1) is usually called the renewal equation because many interesting results of renewal theory satisfy particular cases of equation (2.3.1 1). The solution of the renewal equation (2.3.1 1) in the case when A (x) = A(x) is easily obtainable and is given below. In fact, from equation (2.3.1 1) it can easily be shown that

,

where

We wish to prove equation (2.3.1 la). Taking the L-S.T. of equation (2.3.1 1) and solving it for g(a), we get By inversion, we get

SOME TECHNIQUES OF QUEUlNG THEORY

78

2.3

B A S K RENEWAL THEORV

Key Renewal Theorem [Smith (795811

where

If the function H ( t ) is such that: 1 It is nonnegative for t > 0, 2 It is nonincreasing, 3 1; H(u) du < and A(u) is nonlattice, then

-

lim Jot ~ (- U t )~ M ( u = )

t',

Now rewriting g(a) and using equation (2.3.8), we have

-1 J w ~ ( udu. ) a

0

Another result is the asymptotic normality of the r.v. N(t). For a proof of asymptotic normality of N ( t ) the reader is referred to Karlin and Taylor (1975). Thus if a <- and o2 < are the mean and the variance of U,,it can be shown that

-

The inversion of this gives the desired solution (2.3.1 la). It may be nc,ted tha have used A , ( x ) = A ( x ) while using equation (2.3.3).

that is, N ( t ) is asymptotically normally distributed with mean and variance given by

Limiting (or Asymptotic) Behavior of Renewal Functions

One of thelimiting cases is easy to derive by using the result lirn,,, a f i ) . Thus 1 lim m ( t ) = lim am (a) = -. t-+@-to+ a

f(t)=

Analogous to this, we also have

Residual Life

We approach the topic of residual life by reconsidering the renewal equation for m(t) given in equation (2.3.9). Using the convolution property, we have from equation (2.3.9), m ( x ) = a l ( x ) + joxa(x - u ) m ( u ) du.

t - t ==

This result is known as an elementary renewal theorem. An intuitive interpretation of this theorem is that the average number of renewals in 10, t ] , for large t , is approximately the product of the reciprocal of the mean renewal period l l a and the time t. For nonrigorous nonprobabilistic derivations of the above limiting results and some other limiting results, which are stated below without proof, the reader is referred to Problems and Complements. For rigorous probabilistic proofs the reader is referred to several of the existing sources, for example Feller (1971), Smith (1958), TakLcs (1962), Neuts (1973), or references contained therein. Blackwell's Theorem

If A(u) is a nonlattice distribution function with mean a < w, then ~ ( t ) - - ~ ( t - c )-tC a

for all real numbers c 2 0.

respectively.

as

t+w

Integrating this from 0 to t gives

= A ( t )+

lotA (t - u ) m ( u ) dir

= A , ( t )+

f:

A ( t - u ) dM(u).

(2.3.12)

Consider the modified renewal process { N ( t ) }and let r(t) be the time measured from t , the instant at which we start observing the process to the next renewal instant after time t. Then since W N ( t ) + is the time when the [ N ( t )+ I ] th renewal occurs, we have

SOME TECHNIQUES OF QUEUING THEORY

80

r(t) = WN(t);l

- t,

t >O.

r(t) is called the residual (excess or remaining) life at time t. The distribution of r(t) is given by t

P ~ r ( t ~ < x ) = ~ ~ ( ~ + x ) - j ~ [ ~ - ~ ( t + x - u ) j dX >~O (, u ) t, > ~ . (2.3.13)

We proceed to prove equation (2.3.13). First note that (r(t) < x )

*

{at least one renewal in ] t, t + X]

This implies (since W, = 0 ) that

2.3

BASIC RENEWAL THEORY

first answer, which is normally given, is a/2. This is based on the following arguments. Since the arrival epoch of the person is chosen at random (so that it has a uniform distribution over the interval), it is argued that the expected waiting time should be half the expected waiting time a between the two buses. Thus we may say that lim,,, E(r(t)) =a/2. However, this argument is fallacious, since it ignores the fact that the renewal period is an r.v. The answer is right if the distribution of renewal periods is deterministic, that is, if the buses always arrive after a constant period of length a. Now if we base our arguments on the forgetfulness property of the exponential, then we obtain the second answer Emt+, E(r(t)) = a. Clearly, the two answers are contradictory, but the contradiction can be resolved by more sophisticated arguments, which take into account that the renewal periods are r.v.'s. To this end, by looking at equation (2.3.14), one can easily see that the required expected value (see Problem 12) is given by =

z Ge

P{r(t)<xJ = P ( t < W , < t + x ) +

lim E(r(t)) =

PfWn
t-t-

n=l

= A (t

+x

A

(t)

-

+ n = 1 lot{ ~ ( f+ x - u) -A

81

a a2 -+2 2Q

where a2 is the variance of the renewal periods. Since for the exponential distribution u2 = a 2 , this limit becomes a, which is the answer obtained by using the forgetfulness property of the exponential. Thus the correct answer is a and not a/2. Note that if (and only if) u2 = 0, then the answer is a/2. (t - u)) dFwn(u).

Hence, usingM(u) = C,"=l FFwn(u),we finally get equation (2.3.13). Now we discuss the limiting behavior of the distribution of r(t) when A(u) is nonlattice. Using the key renewal theorem with H(t) = 1 -A(t x), we have from

+

Past U f e

If I(t) is the time measured backward from t to the last renewal instant, then 1(t) = t - W N ( t )

l(t) is called the past life (or age or current life) at time t. The distribution of I(t) is given by ~ ( t ) - ~ ~ [ - ~ ( t - u ) ] ( u ) , 0<x
To prove this, first note that I(t) > x iff no renewals occur in ] t -x, t ],x equivalently iff r(t -x) >x. Thus where we have used the fact that jy([1 -A@)] /a)du = 1. aradox of Residual Life

Let us explain this paradox by means of an example. Suppose that buses arrive at a certain bus stop following a Poisson process, so that the renewal periods are exponentially distributed with mean a. Suppose further that a person arrives at the stop at an epoch t. We discuss here the case when t is large, that is, we assume that the buses have been operating for a sufficiently long time. The question usually posed is this: What is the expected waiting time of the person for the next bus? The

< t, or

P{l(t)<xj = P(r(t -x) < x ) . The result now follows from equation (2.3.13). The case of unity follows by considering the complement of the impossible event l(t) > x , for x > t , noting that P{l(t) = t ) = 1 -A (t). It can be shown, by using the key renewal theorem, that

I-

lim P ( I ( ~ )< X I = t-to

1 -A(u) du a

provided A@) is nonlattice. Thus the limiting distribution when t -t m of the residual life r(t) and the past life I(t), both tend to the common distribution whose

SOME TECHNIQUES OF QUEUING THEORY

82

p.d.f. is [ l -A(u)] /a. It is seen later that this common distribution and its discrete version play important roles in queuing problems. We now discuss another renewal process called an alternating renewal process.

2.3

BASIC RENEWAL THEORY

83

within the sign of integration breaks up into two parts according to whether y or y > t. Thus


Alternating Renewal Process

Consequently, As an example, consider an equipment which is subject to breakdowns. Suppose that initially the equipment is running and that it remains in this state for a random amount of time To. As soon as it breaks down, repairs are started. Suppose that the repair time is Tft. If the cycle of running times and repair times continues, it generates two sequences of r.v.'s To, T,, . . and T;, T i , . . . . Suppose that the two sequences Tn and TA, n 2 0, are i.i.d. as T and T' with c.d.f.3 F and G, respectively. If the sequences ITn) and {TA) are mutually independent, the two-state process defined by the state of the equipment is an alternating renewal process. While this assumption of mutual independence of running and repair times is often made, it is not needed [see Franken (1978)] and is not used in the derivation of equation (2.3.15) below. Also suppose that H i s the common distribution of the r.v.'s Tn T;, that is, H is the convolution of the c.d.fs F and G. The process defined by the state of the equipment referred to above may be considered as an alternating renewal process. The interest in this case might be to find the limiting probabilities such as lim R(t) = lirn P{equipment is running at time t).

.

But as Tft is nonnegative, we must have

Itm P{To > tl(To + Tft) = Y) dH(y)

= P(To

>t

and

To + Th

>t )

Therefore R (t) satisfies the following renewal equation:

+

t+-

which is of the form (2.3.11). Consequently, by using equation (2.3.11a) as the solution of equation (2.3.1 I), we have

t+-

The evaluation of this probability is discussed later. As another example, consider the queuing system M/G/l. Suppose that initially the server is idle and that he remains so for a random time Tft. As soon as a customer arrives, he gets busy and remains busy for a random amount of time T o . As in the last example, the cycle of idle periods and busy periods will generate two mutually independent sequences which can be studied as an alternating renewal process. Similar applications of alternating renewal processes arise in the theories of dams, inventories, and so on. We now return to finding the limit R(t) as t e m . We wish to show that if E(T T') < and H is not a lattice distribution, then

+

-

lim R (t) =

t-t

E(T)

Equation (2.3.16), on applying the key renewal theorem (since H satisfies its properties), leads to

-

lirn R (t) =

t-t

IT [ 1 - F,(y)l G x dH(x)

dy -

E(T)

+ E(T')'

If we suppose that S ( t ) = P(equipment under repair at time t ), then S(t) = 1 R (t).and therefore

+ E(T1)

where E ( X ) is the expected value of an r.v. X. In words it may be stated thus: The limiting probability that the equipment is running is equal to the ratio of the expected time the equipment is running to the sum of the expected running and repair times. To prove equation (2.3.15), we condition on To + T; and get

Since the process regenerates itself at time To + Tft, the conditional probability

It may be noted that the limiting probability remains unaffected whether the equipment was initially running or under repair. From equations (2.3.15) and (2.3.17) we may observe that

Now we apply result (2.3.1 8) to find the mean busy period E(T) in the queuing system M/G/I, in which E(T)/E(I) = p / ( l - p ) , where E(T) and E(I) are the expected busy and idle periods, respectively. Now because of the forgetfulness property of the exponential, the calculation of the idle-period distribution for the

84

SOME TECHNIQUES OF QUEUING THEORY

85

PROBLEMS AND COMPLEMENTS

system M/G/1 is trivial. It is, in fact, the saine as the interarrival time distribution, and therefore E(1) = I/h. Consequently, E(T) =

1 p-h'

-

Using At = t/k and passing to the limit as k -t -, show that

Formulas for the mean busy period E(T) in systems more general than M/G/l, including bulk queues, are obtained in Section 6.7.

le we have now considered several techniques which have been used to analyze queuing problems, there are other techniques which we have not discussed in detail, but which are mentioned here. The first one is the method of collective marks. Though this method gives an interesting probabilistic interpretation of p.g.f.'s, no problems seem to have been solved by this technique which cannot be solved by other techniques. For some details of this method, the interested reader is referred to Kleinrock (1975) and the references therein. Four other methods are the combinatorial technique for which the reader is referred to Takacs (1967) or some other books on queuing theory; the spectral theory approach for which the reader is referred to Prabhu (1965a), Cohen (1980), and Bhat (1968); the random walk method for which the reader is referred to Conolly's (1975) lecture notes and the references contained therein; and finally an algebraic method for which the reader is referred to Kingman (1966). For the use of fluid approximations, diffusion approximations, bounds, and so on, one may consult chapter 2 of Kleinrock (1976) and the references contained therein, and also Newell (1971). For the use of a serniMarkov approach to nonbulk queues, we refer the reader to a book by Cinlar (1975), and for applications to bulk queues to the works of Fabens and Neuts, cited in other chapters. Recently Neuts has been advocating the use of a new class of matrix methods in queuing theory. References to Neuts's book and papers using matrix methods for the study of bulk queues are made in other chapters. An excellent and comprehensive work connected with the mathematical derivation of results leading to the limiting results when t -+ 03, n -t 03, or p -+ I , and many other interesting queuing results is that of Cohen (1980).

P(N(t) = n) = lim k+-

2

+

(hAty(1-

Continuation of Problem 1. One important property of the Poisson process is that the time between occurrences of events is exponentially distributed. Thus if U, = tr - t,-, , r = 1, 2, 3, . . . are interfials of time between successive events, then show that (Ur) are i.i.d. r.v.'s with P ( U r > x ) = e-Ax,

A>0,

x>O,

i-

= 1 , 2,...

.

(b)

If N(t) is the Poisson process with unit arrivals, then show that for u
(c)

Let {Up,r Z I } be a sequence of i.i.d. r.v.'s such that fur@) = pe-"',

u Z 0,

p

> 0.

Define N(t) such that

3

Unit arrival time-homogeneous Poisson process. Here we consider its alternative derivation. Divide the time interval 0 to t into a large number, say k, of subintervals of length At such that kAt = t. Then let the probability of the occurrence of an event in At be XAt o(At). We also assume that the probability of more than one event occurring in At is o(At). Then N(t) will have a binomial distribution. and

(a)

I:(

where N(t) is a nonnegative integer and t > 0. Show that {N(t), t Z 0 ) is a Poisson process with mean pt. Hint: Use N(t) t and the convolutions of U's discussed in Chapter 1. An r.v.X. is said to have a "memoryless" (or forgetfulness) property iff, for anyx>o,y>o, P ( X > x + y l X > x ) = P(X>y). (1 That the exponential r.v. satisfies equation (1) is easy to see. To show the converse, define F$(x) = P ( X >x); then F$(x y) = Fg(x) F$(y). The proof is now based on a well-known theorem in analysis [Hille (1964)l which states: If F$(x), x > 0, is a real-valued function satisfying the functional relation F$(x + y ) = F$(x)F$(y) and is bounded in every finite interval, then either F$(x) vanishes identically or there exists a constant A > 0 such that

+

SOME TECHNIQUES OF QUEUING THEORY

86

4

This shows that the only continuous r.v:'s having the property (1) are exponew tial r.v.'s. The property (1) is also called a characteristic property of the exponential distribution. For a different approach to this problem, see Problem 9 of Chapter 1. Continuation of Problem 1. Let {N(t), t E T ) be a stochastic process with finite second-order moments. The autocorrelation function RN(tl, t2) of the process N(t) is defined for all t,, tz E T as the joint moment of the r.v.'s N(tl) and N(tz). Thus notationally,

Prove that for the Poisson process with unit arrivals

5

Hint: Consider t2 > t l and use the property of independence and stationarity of the Poisson process. It may be noted that RN(tl, t 2 ) = RN(t2, t Continuation of Problems 1 and 4. The correlation coefficient function of a stochastic process N(t) is defined for all t1 , t2 E T by

PROBLEMS AND COMPLEMENTS

7

The order statistics are used extensively in certain branches of statistics. For more details, see Kendall and Stuart (1969). We illustrate their uses in the next two problems. Continuation of Problems 1 and 6. We have already given one important property of the Poisson process in Problem 2. Another one is its relation to order statistics. Given that n events of the Poisson type occur at epochs t, < t2 < , . . < t , in the interval 10, $1, prove that the r.v.'s t,, t z , . . . , tn have the same joint distribution as the n-order statistics corresponding to n independent r.v.'s Xi uniformly distributed over the interval ] 0, t ],that is, prove that

Proof: In view of Problem 6, it suffices to show that the probability that an epoch of occurrence of an event lies in ] x,x dx] ,0 < x < t, given that it lies in ] 0, t] ,is &It, that is,

+

dx

PIX < epoch d x + dx I epoch lies in j 0 , t ] ] = --, t

where o&(,) = E [N2(t)] - E' [N(t)] . For t2 > t l , show that for the Poisson process, the correlation coefficient function is given by

6

Order statistics. Let Xi, i = 1, 2,3, . . . ,n, be a sample of n elements from a population having continuous p.d.f. f(x). Further, let X(, ) be the smallest, X(,) the second smallest,. . . , X(,) the rth smallest, X(,) the largest of the sample of values ( X I , X2, . . . , X,). Clearly, -oo < XC1)<X(2, <Xc3)< . . . <X(,) < m. The r.v.'s X(,), i = 1, 2,. . . , n, are called the order statistics of size n (n-order statistics) of the sample (XI, X 2 , . . . ,X,). Assume that Xi's are i.i.d. r.v.'s and that their joint p.d.f. exists. Show that the joint p.d.f. of the order statistics Xci, is given by

In other words, the epoch of occurrence has a uniform distribution over the interval. To prove this, we first fmd the joint probability that n events occur in the given time interval, ] 0, t ],and one out of the n epochs lies in ]x, x + dx]. Let the interval be divided into three parts, x , dx, and t -x -dx, as shown below: 1 n - 1 -il Events I i I 1 Time 0 x x dx t

+

k t the number of events occurring in the first part be i, i = 0 , I , 2,. . . , n - 1. Then the number occurring in the last part must be (n - 1 - i), so as to have one in ]x,x + dx] and maintain the total of n. Since the events occur in accord with the Poisson process, the probability that they occur in this fashion, given i events occur in ] 0, x] , is the product of the two appropriate Poisson probabilities and thus is given by P(n events occur in ] 0 , t ] out of which one in ]x, x

In particular, if Xi are uniformly distributed with the common p.d.f.,

then

0 < x G t.

Consequently the unconditional probability is

+ dx] Ii in ] 0, x ] >

SOME TECHNIQUES OF QUEUING THEORY

88

PROBLEMS AND COMPLEMENTS

89

P(n events occur in ] 0, t ] out of which one in ] x , x 4- d x l )

(b)

where we have ignored the powers of dx higher than 1 . If this joint probability is divided by the probability that n events occur in 10, t ] ,we get a conditional probability: n P(one out of n epochs of occurrence Pies in ] x , x + d x ] I n lie in 10, t ] )= t dx.

8

It is equally probable that any one of the n events is responsible for causing an epoch of occurrence to all in the dx interval. Thus the conditional probability that an event occurs in d x , given that it is one of the n occurring in 10, t ], 0 < x d t , is dxlt. Since this conditional probability is independent of I ? , it is the unconditional probability sought. Because of this property, the events occurring in accord with the Poisson process are often called random events. (a) Density function of the largest r.v. Suppose that customers come to a service counter which has k parallel service channels, such that only one customer can be served at a time. Each job is distributed among the k channels, which are assumed to act independently. Let the channels have service times X I , X 2 , . . . , Xk which are independently exponentially distributed with a common parameter p. Show that Z = max ( X I ,X2, . . . ,X k ) has a p.d.f. fz(z) and a c.f. $ Z ( w ) given by

fz(z) = k ~ e - ~ ~ ' .

9

(a)

If the groupsize distribution qk follows the geometric distribution q k = qk-'(1 -q), k a l , O < q < l , then it is easily seen that the p.g.f. Po(z; t ) takes the form

m.From this c.f. or other-

p(1)

E(Z) =

-

3 ( 2 )

Var ( 2 ) = where

Note that we have used the forgetfulness property of the exponential. In words, if Xi,i = 1 , 2 , . . . , k, represents the remaining service time of the ith customer, then Z represents the remaining service time of the first of the k customers who are going to leave the counters, and its p.d.f. isf,(z). This property is used extensively in multiserver queuing systems. Problems (a) and (b) can both be generalized to the case when the r.v.'s connected with the service times are exponentially distributed, but with different parameters p, ,p 2 , . . . ,pk. Show that the solution of the differential-difference equations for the bulk-arrival Poisson process (2.2.4a) can be put in the following recurrence relation form, from which it is easy to compute numerically the probabilities Pn(t) of the process:

from which we get

wise, show that

I.1

-( p ) 2 IJ2

X

Po(t) = e-"

(b)

is a combination sign and i =

Density function of the least r.v. As in (a), suppose that customers come to a certain counter which has k parallel' channels. Now the k service channels, instead of serving only one customer at a time, serve independently k customers if they are available; otherwise the channels serve the lesser number of available customers so that, for example, if 5 < k customers are available, only 5 channels will be busy. If k independent exponential service channels, each with parameter p, are busy at any given time, then let X , , X 2 , . . . ,Xk be their remaining service times. If Z = min ( X I ,X 2 , . . . ,X k ) , then show that

SOME TECHNIQUES OF QUEUING THEORY

where L n ( x ) is a special case (a r: 0 ) of the more general form of the Laguerre polynomial L?)(X). This polynomial is related to the generalized hypergeometric function discussed in Chapter 1 by the relation

PROBLEMS AND COMPLEMENTS

Hint: Case 1 : M(t) = j g m ( x ) dx, and so on. Case 2: &(or) = j; e-@'M(t)dt, and so on. 14 Continuation of Problem 13. Show that the result of Problem 13 may also be obtained by conditioning on U, . Hint: M(t)=E(N(t))=JFE(N(t)IU, = u ) d A 1 ( u ) .

The generating function of L,(x) is found by setting cr = 0 in formula (22.9.15) of Abrarnowitz and Stegun (1964),and is (1 --z)-l exp

I:"-;[

-

=

2 ~,(x)z". ,,..

For more details on the Laguerre polynomials, the reader is referred to Rainville (1960) or Abrarnowitz and Stegun (1964). If groups arrive in batches of fixed size, say m , then one can see (c) that Po@;t ) = exp [h(zm - I ) t ] and consequently Pkm(t) = exp (- ht)/k!,k = 0 , 1 , 2 , . . . . 10 Show that for the bulk-arrival Poisson process (N(t),t >, 0 ) ,

E(N(t)lN(O) = 0 ) =

Xqt

Var (N(t)lN(O) = 0) = [u2

+ 41.1 At

where q= CF=P=, kqk and a2 is the variance of the number arriving at each arrival instant. Section 2.3

11 Show that the d.f. of N ( t ) is given by

P [ N ( t ) < n ] = I -A,(t)* A(,)(t).

91

u>t

E(N(t)(U1 = u ) = l+M(t--u),

u
andsoon

This may be stated in words as follows: If the first item did not fail up to time t , then there are no renewals in [O, t ] . On the other hand, if U = u < t , then there is a renewal at u and, on the average, M(t - u ) more renewals will occur in time t - u. In the terminology of Section 2.2, we use the fact that u is a point of regeneration for the regenerative process {NO)]. 15 (a) If the renewal periods of the ordinary renewal process are exponentially distributed with means 1/X, X > 0 , then from the renewal equation for m ( t ) show that m ( t ) = X , and hence P[r(t)< X I = 1 - e-'", x 2 0, t >O. This proves the forgetfulness property of the exponential distribution, and also that in a Poisson process the residual life has the same distribution as every other life. Hint: For finding m(t), use equation (2.3.10). For finding the d.f. of r(t), either (1) take the L.T.of equation (2.3.13) or (2) use the fact that a Poisson process has stationary independent increments. (i) Show that in a Poisson process with rate X, the d.f. of the past life (b) I(t) is given by , e - A , O<xt.

(5) Show further that E(l(t)) = [I - e-"1 /A.

12 Prove that h(u) = [ I -A@)] /a, u > 0 , is a p.d.f. where

Ilint:

(i) Proceed as in (a). (ii) First method: JT x dF,( ,,(x) = ji ~ x e - ' ~

du + te-At, and so on. The contribution te-At comes from the concentration of probability mass e-" at x = t . Second method: Ji e-'" dx, Hint: show

This can be proved by several methods. Perhaps the easiest one is to lim i.+o

J

m

0

e-i..h(u) du = I.

Show that the mean of the r.v. having a p.d.f. h(u) is p2/2a, where pz = J z u2 dA(u). 13 By integrating equation (2.3.9) or otherwise [by taking the L.T. of M(t) given in equation (2.3.6)], show that

which is a renewal equation for the mean value function M(t).

(c)

and so on. Total life. If we define T ( t ) as the total life of the component in service at time t , then clearly T(t) = l(t) + r(t) and consequently one may show by using (a) and (b) that for a Poisson process,

which shows that the total expected life is significantly larger than the expected life 1 / X = E(U,) of any particular renewal period. The significance becomes sharper if we let t -+ in which case E(T(t)) = 2/h = 2E(Un). These facts seem to be paradoxical. However, in reality there is

SOME TECHNIQUES OF QUEUING THEORY

PROBLEMS AND COMPLEMENTS

Proof:

95

Let F(t) =M(t) -M(t

PC).

Then taking the L.T., we get

F(a) =M(a) - e-c"li?(ol). Hence (g)

(h)

lirn @a)

and so on. Show from (f) that the limiting marginal distributions of r(t) and l(t) as t -t m are the same as obtained earlier. Continuation of (f). Show that for a Poisson process with rate A, lirn P{r(t)

t-t-

> u , I(t) > v) = e-A(U+V)

"-+O+

-

(iv) Key renewal theorem.

Proof: m(x) dx, t > 0, (i) Since lirnt+- f(t) = lirn,,o+a~(a), and M(t) = [f, by definition, we have from equation (2.3.12) on taking its L.T.

a:

0

Proof: Let F(t) = j$H(t - v) dM(v). Then taking the get F(a) = @(a) $(a). Hence, and

&?(a) = -

1

joH(t - V)dM(~1)= -a JmZf(v) dv.

am t-t =-

t-r-

,(,).

Show that

t

(i) limM(t)-tm.

+-

""O+

= lim E(a)(l -e-'") a -+o+ 1 -e-C" = lim a *(or) -----a"~+ a

This implies that the limiting distributions of r(t) and I(t) as t + are independent, as they should be. 16. As stated in the text, in (a) below we give nonprobabilistic nonrigorous derivations of some of the limit theorems, using the L.T. and based on the assumption that the L.T. is applicable to the functions under discussion. (a) Show that:

'I(@)

= lirn a%(a)(l -e-'")

(b)

lim &(a)

"0 '+

L.T.,we

= lim @(a:) a:*(cu) "-+a+

provided j;H(v) dv < -. The r.v. X, representing the life of a certain item has a p.d.f. given by

a

Thus Supposing that the renewal process starts with a new item, show that the renewal function M(t) then is given by

lim aii?(a) = 1 + lirn ,-to+ ,+o+

- jo t

Em M(t) = lirn t-rt-r

m (x) dx

Hint: Use the L.T. of the renewal equation for M(t) and then invert. If N(t) is a Poisson process with mean At =y, then show that

17 An important identity.

J a y T d x = 1-

n e-~yk

-.

h=O

where in getting the last step, we have used L'H6pital's rule since by (i) M(t) -t m as t -+ m. The last limit, we know, tends to 1la. (iii) Blackwell's theorem. Show that

k!

Hint: This can be proved by several methods. (i) Integration by parts of the left-hand side gives the right-hand side. (ii) Let

Differentiation gives

EGELLANEOUS PROBLEMS AND COMPLEMENTS

SOME TECHNIQUES OF QUEeiltNG THEORY

96

97

SCELLANEOUS PROBLEMS A N D COMPLEMENTS Integration by parts gives g ( y ) = g(y)--1

+c*c

= 1

1

which proves the result. ) Using Wn < t N(t) Z n and the convolution of exponentid r.v.'s, we get

1,

f(n)(x) d t

)i

Hence, o

(n

- I)!

+

The sum N(t) = N, (t) N2 (t), t > 0, of the two Poisson processes is a Poisson process with mean rate h, + X2. (b) The difference N(t) = N , (t) -N2 (t), t 2 0, of the two processes is not a Poisson process. The process {N(t), t Z 00)has as state space the set {0, A-1,*2, . . . I . (c) In both (a) and (b) {N(t), t > 0 ) has stationary independent increments. [The result stated in (a) can easily be generalized to n independent Poisson processes.] Hint: Use the p.g.f. Continuation of Problem 1. As opposed to pooling the two independent Poisson processes to obtain a new Poisson process, we can also get two independent Poisson processes by branching a given Poisson process, provided the branches are selected independently. Example: consider a Poisson stream of vehicles approaching a fork in the road. Assume that each vehicle, independently of all other vehicles, takes the left-hand road with probability p l and takes the right-hand road with probability p 2 , where p , + p 2 = I. If the rate of the given Poisson process of vehicles entering the fork is h, then the units going into the ith branch follow a Poisson process with rate hp,, i = 1,2. Hint: Let N(t) be the number arriving in time t in the original Poisson process and Ni(t) the number going into the ith branch over the same time interval. Then show that (a)

f

FW,O) =

In many physical situations we are interested to know the nature of the sum or difference of two Poisson processes. As an example, if fNl(t), t Z 0 ) and {N2(t), t 2 0 ) are two independent Poisson processes with mean rates Xi and h2, respectively, then show that:

h=O

k!

Changing n - l to n and putting At = y, gives the desired result. Various generalizations of this identity have been discussed. For details, see Steinijans 71971) and references therein. 18 For the equilibrium renewal process, prove that:

2

n' ( P , ) ~ (p2)nj P(N, (t) = n l , N,(t) = n21N(t) = n) = --$ nl!n2! where n l n2 = n. Remove the conditioning on the value of N(t) by multiplying by the probability that N(t) = n to get

+

nt: (a) show that A(&) = 1/aa. (b) and (c) follow from (a).(d) follows by using (b). 19 Blackwell's theorem. Let H(t) = l/c for 0 < t < c and zero otherwise. Show that Blackwell's theorem follows by using the key renewal theorem. Show that for the ordinary renewal process, the second moment &(t)= E(N2(t)) satisfies the renewal equation

SOME TECHNIQUES OF QUEUING THEORY

98

MlSCELLANEOUS PROBLEMS AND COMPLEMENTS

At any random instant of time in steady state, P,(x) is the joint probability and probability density of the number n of customers in the system, including the one in service, and the elapsed service time x of the customer undergoing service. The time required from the random instant to complete service on the n customers is the convolution of complete service times of the (n - 1) waiting customers and the residual service of the customer undergoing service. Hence the L.T. of the distribution of the virtual waiting time (in queue), including zero waits, is

Since the joint distribution of Nl(t) and'ni2(t) factors into two Poisson distributions, the units going into the two branches are following independent Poisson processes. This result can easily be extended to the case when the number of branches is k instead of 2, using the multinomial distribution instead of the binomial one for the conditional joint probability. 3. Consider a stochastic process N ( t ) = 2 2 f ) Z i , where the Zi's are i.i.d. r.v.'s distributed as Y and X(t) is a stochastic process independent of Y. The process {N(t), t 2 0 ) is called a compound process, to which various names may be given, depending on the nature of the process (X(t), t 2 0). Thus, for example, if {X(t), t > 0 ) is a Poisson process, then {N(t), t B 0 ) is called a compound Poisson process. Suppose that {X(t), t 2-0 ) is a Poisson process with mean rate h, then: Find the c.f., XN(~)(W), of {N(t), t > 0). (a) (b) Find the mean and the variance of {N(t), t 2 0). Show that the process {N(t), t 2 0) has stationary independent incre(c) ments. (d) Find the covariance, cov (N(tl), N(tz)), of the stochastic process {N(t), t 2 0). Answers: XN(~)(U> = exp [ h t ( ~ ~( 111 ~ ). (a) (b)

where Po(z; x ) is defined in Section 2.2. Using equation (2.2.24a) to find the values of P,(&(a);x) and P,@(a); 0), we get

XtE(Y), A~E(Y~).

where

Now in the double integral, first changing the order of integration, then integrating w.r.t. x, we get, on simplification,

We may remark here that a compound Poisson process has a c.f. of the form exp [ht(xEI(u) - I)]. The bulk-arrival Poisson process whose p.g.f. is exp [ht(Q(z) - l)] (discussed in Section 2.2) is therefore a compound Poisson process. Furthermore, the distribution of N(t) can be written in the following explicit form without the use of either the p.g.f. or the c.f.:

4

where Iqnjh* is the n-fold convolution of (q,} with itself, and {q,)Or = 6,o. This shows that the bulk-arrival Poisson process may be seen as a Poisson stream with randomly varying batch size. Waiting time (in queue) distribution for an M/G/1 queue. (a) In this problem we discuss an alternative derivation of equation (2.2.28) and, in fact, discuss the virtual waiting time (in queue) distribution. It will be seen that for M/C/l the distribution of the actual waiting time (in queue) of an arrival is the same as that of the virtual waiting time (in queue) of a fictitious arrival.

(b)

where P = h/p and Po has been determined by using @JO) = 1. Clearly %,(a) obtained here is identical with equation (2.2.28). This confirms our assertion made earlier that the virtual waiting time of a fictitious customer is the same as the actual waiting time of a real customer. The limiting waiting time (in queue) distribution for the queuing system M/G/I may be derived by still another method. If V," denotes the waiting time of the (n + 1)th arrival occurring at a; (with ob = 0 corresponding to the zeroth arrival), then

+ Yn+l --U,+l)+, n = 0, 2 , . . . where V,+ is the service time of the (n + 1)th arrival and U,,, is the interarrival time between the (n + 1)th and the (n 2)th arrivals. The y;+l = (Y;

,

+

SOME TECHNIQUES OF QUEUING THEORY

100

r.v.'s Vn+, and Un+, are independently distributed with p.d.f.'s b(x) and h exp (- Ax), x 0,respectively. Show that in the limiting case when n + -, the L.T. of Vq is the same as that given in (a) above. Hint: ii;,(a,n+l) = E(exp [--orV,"+']) = E(exp [-@(V,"

lSCELLANE0U.S PROBLEMS AND COMPLEMENTS

(c)

(d)

Find the L.T. p'"(cu) of Pa(t) given in equation (2.2.29) and show that lim,,, aPa(a) = A, as it should. Note that this corresponds to the sum of probabilities discussed in equation (2.2.30). Show that the probability Po(y) can be written as

+ Vn+l -Un+l>+l) where the integral is the d.f. of an Em r.v. and represents the probability that all the m customers complete service before the time period y expires. The right-hand of equation (1) may be evaluated by several methods. Show that it is equal to w

by Problem 8(b) of Chapter 1. Now using the fact that P(V," + Vn+l - Un+%< 0) = P(V,"+l = 0 ) and proceeding to the limit as n + -, (A -a)Zq(cu) = hZq(oc)&(a) aPo, and so on. Show that equation (2.2.28) can be written as

5

101

Hint: Different authors have evaluated the expression on the righthand side of equation (I) by various methods. The simplest way is to use first the value of P; and then evaluate the integral (note that Pg E 0). Another way is to replace first the integral by the summation

which on inversion gives w,(t) = (1 - p ) C;f=, pn (R (t)In* where R (t) is the residual service time (see Section 2.3). R ( t ) has the L.-S.T. R(ct), and is given by (1 --B(x)] dx. R(t) = g

and then sum over m. This is done as follows:

e above formula for %,(a) has an interesting interpretation. It shows that the waiting time (in queue) has a p.d.f. similar to that of M / M / l , if time is considered in such a way that the remaining service time is taken as the fundamental unit for all the n potential customers which a customer finds on arrival. Consider an M/M/P queuing system. Show that the mean waiting time (in the system) and the mean busy period for this system are the same, namely, 1/(g - A), but the corresponding variances are different, that is, 11[g(l - p)] ( lp)3 j respectively. and (1 + ~ ) / [ g ~ -

and so on. Still another method, using complex variables, is given by

C

6

+

roblems on GI

7

(a) (b)

Show that ii[p(l - 2 ) ] of equation (2.2.32) represents the p.g.f. of the number of services completed during an interarrival time. Show that the root yl used in equation (2.2.34) is, in fact, real and lies in 10, I[. nt: Use (a) and the Appendix A.2.

(e)

where r is a suitable inversion contour and &(a) = d ( p a). Now first perform the summation over rn and then invert by using partial fractions and inversion tables or by using the residue calculus (see Appendix A.3). Note that once equation (1) is evaluated, one can see that equation (2.2.47), which we have used in the text, is satisfied. In addition, note also that if we use PG instead of P; in equation (I), then the norming constant is automatically ignored if the new value of Po(y)/AC(y) is used in equation (2.2.46). Using Po(y), show that Po = 1 - p.

SOME TECHNIQUES OF QUEUING THEORY

MlSCELLANEOUS PROBLEMS AND COMPLEMENTS

103

The moments of I may be obtained through the relation [implied by equation (2.2.44)]

8

where in getting the last step we have used equation (2.2.32). Continuation. In this problem we discuss an alternative derivation of FI,(r). This approach is based on the discussion of waiting-time distribution for GI/G/l by the integ~alequation technique. Therefore the notation used here is the same as that in Section 2.2. Define the r.v. I,(n) as

(b)

where (x)- = min (0, x). The corresponding relation in the limiting case as n -+

+

9

(c)

(d)

Take the L.4.T. of equation (2.2.49), which is given by

Jr

e-"t dFIu(t), and so on, and a, = ura(u) du with = where TIW(a) al l/h, r = 1, 2, . . . . Expand the right-hand side of %,,(a) as a power series in a and pick the coefficients appropriately.

+

vin)

(e)

Hint:

where S = V- U is an r.v. representing the difference between the service time and the interarrival time, and I is an r.v. representing the actual idle period. Hint: By Section 2.2 and Miscellaneous Problem 8, b'in+') =( ~ 2 ) S,)* and I,(n) = - (vhn) + S,)- imply Vin*') -I&) = + S,. Squaring both sides, and noting that V ~ + l ) ~ , ( = n )0 , gives [v?")] + 1 : (n) = [vP)] + $2 4- 2 Vhn)Sn. Taking expected values, since and S, are independent, we have, in the limit, on using P p ( 1 2 ) = E ( c 1 from (a), the desired result. Continuation of (b). Show that the result in (b) can be written as

VP)

Then I, represents the virtual idle time, and its distribution function is defined as follows: FI",(7) = P(1, > 7)

But as P(Vq V < (u - 7)) = j;-*P(Vq < u - T - v)pe-'*' dv and P(V, < t ) = 1 - 7 %exp [-p(1 - y l ) t ] , substituting appropriately, we get the desired FI,(r). Compare this result with the result obtained in equation (2.2.49). Continuation: moments of I, in GI/M/I and GI/G/l, and some other relations in GI/G/l. (a) Show that the first three moments about the origin of I, in a GI/M/l queue are given by 1 1 E(I,) = --A I.c

The moments of the idle-time distribution and the waiting-time (in queue) distribution are related and may be obtained even for the more general queuing system GI/G/I. For the queuing system GI/G/1 in steady state (p < I), show that

(0

where 02 and 02 are the variances of interarrival times and service times, respectively. Continuation of (c). Specialize the results obtained in (c) for the queuing systems M/G/l and GI/M/l and compare with the results obtained elsewhere. Hint: For M/G/l, E ( I ) = 1/h and E(12) = 2/hZ, and for GI/M/1 use E(I,2) = PGE(12) and (a). Show that the L.T. of the distribution of interdeparture interval for the queuing system GI/G/l, when p < 1 , is given by where D, I,, and V are interdeparture, virtual idle period, and senice times, respectively. Hint: Let D, be the time between the nth and (n + 1)th departures. Thus D, is the service time of the (n + I)th customer plus the virtual idle time: Dn = Vn+, +Iv(n). The proof now follows on using the independence of the r.v.'s Vn+l and I&) and proceeding to the limit as n -+ w. Continuation of (e). Show that the mean and the variance of D are

SOME TECHNlQUES OF QUEUING THEORY

104

(g)

where the various quantities have been defined earlier. nt: Consider the derivatives of &(a) at a = 0 and use (a) and (c). Continuation of (e). Show that for the special case M/G/l

t:

(h)

ForM/G/1, Pi = 1 - p , and by equation (2.2.451,

Continuation of (e). &(a) in (e) indicates that if Vis exponential with parameter p, then D is exponential with parameter h. Conversely, if D is exponential with parameter h, V is exponential with parameter y, implying thereby that the exponential distribution with parameter U , is the only service time distribution for the system M/G/l for which the output process is Poisson with rate A.

Abramowitz, M., and I. A. Stegun (Eds.) (1964). Handbook of mathematical functions. Applied mathematics series 55, National Bureau of Standards, Washington, DC.; also reprinted 1965, Dover, New York. Bartlett, M. S. (1978). An introduction to stochastic processes, 3rd ed. Cambridge University Press, London. Bharucha-Reid, A. T. (1960). Elements of the theory of Markov processes and their applications. McGraw-Hill, New York. Bhat, U. M. (1968). A study of the queueing systems M/G/I and GI/M/l. Lecture Notes in Operations Research and Mathematical Economics, Vol. 2. Springer-Verlag,New York. Burke, P. J. (1975). Delays in single-server queues with batch input. Oper. Res. 23, 830-832. Chung, K. L. (1967). Markov chains with stationary transition probabilities, 2nd ed. SpringerVerlag, Berlin. Cinlar, E. (1975). Introduction to stochastic processes. Prentice-Hall, Englewood Cliffs, NJ. Cohen, J . W. (1980). The single server queue, 2nd ed. North-Holland, Amsterdam. Conolly, B. W. (1958). A difference equation technique applied to the simple queue with arbitrary arrival interval distribution. J. R. Stat. Soc., Ser. B 20, 168-175. (1960). Queueing at a single serving point with group arrival. J. R. Stat. Soc., Ser. B 22,285-298. f 1975). Lecture notes on queueing systems. Ellis Norwood Ltd., Chichester, Sussex, England.

Cooper, R. B. (1981). Introduction to queueing theory, 2nd ed. Elsevier North Holland, New York. Cox, D. R. (1955). The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Philos. Soc. 5 1, 433-44 1. (1962). Renewal theory. Methuen, London. and H. D. Miller (1965). The theory o f stochastic processes. Methuen, London. Dafermos, S. C., and M. F. Neuts (1971). A single server queue in discrete time. Cah. Cent. Etud. Rech. Opt?. 13, 23-40. Doob, J. L. (1953). Stochastic processes. Wiley, New York. Dynkin, E. B. (1965). Markov processes. Springer-Verlag,Berlin. Feller, W. (1968). An introduction to probability theory and its applications, Vol. 1, 3rd ed. Wiley, New York. (1971). An introduction to probability theory and its applications, Vol. 2, 2nd ed. Wiley, New York. Finch, P. D. (1959). A probability limit theorem with application to a generalization of queueing theory. Acta Math. Acad. Sci Hung. 10, 317-325. Franken, P. (1978). A remark on the stationary availability. Math. Oper. Ser. Optimization 9, 143-144. Gani, J. (1957). Problems in the probability theory of storage systems. J. R. Stat. Soc. Ser. B 19,181-206. (1969). Recent advances in storage and flooding theory. Adv. Appl. Probab. 1,90110. Gaver, D. P. (1954). The influence of senice times in queueing processes. Oper. Res. 2, 139149. Gross, D., and C. M. Harris (1974). Fundamentals o f queueing theory. Wiley, New York. Henderson, W. (1972). Alternative approaches to the analysis of the M/G/1 and G/M/l queues. J. Oper. Res. Soc. Jpn. 15, 92-101. Hilie, E. (1964). Analysis, Vol. 1. Blaisdell, New York. Jaiswal, N. K. (1960). Bulk-service queuing problem. Oper. Res. 8, 139-143. (1968). Priority queues. Academic Press, New York. Karlin, S., and H. M. Taylor (1975). A first course in stochastic processes, 2nd ed. Academic Press, New York. Kemeny, J. G., and J. L. Snell(1960). Finite Markov chains. Van Nostrand, Princeton, NJ. Kendall, D. 6 . (1951). Some problems in the theory of queues. J. R. Stat. Soc., Ser. B 13, 151-185. (1953). Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24, 338-354. (1957). Some problems in the theory of dams. J. R. Stat. Soc. Ser. B 19, 207212. Kendall, M. G., and A. Stuart (1969). The advanced theory of statistics, Vol. 1 , 3rd ed. Griffm, London. Khintchine, A. J. (1932). Mathematical theory of a stationary queue. Mat. Sb. 39, 73-84. Kingman, J. F. C. (1966). On the algebra of queues. J. Appl. Probab. 3, 285-326; also Methuen's monographs on applied probability and statistics, supplementary ser., Vol. 6, 1-44. Kleimock, L. (1975). Queueing systems, Vol. 1. Wiley, New York. (1976). Queueing systems, Vol. 2. Wiley, New York. Kosten, L. (1973). Stochastic theory o f service systems. Pergamon, Oxford.

106

SOME TECHNIQUES OF QUEUING THEORY

Krakowski, M. (1973). Conservation methods in queueing theory. Rev. Fr. Autom. Inf: Rech. Opkr. 7, 63-84. (1974). Arrival and departure processes in queues. Pollaczek-Khintchine formulas for bulk arrivals and bounded systems. Rev. Fr. Autom. Inf: Rech. Opdr. 8,45-56. Levy P. (1965). Processus stochastiques et mouvement brownien, 2nd ed. Gauthier-ViUars, Paris. Lindley, D. V. (1952). The theory of queues with a single server.Proc. Cambridge Philos. Soc. 48, 277-289. Luchak, G. (1956). The solution of the single channel queueing equations chaxacterized by a time dependent Poisson distributed arrival rate and a general class of holding times. Oper. Res. 4, 711-732. Meisling, T. (1958). Discrete time queueing theory. Oper. Res. 6,96-105. Moran, P. A. P. (1959). The theory of storage. Methuen, London. Neuts, M. F. (1973). Probability. Allyn and Bacon, Boston. Newell, G. F. (197 1) Applications of queueing theory. Chapman and Hall, London. Palm, C. (1943). Intensitit sschwankungen im Fernsprechverkehr. Ericsson Technics no. 44, 1-189. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Prabhu, N. U. (1964). Time-dependent results in storage theory. J. Appl. Prob. ba 1, 1-46. (1965a). Queues and inventories - A study of their basic stochastic processes. Wiley, New York. (1965b). Stochastic processes. Macmillan, New York. Rainville, E. D. (1960). Special functions. MacmiUan, New York. Regis, R. C. (1973). Multiserver queueing models of multiprocessing systems. IEEE Trans. Comput. 22, 736-745. Romanovsky, V. I. (1970). Discrete Markov chains. Wolters Noordhoff, Groningen, The Netherlands. Seal, H. L. (1969). Stochastic theory of a risk business. Wiley, New York. Seneta, E. (1 973). Non-negative matrices - An introduction to theory and applications. George Allen and Unwin, London. Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge Philos. Soc. 49, 449-461. (1958). Renewal theory and its ramifications. J. R. Stat. Soc., Ser. 3 20,243-302. Steinijans, V. W. (1971). Crossing times in cumulative processes. South Afr. Stat. J. 5, 63-66. Taklcs, t. (1962). Introduction to the theory of queues. Oxford University Press, New York. (1967). Combinatorial methods in the theory of stochastic processes. Wiley, New York. Wishart, D. M. G. (1959). A queueing system with service-time distribution of mixed chisquared type. Oper. Res. 7, 174-179. (1961). An application of ergodic theorems in the theory of queues. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, University of California Press, Berkeley and Los Angeles, 581-592.

The study of bulk-arrival queues may be said to have begun with Erlang's solution of the M / E k / l queue [see Brockmeyer et al. (1948)j since this gives, implicitly, the solution of the ~ ' / n i l / l queue (see Section 3.1). Explicit consideration of bulk-arrival queues [see Gaver (1959)] seems to have begun several yzars after the work of Bailey (1954) on bulk service. We have arranged the chapters in the reverse order, which appears more natural. In this chapter we discuss single-server queuing systems with bulk arrival, and in Chapter 4 queues with bulk service. For interpretation of some bulk-arrival queues as single-arrival queues, or bulk-service queues as single-service (when the server services one customer at a time) queues, the reader is referred to Chapter 6 and the references therein. In "ordinary" queuing problems it is assumed that customers arrive singly at a service facility. However, this assumption is violated in many real-world queuing situations. Letters arriving at a post office, ships arriving at a port in convoy, people going to a theatre, restaurant, and so on, are some of the examples of queuing situations in which customers do not arrive singly, but in bulk or groups, where the size of an arriving group may be an r.v. or a fixed number. Mathematically and also from the practical point of view, the cases when the size of an arriving group is an r.v. are more general, and also more difficult t o handle. Many authors have contributed to the theory of queues with bulk arrival, and in this chapter we discuss some of their works.

THE SYSTEM M X / ~ , / l Let us start with the discussion of a queuing system in which customers arrive in batches according to a time-homogeneous Poisson process with mean rate X. The batch size X i s an r.v. and P(X=m)=am,

r n = 1 , 2 , 3,...

with m

A@) =

C a,zm,

lzl

< 1.

1

We shall assume that 0 < b = ~ ( " ( 1 ) < m and 0 < 0: = E ( X 2 ) - Z 2 < -. This arrival mechanism has been discussed in more detail in Chapter 2. Briefly, in a 107

BULK-ARRIVAL QUEUES

108

small interval of duration A t , the probability of no arrivals at the queue is 1 hA t o(A t) and that of arrival of m customers is ha,A t 4-o ( A t). Batches are served in the order of their arrival. The order of service of customers within batches is not considered in this model. If a batch arrives to find the server idle, a customer of the batch receives immediate service. The processing times or service times of customers who are served singly by a single server are i.i.d. r.v.'s, having a modified Erlangian d.f. B(v) with

3.1

THE SYSTEM M ~ E ~ I

at time t

109

+ A t with that at time t , it can be seen that

+

with b(a) = j,"e-&'dB(zt) = 2; cj{p/(p + a))'. The service times are also independent of the interarrival times and the batch sizes. In other words, service consists of J i.i.d. exponential phases which need not necessarily have a physical meaning. Here J is an r.v. such that P ( J = j ) = cj, 1 < j < s, with 2f=,cj = 1 . The modified Erlangian distribution (random number of phases) is very useful in modeling red queuing systems, as a generalization of the Erlangian distribution (fixed number of phases). The method of phases (Erlangian procedure or its modified form) helps one to replace a non-Markovian process by a Markovian one, as will be seen shortly, and thus to study the queuing system through the use of Kolmogorov equations. For more details, see Chapter 2. The mean service time is easily obtained as

where ?i = 2; jcj. As pointed out in Chapter 2, the phases are labeled in reverse order, and a new customer is taken up for service only when service on all phases of the customer under service is over. The system length process (N(t)), in this case, is clearly non-Markovian. However, it can be made Markovian by considering the vector (N(t), R) where N(t) represents the total number of customers in the system at time t and where, for N(t) > 0 , R denotes the number of service phases to be completed by the customer who is being served. Thus the states of the system are (O), (1, r), (2, r), . . . ,(1 < r < s), and consequently we may define for any time t the following probabilities:*

+ P n + ~ , ~ ( t ) i . l c r Aht a+n P o ( t ) c r A t + o(At), n > l ,

r > l .

From this we get the following differential-difference equation:

Following the procedure used to get equation (3.1.1), we have corresponding to the empty state of the system, the equation

The differential-difference equations (3.1.1) and (3.1.2) are known as the forward Kolmogorov equations of the process. These are to be solved under some initial conditions which we suppose t o be

that is, at time zero there are I units in the system, and the unit under service is in the r t h service phase with probability c,. In the case I = 0, we supposePo(0) = I , but it will be seen that the final result automaticafly includes the case 1 = 0. To solve the set of equations (3.1 .I) and (3.1.2) subject to the initial conditions (3.1.3), we first repeat the definition of the L.T. T(a) off (t) by

Integration by parts of the L.T. of dPn,,(t)/dt gives the relation

Obviously P&t)

= 0, 1 < r

< s. By comparing the state of the queuing system

* I n order to simplify the notation used here, the dependence of Pn,,(t) and P,(t) on the value of N ( 0 ) is not indicated. This convention will be followed henceforth.

= -Glncr

+c~p~,,(a)

(3.1.5)

which has been obtained using equations (3.1.3) and (3.1.4). Applying equation (3.1.4) to the set (3.1.1) and (3.1.2) and using equation

BULK-ARRIVAL QUEUES

110

It may be noted that as x + 1, z + 1, F(1, 1 ; a ) + po(a) -+ a-' , implying thereby that Z:=; 1 Z:;=, P,, ,(t) + P,(t) = 1 so that the distribution of the number in the system remains finite in finite time. The distribution of the number of phases N,(t) in the system M/Ej/l has been discussed by Luchak (1958). However, from the results (3.1.11) and (3.1.12) one can get the distribution of N(t) directly, even for the system M/EJ/l, which is more useful than finding the number of phases in the system. We next show that the distribution of N, (t) in M/EJ/l, obtained by Luchak (1958), is identical with that of N(t) in MX/E1/ I .

(3.1.5), we obtain, after transposing terms,

Next let us define the p.g.f.'s s

The system MX/E1/l. In this system the service times are exponentially distributed. This system may be obtained from equation (3.1.12) by simply letting c, = 6,1. Thus the distribution of N(t), in this case, is given by

1

r

Multiplying equation (3.1.6) by znx', summing over r (1 < r < s ) and n (1 < n < m), and using equations (3.1 S ) , (3.1.7), and (3.1.8) and A(z), we obtain, after exercising some patience, ( a + h + p ( l -x-9-hA(z))P(z,x;cu)

= h ~ ( x ) A ( z ) & ( a ) + C(x)zE

- C(x)(a + +)Po(a) - p ( l -c(x)z-'3

The solution of equation (3.1.9) for p(z,x;a) can be greatly simplified by con+ p -hA(z))-y. sidering equation (3.1.9) first when x takes the value p/{a Definingfl(z; a ) 5 C,"=,pn,, (a)zn, and settingx = y, equation (3.1.9) becomes

++

It can be shown, using RouchB's theorem (see Problems and Complements), that the denominator of the right-hand side of equation (3.1 .10) has exactly one zero inside the unit circle lzl = 1. Let this zero be denoted by zo = zo(a). Since g(z;a) is a proper p.g.f., it must be analytic within the unit circle and in particular at z =zo(a). It follows that the numerator of fl(z;a) must vanish at z =zo(a), and hence that

Also substituting l?ff(z; 4 in equation (3.1.9) yields the relation

The result (3.1.13) is a particular case of Gupta's (1964) result, which is given here in a slightly modified form. However, we observe that Luchak (1958) has obtained the same result for the system M/Ex/l. It is thus interesting to see that the systems M ~ / E11 , and M/Ex/l are equivalent. But it must be remembered that N(t) represents the number of units in the system M ~ / E , / ~whereas , it represents the number of phases in Luchak's model. Intuitively, this may be interpreted as follows: In Luchak's model, a customer on arrival demands a random number X of exponential phases through which he must be served before another customer is taken into service, whereas in M ~ / E , / ~a ,group contains a random number X of customers, each of which is to go through a single exponential service phase. Therefore a single customer in Luchak's model (M/Ex/l) has the same service time distribution as the group in M X / ~ , / 1 .The mean service time in both systems being Z/p (for a customer in M/Ex/l and for a group in M ~ / E , / ~ the ), traffic intensity p (= hZ/p) remains unchanged. Thus the process representing the number of customers at any time t in the system MX/EI/1 is identical with the process representing the number of uncompleted service phases at the time t in the system M/Ex/I, provided that the initial states (customers in the former case and phases in the latter case) are the same. We shall see later that the waiting-time distribution is also the same for M ~ / E/,I and M/Ex/l. If we could invert equations (3.1.12) (that is, find a p.g.f. P ( z , x ; t) which has P(z,x;a) as its L.T.), we would have a p.g.f. for (Pn,,(t)J, the time-dependent state probabilities. Time-dependent (transient, nonequilibrium) solutions for state probabilities are known for some queuing models [see, for example, Ledermann and Reuter (1954), Sack (1963) and Takics (1962)], but even in these cases the transient solutions are complicated and intractable. For most queuing models we have at most an L.T. or other integral transform of the transient state probabilities. In such cases we must rely on asymptotic solutions, valid as r -+ m, or use numerical methods to study the time-dependent solution.

BULK-ARRIVAL QUEUES

112

(3.1.14), (3.1.151, and (3.1.16) yield the relations

The Limiting Distribution of N ( t )

We consider first the traffic intensity p (defined in Section 2.1), which plays a fundamental role in the study of limiting distributions in queues. For the MXIEJI1 queue, the mean total service time for a group of customers is iiF/p. Since the mean interarrival time between groups of customers is 111, we have p = XZF/p. It can be shown (see Problems and Complements) that if p < I , there exist limiting state probabitities for the number in the system, which are independent of initial conditions, so that where in all these results

S

lim t-m

C P , , (t) = ,.=I

tk+mm P, ( t ) = P,

and

2 We define the steady-state r.v. N with P(N = n ) = P,. If p > 1, no proper limits exist, as P, (f) -+ 0 for any finite n. These results are intuitively clear except in the case p = 1, since if p > 1, customers arrive faster than they can be served, and the queue must then grow without limit. Applying the Abelian theorem given in Appendix A. 8 to equations (3.1.1 0) and (3.1.12), we get

3

and

4

wherey=p[h+p-hA(z)]-'. It is easy to see that at a = 0 , C(y) = b(h - h A@)) and hence that the system length distribution has the p.g.f.

where &(a) is the L.-S.T. of B(v). From the normalizing condition P(l -) = 1 we obtain Po = 1 - p, from which we again infer that the limiting distribution exists only when p < 1. For p > 1, all probabilities tend to zero, and hence we say the system degenerates. Next we consider certain special cases of the system M X I E ~ / l . 1

The system M ~ / E ~ /In~ .this case the service time distribution is Erlangian with k phases, each with rate p. Letting c, = 6,.k, equations

The systems M X / E I / l and kfX/LI/1. To obtain the distribution of N for the system kfX/E, / I , simply put k = 1 in equation (3.1 .19). For the system M X / ~ / l in , which service time is constant and equal to b, letting k, p + in equation (3.1.19) so that k/p = b, results in

where we have used limd,, (I + ~ / d =) e~x . The system M ~ / E ~ / I .For this system, in which the size of the arriving batch is constant and equal to k, the distribution of N may be obtained by putting aj = 6ik in equations (3.1.16). Various other cases corresponding to variations in c, may be discussed similarly. The system M / E I / l . This is the simplest system, in which arrivals are single and follow a Poisson distribution with mean rate h and service is exponential with mean 111.1. In this case, the distribution of N may be obtained by either putting a, = and c, = in equation (3.1.16) or a, = and k = 1 in equation (3.1.191, and is given by P(z) = Po(l - pz)-I, where p =A/@. Since p < 1, it follows that p l z l < 1 for lzl< 1. Thus the binomial expansion is valid, and hence

which is a geometric distribution. It may be noted that Po = 1 - p . Although the present derivation of equation (3.1 .I 6) uses the Erlangian method, the result was first obtained by Gaver (1959) by using the imbedded Markov chain technique and renewal-theoretic arguments. In fact, Gaver considers the system hfX/G/l in which the distribution of service time is unrestricted. The present discussion, though restricted to M ~ / E ~ covers / ~ , most of the cases of practical interest. For a discussion of the system M ~ / G / using I the supplementary variable technique, the reader is referred to Miscellaneous Problems and Complements.

BULK-ARRIVAL QUEUES

274

'Ke results (3.1 .I 71, (3.1 .18), and (3:1.19) have been obtained independently by Restrepo (1965). In the case a , = 6 , , they have been discussed by Morse (1958). But while comparing the two results, one must make allowance for differences of notation. The service rate in each phase is k p in Restrepo (1965) or Morse (19581, whereas here it is simply p. A class of queues of the type presented here, but where the distribution of the number of customers in each arrival group is restricted, has been considered by Cohen (1963) using his method of derived Markov chains. ailing-Time (in Queue) Distribution

- The Limiting Behavior of v q ' ( t )

Since the waiting-time distribution in the transient case is cumbersome, we consider this distribution only in the steady state. Let Vq( t ) be the virtual waiting time of a customer arriving at time t , and let Vq( t ) converge to V , in distribution as t + m. By the characteristic property of the exponential distribution, we can suppose that the remaining time of completion of a unit in a given phase recommences at the instant of arrival of a batch. This property will be used whenever the senice time is exponential, or is composed of exponential phases, without repeating this argument in the future. Now the arriving batch must find the system in one of the states 0 , (1, r ) , ( 2 , r), . . . If wql (7) d 7 is the probability that the waiting time of the first customer of an arriving group ends during the interval [7,7 + d ~ , j then

when substituted in equation (3.1.22), leads t o

It can easily be seen from Wq, (0) = 1 that Po = 1 - p , a result obtained earlier. In the particular case when c, = 6 , , , equation (3.1.23) gives E, (a) = (1 - p)/ [l - ( h / a ) ( l - A ( @ ) ) ] , which is equivalent to a result of Luchak (1958) for M / E J / l in the limiting case. It may be remembered here that p will be the same in both systems & f X / E 1/ I and M / E J / l if in M / E J / l the mean service time is taken , These as Zip so that p for this system is h Z / p - the same as in system M ~ / E11. equilibrium waiting time results are valid for p < 1. If p > 1 , the distribution of waiting time does not exist. As an example, consider the case of the simplest system M / E I / l , I t can readily be seen from equation (3.1.23) that W q (a) =

(1 - PNI* + a )

a+p-h

.

Inversion of equation (3.1.24) gives and a result which is found in elementary textbooks on queues and on operations research. Measures of Efficiency

(3.1.21) where 6 (7) is the Dirac delta function whose L.T. is unity. Applying the L.T. to equation (3.1.21) gives the result (3.1.22)

+

where @ = p / ( p a ) and P ( z , x ) is the generating function given in equation (3 .I .I 5). Substituting x = 0 , z = C(@)in equation (3 .I .I 5 ) gives P(C(0). B), which

The expected value of an r.v. connected with a process may not be an adequate measure of efficiency, since the expected value can be the same for several systems with very different input or service rates. Nevertheless, in many physical situations one is interested in the means or other moments of the distributions rather than in the distributions themselves, at least when it is difficult to get sufficient information about the latter. As an example consider mean system length and mean queue length in an M X / E k / l queue, and mean waiting time and mean sojourn time for the first customer in an arriving group for the same queue. First we evaluate the mean number L = E ( N ) in the system. To do this, we find the derivative of P(z) given in equation (3.1.19) at z = 1 by expanding the denominator of P(z) about the point z = 1 and canceling the factor (z - 1) in numerator and denominator. To obtain the expansion of the denominator of P(z), let

116

BULK-ARRIVAL QUEUES

3.2

THE SYSTEM GIrIMI1

317

times o;,, - o; > 0 , n = 0 , 1 , 2 , . . . , are i.i.d. r.v.'s with common distribution function A(u). Let the L.-S.T. of this distribution be denoted by

and let the mean interarrival time be denoted by 1/X, where where p = X kg/@. Expanding the denominator of P(z) in a Taylor series about the point z = 1 and canceling the factor (z - I ) , we get

P(z) = 1 Thus

+

PO z x ( l ) ( l ) 4 z(z - 1 ) ~ ( 2 ' ( 1 ) . . .

+

+

Po [ X ( " ( l )+ 4 X'2)(1)] (1 + x(l' ( I ) ) ~

p ( 1 ) = -

,

which, after using the values of X ( l ) ( l ) ,X ( 2 ) ( 1 ) and , simplification, gives the mean number in the system as e

m

The size X of the arriving batches is such that P(X = r) = 1. The customers are served individually by a single server. If V n is the service time of the n t h customer to be served, then it is assumed that { V n ;n = 1 , 2 , 3 , . . .)is a sequence of positive i.i.d. r.v.'s with the common exponential p.d.f. pe-#', v > 0. It is further assumed that the two sequences {l.;,) and (a;+ - a; are mutually independent-. Let N ( t ) be the number of customers in the system at time t and put N; = N(ok -0), n = 0, 1,2, . . . . In words, N i represents the number in the system, including the one, if any, in service, just before the arrival instant 5;. By using the characteristic property of the exponential distribution, it is easy to see that the chain { N i } is a homogeneous. Markov chain imbedded in the continuous time process ( N ( t ) ) and has a countable infinity of states. Now if D, represents the number of departures (real or potential - potential when there are no customers in the system before an interarrival period ends, otherwise real) during an interarrival period of arbitrary duration, and k j = P(Dn = j), then

If Nq is the number in the queue, then mean number in the queue is given by I

nn

4

=

jOwP(D

= jlinterarrival time = u) dA(u)

The mean waiting time of the first customer of an arriving group may be obtained from E,,( a ) given in equation (3.1.23), and is given by

wql = E ( V q l ) = -G;:)(o) The p.g.f. K ( z ) of the sequence {Kj}is given by and consequently the mean time spent in the system by the first member of an arriving group is given by k

W

=

Wq,+-.

(3.1.28)

!J

Since C(6') = b(a), one can see that equation (3.1.23) agrees with Gaver (1959), and for the special case M ~ / 11, E ~equations (3.1.25) and (3.1.27) agree with Ross (1970).

3.2

which is the p.g.f. of a Poisson r.v. It should be clear from the context that we have supposed (D,, n = 0 , 1 , 2 , . . .) to be a sequence of i.i.d. r.v.'s with the distribution stated above. Define j-1 flj..l = 1k,, j > 1 n=O

THE SYSTEM Glr/M/l

Now one can easily see that the following relation holds between the r.v.'s N;,N;+, ,D,and the fixed number r:

The Limiting Distribution of N,

We next consider a special case of the system G I ~ / M wherein /~ batches of customers of exact size r arrive at epochs 0 = ob, o l , 0 2 ,. . . , a;, . . . . The interarrival t

,

BU LK-ARRIVAL QUEUES

118

where (x)'

=

and then take limits as n

max ( x ,0).

-+

-

to find P-(2). Thus

~ ( ~ ~ n=+~ (r ~)( N n + r - D1n ) +

If PV(n)= P [ N , = jlN< = i] , i, j Z 0 , n Z 1, are the n-step transition probabilities of { N i l , the one-step transition probabilities Pij z P i j ( l ) may be obtained from equation (3.2.1) and are given by

< O)P(Ni+r-Dn < 0 ) = ~ ~ ~ ( z ~ n+ -r -Dn ~ n > l ~O)P(NG ; +r-D, > 0 ) + P ( N i + r - D , < 0). (3.2.5) +E(zO/N;+r-D,

Now

The transition probability matrix P of ( N i l is therefore given by

E ( ~ N -,D n )

=

> 0) + E ( ~ ~ ; - ~ ~ I N ; +< ~ -OD) P, ( N ; + r y D , < 0). E ( Z ~ , - ~ ~ I N , + ~ -> D ,O)P(N;+r-D,

(3.2.6) From equations (3.2.5) and (3.2.6) we have =

2

&+,

kr+,

kr

k

...

kl

k2

ko

0

< 0)

E(~N~= + IZ )r [ ~ ( Z N i - D n- ]E ( ~ N , - D ~INn-+r-D,

...

P(Ni+r-D, Proceeding to the limit as n

-+

< O)] + P ( N ; + r - D n < O ) .

-, we have m

P-(2) = zrP-(z) K ( z - I ) - - z r

Since k j > 0 , it can be seen that the matrix P is:

C

z-"'-'P(N-+r m=o

- D = -m )

Irreducible, since every state can be reached from every other state in a finite number of steps with positive probability.

2

Aperiodic, since the diagonal elements are positive.

3

Ergodic if and only if p = r7 < 1, where interarrivd time) = X/,u.

7 = (mean

service time)/(mean

where m

bm = P ( N m + r - D = - m ) =

In the ergodic case we denote the limiting distribution which exists independent of the initial state of the process by

C

p;k

i=o

.

r+r+m

and therefore

P i = nlim P(N2, = 0) = P(N-+r-D i m

< 0) =

m

b,. m=o The relation (3.2.7) could have been obtained, as is normally done, by using the one-step transition probabilities (3.2.2), but the method used above is more elegant. Now equation (3.2.7) finally gives

P ( N i = N(oA - 0 ) = j). P; = nlim -rm The limiting probabilities are then given by the p.g.f.

where y , ,y 2 , . . . ,7,. are the r roots inside lzl= 1 of the equation

To prove equation (3.2.3), we use equation (3.2.1) to f i d the p.g.f. of N;,

,

To evaluate the constants bm in equation (3.2.8), we first show that equation (3.2.4) has exactly r roots (distinct or coincident) inside the unit circle /zj = 1. Consider absolute values o f f ( z ) = z' and g(z) = - K(z) on the circle lzI = I - 6 ,

3.2

where 6 is positive and sufficiently small. Then I f(z)l = ( 1 - 6)' = 1 - 6r + 0(6), and since K ( z ) is a power series with nonnegative_coefficients, /g(z)/< K ( l - 6 ) = I-6~(')(l)+o(6)=1-6~/h+0(6)=l-(r/p)6+o(6), where p = r h / p . Hence for p < 1 and 6 sufficiently small, If (z)l> Ig(z)l on Izl= 1 - 6 . Since f ( z ) and g(z) satisfy the conditions of RouchB's theorem, it follows by applying that theorem and letting 6 -+O that equation (3.2.4), which is equivalent to f ( z ) g ( z ) = 0 , has exactly r roots in the interior of the unit circle. Let these roots be y,, I = 1 , 2 , . . . , r , and now consider the zeros of the denominator of equation (3.2.8). Since equation (3.2.4) has r roots y,inside the unit circle, the denominator of equation (3.2.8) has r zeros l / y I outside the unit circle. As P-(z) is an analytic function of z for lzl< 1, the function

121

and r roots z = y i l , 1 = I , 2 , . . . ,r , for which lzl > 1. Thus we have an identity

It follows that, in this case, we have

+

must be analytic for lzl< 1. Now let B ( z ) be defined for lzl > 1 by

THE SYSTEM GIrIMII

I

I

UsingP-(1 -) = 1, we finally obtain

Thus in the special case of exponential interarrival times, the generating function P-(z) can be written in a form not explicitly involving the roots y,. The Waiting-Time (in Queue)

Since in this expression all the zeros of the denominator outside the unit circle are also zeros of the numerator, it follows that B(z) is not only analytic for Izi > 1, but is analytic on the whole complex plane. But it is a well-known result of complex analysis that a function which is analytic on the whole complex plane must be a constant, and therefore B ( z ) = constant = B , say. As a consequence,

Let Vql ( t ) be the virtual waiting time in the queue for the first customer of a group which arrives at time t . Denote v:) = Vql (a; - 0). Then if$) is the waiting time of the first customer in the nth arriving batch. We consider the limiting distribution Wil (7) = nlim P ( v ~ )< r ) +If we define

then and using P-(1 -) = 1, we obtain equation (3.2.3). As a simple example, let us consider the systemMr/M/l, in which

+

+

Then Z(a) = X / ( h a), K(z) = h / [ h p ( l - z ) ] of equation (3.2.8) equated to zero gives

which is an equation of degree r

, and therefore the denominator

+ 1 with exactly r + 1 roots, one root at z = 1

We now proceed to prove equation (3.2.1 1). If an arriving batch finds j customers in the system, the waiting time of the first customer in the batch will have the L.T. { p / ( p+ a))'. As the limiting probability of j customers in the system is P;, we have

which gives the required result by using equation (3.2.3). The relations (3.2.3) and (3.2.1 1) are due to Foster (1961). Using the supplementary variable technique, Conolly (1960) has considered e limiting behavior of GIr/M/l in continuous time. Some numerical work on e parameters discussed by Conolly (1960) has also been carried out by Barber

BULKARRIVAL

122

(1964). In this connection, see also Pike (1963). Some properties of the GIr/M/l/Finite have been discussed by Shanbhag (1966). In the lim derived from continuous time and just before arrival instants, we es Chapter 6 a relation between their p.g.f.'s of the queue size. Suzuki (1963) (see Problems and Complements) has extended the re (3.2.3) and (3.2.1 1 ) to the case when the size of an arriving group is an transient behavior of N i for GIr/M/l has already been discussed by Taka but the results are given in implicit form. However, one special case explicit results are available isMr/M/l, which is discussed in the next secti

YSTEM MrIMI1

123

-

PG(n) =

C P i ( n - 1) P r + l - l I=0

P;(n) =

1P;(n - l ) k l + r - j ,

w

x-

O<j
(3.3.5)

I=O

P;(n) =

P;(n - l ) k l + , - j ,

r

< j.

l=i-r

lution of the set of equations (3.3.5) is given by the joint p.g.f.P-(z, w), m

-(z,w)

The Transient Behavior of N, All the assumptions in the present case are the same as in the previous sect' the following exceptions: The interarrival times are exponentially distributed with d.f.

> 0,

A(u) = 1 - e - h u , 2

u 2 0.

The initial number in the system is i. This assumption was the previous section where we considered only the limiting n -t m, in which case the effect of the initial distribution wea

Define

P;(n) = P(N, = N(oL - 0 ) = j )

= Co

C0 P ~ ( ~ ) W ~ Z ~

(

1P

~

q -z

+~

~ O (- 6 ()

+PWZ'+~

,

1

(3,3,6)

tain equation (3.3.6), we proceed as follows. First obtain recursive (3.3.7) for the P;(n), n 2 1 , by substituting and kl+,-j from 3.3.4) into equation (3.3 5 ) :

-,

qP;(n) = P P , ( ~ ) qP;(n)=P;-l(n),

2<j
(3.3.7)

qP;(n)=P;-l(n)-pP;

-,-,( n - I ) ,

j>r+l.

with

P; ( 0 ) = 6 i, that is, we shall assume that the initial state i is arbitrary but fixed. Now

k j = P ( j potential departures during an arbitrary interarrival period P-(z, w ) = =

do- j !

e-(h+~)u du ,

and let

i-1 = 1

Also let p = X/(h (3.3.3) that

+ p),

-

-

j 2 I.

J

P;(n)wn,

Iwl

<

1

c p;{w)z',

lzl

<

I.

-

0

i=O

g equations (3.3.7) by w n , adding, using the initial conditions given (3.3.1), and P J { w )gives a new set of equations. Multiplying the new dding, and using the second p.g.f. P-(2, w ) leads finally to equation ch is completely known except for the constant P;{w). An explicit r P i f w ) may be obtained by Lagrange's theorem (see Appendix A.6)

q = 1 - p . Then it follows from equations (3. k j = pq',

-,

j = O , l , 2 ,...

z kn-= Lk,, 0

x a

py{w) =

= qj.

holds for j = 0 if we define 0- = 1. The definition of It is easy to see that the probabilities Pi(n), n 2 1, satisfy the

1 " P i @ ) = 6 < 0 + - 1~ ~ ~ { x ~ , ~ + -6i,o) ip-'(l 4 n=l x [q(xn,i-xn,i-1)

f

xn,i-xn,i+l])

BULK-ARRIVAL QUEUES

I26

Once again it may be observed that the limiting distribution given in equation (3.3.12) is the same as the one given by equation (3.2.10). In this connection, see also Problem 11. It can be shown that in some but not all queues the continuous time process has the same limiting distribution as the corresponding imbedded ~ a r k o vchain. For further discussion, see Chapter 6. The Waiting-Time (in Queue) Distribution

Let V,(t) be the virtual waiting time in the queue at time t and define v?) = Vq(0;- 0). Then is the time that elapses between the arrival of the (n + 1)th batch and commencement of service on its first member to be served, irrespective of his position in the queue. If win(7)is the p.d.f. of v?),then

v?)

and use the convolution property to get

Using the value of P;(w) from equation (3.3.9) with i = 0, we finally get equation (3.3.14). To prove equation (3.3.15), use Lagrange's theorem as discussed earlier with f(b)= (t - ~HP!: and get

from which we recover fw(n) as given in equation (3.3.1 5). The results of this section are given by BrockwelI (19631, and for the special case r = 1 are contained in Takics (1962), though at several steps they have been incorrectly reported.

and

The Distribution of the Number of Batches Served in a Busy Period

Let fQo(n)be the probability that starting from the empty state (O), the system returns to this state for the first time at the n t h step. This means that fw(n) is the probability that n batches complete their service during a busy period. In mathematical notation we may write, for n > 1,

fw(n)

=

P(N; = 0 , N i f 0 , m = l , 2 , 3 , . . . , n - l I N 6

= 0).

Furthermore, since by definition P i ( 0 ) = 1, we can write P i ( n ) =P(NG = OIN; = 0). Now the p.g.f. of fw(n) is given by

The Limiting Behavior of N: when Service Time Depends on Number i n System

In the system M X / ~ , / 1the , input is through the homogeneous Poisson process with rate A, and the p.g.f. of batch size is A(z) = ; S f a mz m , as described in Section 3.1. The arrivals, in fact, form a compound Poisson process so that if a total of c ( f )customers arrive during ] 0, t ], we have " (At)"

ai(t) = P(c(t) = i) = e - x f {

a

*

i = o , I , ,~. . .

where (ailn*is the n-fold convolution of {ai) with itself, with {ailo* = tiio and {ai)'* = (ai].Also the p.g.f. of ai(r) is where 5. = { ( w ) is the simple root of the equation z = q + p w z r C ' , Iwl < 1, inside the unit circle (zl = 1 . Further from equation (3.3.14) we get

To prove equation (3.3.14), form the generating function of

- ai(tjzi =

e-Xt(l

-A(z)).

(3.4.1) 0

The customers are served by a single server on a FCFS basis. When a group arrives to find the server idle, service on any one member of the group starts instantaneously, otherwise the group joins the queue. The service time V of a departing customer is conditioned on the number of customers at the instant of service initiation. Service times Vn of customers beginning service with the same number n in the system are independent identically distributed r.v.'s with d.f.

BULK.ARRIVAL QUEUES

whose L . 4 . T . is

F,,(a) =

low

em"" dBv,,("

< -.

$

THE SYSTEM MX/G,/I

EE

3.4

jI

From the form of the matrix it is clear that the chain is irreducible, and since the diagonal elements are positive, it is aperiodic. Crabill (1968) has shown that sufficient conditions for ergodicity are

Denote m

a

= A ( ~ ) ( I )=

C ma, 1

I

It can be shown (see Problems and Complements) that equation (3.4.4) can be put in an equivalent form

I

lim sup (pi = hZE(Vi))

<

1

i>O

Let N(t) be the number of customers in the system at time t , and define N i = N(a, 0) so that N: represents the number of customers in the system immediately after the n t h departure. We are interested in the behavior of the stochastic process {N;) when n -+ -. The process {N;} is an imbedded Markov chain with a countable infinity of states. The one-step transition probabilities Pij of the chain for i 2 1 are given by

+

in which p i appears to be in the usual form of traffic intensity. Define

Pi' = nlim P(N; = N(on -+ 05

+ 0)

= j).

In the ergodic case, the probabilities Pi+ are given by the system of linear equations

with

Now using equation (3.4.2), we can change equation (3.4.5) to the form

The p.g.f. of the set of equations (3.4.6) is given by

that is, k,, = P ( j arrive during a service period li were in the system when service began). The transition matrix = (Pij) has the special form

,

where we define

and

To obtain equation (3.4.7), multiply equation (3.4.6) by zi, sum over all j, and use the convolution property. It has not been possible to put equation (3.4.7) in closed form as a function of the {Ki(z)) and a few unknown probabilities. Thus the direct application of this result is limited to some special cases that follow.

3.4

THE SYSTEM ~

~

1

~

~

1

1

131

BULK-ARRIVAL QUEUES

130

1

The System M X / ~ / l . In this case the service time distributions are independent of the number in the system at the time of initiation of service. Thus we may assume, for i 1, 1, kj,i

= kj

= P ( j arrivals during a service period)

This would amount to a situation in which servers work at a slower (or faster) rate when they see an empty queue. We now have what appears to be the simplest special case of iiIX/~,/1,which preserves arrival groups of random size and service time distribution dependent on system size. Thus noting that (aj)"* is the coefficient of z j in the expansion of (alz f (1 - a l ) z 2 ) " , we have from equation (3.4.3)

Then it follows from equation (3.4.8) that

and

where [n] is the greatest integer contained in n, and whence the p.g.f. P+(z), given in equation (3.4.7), gives the solution

kj = P ( i arrivals during a service time /more than one in the system when service started)

The constant Pz may be obtained from P'(1 -) = 1 and is given by

= kj, with p 1 replaced by p, j

> 0.

It can easily be shown by integration that where p = XZb < 1, b being the mean service time. Note that we can replace K(z) given in equation (3.4.9) by b(X - XA(z)).

2

The System MX/Gn/l with Some Modifications. Let us modify this system by simplifying the input distiibution and the service time distribution. Suppose first that each arrival group contains either one or two customers, so that

k . = k. J

1. 1

with p

replaced by p

and from equation (3.4.3)

The p.g.f. Ki(z) given in equation (3.4.8) takes the form This means arrivals are by singlets and doublets with rates h 1 and 1 2 , respectively. Next let the service time distribution of those alone in the system be a different exponential from that governing all other customers, so that

BULK-ARRIVAL

QUEUES

Thus, from equation (3.4.7) we have

P'(z)

=

ZIzKo(z)-K(z)}P,+ + z { K ~ ( z-)K ( z ) ) P ; l / [ z - K ( z ) l ,

which is completely known except for the constants Po+ and Pf.One equation involving these constants is obtained from P f ( l --) and the other from equation (3.4.6). Noting that

the inputs by singlets and doublets. For an independent proof of equation (3.4.9) and its generalization, see Problems and Complements. Ivnitskiy (1975) discussed the transient and the limiting distribution of N(t), the number of customers in the system M : ~ / G , / I , using the supplementary variable technique. In this system, the interarrival rates, the group size probabilities, and the service rates all depend on the number in the system. Sufficient conditions for the existence of the limiting distribution are discussed, and recursion formulas for the L . 4 . T . of the distribution are obtained. However, as one can expect from the discussion of the system M ~ / G , / I , the results are cumbersome. Consequently they are not presented here.

3.5

THE SYSTEM E,f/Er/l WITH GENERALIZED ERLANG a,

we get the first equation as

P,+(l+Po-p)+P;(pl

The Limiting Behavior of N ( t )

- p ) = 1-P

and after using equation (3.4.10), the second equation as

with

The bulk queuing systems which we have considered so far have had either exponential interarrival time distribution or exponential service time distribution. /~ great difficulties Studying more general queuing systems such as G I ~ / G involves if one wishes to go beyond mathematical generalities to a computable solution. Many G I ~ / G / I queues can, however, be approximated with sufficient accuracy for practical purposes by Erlangian queues, such as the E~/E,./I queue discussed below. As before, customers arrive in groups of size X such that P(X = I) = al, I = 1, 2 , 3 , . . . , where the mean ? andithe variance a: of the group size are such that 0 < Z < and O < a: < rn. The service times of the customers are i.i.d. r.v.'s with

and

(See Problems and Complements for the calculation of pa .) Solving the above set of equations for P,' and Pf,we get

which is an Erlangian distribution with mean b satisfying the inequalities 0 < b = j," v d B ( v ) = rlp <m Analogously, the interarrival times of the groups are ii.d. r.v.'s with h

dA(u) =

yi

X i e-'iu du

i=l

where and

The remainingp;, j 2 2, may be obtained iteratively from equation (3.4.6). Most of the results of this section are due to Harris (1970), who has considered certain other particular cases by varying the service time distribution but keeping

This is a generalized k-Erlang distribution which may be thought to be composed of a sequence of k exponential phases or, equivalently, the sum of k independently distributed exponential r.v.'s Xiwith

P(Xi 6 x ) = 1 --e-'ix

Xi>O,

x>O.

BULK-ARRIVAL QUEUES

134

The mean arrival rate for groups is

This distribution Ek is useful in approximating statistical distributions having coefficients of variation between 1 and k-'I2, the first being the coefficient of variation for the exponential and the second for the ordinary k-Erlang distribution. The stochastic process to be studied now is a vector process (N(t), X(t), Y(t)), where N(t) is the number of customers in the system (including the one in service, if any) at time t, X(t) is the number of service phases yet to be completed at time t , and Y(t) is the phase in which the arrival is found at time t . The arrival phases are marked in their natural order, whereas the service phases are in the reverse order. Clearly, using the phase technique, the process can be made Markovian. First we define X(t) 0 when N(t) = 0, and then

The notation and the further analysis can be simplified if we make the transformation 1 < i = (n - l ) r s. Such a transformation has been used by Morse et al. (1954), Jackson and Nickols (1956), and others. With this transformation we can modify the notation and write piTj(t) = Pi,j(t) and po,j(t) = P0,j(t). Obviously, since n represents the number of customers in the system (queue + service), i represents the number of phases in the system. Thus the probabilities Pi,j(t) are the joint probabilities representing the number of phases in the system together with the arrival being in the jth phase at time t . Assume that the probabilities P l j = limt,, Pitj(t) exist, which they do if p = heb < 1, as we expect intuitively. As we are trying to study the limiting behavior of the system, it will be assumed thatp< I. One can now write the basic differential-difference equations, and by proceeding to the limit as t -+m get the following recursive relations. For j = 1,

+

where d = [(i - l ) / r ], [x] being the greatest integer contained in x. For j = 2, 3, ..., k ,

Also k

Po =

C

(3.5.3a)

f'0,j

j=1

the probability that the whole system is empty (no customers in the queue or in service). The difference equations (3.5.1) to (3.5.3) completely define the system under consideration. But these equations are not completely tractable analytically, though numerical work on the computer is possible. Furthermore, after computing P i j , to get the probability of n customers in the system it is essential to calculate

However, one can discuss certain things in more detail and thus come across certain interesting properties of the system. Later on, of course, we shalI show that ciosedform solutions of the above equations are possible for certain particular cases of the system E$/E,/~. We shall also discuss measures of efficiency, such as mean number in the system or in the queue, for the original system E ~ E , / Iwith generalized k-Erlang input. To proceed further, consider first the p.g.f.

Multiplying the set of equations (3.5.1) by appropriate powers of z , using equation (3.5.4) and the convolution property, and simplifying, we get, for j = 1,

Analogously, from equations (3.5.2) and (3.5.31, for j = 2 , 3 ,

. . . ,k,

ULK-ARRIVAL QUEUES

6 36

Before going further, we can find Pj(l),' which helps us in evaluatingP, = Po. = 1 - p , since from equation (3.5.5),

Consequently, Pj(l) = x/hj,

j = 1,2,.

The equation (3.5.6) has an interesting probabilistic interpretation. Since l/X and l / h j are the average times the arrival group spends in the whole arrival channel k is the probability of the group being and its jth phase, respectively, (l/h.)/(l/X) in the jth phase. Next we derive Zj=, Observe from equation (3.5.5) that, since p = r?ix pk(l)/p,

where the product is to be taken as unity whenever 0 > 7,0 and y are nonnegative integers, and y may be zero. Since the p.g.f.'s must converge within the unit circle lzl = 1, the k - 1 zeros zl , . . . ,zk - of the denominator of equation (3.5.9) lying within the unit circle must coincide with the zeros of the numerator of equation (3.5.9), which is a polynomial of degree k - 1 [excluding the factor (1 -z)] . Substitution of the zeros z, , . . . , z k - , into this polynomial leads to a system of k - 1 equations linear in Po,,, Po,, , . . . ,PO,k which, along with equation (3.5.7), can be solved for all the unknown probabilities Po,i, j = I , 2, . . . ,k. When the POsjare known, equation (3.5.9) can be solved for Pk(z), and then equation (3.5.5) can be solved recursively for Pj(z), j = k - 1, k - 2, . . . , 3 , 2 , 1. While all these calculations can be carried out in principle, they are complicated unless k , the number of interarrival phases, is small. Fortunately if we can assume that the arrival group size does not exceed a finite integer q , it is possible to express the generating function in a simpler and more explicit form. Now as D(z) is a polynomial of degree qr + k [since the upper limit of the sum in equation (3.5.8) is q] , it has qr roots which lie outside the unit circle. Let us denote them by zol ,z, , . . . ,zoqr. Since all the zeros of D(z) lying inside the unit circle coincide with the zeros of the numerator of equation (3.5.9), we have

Adding these equations and using the normalizing condition 2!=, P j ( I ) = l immediately leads to

x n XI" p k-1

l=j

Since the set of equations (3.5.5) is linear in generating functions, one can easily obtain the determinant D(z) of the matrix of the set by expanding with respect to the column k. Note that as the coefficients of the elements involved in the expansion are triangular determinants, they are easy to expand. The result is

where A is a constant. When equation (3.5.10) is substituted in equation (3.5.9), we get

[

pk(z) = A

'4

n (Z -zoc)

c=1

1-I

where A can be calculated by using equation (3.5.6). Finally we have

As usual, by Rouchk's theorem it can be shown that the expression D(z) has complex zeros within the unit circle, lzl= 1, and a simple zero, z = 1, on the given circle (see Problems and Complements). Let us now look at the p.g.f.'s Pj(z). The second equation of the set (3.5.5) expresses the dependence of Pj on Pj(z), j = 2 , 3 , . . . ,k. We therefore need to find Pk(z). By Cramer's rule one can find from equation (3.5.5) that

k

-l

The case q = 1 corresponds to the generalized k-Erlang input with single arrivals.

Measures of Efficiency Once again it may be pointed out that the remarks made earlier while discussing ~ / ~here as well. measures of efficiency for the queuing system M ~ / E apply L, and hence L q , may be found by differentiating equation (3.5.5), but this approach requires lengthy and tedious calculations. The following derivation of a

BU LK-ARRlVAL QUEUES

138

-

I t is clear that the calculation of L (or L,), in the general case, requires evaluation of the complex zeros of a polynomial. It is possible to use approximations by considering bounds on L.

formula for L is simpler and more elegant. First, noting that k

k

pjl)(l) = j=1

2 C j=1

iPij = E ( I )

i=o

gives the expected number of phases in the system (queue + service) and then taking the expected value of the transformation I = (N - l ) r X, where I , N, and X are the r.v.'s whose values are i, n, and s, we find that

+

L, = E ( N - 1)* = r-'

The System E,$/Er/1with Ordinary k-Erlang input

As a special case of the E ~ / E , / Iqueuing system with heterogeneous phases discussed above, let us consider interarrival times following the ordinary Erlangian distribution, that is, h j = A, j = 1 , 2 , 3 , . . . . In this case one can easily see from equation (3.5.8) that D(z) turns out to be

where E ( X ) is the expected number of unserviced phases of a customer. E ( X ) may be found as follows:

Y being an r.v. whose value is j, and P ( X = s IX > 0) = I/r imply E ( X ) = E(XIX

> O)P(X > 0)

=

=

If the size of the arrival group is bounded above by q, then one gets the corresponding p g f . for Pk(2) from equation (3.5.1 1). The mean number of customers in the system, in this case, may be found from equation (3.5.1 3 ) and is given by

-

i - p = P @ + 1) 2 r

j=l

Consequently, Also L is given by equation (3.5.12). But to evaluate L we first need to know Pj(')(l), which can be obtained by differentiating the second equation of the set of equations (3.5.5) with hi = A , j = 2 , 3 , . . . ,k. Now

,,z; and therefore using L = L,

+ p, and using this equation to express P { ' ) ( I ) , P i X ) ( 1 )., . . , ~ d ' ? ~ ( in l ) terms of P~')(I),we obtain

Fainberg (1974) obtains, equation (3.5.12) by a method similar to that given above and also states without proof an alternative formula for L ,

Finally, using equations (3.5.1 2) and (3.5.1 5 ) , we get

If the size of the arrival group is bounded by q, then L may be expressed in terms of the qr zeros of D ( z ) lying outside the unit circle. From equation (3.5.13) by using equation (3 S .1 I), we obtain

L =

P - (a,' 2 +a2 + E ) + ; 2a

I:( . ~ ~ , - 1 ) - ~ .

qr

cs 1

If r -+m, p +- SO that r / y -+ 1, then equation (3.5.16) gives L for the system E;/D/~ which has constant service time.

140

BULK-ARRIVALQUEUES

Customers are served one at a time, and their service times are independent and general with the p.d.f.'s e Transient Behavior of the Joint Distribution of the Customers in the System

In previous sections we have considered queuing systems in which customers, all of the same type, arrive to get service at a given service facility. However, situations do arise when customers of more than one type, say j(> 1) types, arrive at the service facility and are served according to some kind of priority discipline. While several kinds of priority discipline have been studied, we shall be concerned only with head-of-the-line priority discipline, which is defined later. Normally, a priority index i, 1 < i < j , is associated with each type of customer where 1 indicates the type with top priority and j the lowest, that is, as the index increases, the priority rank decreases. Since our discussion will be confined to the case j = 2, we can without ambiguity refer to customers of priority type 1 simply as priority customers, and to customers of priority type 2 as nonpriority or ordinary customers. Our discussion of priority queues will make frequent use of busy-period processes, discussed in Chapter 2, and of completion-time processes, which we now introduce. The interval of time starting from the instant the service of a customer begins, and ending when the server is free to take up the next customer of the same type, is defined as the completion time of the customer under consideration. Thus for queuing systems with two priority types, the completion time for a priority customer is simply his service time, and is of no special interest. The completion time for each nonpriority customer may be defined as the sum of his service time and the time it takes to clear the system of any priority queue that may have formed during his service time. In other words, this is the interval of time that elapses after the start of service of a nonpriority customer until the next nonpriority customer can begin service. These ideas will be used when we discuss the waitingtime distribution of a nonpriority customer. Introducing bulk arrival or service into priority queues, particularly when the number of priorities is greater than 2 and the interarrival-time or the service-time distributions are general, creates tremendous analytical difficulties and thus makes mathematical modeling of such queuing systems impractical. This is the major reason that not much seems to have been done in the direction of priority bulk queues. The difficulty might be better realized if we consider the queuing system M ~, M$ X /GI, G2/ I . This is a queuing system in which two types of customers, to be called priority and nonpriority customers, arrive according to two homogeneous independent Poisson processes with mean rates hl and X 2 . Both types of customers arrive in batches such that the batch sizes X1 and X2 are r.v.'s with distributions P ( X , = r) = al(r), (3.6.1) r = 1 , 2 , 3, . . . .

The corresponding p.g.f.'s are denoted by A l ( z ) and A 2 ( z ) , and the means by 0 < Zl < m, 0 < Z2 < m. The numbers in different batches are independent.

and the means b I , b2 for the priority and nonpriority customers, respectively. The queue discipline is FCFS within each type, but a priority customer is always served before a nonpriority one. However, if a priority customer arrives to find a nonpriority customer in service, he cannot preempt the nonpriority customer who is undergoing service. The service on the priority customer beging only on the completion of service on the nonpriority customer. This type of queue discipline is known as nonpreemptive or head-of-the-line priority discipline and was introduced by Cobham (1954). For nonbulk priority queuing systems and other priority disciplines which we do not discuss here, the reader is referred to Jaiswal (1968). To discuss this system, we use the supplementary variable technique discussed in Chapter 2, and define the following state probabilities. 1

Pm,,(v, t ) d v t o(dv), m > 1, n > 0, is the joint probability that at time t there are m priority and n nonpriority customers in the system, and a priority customer is being served with elapsed service time between v and v+dv.

2

Qm,,(v, t ) d v + o(dv), m > 0, n > 1, is the joint probability that at time t there are m priority and n nonpriority customers in the system, and a nonpriority customer is being served with elapsed service time between v and v + dv.

3

Po(t)is the probability that the system is empty at time t ,

With the inclusion of a single supplementary variable v, the process becomes Markovian and thus can be studied by writing the differential-difference equations. Following the usual procedure (discussed in Chapter 2), we get the set of equations given below:

BULKARRIVAL QUEUES

146

Since by the normalizing condition we must have P ( 1 , l ) = 1, it follows that Po = 1 - p, - p 2 , where pi = h ibi ii,,i = 1.2. The limiting distribution exists if p, + p2 < 1. From equation (3.6.18) one can obtain the expected values of the number of priority and nonpriority customers as

Now to proceed any further, we first need to know the values of C2(I), C, ( I , w), where Cz (z) is given by equation (3.6.1 9) and ~ , ( z , w )= lim a C l ( z , w ; a ) CY-tO

-C2(z' Consequently,

F

respectively, where Ei(v2) = v2bi(v) d v and P i = [dZAi(z)/d~'],.I for i = 1,2. Next we discuss the limiting waiting-time (in queue) distribution of priority and nonpriority customers.

1 - ( l i z ) b ; IXl(1 - A t ( w ) ) + h z ( l -A2(z)>l 1 - ( l / o ) b , [ h r ( l -Al(w))+ A2(i -A,@))]

'

C2(1) = h 2 iiz where we have used Po = 1 - p , - p , and

The Waiting-Time (in Queue) Distribution

Our discussion of waiting time in the M?, M ~ / G , G , z / l queue makes use of (and the reader is therefore referred to) an outline given as Miscellaneous Problem 3 of this chapter, and the discussion in Chapter 2 of the busy-period distribution for the simple MiGI1 queue.

Finally, replacingA , ( a ) by o in P(1, o;v), P'(1, w; V ) is given by

P'(1, w; v) = C;(l, w) exp [- v(Al - A , w] f(v)

Priority Customer

where First we discuss the distribution of waiting time (in queue) for a priority customer, which is easier to discuss than the one for a nonpriority customer. If we identify a priority group with a single priority customer, then as its (the group's) service time is the total service time of the members constituting the group, the L.-S.T. of this service time is A (SI(a)) = 6;(a), say. The mean arrival rate of groups is h l , and its (the group's) "group size" p.g.f. is simply a.To facilitate further discussion, it is convenient to use primes (or some other notation) on the functions that are altered in the preceding discussion, and the notation' for probabilities immediately after a departure of a group. Thus the probability that a departing priority customer (group) leaves behind him m priority customers (groups) is, in the new notation,

and substituting in equation (3.6.20), PW(w) = C'l(l, w ) b ; ' ( h ~-X1o)/hlw. (3.6.21) The customers (groups) left behind by a departing customer (group) are just those that arrive during the time it spends in the queue and in service. Since arrivals are random, it follows that P"(w)

where h, is the normalizing constant, t is omitted in the above notation since we are dealing with the limiting case, and

ii;, (XI -X~w)&'(h,

-Xlw)

(3.6.22) where iijbl (a) is the L.-S.T. of the distribution of queuing time for a priority p (i.e., for the first customer of the group). Equations (3.6.21) and (3.6.22)

%;I

=:

(a) =

PO&+ a2 z211 -&(a)] hl &;(a)-hl + a

(3.6.23)

7 48

BULK-ARRIVAL QUEUES

If the customers within an arrival g o u p are selected randomly, then the probability of a priority customer being j t h in its arrival group is

and the p.g.f. of I x j ) i s

Now as the waiting time (in queue) of this j t h customer in a group is the sum of the queuing time of the first customer of the group together with the service times of the first j - l customers of the group, it follows (since we are dealing with priority customers) that the L.-S.T. of the distribution of waiting time (in queue) for a random priority customer is given by

3.6

THE SYSTEM

MF,M~S~IG,, G,II

149

priority group. Consequently, if in the above equation &(a) is replaced by A , (5,(a)), we have To consider the walting-time (in queue) distribution for a nonpriority customer, we identify, as in the priority case, each nonpriority group with a single nonpriority customer, so that the service time of the group is the total service time of its members. The L.-S.T. of the group's service-time distribution will then be, say, Az(b;(a)) = 6;(a). We shall use double primes t o identify the corresponding functions which we use for a group of nonpriority customers in the present case. As we did for priority customers, we first consider the waiting-time (in queue) distribution of the first served customer of a nonpriority arrival group. If the first customer of a nonpriority group is served only in the absence o i priority customers, the order of service of priority customers and nonpriority customers, other than the first, does not affect the waiting time (in queue) of any first customer. Hence we can use the previous discussion by taking A z ( z ) E z and assume that the nonpriority customers in a group are served without interruption, so as to give a service-time distribution (for the entire group) whose L.-S.T. is 5l(a). Consequently, the number of nonpriority groups present in the system when the service on the first customer of such a group is about t o start has the p.g.f.

where we have used equations (3.6.23) and (3.6.24). Next we consider the equivalent distribution for nonpriority customers. onpriority Customer

To find the equivalent distribution for nonpriority customers, we need to use the concept of completion time for each nonpriority customer, which has been defined earlier. Given that the service time of a nonpriority customer is v, the probability of n priority groups arriving during this time is, since the arrival process is a Poisson process, e-hlY(hlv)"/n!. The conditional L.-S.T. of the distribution of the time taken to clear the queue which has built up during the time v is [g(a)j ", where g(a) is the L.-S.T. of the busy-period distribution for a queue of priority customers, considered in isolation. Thus the L.-S.T. of the distribution of completion time F(a) is given by

where we have used the limiting form of equation (3.6.15). The multiplicative constant l / X z is the normalizing constant, as in the priority case. The groups present at the start of service are the ones whose service is about t o start and those that have arrived during its queuing time. If the queuing-time distribution of the initial customer (group) whose service is about t o start has L.-S.T. G;: (a), then the p.g.f. of the number of nonpriority groups arriving during its (the group's) queuing time is zG;:(h2 -h2z), where the factor z corresponds t o the initial customer (group). But this p.g.f. is also given by equation (3.6.27). Thus we have

which gives

where we have used equation (3.6.19) with A;'(z)=z and wf'(cr) satisfies the equation

Arguing as above (or independently as in Chapter 2), one can also obtain g(cr). In this case the service time of the first customer will be the service time of a

and

6l(a) =A,(b;(ac)),

Now as the queuing time of the j t h customer of a nonpriority group is the queuing time of the first customer of the group together with the completion

BULKARRIVAL QUEUES

150

times of the first ( j - 1) customers, the i.-S.T. of the distribution of a randomly selected nonpriority customer is

wq2(a)

= G'p)

1 --Az(F(cr)) a2(l - F(Q))

(a) is given by equation (3.6.28). While explicit results do not seem to be where accessible, moments can be obtained as usual. In particular, the expected queuing times for the priority and nonpriority customers may be found from equations (3.6.25) and (3.6.29), and are given by

It may be noted that once again Little's formula is satisfied. The results of this section are due to Hawkes (1965). For single arrivals, that is, when Al(z) = z and A2(z) = z, the joint distribution of the number in the system has been discussed by Jaiswal (1968). In the case of single arrivals, Miller [see Jaiswal (1968) for details] has also discussed this problem using the imbedded Markov chain technique. By some minor modifications in Hawkes's work, one can obtain similar results for the case of preemptive priorities.

PROBLEMS AND COMPLEMENTS

151

semi-Markovian. A more general single-server queuing system (with bulk arrivals) than that of Neuts (1977) has recently been examined by Ramaswami (1980), using the matrix-geometric method [see Neuts (1981)j. In this paper Ramaswami discusses, among other things, the busy period, the busy cycle, the steady-state queue lengths (at departure and random epochs), the virtual waiting-time distribufion, and the moments of some of these distributions. He also discusses the computational aspect of several formulas which he investigates. Other contributors whose work we describe briefly are Chaudhry (1974) who analyzes a bulk-arrival queuing system with intermittently available server; Foster (1964) who discusses some buIk-arrivaI or bulk-service queuing systems using renewal theory; Gupta and Goyal (1965) who discuss the queue length process for the bulk-arrival queuing system M ~ / H E , / ~with hyperexponential service times; Keilson (1963) who considers, among other things, the asymptoljc behavior of the queue MX/A4/1 using Bessel-type functions; Kerridge (1966) who gives a numerical method for a typical simple bulk-arrival queue; Kleinrock et al. (1971) who discuss a processor-sharing queue with bulk arrivals; Moore and Bhat (1972) who give a computational approach to some mean value functions in the bulkarrival queuing system M ~ / E ~[see/ ~also Moore (1975)j ; Rosenlund (1973) who considers the L.4.T. of the joint distribution of the length of time and the number of customers served during a busy period of the bulk-arrivaI queuing system with finite waiting space. Several authors, including Fainberg (1974) and Yao (1981), have given bounds and approximations for various bulk-arrival queuing systems.

BULKARRilVAL QUEUES

PROBLEMS AND COMPLEMENTS

$52

Consequently,

and thus g(z) has no singularities inside B. Also on B,

I* <- P ~ c r + h + ~ - h A ( e ~ ) ~R e c r + p

(b)

<1

if Recr

153

> 0. 2

Thus on B, 1 f (z) -g(z)I < 1 = If ( z ) \ Consequently, as all the conditions of Rouche's theorem are satisfied, g(z) has one zero inside the unit circle, since f(z) has one. For the system M X / E ,/ l in equilibrium, show that

Moreover, for p > 1 , it is clear that Po = lim,, , + cupo(cu) = 0, since thenz(cu)+z such that O < z < 1 . This proves the existence of the limiting distribution for M X / ~ , / I w h e n p < 1. If Nq denotes the number in the queue (excluding those, ifxany, in service) when the system M X / E J / l is in statistical equilibrium, then show that the p.g.f. of Nq is given by

Show further that

This means that for p 2 1, all the probabilities Pi, j > 0, are zero and for p < 1, the Pj's are given by the p.g.f. (3.1.19) when k = 1. Proof: Consider the zeros of the denominator of equation (3.1.19) when k = 1, that is, consider the roots of the equation

Clearly the right-hand side of this equation is the p.g.f. of the number of arrivals during a service time of a customer. Let us call this K(z). Now applying a result from branching process theory (see Appendix A.2), we can say that z = K(z) has a real root z such that 0 < z < 1, iff ~ ( ' ) ( 1 )= hZ/p = p > 1. If such a root exists, then as a + 0 , z(a) + z , where 0 < z < 1, otherwise (when p < I), z(a) + 1. Besides, differentiating the equation z(c~)[a+h+p--XA(z(cu))] = p

3

: and Cb2 are the coefficients of variation of group size and service where C time, respectively. When the system ~ ~ 1 is ~in ~statistical 1 1 equilibrium, show that the L.T. of the distribution of waiting time for the first customer of an arriving group when the group has to wait is given by

where Qo = 1 -Po is the probability that the arriving group finds the system busy. In particular, show from this or otherwise that for the systems M ~ / E ~ / ~ and M X / ~ / 1respectively, ,

and

[obtained by setting the denominator of equation (3.1.13) equal to zero], with respect to cr and letting cu -+ 0, we get where b is the constant mean service time of a customer, and p = hZb in equation (1). The result (I) in its unconditional form and when a, = 6,, , may be compared with M/D/c (when c = 1), which has been considered by many authors, including Erlang (1920) and Crommelin (1932). An extension

B U L K A R R l V A L QUEUES

PROBLEMS AND COMPLEMENTS

queues. For the system M ~ / E in ~ statistical / ~ equilibrium, the mean waiting time spent in the queue may also be obtained as follows. If a group arrives and finds the system in state (n,r), the mean waiting time in the queue for the first member of the group is [(n - 1) k r ] 111, and consequently this quantity for any group is given by

4

n

+

> 0.

Ce

P C , = Pi,1/ CP,,,,

i = 1,2,3,. . .

1

is the probability that there are i- 1 customers in the system immediately after departure of a customer. Hence show that the p.g.f. of {P,') is given by k

+-

= -La I*

xPo(A(z) - 1 ) P P [ I z / ~ (A XA(z))] - 11 ' Obtaining the constant Po by P+(1)= I , show finally that P'(z) =

-

where p ( l ,x) may be obtained from equation (3.1.18). Consequently it can be shown that

cpl

x=I

P'(z) =

P @ + 1) = -. 2

This result is obtained independently through the imbedded Markov chain in Section 3.4. For another proof, see Miscellaneous Problem 1.

Thus

wql

=

%+ 2 I*

I*

7

5

After substituting for L, from equation (3.1.26) into the above equation, one gets equation (3.1.27). ~t may be observed that if S is the number of unserviced phases of the unit under service at the instant of arrival of a group, then E ( S ) = P ( +~ 1)/2 For an independent elegant proof of this, See Section 3.5. the system M ~ / E ~ / Iin statistical equilibrium, if the size of an arrival group follows a geometric distribution, that is, if aj = (1 -a)a" -' ,j = 132, 3 , . . . , then show that

%,(4 =

and

(1 - - P ) I A @ ) - I I ~ W ( A- A A ( Z ) )- 11 '

p(1 - a ) - h e ,u(1 - a )

p(l --a)+& ~ ( - al ) - A $ - @

p(1 -a)-X mqc(O0 = & + I * ( ] - a ) - A 1 a ) 1 -az P(z) = I* -z(X + W ) (1 - a )

(a)

The system GIIMI 1 . Show that when r = 1 in G I r / M / l , the distribution of the number in the system is geometric, and that the distribution of waiting time (in queue) for those who have t o wait is expo-

(b)

The system ELIEIl. distribution, namely,

Here the interarrival times have an Erlangian

a ( u ] = 1 - k - 1 (""U)i' j=o j !

+

with mean 1/A. In this case T ( a ) = [ Ak/(X k a ) ] k , and so K ( ~=) [ 7 / ( 7 + (1 - z ) k - ' ) l and therefore the denominator of equation (3.2.8) equated t o zero gives

1 - z r [ T ( T+ ( 1 - z - l ) k - l ) - - ' ]

= 0

.

'

is defined in Problem 3. It may be noted from the expression waiting time in the queue for the first member i ~ , , ( ~ that ) the arriving group is exponential with parameter I*(l -a) -A. The distributi

( k ~ ) ~ z ' +-{(I '

+7k)z -lIk

= 0.

This equation of degree r + k has exactly r + k roots, 1/71> 1 / 7 2 , , . . , I / ? r > ~ / Y I Ol /, y 2 ~ ,.. . ,117 ( k - l ) o

BULK-ARRIVAL QUEUES

I

PROBLEMS AND COMPLEMENTS

are used in the discussion of the waiting-time problem:

the roots l / y l , I = 1 , 2 , . . . ,r, being inside and l / y j o , j = 1 , 2 , . . . , k - 1 , being outside /z[= 1. By virtue of this we may write

s

Therefore we have

where ~ ( x ) = f i (x-yi),

The normalization finally gives

Ti,

i = l ,

..., r ,

1=1 I

(c)

For the system GIr/M/l, show that the mean waiting time (in queue) of the first member of an arrival group (including zero waits) is given by Wi1 = [ S - r] /p, where r

s (d)

8

=

1 I l K l -rdZ I

are the r roots inside 121 = 1 of the equation (3.2.4), and F(')(x) indicates the derivative of F(x). Hint: For details of the proof of the first identity, see Miscellaneous Problem 1 of Chapter 6. On the same lines one can prove the second identity. In the limiting case of Glr/M/l , show the following: (a) The p.d.f. of the waiting time (in queue) of the first member of a group of size r is P(waiting time in queue = 0) = P, = F ( 1 )

=1

Let P = II[==, (1 - y,)-'. Show that the probability that an arriving batch finds the system empty is l/P. Use this t o show that the mean waiting time (in queue) for the first member of an arriving group (excluding zero waits) is given by E(VQl1 Vql > 0) = ( S -r)P/(P1)p. These results have also been derived by Conolly (1960), using the supplementary variable technique. Barber (1964) has calculated numerical values of P and S for the system D r / M / l . For more discussion on the system GIr/M/l, see Chapter 6 and problems therein.

The moments of the wait (in queue) may be obtained either directly or by using the transform of the above density. In particular, show that

Continuation of Problem 7(b). Show that when k is odd, all the yjo's are complex, and when k is even, exactly one yjo is real and positive. In particular, show that for k = 2,

Hint:

wGl (x) =

2

P(n exponential services completed in time

n=l

4 7 0 -PI($ -2) P-@l = L b 2 z r t 2 - {(I + 27)z

1-yz

- 1j2

1-7

where y-"s the unique real zero of 4 ~ ~ 2 -{(I " ' ~+ 2712 - 1)' lying within j 0, 1 I. " ~ Hint: Consider the intersection of the pair of curves y = ( k ~ ) ~ z and y = {(I 7k)z - 1)' for realz.

+

9

The system GIr/M/l. This problem deals with the waiting-time distribution for the system GIr/M/l. First consider the following identities, some of which

(b)

The transform of the p.d.f. of the waiting time (in queue) of the second member of a group of size r, since its wait (in queue) is the convolution of the wait (in queue) of the first member and its (the first member's) service time, is iii,, (a) p/(p a). This procedure can be easily extended

+

Show further that the mean waiting time (in queue) given by

wi in this case is

I1

Hint: Use arguments used in Miscellaneous Problem 3. (a) Continuation of Problem 10. Show that the mean waiting time (in queue) Wi of a random customer of an arriving batch is gken by

>.

10

where mz = 2 f j 2 a j and

Consider the limiting distribution of N i in G I ~ / M / Iwherein the size X 0 arriving batches is not fixed but rather follows a probability distributio given by P(X = j ) = aj, j = 1 , 2 , . . . , I . Show that corresponding to eWat (3.2.1), we have the relation N;+, = (N; + size of a batch arrived at instant cr:, -D,)+.use this relation to show that the transition probab matrix (t.p.m.) for the chain N,, for i > 0, is given by

p.. =

0 # j

Pl+i-j,

P-(z) =

Z =

i = O

+ . ..+alki-j+l, a l k l + i - j + a l - l kl+i-j-l + . . . + a j - i k o >

i

O+j
kzllC1= x zf jajlP < 1, it is possible to prove the existence If limiting distribution for the process (N, ) when -+ show that, in the limiting case, the distribution of: (i) N;hasap.g.f.

*

-

co

=

1-Ym n -----I

I

jaj.

j=l

a,kl+i-j +al-1 kl+i-j-1

P-(z) =

(I -z)(l -p)Z Z - Z { ~ +p ( l -A(z)))

where

< 14-1

where

=

z=2:jaj.

In the particular Case of Problem 10 when the interarrival times are distributed, that is, for the system MX/M/I, show that the p&f. 0 f N - can be expressed in the form

t+i-I

i - z ~ , ,

fil+i-j

(b)

l-yrnz

the where y,, y 2 , . . . ,7, are the reciprocals of the roots circle, lz 1 = 1 , of the equation A(z) ~ ( z - )' = 1, with A(z) = 'j and K(z) = F(p(l - z)) = 2; kjzJ-

':

the

It is interesting to see that for the system M ~ / M / I P-(2) , =P(z), where the ter p&f. may be obtained from equation (3.1.19). This is true because the process is Poisson. [In view of the input process being Poisson, the relation = P(z) is true even for the more general queuing system M X / ~ /.Il This that the limiting distributions of the number in the system considered arbitrary instant and at an instant just before an arrival are the same, when arrivals are in groups of random sizes and the interarrival times and the service es are exponentially distributed. In fact, when the arrival batch sizes are fixed, tions between generating functions of the queue size considered at an arbitrary ant and at an instant just before the arrival of a group and just after a departure ve been considered in several papers of Foster, Foster and Perera, Chaudhry, d ~haudhryand Templeton. Though some of the results of these authors have discussed at appropriate places in various chapters, most of them and other d material are the subject matter of Chapter 6. n 3.3 Show, by eliminating

P<(w) from equation (3.3.6), that P-(2, w) is

BULK-ARRIVAL QUEUES

161

Section 3.4

\

given by

13

PROBLEMS AND COMPLEMENTS

15

For the s y s t e r n ~ ~ / ~ show ~ / lthat , for i > 1 ,

16

that is, Ki(z) is related to the L.-S.T. of the service-time distribution. For the system MX/6,/1, show that for i > 0 ,

K ~ ( z )= 6i(X -AA(z))

Consider the transient behavior of N, in the system Mr/M/l. Show that the first two moments of N , about the origin are given by

pi =

x

jkj,i = hZE(Vi)

i=l

and

where P i and E ( N - ) refer to the limiting case when iz

-+ m

and are given by

1

,

17 18

Hint:

19

E(N,) = coefficient of w n in

Since po < m, the condition of the existence of a limiting distribution can be changed to lim supi > {pi) < 1, as stated in Section 3.4. Hint: To get pi, i > 0, use either equation (3.4.1) or problem 15. Continuation of Problem 16. By using Problem 16, show that p, has the value stated in equation (3.4.1 1). For MX/6,/l of Problem 15, show that lim supi > (pi) < l reduces to ( A , + 2h2)/,u s p < 1, regardless of the value of p, . For special case (b) of MX/6,/1 as given in equation (3.4.10), show that the expected number in the system is given by

,

,...

14

where P-(z, w ) is given in equation (3.3.6). Continuation of Problem 13. In the same system MT/M/l,show that the m t h moment about the origin of the waiting-time distribution is given by

+

(I

-PI

I

((~11- ~ 1 2 ) ( ~ :+ ~ 1 2 + ) p12(p2 + P + ~ 0 2 ) )

+ C [ ( P I -p)(1 - p ) + ( p 2

P11 + P I Z ) ( ~--P)

-(p2 ++poZ)(l -PI)] and in particular,

where L+=E(N+), p, = X2/p, and other quantities have been defined earlier. Hint: Differentiate the requisite P+(z) and use K j ( z ) as given in Problem 15.

1

I

Section 3.5

20 Hint: Take the L.T. of w&(r) given in equation (3.3.131, etc. I t may be seen that the moments discussed in Problems 13 and 14 can calculated numerically once we know P,(n), which can be determined fr equation (3.3.9).

Show that the right-hand side of equation (3.5.8) has k - 1 zeros within the unit circle, lzl = 1, and the simple zero, z = I , on the unit circle. Hint: Take the circle lzl = 1 + E(E> 0, E -+ 0), consider f(z) and g ( z ) as the first and second terms of equation (3.5.81, and apply RouchB's theorem. To show that z = 1 is a simple zero, note first that D(z) has no zeros on

B U L K - A R R I V A L QUEUES

162

the unit circle, lzl = 1, except z = 1, since I f(z)l on it. Besides, since

PROBLEMS AND COMPLEMENTS

Ent:

f Ig(z)/ at all other points

L can be obtained by one of the following methods:

(i) By using equation (3.5.1 6).

By using P(z) and the transformation used in Section 3.5. By using equation (3.5.13) if k = 1 and hl A. Keeping in mind the notation used in Section 3.2, verify that L as obtained here agrees with the L as given in equation (3.1.25). Show that for the system M X / ~ / 1 ,

(ii)

(iii)

21

22

z = 1 is the simple zero on lzl= 1 of D(z). For k = 2, verify equation (3.5.13) by using equation (3.5.12). . Hint: Solve the set (3.5.5) for Pl(z), Pz(z). Find p i l ) ( l ) , ~ $ ' ) ( 1 ) Use Po,, + Po, = 1 - p and p = r2hl A2 /p(hl 4- hi), wherever necessary. One can also verify by direct differentiation of equation (3.5.5). For k > 2, the method is straightforward though laborious. (a) The system E ~ E , . / Iwith generalized 2-Erlang input. Using equations (3.5.9) and (3.5.7), or the set (3.5.5) with k = 2, show that

Show further that, in this case, D(z)/(l - z ) given in equation (3.5.8) has a single root within lzl = 1. If this root is z o , then show that 0
(b)

24

The system Ek/M/I. Let the arrivals be by singlets, the interarrival times follow the ordinary Erlangian Ek,and the service times follow an egrponentid distribution. Define the following generating function:

Show that

This equation is true for all Izl =G1 and 1 y l < 1, and in particular for z = y k . Consequently we can write

-Xa+

23

[ h l ( l -p)\p0'2*

Further, if h l = X 2 = A, then show that this value of L agrees with the one which can be obtained from equation (3.5.16). Discuss L for the system E$/D/I with generalized 2-Erlanginput and constant service time. (a) The system M~/E,./I. Take k = 1 and let X 1 = h , PI ( z ) -P(z) and Po =PO, Show that the first equation of the set (3.5.5) (other equations being redundant in this case) gives uniquely the p.g.f. of the number of phases in the system (queue + service) as

where Po = 1 - p, p = hrZ/p. In this case L is given by

Here L is given completely in terms of parameters of the input and the service distributions.

Using Rouchk's theorem, show that the denominator of P ( y ) has one zero, say Y O ,such that IYOI > 1. Using this fact, the convergence of P(y) within and on the unit circle, 1 yl = 1, and the normalizing condition, show that P ( y ) can be written as

BULK-ARRIVAL QUEUES

364

Now as P,, represents the joint proljabilit~that there are n customers in the system (queue + service) together with the arrival being in the jth phase, Pn = ~ i kP,,, ~ gives the probability of n customers in the system. Show that the distribution of the number in the system is given by Po = 1 -p, P, = P(Y; I ) ~ ; " ~n, 2-1, p = hlkp, and therefore L = [k(yo - I)]-'. L can also be found from equation (3.5.16), which gives an expression not involving the root y o . Hint: Po = sum of the coefficients of y , y 2 , . . . , y k in P(y), P, = coefficient of y nk*i in P(y), and use 1 - (1 (AIM))+ ( ~ / p ) y $ + l= 0, wherever necessary. Here we have found directly the distribution of the number of customers in the system. Jackson and Nickols (1956) obtained the same results for Pn and L by first considering the number of phases in the system and hence obtaining the distribution of the number of customers in the system. Accounts of Jackson's work may be found in textbooks by Saaty (1961), Prabhu (1965), and Gross and Harris (1974). 25 (a) Continuation of Problem 24. Find the waiting-time distribution for the systemEk/M/l with ordinary k-Erlang input. To consider the waitingtime (in queue) distribution for the system Ek/hf/l, we first need to know the probability that an arrival when it enters the queue finds n customers in the system (queue service). Let this probability be Pi. In other words,

yt +

(b) Show that the conditional waiting-time density w,,(r)

for those who

have to wait is

dP(Vq ,(r / Vq > 0) =y$w,(r), 7>0 dr which is an exponential distribution. This assertion holds even for queuing systems with general input and exponential senice times [see Smith (1953) or Problem 7(a)]. For numerical work on particular cases such as M/M/I, DIM11 of the model Ek/M/I, the reader is referred to Jackson (1956). Continuation of Problems 24 and 25. Verify Little's formula L = h,W- for the system Eh/M/I discussed in Problems 24 and 25, where he is the effective arrival rate and W-is the average time spent by an arrival in the syskem. Hint: Note X, = h/k. The system E2/M/1. Consider the system E2/M/l with generalized 2-Erlang input and exponential senice time. Show that for such a system, ~ 2 7 .= )

Lq = where

P 1 -P

+

P i = P(N = nlarrival about to join the queue) = P(N = nl arrival in phase k of the arrival channel)

where we have used equation (3.5.6). Thus by using the expression for PnSk- the coefficient of y n k c k in P(y) - we find that

Hint:

Use the root zo of the quadratic equation

which lies in ] 0, 1 [. Continuation of Problem 27. ordinary 2-Erlang input,

Show that for the system E,/M/I with the

Consequently, if W;(T) is the p.d.f. of the waiting time (in queue) Vq of an arrival, then W&r) = P(Vq /4 r), and therefore

pnrn-l

wq(r) =

---Pi (n-I)!

exp (- 117)

Find the waiting-time (in queue) distribution for the queuing system E?/E~/I with generalized k-Erlang input. (a) Show that the L.T. of the waiting time (in queue) of the first customer of an arriving group is given by G,, (or) = (kk /QP,(~/(M or)), where Pk(z) is given by equation (3.5.91, and the corresponding mean waiting time is given by

+

where we have used the expression giving the root y o .

BULK-ARRIVAL QUEUES

N~~~ that Hint:

wi1can also be obtained without using G,,

Using Problem 25, we get

"[

( T I * ) ( n - l ) r + s - el - P T

{(n - 1)r + s -

P~n-l)r+s.h s=l

where Po = 1 - p l system is L1 = Pl

wql ( 7 ) d7 = -= PO,k 6 ( 7 ) d~ h

+ n=l 2

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

and the corresponding expected number in the

+(Xl~lj2E,cv.>+(Pllzl)~l 2(1 - - P I )

Hint: For getting P(w), use h2 E 0, z 0, and b2(v) 0 in equation (3.6.181, and for getting L , ,use h2 0 in the value of L , . (b) Show that the L.-S.T. of the waiting time (in queue) of priority customers alone is given by

--

BULK-ARRIVAL QUEUES

768

beginning of the time period T; and v'is the number of customers arriving during the delay of beginning of service for this group. The process (N;) is an imbedded Markov chain, and when p = hzb < 1, the limiting probabilities P,? exist and have the p.g.f. P+(z) = P,f[A(z)Z(h- XA(z))

- 11 [z{ 116(h - XA(z))} - 11-\

IzI

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

I69

==

B, (x) =

P(service time of the nth arrival group d x l n t h arrival group is of size r)a,

< 1,

+

with Pi = (1 - p)/Z((l Xd). Finch (1959) has shown the existence of the limiting distribution for a more general queuing system GI/G/l when p < 1. His arguments can be used to show the existence of the limiting distribution for GIX/G/l when p < 1 . The case under discussion is a particular case of G I X / ~ / 1 . Hint: Taking the p.g.f., we have

Noting E(zN; IN; > O)P(N; > 0) = E ( z ~ ; ) -P(N: = 0) = P+(z) -Po+,where P"(z) = ZF-, P:zn, P; = P ( N i = O), using independence of the r.v.'s involved and letting n -+ w, we get the required p.g.f. As a particular case, show that if P ( T = 0) = 1, then P+(z) reduces to equation (3.4.9). Further show that if P(T = 0) = 1 and a, = 6,,, i = 1, 2, 3,

{B(x)]'* = B(x), and A(x) and B(x) are the interarrival-time and the service-time distributions, respectively. We now explain the use of the above integral equation by discussing the waiting-time distribution W, (7) for the queuing system MX/G/1. Note here that since the arrivals are by Poisson process, the notation Wil (7) can be replaced by Wq, (7). To discuss this problem, we need to define the following additional transforms:

Further, define the function W,*, (7) by the following equation:

2

where p = hub. (a) Continuation of Problem 1. It is possible to discuss the limiting waitingtime (in queue) distribution of the queuing system of Problem 1. If p < 1, the limiting waiting-time (in queue) distribution of the first member of an arriving group, namely, W& (7) = lim,,, P(Vq, (n) d r ) exists, where Vql(n) = Vql (a; - 0), Vq, ( t ) being the virtual waiting time at the instant t. Following arguments due to Lindley (1952) (refer also to Chapter 2) and Finch (1959), it can be shown that the distri/ ~a proper distribution bution, W i l (7) of the queuing system G I ~ / G is and is the unique solution of the integral equation

where, since arrivals are random,

Since B1(7) = 0, for 7 < 0, SO for cedure discussed in Chapter 2:

< 0, we have, following

the pro-

where C > 0. Thus we can rewrite equation (MI) as

But using equation (MI) one can see that C = j?, WqI (AX) dS(x), which we assume to hold. Consequently,

BULK-ARRIVAL QUEUES

MISCELLANEOUS PROBLEMS A N D COMPLEMENTS

From equations (MI) and (M2) we get, respectively, Now as the sequence of groups can be taken to form a renewal process with the size of the group playing the role of the waiting time, one can use the discrete form of the distribution of past life (Chapter 2). Thus if yi is the probability that the customer, though selected randomly within the group, is served jth in its group, then From equations (M3) and (M4) and usingT(a) = A(c(a))h/(h -a), show that Gql (a) = C[(h

- a)a(a)

where b(a) =

. j

- h] [h - a - XA(6(a))] -'

m

But since xj is the equilibrium (discrete) density at j - 1 of the past life (see Chapter 21, we have i-I

e-'U dB (x). U

The constant Cis evaluated by using the normalizing condition GqI(0) = 1 and hence C = - 1- P l+hd'

(b)

Consequently, taking the L.T. of equation (M5), we get Gq(cr) = E(e-cYVq) =

&'(e-"Vql

) E(e-a

2;;:

si)

If P(T = 0) = 1 , then show that the L.-ST. of the waiting-time (in nueue) distribution of the first member of an arrival group is G,! (a) = 1- (1 - p) a/ [A - a - h~(F(a))]. It is interesting to observe that if we identify a single customer with a group whose total senice-time distribution function has L.-S.T. equal to A(6(a)), then Gql (a) could be obtained from an ordinary M/G/1 queue with 6(a) replaced by ~ ( 6 ( a ) ) .For example, see Miscellaneous Problem 6. This is due to the input being by the Poisson process. Consequently, these observations apply equally well to the problem discussed in (a). Continuation of (b). Show that the mean waiting time (in queue) of the first member of an arrival group is I

(c)

where Cz and C$ are the coefficients of variation of group size and service time, respectively.

3

Continuation of Problem 2: waiting-time (in queue) distribution of a randomly selected customer of an arrival group. Let V , be an r.v. representing the limiting waiting time (in queue) of a customer selected randomly from a group so that its d.f. is Wq(r) = P(Vq < 7). If j B 1 is the position of the customer selected within the arrival group, then

It may be noted that this agrees with Problem 3 1(b) (Section 3.6). Continuation of Problems 2 and 3. Show that the expected waiting time (in queue) of a random customer of an arrival group is

B ~ ~ ~ - ~ ~ QUEUES R I V A L

MlSCELLANEOUS PROBLEMS AND COMPLEMENTS

172

'173

which may be obtained by adding b iZ(l + C): - 11/ 2 to the expression given in Problem 2 (c). It may be observed [in view of this relation, which could be obtained from Mscellaneous Problem 5(a), that is, from equation (M6)] that Little's formula L , = hew;, where he = h a , holds for the system M X / G / l . In the slightly less general case M X / E r / l , the formula can easily be verified by l o o h g at the value of L , found in Problem 2. Quite a number of authors have given derivations of Little's formula, including Jewel1 (1967), Eion (19691, Maxwell (1970), and Stidham (1974) among others. Ifa,=6i,,Z= 1 , 2 , 3 . . . ,inadditiontoP(T=O)= 1,thenshowthat

5

Most of the results discussed in Problems 1 to 4 are due to Sahbazov (1962). Following Burke (19751, we have corrected the error in the orignal derivation of $,(a) by Sahbazov. The question discussed in Problem 3 has also been discussed by Soriano (19661, though under a different notational garb. l The system M X / ~ / 1 . In this problem we discuss the system M X / ~ / using the supplementary variable technique. The present solution generalizes the solution in Section 3.1 a11d derives some of the above problems as particular cases. A comparison between the solution to this problem and the Erlangan solution given in Section 3.1 thus shows both the usefulness and the limitations of the Erlangian technique (or its modified form). Section 3.1 also shows the ~ /M~/ E x / l systems. formal identity between the M ~ / E and In view of this, one might question the need for Section 3.2. &though for some purposes the Erlangian technique has been superseded by more general methods, we consider it to be stiU worthy of study for several reasons, in particular its relative simplicity for some problems requiring explicit solutions, its wide use in the literature, and its continuing usefulness in some models which yield most easily to a combination of two or more techniques. Since the procedure and steps used are quite similar to those discussed in Section 2.2 and also the notation is the same, we do not give the details here, but rather give the results with brief descriptions. Chaudhry (1979), while discussing the advantages of the supplementary variable technique over some other techniques, unifies several results for the queuing system M X / ~ / and 1 gives the correct expression for L, which has been incorrectly reported by Krakowski (1974), using the conservation principle. (a) Show that the limiting p.g.f. of the number of customers in the system

(b)

equation (2.2.25), or by following the procedure discussed in the derivation of equation (2.2.25). Continuation of (a). Show that the limiting p.g.f. of the number in the system considered immediately after a departure instant (imbedded Markov chain) is given by

Hint: Use the procedure used to get equation (2.1.27). It is interesting to see the relation between P(z) and P+(z), which is

(c)

In Chapter 6 we discuss many such relations. Waiting-time (in queue) distribution for the first customer of an arrival group. This is essentially Miscellaneous Problem 2(b) whose discussion depends on the use of the integral equation technique. The following alternative derivation, which depends on the use of the imbedded Markov chain technique, is more elegant. For if we identify a group with a single customer, then its (the group's) service time is just the total service time of the members constituting the group. Consequently, the group (customer) will have as its group size p.g.f. A(z) = z. The mean arrival rate wiU be A, and the L.-S.T. of the group's service time distribution will be S,(a) = ~(b(oi>). Using this information, we have from (M7) P,'(z)

=

(1 -p)(z

- I)$,(& - hz)

z - gg(h - hz)

where we have used ii = 1. e ( z ) may now be interpreted as the p.g.f. of the number of customers (groups) left behind by a departing customer (group). This p.g.f. may also be obtained as follows. Since the customers (groups) left behind a departing customer (group) are those that arrive during the time it spends in the queue and in service, and since arrivals are random, it follows that

wq,

(a) is the L.-S.T. of the distribution of queuing time for a where customer (that is, for the first member of a group). From equations (M10) and (M9) follows:

where Po = 1 - p = I - (ha/@), A(z) = Z L l a$, and b(or) is t L.-S.T. of the service time distribution. nt: We could obtain this by replacing z by A(z) in F(h -Xz)

which is essentially the result contained in Problem 2(b).

BULK-ARRIVAL QUEUES

3 76

Chaudhry, M. L. (1979). The queueing system M X / 6 / 1 and its ramifications. Naval Res. Logist. Quart. 26,667-674. Clarke, A. B. (1956). A waiting line process of Markov type.Ann. Math. Stat. 27,452-459. Cobham, A. (1954). Priority assignment in waiting line problems. Oper. Res. 2, 70-76. Cohen, J. W. (1963). Applications of derived Markov chains in queueing theory. Appl. Sci. Res. 10B, 269-303. Conolly, B. W. (1960). Queueing at a slngle serving polnt with group arrival. J. R. Stat. Soc. Ser. B 22, 285-298. Crabill, T. B. (1968) Sufficient conditions for positive recurrence and recurrence of specially structured Markov chains. Oper. Res. 16, 858-867. Crommelin, C. D. (1932). Delay probability formulae when the holding times are constant. P.O. Elect. Eng. J. 25, 41 -50. Eilon, S. (1969). A simple proof of L = AW. Oper. Res. 17, 915-917. Erlang, A. K. (1920). See Brockmeyer, E., et al. (1948), chap. I. Fainberg, M. A. (1974). Servicing a nonordinary flow by a one-channel system with waiting. Eng. Qbern. 12, 82-89. Finch, P. D. (1959). See Chap. 2. Foster, F. G. (1961). Queues with batch arrivals I.Acta Math. Acad. Sci. Hung. XIl, 1-10. Gaver, D. P. (1959). Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Stat. 30, 698-720. Gross, D., and C. M. Harris (1974). Fundamentals of queueing theory. Wiley, New York. Gupta, S. K. (1964). Queues with batch Poisson arrivals and a general class of service time distributions. J. Ind. Eng. 15, 319-320. Harris, C. M. (1970). Some results for bulk-arrival queues with state-dependent service times. Manage. Sci. 16, 313-326. Hawkes, A. 6 . (1965). Time dependent solution of a priority queue with bulk arrival. Oper. Res. 13, 586-595. Ivnitskiy, V. A. (1975). A stationary regime of a queueing system with parameters dependent on the queue length and with nonordinary flow. Eng. Cybern. 13, 85-90. Jackson, R. R. P., and D. G. Nickols (1956). Some equilibrium results for the queueing process Ek/M/l. J. R. Stat. Soc. Ser. B 18, 275 -279. Jaiswal, N. K. (1968). Priority queues. Academic Press, New York. Jewell, W. S. (1967). A simpler proof of L = AW. Oper. Res. 15,1109-1116. Krakowski, M. (1974). Arrival and departure processes in queues. Pollaczek-Khintchine for muIas for bulk arrivals and bounded systems. Rev. Fr. Auto. Inf: Rech. Opkr. 8, 45-56. Ledermann, W., and 6. E. 11. Reuter (1954). Spectral theory for the differential equations o simple birth and death processes. Philos. Trans. R. Soc. London. Ser. A 246,321-369. Lindley, D. V. (1952). See Chapter 2. Littie, 3. D. C. (1961). A proof for the queueing formulaL = AW. Oper. Res. 9, 383-387. Luchak, 6. (1958). The continuous time solution of the equations of the single channei q with a general class of service-time distribution by the method of generating funct J. R.Stat.Soc.Ser. B 2 0 , 176-181. Maxwell, W. L. (1970). On the generality of the equation L = AW. Oper. Res. 18, 172-173. Morse, P. M. (1955). Stochastic properties o f waiting lines. Oper. Res. 3, 255-261. (1958). Queues, inventories and maintenance. Wiley, New York. Morse, P. M., H. N. Garber, and M. Ernst (1954). A family of queueing prob1ems.J. Oper. Soc. Am. 2,444-445. -

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REFERENCES

177

Pegden, C. D. and M. Rosenshine (1982). Some new results for the M/M/I queue. Manage. Sci. 28, 821-828. Pike, M. C. (1963). Some numerical results for the queueing system D/Ek/l. J. R. Stat. Soc. Ser. B 25,477-488. Prabhu, N. U. (1965). Queues and inventories, a study o f their basic stochastic processes. Wiley, New York. Restrepo, R. A. (1965). A queue with simultaneous arrivals and Erlang service distributions. Oper. Res. 13, 375-381. Ross, S. M. (1970). Introduction foprobability models. Academic Press, New York. Saaty, T. L. (1961). Elements o f queueing theory with applications. McGraw-HiU, New York. Sack, R. A. (1963). Treatment of the nonequilibrium theory of simple queues by means of cumulative probabilities. J. R. Stat. Soc. Ser. B 25,457-463. Sahbazov, A. A. (1962). A problem of service with nonordinary demand flow. Sov. &fa&. 3, 1000-1003. s Shanbhag, D. N. (1966). On a duality principle in the theory of queues. Oper. Res. 14, 947949. Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge Philos. Soc. 49, 449-461. Soriano, A. (1966). On the problem of batch arrivals and its application to a scheduling system. Oper. Res. 14, 398-408. Stidham, S. (1974). A last word on L = AW. Oper. Res. 22,417-421. suzuki, T. (1963). Batch arrival queueing problem. J. Oper. Res. Soc. Jpn. 5, 137-148. Takdcs, L. (1962). Introduction to the theory of queues. Oxford Univ. Press, New York.

Additional References Bhat, U. N. (1968). A study o f the queueing systems M/G/l and GI/M/l. Lecture Notes in Operation Research and Mathematical Economics, vol. 2. Springer-Verlag, New York. Chaudhry, M. L. (1974). Transient/steady-state solution of a single channel queue with bulk arrivals and intermittently available server. Math. Operationsforsch. Stat. 5, 307-315. Fainberg, M. A. (1974). See references. Foster, F. G. (1964). Batched queueing processes. Oper. Res. 12, 441-449. Gupta, S. K., and J. K. Goyal(1965). Queues with batch Poisson arrivals and hyper-exponential service.Naval Res. Logist. Quart. 12, 323-329. Kambo, N. S., and H. S. Bhalaik (1982). Bulk arrival heterogeneous queueing systems. Opsearch 19,97-105. Keilson, J. On the asymptotic behaviour of queues. J. R. Stat. Soc. Ser. B 25,464-467. Kerridge, D. 11966). A numerical method for the solution of queueing problems. New J. Stat. Oper. Res. 2, 3-13. KPeinrock, L., R. R. Muntz, and E. Rodemich (1971). The processor sharing queueing model for time-shared systems with bulk arrivals. Networks 1, 1-13. Lippman, J. A., and S. M. Ross (1971). The streetwalker's dilemma - a job shop model. SIAM J. Appl. Math. 20, 336-342. Mohanty, S. G., and J . L. Jain (1970). On two types of queueing processes involving batches. Can. Oper. Res. Soc. J. 8, 38-43. (1971). The distribution of the maximum queue length, the number of customers served and the duration of the busy period for the queueing system M/M/l involving batches. INFOR 9, 161-166.

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Moore, S. C. (1975). Approximating the behavior of non-stationary single server queues. Oper. Res. 23,1011-1032. a n d U.N. Bhat (1972). A computational approach to some mean value functions in an M ~ / E ~queue. /I Tech. Rep. CP72015 CS/OR Centre, SMU (U.S.A.). Murari, K. (1969). A queueing problem with arrivals in batches of variable size and service rate depending on queue length. Z. Angew. Math. Mech. 49, 157-162. (1972). Time dependent solution of a queueing prablem with correIated batch arrivals and general service time distribution. Metrika 19, 201-208. Nakamura. G. (1968). Analysis of a discrete-time queueing system with bulk arrival. Electron. Commun. Jpn. 51-A, 27-32. Narasimham, G. V. L. (1968). A note on the asymptotic distribution of the traffic-time average in GI/G/m with bulk arrivals. J. Appl. Prob. 5, 476-480. Neuts, M. F. (1977). Some explicit formulas for the steady-state behavior of the q p u e with semi-Markov service times. Adv. Appl. Prob. 9, 141-157. (1981). Matrix-geometric solutions to srochastic models - an algorithmic approach. The Johns Hopkins Univ. Press, Baltimore. and S. Chakravarthy (1981). A single server queue with platooned arrivals and phase type services. Eur. J. Oper. Res. 8, 379-389. Ramaswami, V. (1980). The N/G/1 queue and its detailed analysis. J. Appl. Prob. 12,222-261. Rosenlund, S. I. (1973). On the length and number of served customers of the busy period of a generalized M/G/l queue with finite waiting room. Adv. AppI. Prob. 5, 379-389. Shanbhag, D. N. (1969). A queueing system with several types of customers. Ann. Inst. Stat. Math. 21, 367-371. Sharda (1973). A queueing problem with batch arrivals and correlated departures. Metrika 20, 81-97. Sharma, S. D. (1975). On continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures. J. Appl. Prob. 12, 115-129. Soriano, A. (1962). On the problem of batch arrivals and its applications to a scheduling problem. Doctoral dissertation, The Johns Hopkins University, Baltimore. Takics, L. (1962). See references. (1967). Combinatorial methods in the theory of stochastic processes. Wiley, New York. Van Hoorn, M. H. (1981). Algorithms for the state probabilities in a general class of single server queueing systems with group arrivals. MQtZQge.Sci. 27, 1178-1 187. Yao, D. D. W. (1981). "Contribution to the analysis of bulk arrival queueing systems". M.A.Sc. thesis, Department of Industrial Engineering, University of Toronto.

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-

-

In Chapter 3 we have considered single-server queuing problems in which arrivals occur in bulk. We now turn t o queuing situations in which arrivals occur singly, but service is in bulk. Many transportation processes involving buses, airplanes, trains, ships, elevators, and so on,allhave a common feature of bulk service. Ibmay happen that the server (or carrier) has a fixed maximum capacity, say B,or else the server may take units depending upon the number already present with the server. Such systems may serve as a model for a shuttle or automatic elevator. At times it may also be necessary t o change the capacity at the time of initiation of service. For example, consider a train to which additional cass may be added at the starting station when the queue length at that station reaches or exceeds a certain number B, or may be removed if the queue length drops betow A k but G B , the entire queue is served; if the number in the queue >B, then the first B customers will be taken into service. This description may well fit, at least approximately, the operation of an unscheduled car ferry or a single ground-floor station of an elevator. Another application of this could be t o traffic flow where a minor road merges into a major road. The traffic flow on the major road is interrupted by a traffic light after a certain period of time if at least k cars have activated a trip plate on the minor road. Otherwise the light stays green until k cars have arrived. The duration of the red signal is such as to allow at most B cars per cycle to merge into the traffic on the major road. If the successive service times are taken as the times required for successive batches of cars to merge, together with the fixed length of the green signal on the major road, then the study of the queue on the minor road may well be described by the above model. Such a bulk-service system has been called a system with general bulk-service mle as opposed to other types of bulk-service rules. Another illustration arises if we consider places like an amusement park, a museum, or an art gallery where there are guided tours. Suppose that a guided tour is not started until there are at least k prospective customers (visitors). This introduces what we normally call a "cporum," which occurs in other service systems as well. In another type of bulk service, the server always takes a fixed number of customers. Such bulk queuing systems have some intrinsic interest, as well as some practical applications, and can be considered as a special case of the general bulk-service rule.

BULK-SERVICE QUEUES

180

There is a close analogy between the'theory of queues and the theory of reservoirs and dams, a still closer one between singie-server bulk queues and models of storage and inventories. We pause briefly to point out this analogy by giving a few references and examples. For the analytical probabilistic study of models of storage and dams, the interested reader is referred to books such as Moran (1959) and Prabhu (1965). In models of storage of inventories, which may be considered to consist of discrete units (such as storage of newspapers by a newsboy or cigarette boxes by a tobacconist), or to be measurable on a continuous scale (such as electricity, water, or even wheat), the output is random, but the inpat may be somewhat controlled, whereas in models for dams the input is random, but the output may be controlled. In the theory of queues, both the input as well as the output are, in general, random. Clearly, the element of randomness appears in all three subjects - theories of queues, storage, and dams. In addition, a large number of problems arise in the three studies depending on whether the queue (dam or store) is of finite or infinite capacity, and on whether the queue length (commodity stored or quantity in dam) and the time variable are considered as discrete or continuous. The mathematical theory of dams of "infinite capacity" is closely related to the theory of infinite waiting-space bulk-service queues of the type ~ e o m / G ~ (see /1 Section 4.4), if we identify the content of the dam at each instant of release with the number B being served at the corresponding service instants. For details on the discussion of models of this kind for dams, see Moran (1959). A different analogy exists between the queue D/G/1 (a special case of the GI/G/l queue discussed in Chapter 2) and dams of infinite capacity. For details, we again refer the reader to Moran (1959) or Smith (1953). Stidham (1974) considers a queue (or stock in inventory) which builds up until it reaches a certain level, say I, and then is instantaneously cleared and the situation allowed to repeat itself. Assuming that the epochs of clearance are regeneration points, Stidham studies the properties of the net quantity in the system at a given instant of time. Ghosal(1970) approaches some bulk-service queuing and inventory problems from a unified viewpoint. The introduction of some kind of dependency, queue disciplines other than FCFS, correlation, and so on, make bulk-service queuing systems more cumbersome and less tractable analytically. If such a model is tractable at all, computational work is likely to become more tedious and time consuming even on large computers. In view of this, we do not consider such systems here although the lists of references and additional references include papers on some such systems, on which analytical work appears to have been done. In this chapter we limit ourselves to an analysis of bulk-service queues with independent arrivals and FCFS queue discipline.

customers, instead of being served singly, are now served in batches of maximum size B. Entry into service is instantaneous so long as customers are waiting in the queue, otherwise the server waits for a customer to come and starts service as soon as the customer arrives. In the study of M/G/l in Section 2.2 we obtained the distribution of the number in the system. For the present system it is more convenient to obtain the distribution of the number in the queue, which immediately leads to the imbedded Markov chain results and the waiting-time distribution. Let Nq(t) be an r.v. representing the number in the queue (waiting for service) at time t. The process {N,(t), t > 0), though non-Markovian, can be made Markovian by the introduction of a single supplementary variable X(t), which may be taken as the elapsed service time of the group under service at time t . We study the steady state of the vector process {Nq(t), X(t)) as t + -. Since the analysis runs parallel to what has been done in Section 2.2, details are omitted. Let tp now define the following probabilities:

1

~ ( x is ) the conditional service rate, so that the service-time density b(x) and the d.f. B(x) are given by

and B(x) = l

In the system M / G ~ the / ~ customers arrive singly by a Poisson process with rate h and wait in the queue until served. The service-time distribution is arbitrary. The

~:e(t) d t

]

2

P,,l(x,t)dx=PINq(t)=n,x<X(t)<x+dx],

3

Thus Pn,,(x, t)dx is the probability that there are n in queue [not in system (queue + service)] and the elapsed service time lies in ] x , x + d x ] . Poo(t) = P[N,(t) = 0 and the server is idle] .

n>0.

Let customers arrive at the arrival epochs 0 = a;, a;, . . . , a;, . . . and let ao, a,, a2, . . . , on, . . . be the departure epochs. In addition, we shall use the following notation:

lim P,(t)

of N q ( t )

[-

with

t -+e Limiting Behavior

- exp

=

P,

"-

Pj = lim P[N,(t) t

= j]

.

Note that Pi is the limiting probability (as t -+-) of j customers in the queue at a random epoch of time.

BULKSERVICE W

182

E

4.1

THE SYSTEM I M / G ~ / ?

Assuming that the various limiting probabilities exist, as they do when P = h/(Bp) < 1 , we find the p.g.f. P(z) =

m

b(u) =

2 pjZ'

e-@b(x)dx.

.P.(z) = j o W p q ~ ; x ) d x

i=o

of { p j )from the following partial differential-difference equations:

+i

= P4(z;0)

co

0 = -wm

P0,1(x)W~

1 -b(h-Xz) h(1 - z )

(4.1 .E

P(z) = Pm +P,(z). = -(h+g(x))P0,,(x)

"*l(x) ax

(4.1.5

+

a p n ~ l ( x ) = - ( h + g ( ~ ) ) P n , ~ ( x ) + ) i P n - l , I ( ~ ) n. > O

ax

(4.13)

Equation (4.1.9) gives the p.g.f. of the number in the queue, but i n v o l v e s ~ unknown probabilities Jr P,, 1 ( x ) v ( x ) d x , n = 0 , I,, . . . ,B , where we have use equation (4.1 .I). Evaluation of these unknown probabilities is discussed later.

which are to be solved under the boundary conditions If N(t) is the number in the system at time t , then

ce

P ~ + B , I ( X ) V ( ~ ) n~ > ~ ,O

Pn,l(0)=j

Po,i(0) =

f 1-

k=l

&+ =

is means that {<+I is the limiting distribution (as n e m ) of the number ( ustomers in the system immediately after a departure epoch. We have, since there will be j customers in the system just after a departure nd only if there were j customers in the queue before the departure,

Pn,i(x)~i(~)~~+~oo

0

and the normalizing condition

z m

pm

+

lim PIN(@,, +O) = j ] .

n-+-

ce

m

p n , ~ ( x ) d x= 1.

P:=D!

&,,(x)q(x)dx,

SO

(4.1.11

here D is a normalizing constant. The p.g.f. P+(z)= Ey=oPT~?is then given by

Define the p.g.f. Pq(z;x)by

x m

Pq(W) =

n =O

P+(z) = D

Pn,,(x)zn.

0

Multiplying equations (4.1.1) to (4.1.3) by appropriate Powers of equation (4.1.7), we get P,(z;x) = P,(z;O)(l -B(x)) exp

imj=of $ 4 , ,(x)g( x )dx

1-W

= DPq (z ;0 )b(X - Az)

and using

- b(X - hz) ,;I:=o 4+(zB- 2) -

--z)xl.

I

zB - 6(X - Xz)

Similarly, from equations (4.1.4) and (4.1.5) and the above equation which gives

jOw~,(z;x ) q ( x )dx = b ( h - k ) P q(z ;0 )

(4.1.1

B

(4.1.1

1st putting the value ofPq(z;0 ) from equation (4.1.8) in equation (4.1 .I 1) ar using the normalizing conditions Z:y=o = 1 and Poo + P q ( l ) = I , one c;

D = [(I -PW)p]-' .

one finds

f

P4(z;0) =

k=O

~ J ? ~ , I ( x )- (- z ~ ) v ( x ) ~ x zB - 6 ( ~ XZ)

view of this, we can put equation (4.1 .lo) in the form

:1

Pj, I ( x )(x)dx ~ = &(I - Pm)P,+.

(4.1.1

BULK-SERVICE QUEUES

Before proceeding further, it would be appropriate to discuss the evaluation of the unknown probabilities in P*(z). It can be shown by RouchC's theorem that the denominator of equation (4.1.12) has B - 1 zeros inside and one on the unit circle, lzl = 1. As P+(z) is analytic within and on the unit circle, the numerator must vanish at these zeros, giving rise to B equations in B unknowns Po+&, . . . , Pi-,. In order that P+(z) be uniquely determined, it is necessary to show that these B equations are linearly independent. This condition is satisfied for the system M/E:/~ in which the service-time distribution is k-Erlang, the convolution of k independent exponential distributions with common mean 1/p.

4.1

THE SYSTEM M / G ~ / ~

Since [zil < 1 and zi #zj, I < i < j < B - 1, A # 0, and hence P+(z) is uniquely determined. For details of the evaluation of A see Problems and Complements. Relation between Po,, and P i

Now we return to M / G ~ / Iand the evaluation of the unknown probabilities are known, we can find these probabilities by using equation (4.1.13), since Pm may be expressed in terms of Po+, which is known. From equations (4.1.1) and (4.1.1 3) we can easily get (for M/GB/l, in which H-' is the mean service time and Bp = X/g)

Jom Po, (x)v(x)dx, jomPl, (x)q(x) dx, and so on. Once Po+, . . . ,PJf

D+

this case we can show that Pi(z) is unique. For ~ l ~ k " the / l , denominator equation (4.1.12) becomes, since now P = (kh)/@H),

poo =-. 1 0 Bp +Po+

J~

zB which has B - 1 simple zeros zi, i = 1 , 2 , . . . ,B - 1, inside iz 1 = 1 together with one simple zero zB = 1 on lz I = 1. That the zeros in or ,on the unit circle are is easy to see. For if z = f is a multiple zero with IT1 < 1, -k

and, differentiating both sides, -B-1

z

=

excluded by hypothesis. the B equations containing the B unknowns are B-I

C

Pi+(z~-zj) = 0,

i = 1 , 2,..., B - 1

j=O

(4.1.14) s

This completes the discussion of the number in the queue for the system M/GB/l. For more details and references to numerical work, see Problems and Complements. It may, however, be remarked here that since arrivals are by a Poisson process, the number in the queue as seen by an arrival and by a random customer must be the same. The above results and derivations are due to Chaudhry and Templeton (1981). In the special case when G = ER,that is, when the service is through a random number R of exponential phases, similar work has been done by Jaiswal (1960a). For correction to equation (29) of Jaiswal(1960b) see Problems and Complements. The result (4.1 .I 2) was first presented by Bailey (1954) through the imbedded Markov chain technique (see Problems and Complements). In Bailey's case the queue length is measured at epochs just before service is to start, or equivalently just after service ends. In his analysis, Bailey assumed that the server is "intermittently available," and continues to serve even when there are no customers waiting. Thus in such cases service is virtual; otherwise it is real. However, it can be seen that the result (4.1.12) still holds even if the server interrupts service whenever he finds no one waiting in the queue, and resumes service as soon as the subsequent arrival comes, as assumed by Jaiswal. Bioemena (1960) also made this additional assumption, and using the method of collective marks due to Van Dantzig (1947, 1948, 1955), showed that equation (4.1.12) holds. The system M J G ~ / Ihas also been discussed by Novaes and Frankel (1966) as a particular case of their system M / G ~ J Iwith balking. They reproduce the analysis due to Bailey, but give a different approach for the numerical analysis of the problem.

together with B-I

(B-j)PT j-0

= B(l

-PI.

The Waiting-Time (in Queue) Distribution

Let Vq(t) be the virtual waiting time in queue for a customer who may occupy any ~ositionfrom 1 to B in his group, and consider the limiting behavior of Vq(t) as t j m .In the limiting case, V, is the time spent in the queue by an arriving tomer. Let ~ ~ ( be 7 the ) p.d.f. of Vq so that wq(r)dr is the probability that aiting time of a customer, who may occupy any position from 1 to B in his , lies in [r, T dt] . Since we are considering a queue with FCFS queue

+

186

BULK-SERVICE QUEUES

discipline, the number of arrivals during the waiting time (in queue) of the customer under consideration must equal the number in the queue immediately after the start of service on his group. Since arrivals are random, it follows that 2

waiting in the queue, the input process stops after accepting the arriving customer; it does not restart until M customers only are present in the system, that is, until the current service period is completed. The customers are served in batches of variable capacity, the maximum capacity of the server still remaining B.

where %,(a) is the L.-S.T. of the waiting-time d.f. W,(t), defined by

From equations (4.1.8), (4.1.1 I), and (4.1.1 5), we get, using also a = X - X z ,

It is now an easy matter to find the distribution of waiting time for those who have t o wait. For if wC(r1r>O)dris the conditional probability that a customer waits in the queue between and 7 dr, given that V , > 0 , then from equation (4.1.16) on removing the additive constant Pw (corresponding t o the zero wait) and the multiplicative constant (1 -Poo), we get

+

It may be observed that once P,(z) is known, it is easy t o find @,(a) given by equation (4.1.16) or @,(a1V, > 0) given by equation (4.1.17). For the system M / E g / l , Jaiswal (1960a) derived equation (4.1.17) by a cumbersome though straightforward method, which has already been explained in Chapter 3 while discussing the system MX/ER/1. Downton (1955) derived the same equation through the imbedded Markov chain technique, using Bailey's (1954) results. In the above approach, which gives immediate results, we use neither Downton's technique nor Jaiswal's method. Little's formula is satisfied (see Problems and Complements) as it should be since his formula is independent of the distributions involved, though it requires careful interpretation in some situations.

4.2

T H E SYSTEM M / G B / l / MWIT CUSTOMERS SE BATCHES OF VARIABLE CAPAClTY

The Limiting Behavior of N;

Consider the following modifications of the system M / G ~ / discussed I in the last section (more specifically, see Problems on that section). Everything remains the same as in M / G ~ / I save , the following:

Although the infinite waiting-room queues serve as a good approximation to queues with finite waiting room in light traffic (p < I), the finite waiting-room queues give better results in heavy traffic (p -+ 1) or where p > I . In practice, all queuing systems have finite waiting room, and often this waiting room is small, so that for practical applications there is a real need for a theory of finite waitingroom queues. Let now oo = 0, o l , o,, . . . , o n , . . . be the instants of departures of successive batches, oo corresponding t o the initial departure. The service times (V,) are i.i.d. r.v.'s with d.f. B(v). The sequence (V,) is independent of the arrival process, which is Poisson with rate A. If Y , customers are already present with the server at epoch a,, then the server takes min (B - Y,, whole queue length) customers at on. We are assuming that the service batches are not accessible in the sense of Bhat (1964). That is, if a customer arrives to find m , 0 < m < B, in service, he cannot be accepted into the batch already undergoing service, but has to wait until the next service instant. Suppose that {Y,} are i.i.d. r.v.'s with distribution given by b,, O < m < B P(Yn = m) = 0, m>B s o t h a t ~ fb, = 1. Also suppose that {Y,) are independent of the arrival process. Let N(t) be the number in the system at any time t. Define N: =N(o, + 0 ) and let P,+= limn+, P ( x =j). Denote g), = P(Yn <j) = I;', b, and let Gj(z)= 26 b,zm. Then QjB(z) is the p.g.f. of the sequence {b,), with QjB(l) = (PB and g), = bo. The service initiation is instantaneous as long as there are customers to be served, but for an epoch at which there are no customers remaining to be served, neither in the queue nor with the server, the server relaxes. The next increase in the index n occurs at the end of the service interval that begins with the first subsequent arrival. Now the process (N:) is a time-homogeneous Markov chain which is discussed is thought of as the number of customers arriving during a service below. If +,+, period ending at on+,, then kj =

, jfi P ( i n + , = jlservlce period = v)dB(v)

BULM-SERVICE QUEUES

4.3

OF EXACTLY SlZE k The Limiting Behavior of

N:

Similarly,

In the present case the input is by Poisson process with rate X, but customers are served in batches of exactly size k by a single server in the order of arrival. If fewer than k customers are present in the queue at the instant when the server becomes free to serve a new group, the server remains idle until there are k customers and then restarts. Let Vn be the service time of the nth batch to be served. Suppose that fVn) is a sequence of i.i.d. r.v.'s such that

After some simplifications, left-hand side = right-hand side gives

and p = Ilb. Define p = X/kp. The sequence (Vn] is also with b =J;vdB(v) independent of the sequence of interarrival times. Let N(t) be the number of customers in the system at time t , including the batch under service. Further, let on be the instant at which the nth batch (of size k ) departs from the system on completing service. Put N c = N(on 0) and let

+

Pj+ = nlim P ( x = j), -+This implies equation (4.2.4). If expression (4.2.4) for P+(z) can be expanded in a power series in some suitable region of convergence, then the second term will be a power series in z with powers of z at least M B , which implies that P;, j = 0 , 1 , 2 , . . . ,M, is the coefficient of zi in the first term. Let us denote the first term of equation (4.2.4) by Q(z). Thus

+

j = 0, I , 2 , . . ,

As discussed in earlier sections, the distribution {Pi+}exists and is independent of the initial state of the system if and only if p < 1. The probability kj that the number of customers arriving during a service period is j remains the same as in Section 4.2. Define

We show that P+(z) is given by When the constants Po+, P:, . . . ,Pg are evaluated, as usual, by RouchC's theorem and the analyticity of Q(z), the function Q(z) becomes completely known. The results of this section are due to Singh (1971-72) but the arguments used are different at least for calculating the last probability P&. An imbedded Markov chain analysis of a more general bulk-service queuing system than the one discussed above has been carried out by Dick (1970). In his system, besides the assumption in the above system, Dick (1970) assumes that (I) the service does not start until there are at least k in the queue, (2) capacity of the server is B, and (3) when the waiting room is full (that is, when there are M in queue), the customers' arrival rate changes from h to Ap. The rate Ap has an interesting interpretation in that if 0 < p < 1 , the rate reduces, that is, 100(1 -p)% of all arrivals balk, whereas if p > 1, the faster rate h p could be interpreted to mean that a crowd of M in queue attracts more customers. The case when the waiting room is infinite is discussed later.

where zl ,zz, . . . ,zk are the k - 1 roots in z of the characteristic equation zk = k(z) inside the unit circle, / z1 = 1, and

We proceed to prove equation (4.3.1). Clearly, the process (N;)is a Markov chain with transition probability matrix described by the relations: where J/, is the number of customers who arrive during the service time of the nth batch. Now the p.g.f. of the r.v. (N,+- k)' is

3 92

But

E(Z *-~)+

BULK-SERVICE QUEUES

> X)P(N; > X ) + E(ZOIN;4 OP(N; < k j (4.3.2) = z - k ~ ( ~ N L ~ ~>, '~ ) P ( N ;> ~ ) + P ( N ; < k).

=

~ ( 1 ~ L I- N k;

EGG)= E(Z%\N,'

distribution. Since the arrivals are random, the number of new arrivals during the waiting time in the queue and the service time of a customer must be equal to the queue size at a departure epoch in the limiting case. Thus as in Section 2.2,

k

= m)P(N: > k)P(N; > k ) + m=O E(zG~N:

=

m). (4.3.3)

Using equation (4.3.3) in equation (4.3.2), we get a new form of the p.g.f. of (N,* - k ) * Letting n -* i". in this new form gives us the limiting p.g.f. of (N* - k)*

Equation (4.3.5) follows immediately from equation (4.3.6). It may be remarked here that Cqk(a)given in equation (4.3.5) has an interesting interpretation in that it gives the waiting-time (in queue) distribution for the system E k / G / l .For details, see Chapter 6 .

E SYSTEM GEOM/G~/I 34

The Limiting Behavior of N,f

whence, since Ji and (N* - k)* are independent, we have the relation which, on simplification, reduces t o

(4.3.4) can be put in the form (4.3.1) by considering the zeros of the Eauaaon denominator within the circle, lz I = 1 + 6 . The result (4.3.4) is the same as equation (4.1.12) (B is now k). ~ l t h o u * the underlying processes are different, the difference does not show up at the instants examined in the Markovchain. For the transient solution of the M / G ~ system, / ~ the reader is referred t o T a k r s (1962). Using the theory of semi-Markov processes, the same system has also been considered by Fabens (1961). See also corrections to Fabens's results by Fabens and Perera (1963).

This section deals with the same problem as discussed in Section 4.2, the only difference being that time is considered as a discrete r.v. rather than a continuous one. In the discrete time queuing system the events occur only at definite time points called "time marks" by Meisling (1958). Thus let the arrivals occur at these time marks, which we suppose are regularly spaced with an interval of magnitude At. The service is initiated at one of these time marks and completed at another time mark. A service period may consist of several of the intervals At with a given probability distribution. The system is discussed under the following assumptions: 1

2

The service is in batches of maximum size 3. The server is continuously busy, so that the service is virtual in case no customers are available at the time of initiation of a service period, otherwise it is real.

BULK-SERVICE QUEUES

194

marks, then Sk = P(V = kAt), k = 0 , l ; 2 , . . . ,with Z;P Sk = 1 . Note that here the service times have a discrete distribution as opposed to a continuous one, and the interval under consideration is open on the left and closed on the right. One can easily see that E ( V ) = At Z;P kSk and that E(V(V - At)) = (At)' Z r k(k - l ) S k . Also p , the traffic intensity, is defined by p = XE(V)/B = p Z; kSk/B. We assume p to be less than unity, so that the system has a limiting distribution. Let now N ( t ) be the number in the system at time t . Define N; = N(un + 0). The process (N:) is an imbedded Markov chain and thus can be analyzed as in the , P(N; = j), then the p.g.f. of Pj+is given by previous sections. If Pif = limn ,

P(V = kAt) =

sk =

and consequently and

E ( V ) = yo,

I

1,

if

0,

otherwise

k=ko

E(V(V - At)) = V,2 - yoat

whence the denominator of equation (4.4.1) becomes zB(pz + q)- vo'Af

- 1.

(4.4.3) Now we show that the zeros of expression (4.4.3) within the unit circle are simple if 1 > p > p . For if it were not so, we should have, for some z , = where

is the p.g.f. of the number of arrivals during a service period. Clearly,

We get equation (4.4.2) by taking the p.g.f. of k j . Also as

Dividing these two equations gives z = (1 -p)/(p - p ) , which requires p > 1 for lz/ < 1, but this is excluded by hypothesis. Thus expression (4.4.3) has B - 1 simple zeros inside the unit circle and one simple zero z = 1. Now since P+(z)is convergent in 12 1 < 1, the numerator of equation (4.4.1) must vanish at z = z j , where ( z j ) ,1 < j
O G i G B and

i

= B-Bp,

o

> j+B

equation (4.4.1) is obtained by taking the p.g.f.of

It can be seen that the determinant of the matrix of coefficients of P' does not vanish. This implies that the set of equations involving P" is linearly independent. Hence P+(z),for this case, is completely determined and is given by

As in earlier sections, one equation involving the B constants Po+,P:, . . . ,Pi-, is determined by the normalizing condition P'(1-) = 1, and the others can be determined by applying RouchC's theorem (see Appendix A.5) to the denominator of equation (4.4.1) by taking the circle / z (= 1 6 , where 6 > O is sufficiently small. Rouchk's theorem shows in this case that there are B - 1 roots zj, j = 1,2, . . . ,B - 1, other than z = 1 within the unit circle. This gives rise to B - 1 more equations involving the constants. More details regarding the zeros o f the denominator are given in the examples that follow.

+

1 Constant Service Time. a constant. Thus

C Pif(B-i)

Let the service time V be equal to koAt = V o ,

B-1

z

P*(z) = zB(pz

Pi+(zB--zi)

+

q)-vo'At

-I

(4.4.4)

Mean and variance of system length may be more easily calculated if we express equation (4.4.4) in terms of roots inside the unit circle. As the numerator of equation (4.4.4) can be written as

BULK-SERVICE QUEUES

196

P"(z) given in equation (4.4.1) becomes

P+(l-) = 1 gives the constant

and finally

The denominator of equation (4.4.9),

From equation (4.4.5) one can find the mean and variance of the system length distribution. To get the mean, we proceed as follows. Let the numerator and the denominator of equation (4.4.5) be denoted by @(z) and $(z), respectively. Clearly @(z)and $(z) vanish at z = 1, but as ~'(1-) = I , we must have 4 y l ) = $yl).

has B - 1 simple zeros inside the unit circle, one simple zero z = I , and one simple zero ZB say, outside the unit circle. Then we can write equation (4.4.9) as

(4.4.6)

Also differentiating equation (4.4.5) with respect to z and taking the limit as z -+ 1-, we get a result which becomes indeterminate even after using L'Hbpital's rule once because of equation (4.4.6). Using L'H6pita17srule a second time, we have

where Cis a constant to be determined. Since P+(1-) = 1, C = I - z, . Therefore,

From equation (4.4.1 1) we can get

Similarly, one can determine the variance of system length as

X3(G

- At - Vo)(Vo - 2At) - B(B - 1)(B - 2)

(4.4.8) 3B(1 - p) Note that C is taken to be zero when B = 1 , and that we have assumed that the roots are simple. If the roots are not simple (repeated), a slight modification to the above procedure as indicated by Wishart (1956) is necessary. In the next example no such modification seems necessary, for the roots are simple without any constraint on p except that p < 1. 2 Geometric Distribution of Service Time. Let Sh = d k ( l -d), 0 < d < 1, -I-

k = 0 , 1 , 2 , 3 , . . . .Since

By the transformation z = I/y we can change expression (4.4.10) to an expression having zeros which are reciprocals of those of expression (4.4.10). Then the zero y B whose modulus is less than 1 will be the only zero of the transformed expression within the unit circle, which can be determined either numerically or by Lagrange's expansion (see Appendix A.6). It is interesting to observe that the imbedded Markov chain results for the system M / D ~ / (Problem I 5 ) can easily be obtained from the discrete time results for the system ~ e o m / ~ given ~ / l by equation (4.4.5) by assuming p -t 0 and At -+ 0 so that the limit of the ratioplat + a constant = X. Similarly G e o m / ~ e o r n ~ / l becomes M/MB/l if in addition we suppose that I/M= E(V) = Atd/(l - d). In this connection, see equation (4.4.9) and Problem 3. For in the first case, we can see q)] / that (pz + q)-vo'At -+ exp [Bp(l -z)J and in the second case [I -d@z (1 - d ) -+ [(I + Bp) -Bps]. This equivalence, in fact, is based on the fact that the binomial distribution tends to the Poisson distribution and the geometric distribusion tends to the exponential distribution.

+

The results for B = 1 are due to Meisling (1958), and for the general value of B they are due to Natarajan (1962). However, the arguments given here are more complete, at least for the examples considered above, than those given by Natarajan ( 1 962) who assumed throughout that the roots are simple.

BULKSERVICE QUEUES

~h~ Waiting-time (in Queue) Distribution

the For the discrete case the Waiting-time distribution has been obtained by remits from the continuous-time case. TO get it, consider the waitingetime as to write bution discussed in equation (4.1.17). Transform equation (4.1.16) so

4.5

THE SYSTEM M/MB/I/@~

customers. 1 - e-hv The service times of successive batches are i.i.d. r.v.ls with d.f. ~ ( = ~ O < v < OO. The waiting room is of finite capacity m. Let N(t) be the number of customers in the system at any time t and define N.* = W a n + 0) and Pr(*(")= P ( K = i ) , j = 0 , 1 , 2 , . . . ,MB. if km is the ability that there are m potential arrivals during a service then

and for the geometric service time case, (4.5.3) Using equations (4.5.1) and (4.5.21, equations (4.5.3) lead after simplification to P:(n

+I)

= (PE;(n)+ . . . + P:(n))q

1

BULK-SERVICE QUEUES

200

N;


'

XMB(w)= YMB( ~ 1 x(w) 0 + 1.

C

(4.5.9)

The normality condition 2Xj(w)= 1/(1- w)gives

B-1

Pj+(l)= 1,

Pj+(l)= 0,

j 2 B.

j =O

Because of the above remark, it is sufficient to determine the distribution P/(n) subject t o the initial condition Pof(l)= 1, and it will be supposed that this is so throughout our discussion. we define In order t o solve the set of equations (4.5.4)

is completely known if the Y's are known. To find and consequently the set (4.5.9) the Y's, write

Then from equations (4.5.8) we obtain, for a suitable domain of example, Izl < p , Iwl< 1,

Then from equations (4.5.4) with Pl(1)= 1 we obtain

5 and w ,for

Y(o,Z)= (1-z)[l - zp-yl - qwzB)]-l

Pof{w)= 1 + q w{P,+{w)+...+P;{w))

<

P~{o)= ~ w P & B { w ) + P P ~ - ~ { ~ ) - P ~ 1I , ~j,< MB-B

(4.5.10)

where Fk(w)is given by

Fo(w) = 1

[xj being the integral part of x. From equation (4.5.1 I), q(w)= e(w)- Fj- (w), j > I , and as CE, FMB(u),we obtain, from equations (4.5.6) and (4.5.9),

It follows from the third equation of the set (4.5.7) that C: Xj(w)= xB(~)/q, and hence that the first equation can be derived from the second and third equations, together with the normality condition

$+@)

Yj(w)=

= 60,j ~ (-a)-' 1 [FMB-~(~)-FMB-~-~(w)] [FMB(U)]-', j =

0,1,2,. ..,MB-I

(4.5.13)

P&B {w)= w (1 - w)-' [FMB(w)] . for further Thus we consider the second and third equations of the set (4.5.7) discussion. Introduce an infinite sequence {E;.(w)} defined by the equations

Equation (4.5.13) gives the generating function of the transient probabilities P/(n). The limiting behavior is considered below. Various special transient and limiting cases are discussed in Problems and Complements. The Limiting Behavior of N:

Then it is easy t o show that X's and Y's are related as follows:

The limiting behavior may be obtained by rewriting equations (4.5.3) with n suppressed and then solving the resulting equations along the lines of the above by (1 - w)and taking transient solution, or else by multiplying equations (4.5.13) the limit as o 1. The requlred probabilities, in this case, are -+

Pj'=(FMB-j-FMB-j-l)[FMB]-',j E 0 , 1 ,,2. . . ,m-1 (4.5.14)

P & ~=

[F~Bl-l

BULK-SERVICE QUEUES

202

where Fo F o ( l ) = 1, Fk = Fk(l), k 2 1, and F k ( u ) is given by equations (4.5.12). The results of this section are due to Finch (1962).

This section, like Section 3.7, gives brief descriptions of some work we have not discussed in detail. Publications cited here are listed in Additional References at the end of this chapter. The first and foremost reference here is the work of Bhat (1968), who provides a combinatorial analysis of the transient and limiting solution of the queue GI/&f/l with bulk service, and gives references to work on bulk-service queues before 1968. Chang (1970) discusses many examples of the application of bulk-service queues to computing systems. Another application to computing systems is given by Frank (1969). Chaudhry and Templeton (1968) discuss a different kind of singleserver bulk-service system in which time is discrete and arrivals are correlated. A bulk-service queue with single intermittently available server has been investigated by Chaudhry and Lee (1972). A paper by Crane (1974) deals with a linear network of N 1 terminals served by S vehicles of fixed capacity B. Murao and Nakamura (1 973)investigate some relations between the queuing systems GI/D/c and GI/DC/l. Jain (1971) investigates, among other things, the joint distribution of the length of a busy period and the number served during the busy period for the bulk-service queuing system G1/Mk/l. A queuing system with "concentrated service," loosely connected with a bulk-service queue, is studied by Fukuta (1969). In a concentrated service system, an idle server will delay the start of service until 1 > 1 customers are waiting, and will then serve customers one by one until the system is empty. A service discipline of this type may be useful in reducing start-up costs. Another prolific researcher on the theory of queues is Marcel Neuts. In the sixties, besides other studies of queuing systems, he and his collaborators published several analytical results on bulk-service queues using the semi-Markov approach.

PROBLEMS A N D COMPLEMENTS

Problems. Some properties of a bulk-service queuing system wherein a car ferry operates at regular intervals have been investigated by Stuart and McMahon (1966). This latter system could also have applications to a plant of fixed capacity, processing work that can arrive only at regular intervals. Kashyap (1966, 1969) discusses a bulk-service queuing system which he calls a double-ended queue, in which taxis and customers wait for each other, with the former serving customers in bulk. The model is interesting, but the analytical results appear to be v e y complex. Dave (1971) develops an analyticalmodel for a priority queue under the preemptive resume rule, with bulk service for ordinary (nonpriority) customers. A few papers consider optimal control of bulk-service queues. See Deb and Serfozo (1 973), Frank (1 969), Handa (i97 I), and a recent paper by Weiss (1 979). As in Chapter 3, we now list some other contributions related to single-server bulk-service queues: Abramov and Tsvirkyn (1968), Alagaraja (1976), Bahary (1969), Bahary and Kolesar (1 972), Borthakur (1 971), Craven (1963a, b), Georganas (1976), Goyal (1967), Harris and Markin (19721, Ivnitskiy (1971), Kotiah et al. (1969), Mohan and Murari (1 9721, Mohanty (1 972), Murari (1 969), Nakamura and Murao (1968), and Sharda (1981).

+

Section 4.1

1

(a)

The system J4/GB/l : alternative solution procedure. Consider Bailey's (1954) problem discussed in Section 4.1. Let Nz be the number of customers waiting in the queue just before service interval n + 1 begins. Show that

-

where $,+, is the number of customers who arrive during the service interval ending at on+,. Define &+= limn , ,P ( x =j ) , which exists if p AbIB, the traffic intensity, is less than 1. Show that the p.g.f. of P,? is given by equation (4.1 .12).

BULK-SERVICE QUEUES

PROBLEMS A N D COMPLEMENTS

Continuation of Problem 1. For the system I M / E ~of/ ~Problem 1, that ~X z ) = is, when the service-time distribution is k-Erlang, show that 6 ( I (h/p)(l - z)] and consequently

2

+

-'

Show also that P+(z) can be put in the form

where zj(B

Hint: Use equations (4.1.17) and (2). Here, following Downton (1956), we have changed the notation of Problem 2 so that the service rate in each phase is now kp. The mean service time of a customer is therefore I/@,as required for validity of equation (4.1.17). If we define $(a) = In %,(a), the conditional mean and the conditional variance may be found from equation (3) and are given by

<j 01 = G(~)(o)= h2

outside the unit circle. Through the use of the cumulant generating function or otherwise, show that the mean and the variance of queue size are given by

B+k-l c j

=

~

I)(k - 5 ) + (k -12k2 (zj p2

The system M / D ~ / I . In this case show that B-1

(B -BP)(z - 1) .fl [(z -zi>l(l - zi)] P+(z> =

r=1

zB exp [Bp (1 - z)]

-1

where P = XbIB and b = lim,,,,, , ,klp. Hint: Write the numerator of equation (1) as

Show that the mean can also be calculated from equation (1) and is given by and use P+(1-) = 1. Using equation (4), prove that

3

Mint: Use P'(1-) = 1. / l , that Continuation of Problem I . In particular, for the s y s t e m M / ~ ~ show the queue size distribution forms a simple geometric series whose common ratio is the unique root y of the equation h - ( ~ + p ) z + p z ~ +=~ 0

4

suchthatO
Also show that

For the systernM/~i?/1show that where { y , )are the Broots of yBeab(' - Y ) - 1 = 0 within and on 1y I = 1, whence

where { z j ) are the roots with modulus greater than unity of the equation

206

BULKSERVICE QUEUES

and

From this show that 6

Hint: Use equation (3). Show that

where y is the unique real root outside the unit circle of the characteristic equation with k = I , defined in Problem 2, and

A =

Jaiswal (1960b) obtains relation (6) and states that it9holds for the However, it should be pointed out here that his relation system hf/E;/l. is true only for the system M/E?/I and not for the system M/E;/I. Equation (5), in fact, gives the correct relation even for the more general system A4/GB/1 of which Jaiswal's s y s t e m M / ~ is ~ aI ~special case. Continuation of Problem 7: the system M/G/l. For this system find independently PJz) and P"(z) [without using equation (4.1.12)j and then show that the result given in Problem 7(b) is true. The system IGl/G/I. If Pn is the probability that there are n customers in the system M/G/l (as opposed to n customers in queue) and P(zj = w Pnzn is the corresponding p.g.f., then show that for the system MIGI 1,

Hint: To get the second determinant from the first, proceed as follows. Subtract each column of the first determinant starting with the second from the column to its left. This leads to a new determinant. After taking common factors out, we get the second determinant. To get the final answer, note that the second determinant vanishes if zi = 1 or zi = zj, i f.j . Therefore

a

Hint:

-'

7

where C may be found to be (- l)B . (a) The system M / G ~ / I . Show that the following relation between the p .g.f .'s P,(z) and P+(z) holds: (-z-) P+(z)

-

~ - ~ ( A - X Z ) 1-Poo (I -z)6@ - Xz) Bp

If B = 1, then first show that Poo= 1 - p , and consequently from equation (5) that

9

&(z) =

1 pnrlzn= 2-1 n2= o n=o

and so on. The result P(z) = Pc(z) was first obtained by Khintchine (1932) and later by other scholars. The above derivation is due to the authors. For references and more details on such types of relations refer to Chapter 6. The system M / E ~ / I . In case the service times are exponential with mean I/@,one can get explicit results for the unconditional waiting-time distribution. Show that in this case,

inversion of which gives If the service times are exponential with mean I/&, then

P(wa1ting time in queue = 0) = P,

8

a

rc. 0

BULK-SERVICE QUEUES

=

lim

at-0

(

PROBLEMS A N D COMPLEMENTS

17

w

exp

-0

213

If P:(l) = 1, then the generating functions Ph+{w),0 < w < 1, 0 < k <MB, are given by

where we replace the probability mass function (p.m.f.) {Sk) with the continuous p.d.f. of service time, thus replacing summation by integral. The integrand being indeterminate when At -t 0, we employ L'Hbpital's rule and get

where yj(w), j = 0, I , . . . ,B, are the B + 1 distinct roots of g - z -kpzB+l = 0, and

16

Show that the equation, for 0 < w < 1,

(a)

p

- z fqwzBC"

0

has distinct roots, and further that it has only one root z = y(w) within the unit circle, Iz 1 = 1. This root is given explicitly by

)

Hint: A repeated root implies w = (BP-')~{~(B+ I)"+')-' > 1, which is a contradiction. Hence roots are distinct. That only one root i s within the unit circle follows from RouchC's theorem, and the explicit expression for this root follows from Lagrange's theorem (see Appendix A .6). If B = 1 , show that equation (10) becomes

Hint:

18

19

Hint: Letp-z +qwzB+l = p ( l -zyil(w))(l -zy;'(w)). . . (1 --zyB1(w)). Use partial fractions for the function Y ( w ,z) given in equation (4.5.10). Continuation of Problem 17. If P:(1) = 1, M = -, the limiting (M -+ -) generating functions Ph+{w), k > 0,are

where {y(w))j, j > 1, is given by equation (10) of Problem 16. Hint: Let M + m in Problem 17. Continuation of Problem 18. Show that the expllcit results for the limiting (M-t m) probabilities P&) are given by P:(n) = 1 - c~c," ~ in >,2, and Ph+(n) = C~Z," n > 2, k > 0. Here

~ d ,

Use the binomial expansion Hint: Expand the function given in Problem 18 and use equation (10) of Problem 16.

Note that when B = I , y(w) could have been more easily obtained b! solving the quadratic equation

PROBLEMS AND COMPLEMENTS

where y ( w ) is given by equation (1 1) of Problem 16 and ( ~ ( w ) ) 'i,> 1. is eiven bv the series expansion (10) of Problem 16 with B = 1. The probabilities " Pi+(n)are given by the equations

From this show that

where y is the unique real root outside the unit circle of the characteristic equation with k = I , defined in Problem 2, and

where

Jaiswal (1960b) obtains relation (6) and states that it Folds for the system M / E ~ / IHowever, . it should be pointed out here that his relation is true only for the system M/EfS/l and not for the system M / E ~ / I . Equation (5), in fact, gives the correct relation even for the more general I which JaiswaI's system M / E ~ /isI a special case. system M / G ~ /of

The second sum in T! is zero for (k

,"'

x

Proof:

+ j ) < (M + I), and the r, are given by

Continuation of Problem 7 : the system M / G / I . For this system find independently Pq(z) and P+(z) [without using equation (4.1.12)] and then show that the result given in Problem 7(b) is true. The system M/G/I. If Pn is the probability that there are n customers in the system M/G/I (as opposed to n customers in queue) and P(z) = XE='=, Pnzn is the corresponding p.g.f., then show that for the system WGIl,

From equations (4.5.1 3), with B = 1, we have

M

~ ~ ; { W ) = W ( ~ - W ) - ' F ~ - ~ ( W ) [ F ~ (l U C )j ]<- ~ M, .

Hint:

k=j

Therefore to prove equation (173,it is sufficient to prove the following: ( y( W ) ) 2 ~2+- 21 1 . FM-j(w)[FM(w)] = (l( W ) )[ I~- (pel q ~ ) ~ +

'-'

[I - ( p - 1 q ~ ) M + 1 ( y ( w ) ) 2 M + 2 ] - 1 where Fk(w) is given by equation (4.5.12) with B = 1. By the definition of Fk(w), we have from equation (4.5.12) with B = 1, and x = (p-1qw)112z, m

F ~ ( W ) ( ~4-w' ) - k 1 2 ~ h= ( 1

- 2x cosh 8 + x2)-' ,

k=O

where (pyw)-1'2 = 2 corh 0 . But x sinh 8 ( 1 - b cosh ff 4- x2)-' = EL x sinh k8, implies (1 - 2x cosh 8 x2)-' = Eh= xk-I (sinh %)-I sinh k f f and hence

+

F k ( u ) = (p-l q ~ ) h '(sinh 2 19)-hi& (k Thus

+ 1)O.

and so on. The result P(z) = P + ( z ) was first obtained by Khintchine (1932) and later by other scholars. The above derivation is due to the authors. For references and more details on such types of relations refer to Chapter 6. The system M / E ~ S / I . In case the service times are exponential with mean 1/p, one can get explicit results for the unconditional waiting-time distribution. Show that in this case, iii, ( a ) = P,

+

~BPYP, Xy-X+a

inversion of which gives P(waiting time in queue = 0) = P,

BULK-SERVICE QUEUES

PROBLEMS AND COMPLEMENTS

where y is defined in Problem 3. For another independent derivation of this, see Miscellaneous Problems and Complements. / ~ that For the system M / G ~ show L, = (1

- P0,)L+ +

h

p(1

-Pw) 12

where L, = P$)(~),L+ = ~ + ( ' ) ( 1 ) ,and a2 is the variance of the service time Observe that if the service is exponential, then L q = (1 - Pw)LC. Further more, if B = 1, then L is the average number in the system, and equals L+,a it should. Besides, it can be seen from equation (4.1.15) that hW, = L, which shows that Little's formula is satisfied for the system M / G ~ / So ~. numerical work on the system M/E?/~ has been done by Downton (195 1956).

209

The analysis for the infinite case was carried out by Jaiswal (1961). He obtained the same results through the phase technique, which can also be used t o give the transient solution, if desired. Both Jaiswal (1961) and Bailey (1954) assumed that the server is continuously busy, even when there are no customers to be served. However, the limiting solution holds good even under the relaxed assumptions as stated in Section 4.2. Murao (1970) discusses the limiting results for the same queuing system discussed by Jaiswal, with servicetime distribution deterministic and batch size dependent on queue length. The system M / G ~ / I / M . This is a bulk-service system with the "service capacity" fixed. Thus we have

and hence @B-i = 1 = @+,-{(z) = QB(z). Therefore the expression (4.2.5) for Q(z) reduces t o

Section 4.2

The system M/GB/1 with variable capacity. Considerable simpli occurs in the theory of queues, particularly in bulk queues, if we assu capacity of the waitingroom t o be M = m, although in practice this is t o be true. The transition probability matrix (t.p.m.), in this case, is in. and may be obtained from M / G B / l / M with columns and rows correspondi to M - 1, M omitted. The Markov chain corresponding to this new matri irreducible and aperiodic, and is ergodic if p

??

X"(V

B - @$)(I)

<

It may be remarked that equation (7) holds even if the power series for k(z) is extended t o infinity, for this extension would not affect the first M + 1 probabilities P:,P:, . . . ,P&.This problem has also been discussed by Lwin and Ghosal (1971) following a procedure similar t o that of Finch (1958) who discussed the same problem with B = 1. For a detailed account of the latter case, see also Gross and Harris (1 974). Continuation of Problem 12: the system M/D/l/M. equiIibrium, since B(v) =

where E ( V ) is the mean service time. This restriction on p is a necessa condition for the limiting distribution to exist; for hE(V) is the avera input per service period and

1,

vZl/p

0,

otherwise

For this system in

B

B-@$")(I)

= B-xmb,

= Bbo+(B-l)bl+...+(B-B)bB

0

is the mean service capacity. For queuing systems in the steady state, t average input must be less than the mean service capacity. . this case is P z j in Show that the p.g.f. P+(z) =

where

B-1

2 P T [ ~ ~ -@ z ~~@- ~~- ~ ( z ) ]

P+@) = where k(z) = ZT=o kjzj.

i=O

zB/k(z) - @B(z) Continuation of Problem 12: the system M/M/l/M.

For this system in

PROBLEMS A N D COMPLEMENTS

equilibrium, b(v) = ~ e - ~ v' ,d 0 , 'and k j = p'q, where p = X/(h + p), 4 = 1 - p. Using k ( z ) = (I + p - pz)-l ,we have, from equation (7) (8) Q(z) = P;[1 -pz]-" where p = Alp. Two cases arise: (i) If X f y, expand the above expression within J z This gives j=0,1,2

partial fractions, and consequently we can pick up the desired coefficients. Thus

< E = min(1, p/;\).

where we note that

,...,M and

and normalization gives P,' = [ I - p ] [ l - p (ii)

M+11-1.

(9)

A i = lim Z+Lf

If h = y, expand equation (8) within the region J zI < 1 . Hence Pj+=P,+, whence

e+= M +1 1 -3

21 1

4k(l - z ) ( q - z ) qk -z(1 --pz)k

j = 0 , 1 , 2 , . . . ,M The normalization gives j = 0 , 1 , 2 , . . . ,M.

that we could have obtained this distribution from equation (9) when The system M/Ek/lIM. For this system B = 1 in M / E ~ / ~ and / M the service time distribution is Erlangian with dB(v) = vkvk-' e-&'/(k - I)!, y > 0 , and k >, 1 being an integer. In this case kj is a negative binomial probability given by k+j-l pjqk,

j = 0, 1,2,. . .

and consequently

A problem of a similar nature occurs in the theory of dams. For this we refer the reader to Chapter 6 of Prabhu (1965). Section 4.3

Since the system discussed here is related to those studied in Chapter 6, for problems related to this section see Problems and Complements in Chapter 6.

Section 4.4

15.

where p,q,k(z) are defined in Problem 14. Thus we have from equatio (7) of Problem 12, P,'(l -z)qk . Q@) = 4k (1 - pZ)k Clearly qk - z ( l -pz)k = 0 has one root equal to unity = zo, say. the other roots be z l , z 2 , . . . ,z k . We consider further the case w the roots z l , z 2 , . . . ,zk are all simple, and none are equal to unit However, the case of repeated roots can be treated similarly. If zo = and z l , z 2 , . . . , z k are all simple, then Q(z) can be broken up

(b)

Derive equation (4.1.1 2) from equation (4.4.1). Hint: Since b(a) =E(e-"V) = C= ;o f ? - " k A t ~ k ,k ( z ) = b(-ln(pz 4- q)/ At). If At ' 0 , p 0 , such that p/At -+ X , a constant, then show that k ( z ) F(A - Xz). It is thus interesting to see that under the above limiting process, as binomial input tends to Poisson input, the servicetime distribution becomes continuous, and thus we get equation (4.1.12) as a limit from equation (4.4.1). All the results deduced from equation (4.4.1) hold good under the above limiting process. Solution: -+

+

BULK-SERVICE QUEUES

212

PROBLEMS AND COMPLEMENTS

17

213

If Po+(l)= 1, then the generating functions P${w), 0 < w < 1, 0 G k G MB, are given by

where we replace the probability mass function (p.m.f.) {Sk) with the continuous p.d.f. of service time, thus replacing summation by integral. The integrand being indeterminate when At + 0 , we employ L'HBpital's rule and get

where yj(w), j = 0, 1, . . . , B , are the B = 0, and

+ 1 distinct roots of p - s+ pzB+'

Section 4.5

16

(a)

Show that the equation, for 0 < w

< 1,

p - - z + ~ ~ z ~= +0 '

has distinct roots, and further that it has only one root z = y(w) within the unit circle, iz 1 = 1. This root is given explicitly by

Hint: A repeated root implies w = ( ~ p - ' ) ~ { q ( B+ I)~+')-" 1, which is a contradiction. Hence roots are distinct. That only one root is within the unit circle follows from RouchC's theorem, and the explicit expression for this root follows from Lagrange's theorem (see Appendix A.6). If B = 1 , show that equation (10) becomes

= (2qw)-l [ l - (1

Hint:

18

19

Hint: ~etp-z+q~~~~~=p(l-z~~'((w)(l-zy;~(w))...(1-~~~~(~)). Use partial fractions for the function Y(w, z) given in equation (4.5.10). Continuation of Problem 17. If Pz(1) = 1, M = -, the limiting (M-+ m) generating functions Ph+{w), k > 0, are where {y(o)Y, j > 1, is given by equation (10) of Problem 16. Hint: Let M + m in Problem 17. Continuation of Problem 18. Show that the explicit results for the limiting (M -+ m) probabilities Ph+(n) are given by P:(n) = 1 - Z~Z: A;, n > 2, and P,'(n)= &?:~A!,n>2,k>0.~ere

- 4pqw)"2 1

Use the binomial expansion Hint: Expand the function given in Problem 18 and use equation (10) of Problem 16.

Note that when B = 1, y(w) could have been more easily solving the quadratic equation

BULK-SERVICE QUEUES

214

[ l - ( p - Q ~ ) M + 1 ( y ( ~ ) ~ M + 2 ] - 1 1,

<j <M

(12)

where y ( w ) is given by equation (1 I) of Problem 16 and ( ~ ( w ) ) 'i, > 1, is given by the series expansion (10) of Problem 16 with B = 1. The probabilities P:(n) are given by the equations

21

Substituting cosh8 = (pqw)-'j2/2 in the identity e-@ = cosh8 -(cosh28 - 1)"' and using equation (1 1) of Problem 16, we obtain e-e = ( p - ' q ~ ) ' / ~ y ( w )Putting . this expression for e-@ in equation (15) gives equation (14) and hence equation (12). Expansion of equation (12) as a power series in y ( w ) and using equation (10) of Problem 16 with B = 1 gives equation (13). Discuss independently the limiting (n -+-) behavior of the system M/AfB/1 and show that

where

if

lim Ph+ =

X

> Bp

M-+m

where y is a root of the equation p - z + qzB+" 0 inside the unit circle. Hint: To do this, consider equations (4.5.4) with n suppressed. Define Xj = PGB - j , 0 <j G M B , and the infinite sequence (q)satisfying The second sum in

Proof:

Yo

Tik is zero for (k + j) < (M + I), and the r',are given by

=

1

From equations (4.5.1 3), with B = 1, we have

M

~ P , t { ~ ) = w ( l - - w ) - ' F ~ - ~ ( w ) [ F ~ ( w ) 1~ <- 'j ,< M . k =j

and

Therefore to prove equation (1 2), it is sufficient to prove the following:

F M - j ( w ) [ F M ( w )-' ] = ( y ( w ) ) j [1 - (p-' q ~ ) '-J' ~ (Y+(w))

2M+2-2j

[I- (p-"q~)~+~(y(w))~~+~]

3. (14)

where F k ( w ) is given by equation (4.5.12) with B = 1. By the definition of Fk(w), we have from equation (4.5.12) with B = 1, and x = ( ~ - ' q w ) ' ~ z ,

where (pqw)-1'2 = 2 cosh 8. But x sinh 0 (1 - ?x cosh 6 + x2)-I = 2 6 xk sinh kg, implies (1 - 2x cosh 8 x2)-' = C;fZl xk-' (sinh 0)-' sinh k8 and hence

+

Fh (w) = (P-' q wlkl2 (sinh 8)-' sinh (k Thus

+ 1)0.

Now wrlte p - Z + q z B C t = p ( l - z ) ( l - y ; % ) . . . (1 -yglzz). One can see that the roots 1, yl , . . . ,y ~of, p - z + q ~ B c l= 0 are distinct, except when X = Bp, in which case z = 1 1s a double root. When X = Bp, we suppose that y = I . From equation (1 7 ) Y ( z ) = EBB, A j ( l - zyyl , where A, = ~ , n.a .(I - ?.] y 71a )-1 .Thus Y k = Ef=' ~ j y ; and

k = 0 , 1 , 2 , . . . , MB,

X f

Bp.

(1 8) If X = Bp, the expression for Ph+is valid if we replace the indeterminate ratio (1 -- y ; M B - l ) ~ l - y-l )-1 by itslimit at yl = I,namely,MB-t 1 .

216

BULK-SERVICE QUEUES

MlSCELLANEOUS PROBLEMS A N D COMPLEMENTS

The limiting case when M -t may now be derived from equation (1 8). By means of RouchB's theorem, one can easily see that the equation p -z + pzB+' = 0 has only one root inside the unit circle if X
p.. =

+

1

(e+]

The normalizing condition P(1-) = 1 leads to

2

Jc

< i < j+

where kj = P ( j arrive during a service period). Show further that the probability sequence has the p.g.f. given by

which is a classical result and has been derived by many authors. See, for example, Morse (19%) and Finch (1 958).

Consider the following oversimplified analysis of a computer core storage system. Suppose that the input to the computer core storage system comes from a large number of sources and thus can be reasonably approximated by a Poisson process with rate X. Suppose further that during each service period, one or two storage requests can be serviced, depending on the size of the queue. If the queue size at the completion of service is 1, the next batch taken for service is 1. However, if the queue size is 2 2 , the next batch taken for service is either 1 or 2. The selection of the batch size 1 or 2 is assumed to be independent of the previous selection. The probability that a batch will be of size i ispi(n),i = 1, 2,wherepl(l)= l , p 2 ( l ) = O,andpl(iz)=p, p2(n) = q if n 2 2, with p + q = 1, n being the size of the queue at the instant of initiation of service. Any new arrivals during the service period will be assumed to join the queue. The service periods are assumed to be i.i.d. r.v.'s with common d.f. B(v). Let 6(v) = Jow e-a' dB (v), with bl = Jow vdB(v), and bz = v2 dB(v). If N(t) is the number in the system at any time t , then let N: = N(o, 0), where a, is the nth departure instant. Define Pj+ = limn , , P(Ni = j), j = 0, 1 , 2 , . . . , which limit exists under certain con. ditions discussed later in Miscellaneous Problem 2. The process {NiJforms a homogeneous Markov chain. Show that the transition probability matrix P = (Pi!) of the chain is given by

2

i = j+2

where p = p/q = Alp. Using this value of Fk and equation (4.5.14), show further that

P

pkj++, +qkj++,,

217

3

4

5

which gives one equation involving the unknown constants P$,Pf. Discuss how to get a second equation involving P$,P;. Find the expected queue size L' in terms of the given constants and bl ,b2. Observe that p = 0 , q = 1 and p = I , q = 0 reduce P+(z) to particular cases M / G ~ / and I M/G/l, respectively, of Bailey's bulk-service system discussed in Problem 1 and Section 4.1. The latter case is the well-known PollaczekKhintchine-Kendall formulation of M/G/l. Discuss the ergodicity condition for Problem 1, that is, find the traffic intensity p and show that the condition reduces to the one for M/G2/1 if p=O,q=1andtotheoneforM/G/1ifp=I,q=0. Continuation of Problem 1. If B(v) = 1 - e - p V ,1.1 > 0 , v > 0, then show that the denominator of P+(z) . . has three zeros. one of which lies within the unit circle, izl = 1. Show further that this root is - [@/A)' + 4q(~(/X)]"~(fiiX)1/2. Continuation of Problem 1. If in Problem 1 the service times are constant and equal to p-' = per second, find the characteristic equation for this case and solve it using a digital computer or otherwise. Draw the curves for L+ for various values of hip and p and show that the optimum (minimum) for L+ occurs at p = 0, q = 1 . Observe the effect on L + for light and heavy traffic conditions, that is, for small or large values of h/p. Continuation of Problem 1. If in Problem 1 we suppose that the service time required to handle two requests simultaneously is longer than that for one request, two different service-time distributions must be used. Let therefore the service time d.f. in the former case be Bl(v) and in the latter case B,(v). If P+(z) is the p.g.f. of the number in system in this case, show that

MISCELLANEOUS PROBLEMS A N D COMPLEMENTS

219

Since P+(z) is analytic for 121 < 1, the constants are determined by using RouchC's theorem. Continuation of Problem 1. Show that the waiting time in the queue for the bulk-service system of Problem 1 has the L.-S.T.

Terminal

Loaded carrier

<

I

Units

Figure 1

Units

A series of stations on a single loop.

t

Units

where P+(z) is defined in Problem 1 and 6(4 is the L 4 . T . of the servicetime distribution. In particular, show that for the system M/G/l (that is, when p = 1 and q = 0 in the original system of Problem I),

where Pi = I - p . Consider a grossly oversimplified or a hypothetical model of a real-world transit system wherein carriers with common fixed capacity provide service at a series of stations o n a single loop, as shown in Figure 1. In this system, fixed capacity carriers leave the terminal area empty. Each carrier visits every service station while serving all customers waiting, up t o the capacity of the carrier. Carriers are equally spaced in time with no early turnoffs or passing permitted. The capacity of each station is infinite. All customers remain on board the carrier until it reaches the terminal where all customers are unloaded. The empty carrier is then free to run the route again. Customers who are not sewed by the first carrier passing their station after they have entered the queue are lost to the system. Such customers may be thought to seek an alternative mode of transportation in order to get t o work on time. A simplified model of the type considered here might be used as a first approximation to a problem faced by suburban transit facilities during certain peak hours of the day. Let now the parameters of the system under consideration be specified as: t = time spacing between carriers, in minutes n = number of stations to be served B = capacity of each carrier hi= mean arrival rate of customers seeking service a t station i, per minute

Suppose customers arrive at station i following a Poisson distribution with mean rate Xi such that

p ( x , hit) = probability that x customers arrive in a t-minute interval at station i

With these assumptions it is possible t o derive probability distributions for a

Note that for n = 1, we have

variety of r.v.'s of interest. First suppose that we are interested in finding the probability distribution for the number of customers on board a carrier at any point along the route. To do this, we need t o consider the sum of r.v.'s X;. reuresenting- customers present for service at each station thus far served. Then the variable Yn+ which represents the total number of customers on board as a carrier approaches the terminal is easily seen t o be given by .>

P2D) =

.

(

P(Y,WI),

O
P@,tp~),

y

= B.

Show that the mean and the variance of Y n + ,are

The unused carrier capacity Un+ may be represented by Un+, = B - Y n + l . As customers are assumed t o leave the system if not served by the oncoming carrier, each Ximay be taken t o be identically distributed for every carrier that passes through the system. The sum C?=,Xi, being a sum of i.i.d. Poisson r.v.'s, is therefore a Poisson r.v. with mean tpn where pn = C:=, Xi. The following notation associated with the Poisson distribution is used for further, results:

*

(b)

where m represents the mean. Now it is easy to verify the following identitie

f

Show further that the mean and the variance of L are

jp(j,m) = mP(k - 1,rn)

j=k

x

Hint: Use equations (M6) and (M7). Units are lost to the system whenever an approaching carrier is unable t o provide service. This occurs every time the total number of customer arrivals during the passage of a carrier around the loop exceeds carrier capacity. If L is the number of customers lost per carrier, then show that the distribution of L is given by

( B1, tfin) var ( L ) = tpnP(B,tpn)(l - 2B) + ( ~ P , ) ~ P -

k

ip(i,m> = m [ l - - W , m ) l

j=O

x k

j2p(j, m ) = m [ 1 -P(k, m ) ] + rn2 [ I -P(k

- 1 ,m)]

L should not be confused with the usual meaning of mean system length. Hint: To derive E(L), use a modified form of equation (M5). It may also be derived from the following necessary condition for equilibrium: On the average, the average number arrived during an interval t must equal the sum of the average number lost and the average number served. Thus

j=O m

j2p(j, m ) = mP(k - 1, m ) + m2P(k - 2, m).

j=O

(a)

Show that each carrier will arrive at the terminal of an n-station sys with the probability distribution of the number of passengers given b

Pn+l(.Y) = P ( y n + ~= Y ) =

0,

Y < O

p0,tpn),

y = 0 , 1 , 2 , . . . ,B - 1

W ,&,I,

Y =B

0,

y

> B.

(c)

The unused capacity u on a carrier is represented by equation (M2). Show that this r.v. has the distribution given by the probability mass function (p.m.f.) hn+,(u),where

BULK-SERVICE QUEUE

u

8

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

> B.

For numerical calculations one may use the tables for p(k, m ) and P(k, m) which are reproduced in standard handbooks, such as Burington and May (1958). The above problems are based on the work of Griffin (1 966). In this system arrivals are by the Poisson process The system with mean rate A. The service times are i.i.d. r.v.'s with p.d.f. b(v), finite mean l / y and finite variance a'. The service is in groups with the rule that the server does not start service until there is (at least) a fixed number k of customers in the queue, and the maximum capacity of the server is B. I other words, service is in groups of sizes m such that k < m < B. We say that the server has quorum k and capacity B. Using the notation and procedure of Section 4.1, show that the steady. state partial differential-difference equations for the system M / G * . ~ / Iare

,

w

0 =

-w,+j P o , I ( x ) V c 4 ~

O = - ~ , o + ~ r ~ , , o + ~ ~ ~ r , l ( x ) ~ ( x 1) d < xr 4, k - 1

(Mi0

aPO,l(X) ax = -(X+~(X))P~,~(X) apn31(X) = -(A

ax

+ v(x))P,,

l(r)

+ h ~ , - ~(XI,,

n

>o

and p = X/Bp< I . There are several points to note about the expression in equation (M12). The most important one is that there is no explicit dependence on k, the minimum number in a batch. In view of this, the p.g.f. of P,, "takes the same form for the systems M'/Gk/l and M/GB/l, where the service batches are of fixed size k (= B) and variable up lo B, respectively. This implies that the imbedded Markov chain considered at departure epochs (see Problem 11) would be the same for the three systems M/Gk/1 , M / G ~ / IandM/Gk,B/l. , Since equation (M12) is the same as equation (4.1.8) forlMjGB/1,evaluation of the constants involved is as discussed in Section 4.1. Explicit results are available only in the special case when G E, which is discussed later. In other cases one has to do numerical work [see Holman (1977)]. The distribution of the number in the system, among other things, for the more general queuing system M / G ~ , ~in/ Iwhich service times may depend on the batch size has been discussed by Neuts (1967) using the theory of semi-Markov processes. The results, however, are given in forms that are computationally inconvenient. It may be noted that the special case when k = 1 is discussed in Section 4.1 and the one when B = k is discussed in Problem 9. (a) Continuation of Problem 8. The system M/Gk/l. If B = k, then show that equation (MI 1) becomes

9

where Pn,o = P(n in queue and the server is idle). Other probabilities ha been defined in Section 4.1, and the boundary conditions are B

po,i(o) =

jmp,,I(X)ri(x)dX+*k-i,o r=k

. Pn,l(O)=)

0

w

P~+B,~(x)v(x)~x, n > O -

Show that the p.g.f. of the number in the queue is k-1

P(z) = where

C

r=O

P7, OZ' + P&)

(b)

The distribution of the number in the system M/G'/~. Once we know the p g f . of the number in the queue, it is easy to find the p.g.f. of the number in the system. Show that the p.g.f. of the number N in the system is

Take the p.g.f. of N , , etc:, and use the result of Problem 9(a). For one alternative proof of equation (M14) see Chapter 6; for still another one, see TakPcs (1962). By taking a look at equations (M13) and (M141, one may observe that P&) = P(z)6(h - Xz). This has the interpretation that the p.g.f. of N equals the p,g.f. of N, times the p.g.f. of the number that arrive during the service of a group. Furthermore, the probability that the server is idle may be obtamed from equation (M13) or (M14) by using the normalizing condition. Show that it is given by

(d)

Now substituting for P,(z) and using h - X z = a gives the desired result. Takacs (1962) obtains the transform @(a) by following a rather complex procedure. The procedure adopted here is unclassical and the result for @(a) agrees with the one obtained by Takdcs, except that his resuIt has a sign error. In the case k = 1 , the result may be compared with the now familiar result for the waiting time for M/G/l. Show that in the special case M/Mk/l the distribution of t; is given in explicit form by

BULK-SERVICE QUEUES

(b)

(c)

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

Show that

The results of this problem are due to Borthakur (19711, but our procedure in deriving the results is slightly different. Medhi and Borthakur (1972) have also found the p.g.f. of the number in the queue for the two-server queuing system M / M " ~ / ~ . The system M / M ~ * ~ /The ~ . results obtained in (a) are simplified if the capacity B of the server is assumed to be infinite. Show that in this case P,,, = (1 - y-r-l )(k+(h/p)y-k)-l, O G A k - 1 , and Pn,l = (X/p)y-" Po?,, n > 0, where y is given by y = (A + p)/h. Since the distribution is known completely, one can easily get numerical values of probabilities and expressions for moments. In particular,

(c)

Show that for the system M / G ~ . ~ / ~

(d)

In the special case G = M , this relation takes the simple form P,(i P'(z) = l - c;=;:Pr, . Use (c) to show that

(e)

where L+ = P+(')(I). In the special case when B = k, show that tf becomes L, - L" = (k - 1)/2 - kp. Hint (special case): Differentiate equation(MI3) once to get Pr, a second time to get Z~L;rP,,,, and substitute their values in the 1 found earlier. Rewrite equations (M9) and (MIO) in terms of P: as foIlows:

,

$z,"

11

(a)

(b)

Continuation of Problem 8: the imbedded Markov chain of the system that for the system M / G ~ , ~ / I , M / G ~ , ~ / I Show .

This system is discussed by Holman and Chaudhry (1979) and Holman et al. (1981) and, in more detail, by Holman (1977). Some of Holman and Chaudhry's findings concerning this system are reported in Chapter 6. For the special case when k = 1, see Chaudhry and Templeton (1981), and when B = k see Section 4.3. Observe again that equation (M18) has no dependence on k. It may be further observed that though the underlying processes for the three systems h'/Gk/l (if k = B), M/GB/l, and ~ G ~ , differ, ~ / the I difference does not show up at the instants examined in the Markov chain. In his paper concerning the MX/G Y / l queuing system, Bhat (1964) has made this same observation. However, he has not considered the present M/GksB/lqueuing system. Hint: Define Pz = D:J Pn, (x)q(x) dx, where D is a normalizing constant, and proceed as in Section 4.1. Show that D, defined in (a), is given by

This glves a set of k equations expressing k unknowns Po,, PI,, , . . in terms of P:, O < i < k - 1, the evaluation of wh~chhas bet discussed In Sectlon 4.1. Solving equations (M19) and (M20) t Cramer's rule, one finds rather complicated results even for small valu of k such as k = 3 or 4. However, the following procedure IS mo elegant. Multiply equation (M19) by k and equatlon (M20) by (k - r Add and simpllfy to get

Ph-

Now add equations (MI 9) and {(M20) for i = 1 to j) and use equatic (M21) to get

BULK-SERVICE QUEUES

228

MISCELLANEOUS PROBLEMS A N D COMPLEMENTS

229

In the special case B = k , the expression for Pj,, can be simplified. Show that when B = k , then

12

An alternative proof of this for the system M/G'/~ is given in Chapter 6. Furthermore note that when k = 1 , the relation between Poo and Pi turns out to be the same as the one discussed in Section 4.1. Hint (B = k case): Use equation (MI 8) with P'(1) = 1. Continuation of Problem 10: the waiting-time (in queue) distribution. Nair and Neuts (1972) have discussed the virtual waiting time when service times may depend on the size of the batches to be served. The expressions are unwieldy, and a look at their paper shows that even if the service times do not depend on the size of the batches t o be served, numerically manageable results do not appear t o be possible. Analytically explicit results are available only for the actual waiting-time (in queue) distribution for the simpler queuing system IM/M'*~/I,which we proceed to discuss in this problem. The service discipline is FCFS, and the other assumptions are the same as in Problem 8 , except that now G =M. Further, it is assumed that the servicetime distribution is independent of the batch size. The discussion given below is similar t o that of Medhi (1975). The distribution of the number in queue for the two-server queuing system M/MkvB/2has been investigated by Medhi and Borthakur (1972). We are interested here in the limiting distribution of Vq(t) as t +-. Let Vq(t) converge t o Vq in distribution as t -+-. Let wq(r) be the p.d.f. of V,. Further, let us define the following functions:

where y (introduced in Problem 10) is the root outside the unit circle, / z I = 1 , of the equation Alp = (1 - z - ~ ) / ( z - I ) , an equation which is often used in changing the results from one form to another. Hint: An arriving customer will find the system in one of the following classes of states: X

(ii)

(s,O),

O<s
(iii)

( n ,

iB+k-l
(iv)

(

iB
1

i = 0,1,2,..

In case (i) the arriving customer immediately goes into service and thus P(Vq = 0) = Pk-,,,. In case (ii) the arriving customer has t o wait for the arrival of k - 1 - s customers before service o n his group can be started. The time required for k - 1 - s arrivals has an Erlang distribution with p.d.f. f(X, k - 1 -s;r). In case (iii) the arriving customer has t o wait for the completion of service on i + 1 groups before service on his group can be i + 1 ;7). started. The time required for this has an Erlang distribution f(1, In case (iv) the arriving customer has t o wait until k - 1 - s customers arrive where n = iB + s, 0 < s < k - 2, and the sewice on i I groups is over. The time required for this to happen is the maximum of two independent Erlang r.v.'s with parameters (A, k - 1 - s) and (p, i + 1). Let us define this maximum by Y = man (Erlang r v . with parameters h, k - 1 - s and Erlang r.v. with parameters p , i + I ) .

+

Since the two Erlang r.v.'s under consideration are independent, the distribution function Fy(y) and the corresponding p.d.f. fY(y) of Yare given by This is the p.d.f. of an Erlang distribution with L.T. [v/(v bution function corresponding t o f(v; I ; t ) is

+

+ ol)] '. The distri-

with L.T. [v/(v a)]'/a. Show that the distribution of Vq is given by P(Vq =OO)=Ph-l,O and

From the above discussion one gets

BULK-SERVICE QUEUES

MlSCELLANEOUS PROBLEMS A N D COMPLEMENTS

Hint:

Consider three mutually exclusive classes of states:

(i)

(r,O),

r = 0 , 1 , 2 ,...,k - 1

(ii)

( n l

n=iB+r r = 0,1, . . . , k - 1 ,

(iii)

Note that the integral of wq(7) from 0 t o is 1 - P k - l , O , as is expected. Now there are two ways t o proceed: Use the probabilities given in Problem 10(a), simplify, and get the desired result. Use the probabilities, take L.T., simplify, and then invert. However, in this case we need to use the following expression for the L.T.:

n=iB+r, r = k , k + l , ...,B-1,

i = 0,1,2,

Then use P({ = 0) = ~,h;iPr, and

(b)

and simplify t o get the required result. If B = k = 1, then show that the distribution of { is the same as that of V q ,which is given by

(a)

A typical bulk-service problem M/EIEZ . . .E k / l .

Show that the first and the second moments of V , are given by

14

from which one can obtain the variance of V,. Using Problem 10, show that Little's formula L, = XWq is satisfied. The system E , / M ~ . ~ / I . This system differs from the one discussed above and in Problem 1 0 in that the interarrival times are now r-Erlang distributed. The distribution of the number in the system, the waitingtime distribution, and the computational aspects of some measures of efficiency have recently been discussed by Easton and Chaudhry (1982). Continuation of Problem 12: the distribution of occupation time { ( t ) . Let {(t) converge t o 5 in distribution as t + - and let w(7) be the p.d.f. of 5. Show that the distribution of 5 for the queuing system M / M ~ , is ~ given / ~ by

(n,)

i = 0 , 1 , 2, . . .

In this problem we discuss the steady-state waiting-time (in queue) distribution of the queuing system in which arrivals are by the Poisson process with rate A. The service is performed in a manner discussed in Problem 8(a) of Chapter 2, and thus the p.d.f. of service times is the one given in that problem. Assuming that the queue discipline is FCFS, show that Wq = E [ V q ]= ~ ( ~ ) / [ p-~p)] ( l and var [Vq j = 2 ~ ( ~ ) / [ 1 . ~ ~p)]( 1+ where p = (A@lf/p) < 1, Vq is the steady-state waiting-time (in queue), and @ ( ' ) , I = 1 , 2, . . . , is defined in Problem 8(a) of Chapter 2. Hint: In this case the waiting-time (in queue) distribution is the same as equation (2.2.13). Consequently t o get moments, use equation (2.2.13), and for notations use Problem 8(a) of Chapter 2. If in Problem 8(a) of Chapter 2, we now assume that service times X1 ,X 2 , . . . ,Xk are independently but not identically distributed so that E(Xi) = l/pi,i = 1, 2, . . . ,k , then show that

e,

(b)

BULK-SERVICE QUEUES a n d t h e characteristic function of Z is

w h e r e E , m , n = 1 , 2 , 3 , . . . , k. Find E(Z), var (Z), a n d verify t h a t w h e n pi = p, f o r all i, t h e y agree with the results f o u n d in Problem 8(a) of Chapter 2.

If the service-time distribution f o r Problem 14(a) is the o n e given in Problem 14(b), t h e n s h o w t h a t , f o r k = 3,

+ (PI + P21 + l3 P3

where

C = I -U(Z)

with U ( Z ) < 1. Discussion o f W , a n d var [ V q ] f o r a n arbitrary value of k is n o w straightforward, [Fujisawa (1962).]

Bailey, N. T. J. (1954). On queuing processes with bulk service. J. R. Stat. Soc. Ser. B 16, 80-87. Bhat, U. N. (1964). Imbedded Markov chain analysis of a single server bulk queue. J. Aust. Math. Soc. 4,244-263. Bloemena, A. R. (1960). On queuing processes with a certain type of bulk service. Bull. Inst. int. Stat. 37,219-226. Borthakur. A. (1971). A Poisson queue with a general bulk service rule. J. Assam Sci. SOC. XIV, 162-167. Burington, R. S., and D. C. May (1958). Handbook o f probability and statistics with tables.

REFERENCES Handbook Publishers, Sandusky, Ohio. Chaudhry, M. L., and J. G. C. Templeton, (1981). The queueing system M/GB/1 and its ramifications. Eur. J. Oper. Res. 6,57-61. Dick, R. S. (1970). Some theorems of a single server queue with balking. Oper. Res. 18,11931205. Downton, F. (1955). Waiting time in bulk service queues. J. R. Stat. Soc. Ser. B 17, 256-261. ---- (1956). On limiting distributions arising in bulk service queues. J. R. Stat. Soc. Ser. B 18,265-274. Easton, G., and M. L. Chaudhry (1982). The queueing system Ek/@ b / l and its numerical analysis. Comput. Oper. Res. 9,197-205. Fabens, A. J . (1961).The solution of queueing and inventory models by semi-Markov processes. J. R. Stat. Soc. Ser. B 23,113-117. --- and A. G. A. D. Perera (1963). A correction to the above. J. R. Stat. Soc. Ser. B 25,455-456. x Finch, P. D. (1958). The effect of the size of the waiting room on a simple queue. J, R. Stat. SOC.Ser. B 20,182-186. ------(1962). On the transient behavior of a queueing system with bulk service and finite capacity. Ann. Math. Stat. 33, 973-985. Foster, F. G., and K. M. Nyunt (1961). Queues with batch departures. Ann. Math. Stat. 32, 1324-1332. Fujisawa, T. (1962). On a waiting-line process with a particular bulk service. Yokohamo Moth. J. X , 45-51. Ghosal, A. (1970). Some aspects o f queueingand storage systems. Lecture Notes in Economics and Mathematical Systems 23. Springer-Verlag, New York. Griffin, W. C. (1966). A simplified model for bulk service at a series of stations. J. Ind. Eng. 17, 430-436. Gross, D., and C. M. Harris (1974). Fundamentak o f queueing theory. Wiley, New York. H o h a n , D. F. (1977). Some problems in the theory of bulk queues. Thesis, Royal Military College of Canada, Kingston, Ontario. -and M. L. Chaudhry (1979). A unified approach to some results for the queueing system M / G ~ ' ~ Presentedat /I. the XXXIV International Meeting of the IMS, Honolulu, June 18-22. ---- and A. Ghosal (1981). Some results for the general bulk service queueing system. Bull. Aust. Math. Soc. 25,161-179. Jaiswal, N. K. 11960a). Bulk-service queueing problem. Oper. Res. 8 , 139-143. --- (1960b). Timedependent solution of the bulkservice queueing problem. Oper. Res. 8,773-781. ----- (1961). A bulk-service queuing problem with variable capacity. J. R. Stat. Soc. Ser. B 23,143-148. Khintchine, A. Y. (1932). Mathematical theory of a stationary queue (in Russian).Mat. Sb. 39, 73-84. Lwin, T., and A. Ghosal (1971). On the finite capacity treatment of Bailey's queueing model. Calcutta Stat. Assoc. Bull. 20, 67-76. Medhi, J . (1975). Waiting time distribution in a Poisson queue with general bulk service rule. Manage. Sci 21, 777-782. ----and A. Borthakur (1972). On a twoserver Markovian queue with a general bulk service rule. Cah. Cent. Etud. Rech. Opkr. 14, 151 -158. Meisling, T. (1 958). Discrete-time queuing theory. Oper. Res. 6 , 96-105. Mercer, A. (1968). A queue with random arrivals and scheduled bulk departures. J. R. Stat. Soc. Ser. B 30,185-189.

234

BULKSERVlCE QUEUES

Moran, P. A. P. (1 959). The theory o f storage. Methuen, London. Morse, P. M. (1958). Queues, inventories and maintenance. Wiley, New York. Murao, Y. (1970). An analysis of the bulk service queue with stochastic variable batch size (depending on queue length) (in Japanese). Keiei Kagaku (Jpn.) 14, 34-51. Nau, S. S., and M. F. Neuts (1972). Distribution of occupation time and virtual waiting time of a general class of bulk queues. SankhyZ Ser. A , 17-22. Natarajan, R. (1962). Discrete-time bulk service queueing process. Def. Sci. J. 12, 318-326. Neuts., M. F. (1967). A general class of bulk queues with Poisson input. Ann. Math. Stat. 38, 759-770. Novaes, A., and E. Frankel (1966). A queueing model for unitized cago generation. Oper. Res. 14,100-132. Prabhu, N. U. (1965). Queues and inventories - a study of their basic stochastic processes. Wiley, New York. Singh, V. P. (1971). Finite waiting space bulk service system. J. Eng. Math. 5, 241 -248. (1972). Addendum to theabove paper. J. Eng. Math. 6 , 8 5 4 8 . Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge PhiZos. Soc., 449-461. Stidham, S. (1974). Stochastic clearing systems. Stochastic Processes Appl. 2, 85-1 13. Takics, L. (1962). Introducrion to the theory of queues. Oxford University Press, New York. Van Dantzig, D. (1947). Kadercursus statistiek. Stencilled notes of lectures given at the University of Amsterdam. (1948). Sur la mbthode des fonctions g8nkratrices. Colloques Internationaux du Centre National de la Recherche Scientifique, 13,29-45.

-(1955). Chaines de Markof dam les ensembles abstraits et applications aux processus,

avec r8aions absorbantes et au problkme des boucles. Ann. Inst. H. Poincard, Sec. B 14, 145-149. service-time distribution. Ann. Math. Stat. Wishart, D. M. G. (1956). A queuing system with 27,768-779.

Additional References Abramov, A. Kh., and A. D. Tsvirkyn (1968). On the optimal designation of speed service. Autom. i Telemekh. (USSR) 2 , 7 6 4 0 . Alagaraja, K. (1970). A queue with particular bulk service having dependent service times. Yokohama Math. J. 18,15-22. Babary, E. S. (1969). Multi-level bulk service queues. Thesis, Columbia University, New York. and P. Kolesar (1972). Multi-level bulk service queues. Oper. Res. 20,406-420. Bhat, U. N. (1968). See Chapter 3. Borthakur, A. (1971). An additional special channel Poisson queueing system with a general rule for bulk service. Inst. Math. Stat. BulL, 72t-8. Chang, W. (1970). Single server queueing processes in computing systems. Int. Bus. Mach. Syst. J, 9,36-71. Chaudhry, M. L., and A. M. Lee (1972). Single channel constant capacity bulk service queueing process with an intermittently available server. INFOR 10, 284-291. and J. G. C. Templeton (1968). On .the discrete-time queue length distribution in a bulk service system considering correlated arrivals. Can. Oper. Res. Soc. J. 6 , 7 9 4 8 . Crane, M. A. (1974). Queues in transportation I1 - an independently despatched system. J. Appl. Prob. 11,145-158.

REFERENCES

Craven, B. D. (1963a). Some results for the bulk service queue. Aust. J. Stat. 5, 49-56. (1963b). Asymptotic transient behaviour of the bulk service queue. J. Aust. Math. SOC.3,503-512. Dave, H. B. (1971). A bulk queue with priorities. M.A. Sc. thesis, Department of Industrial Engineering, University of Toronto. Deb, R. K., and R. F. Serfozo (1973). Optimal control of batch service queues. Adv. Appl. Prob. 5,343-361. Frank, H. (1969). Analysis and optimization of disk storage devices for time sharing system. J. Assoc. Comput. Mach. 4,602-620. Fukuta, J. (1969). Concentrated service queue with limited unit source. J. Oper. Res. Soc. (Jpn.) 12, 21-35. Georganas, N. D. (1976). Buffer behavior with Poisson arrivals and bulk geometric service. IEEE Trans. Commun. 24,938-940. Goyal, J. K. (1967). Queues with hyper-Poisson arrivals and bulk exponential serGce. Metrika 11,157-167. Handa, J. M. (1971). Optimal control of service rate and service batch size in single channel stochastic service systems. M.A. Sc. thesis, Department of Industrial Engineering, University of Toronto. Harris, C. M., and P. G. Markin (1972). A note on feedback queues with bulk service. J. Assoc. Comput. Mach. 19,727-733. Ivnitskiy, V. A. (1971). A single channel bulk service system with queue. Eng. Cybern. 6, 1066-1077. Jain, J. L. (1971). Some contributions to the theory of queues and dams. Ph.D. thesis, University of Delhi. Abstracted in Opseurch 8, 293-294. Kashyap, B. R. K. (1966). The doubleended queue with bulk service and limited waiting space. Oper. Res. 14, 822-834. (1969). The doubleended queue with batch departures. Advancing Frontiers in Operational Research - Proc. Int. Seminar, New Delhi, 1967. Hindustan Publ. Corp. (India), New Delhi, 1969, 139-143. Kotiah, T. C. T., J. W. Thompson, and W. A. O'N. Waugh (1 969). Use of Erlangian distribution for single-server queueing systems.'J. Appl. Prob. 6. 584-593. Mohan, C., and K. Murari (1972). Time dependent solution of a correlated queueing problem with variable capacity. Merrika 19, 209-215. Mohanty, S. G. (1972). On queues involving batches. J. Appl. Prob. 9,430-435. Murao, Y., and G. Nakamura (1973). On the relation between C/D/s and G/D/l (bulk service with sizes). Keiei Kagaku (Jpn.) - . 17.. 26-31. Murari, K. (1969). A queueing problem with correlated arrivals and general service time distribution. Z. Angew. Math. Mech. 49, 151-156. Nair, S. S., and M. F. Neuts (1972). Distribution o f occupation time and virtual waiting time of a general class of bulk queues. SankhyESer. A 34, 17-22. Nakamura, G., and H. Murao (1968). A solution on bulk service waiting queues (in Japanese). Denkitsushin Kenkyujo Kenkyu Jitsuyoka Hokoku 17, 1599-1618. Neuts, M. F. (1965). The busy period of a queue with batch service. Oper. Res. 13,815-819. -(1 981). Matrix-geometric solutions to stochastic models - an algorithmic approach. The Johns Hopkins University Press, Baltimore. Runnenburg, J. Th. (1965). On the use of the method of collective marks in queueing theory. Proc. Symposium on Congestion Theory, W. L. Smith and W. E. Wilkinson, Eds. University of North Carolina Press, Chapel Hill, 339-438. Sharda (1981). A limited space correlated queueing problem with departures in batches of variable size. Cah. Cen. Etud. Rech. Opkr. 23, 87-96.

------

Stuart, I. M., and G . B. McMahon (1966). A queueing system with bulk service. Oper. Res. 14, 728-731. Weiss, H. J . (1979). The computation of optimal control limits for a queue with batch services. MQnQge. k i . 25, 320-328.

In Chapter 3 we discussed single-server queuing systems with bulk arrivals, and in Chapter 4 a similar analysis was carried out for queuing systems with bulk service. In this chapter we discuss a different class of queuing systems knowfi as multichannel (or multiserver) queues having more than one server in parallel, and either bulk arrival or bulk service. Multichannel queues constitute an important class of queuing processes and have broad practical meaning. Examples of such systems may be found in banks, telephone exchanges, hospital admission systems, seaports, toll booths, military tactics, courts of law wherein servers are judges and customers are cases, and information transmission systems wherein messages containing a (small) random number of characters (a batch of characters) arrive according to a Poisson process and must be transmitted t o some destination. It will be seen that multichannel queues are more complex t o deal with than single-channel ones, and this complexity increases when customers either arrive or are served in bulk. Some cases arise in which the number of channels is so large (infinite for all practical purposes) that waiting times are negligible and only the level of server occupancy is of analytical or practical interest. An intensive care unit in a hospital, where delay in service may result in loss of a life, may provide an example of such a situation. Other examples are self-service stations such as large parking lots, theaters, or auditoriums. In addition, a large telephone exchange, the number of claimants drawing unemployment insurance or workmen's compensation, may be considered t o fall under the category of infinite-server queues. Analytically, infinite-server queues are easier to handle than their counterpart finite-server queues. Both the finite- and the infinite-serverqueues having either arrivals or service in bulk, are discussed in the present chapter. The reader will find that the algebra, at times, becomes rather involved. We wish we were able to find some guiding thread, but we are sorry to say that we were not as successful in unifying the various models or simplifying the algebra as we were in the other chapters. Whenever we did succeed in simplifying some algebra, we have indicated this by our remarks at various points.

E SYSTEM M X ( t ) / M / o oW I T

E INPUT RATE X ( t )

The Transient Behavior of N ( t )

In the system under consideration the customers arrive in groups following a

MULTICHANNEL BULK QUEUES

240

5.2

THE SYSTEM M X / m

The result (5.1.9) is due t o Abol'nikov (1968). For some results which are given in implicit form for the more general system M X ( t ) / G / m ,see Shanbhag (1966). = A (z)F(z ;(u)

5.2

THE SYSTEM M X I M / c

and transposing we get

The Transient Behavior of N ( t )

In contrast t o the previous section we now have constant input rate. The input process is a Poisson process with constant rate h for all values of t . For simplicity we now assume N(0) = i, that is, initially there are i customers in the system. All other assumptions are unchanged. The group size X is an r.v. with distribution given by a m = P ( X = m ) , m 2 1. X has mean ii, O < i i = X:=lmam < m ,variance 02, 0 c. In short, the present system may be considered t o be a c-channel analog of the corresponding infinite-channel system treated in the previous section. Let Pn(t) P(N(t) = n) be the probability that there are n customers in the system at time t. It is not difficult t o see that the process (N(t)) is a Markov process with an enumerable number of states. The differential-difference equations for the present system are

pcd )(t)

=

- XP, (t) + pP1 (t)

+ (n + 1)pPnc1(t),

1
(5.2.1)

Define the generating function p(z; or) = X$o~n(or)zn, where Pn(or)= Pn(t)dt, Re a 2 0. Taking the L.T. of equation (5.2.1), multiplying by appropriate powers of z and summing over all n, we obtain, after some algebra,

where X(z) is defined in Section 5 .I. By using RouchC's theorem it is simple t o see that the denominator of equation (5.2.2) has a unique root within the unit circle, Iz I = 1, which we denbte by zo = z(or). The numerator must vanish at z o , for otherwiseP(z;or) would not exist. Putting z = zo in the numerator of equation (5.2.2), we obtain

The relation (5.2.3) together with the first c - I equations of the system (5.2.1) offers the possibility of finding all the Pn(a), n = 0, 1, 2, . . . , c - 1, and along with them also the function &z; a). Some further analysis is carried out by expressing Xglb (c - n)~n(or)znin terms ofpo(a). Since this cannot be done by directly using the system (5.2.11, we proceed on the following lines. Let us consider the system M ~ / Mof /Section ~ 5.1, which differs from the present system only in the presence of an infinite number of servers. Let the probabilities and generating functions ^of Section 5.1 now be denoted by placing a caret over them, that is, let them be Pn(t), P(z; t), and so on. Taking the L.T. of the equation (5.1.4) with respect t o t with h(t) = h, we have

where we have used the initial condition

P (z; 0)

= zi.

(5.2.5) Comparing the systems (5.2.1) and (5.1 .I) with h(t) = h, it is easy t o observe that

n expressed by exactly the same form in terms ofpo(or) as is

where we have used Pn(0) = a,,. Using

p l : h

0

I 3

w

-

Cal

I?

R

N

3

II 'a

".

n N

R

V

Cal

i .

R

w

I Y

A

-.N R

w

MULTICHANNEL BULK QUEUES

5.2

THE SYSTEM M ~ I M I C

+ 6. For the systemMX/M/2,

p being Xdip and a; = ~ ( ~ ) ( 1)

= lim

a [z(a)]

'+'

-

a+o* R(z(a); a)/J(l - z(a))

p>l

0, c(l-p),

(5.2.12)

r where r = lirn,,,,

-

c/J(l - P M ) [c/J - h X ( z ) l r

P(z) = lim aP(z;a) = a+o+

'

Hence it is clear from the analysis carried out above that the limiting distribution of N ( t ) exists iff p < 1. For certain purposes, however, it is better t o derive P(z) from equation (5.2.2), that is, lim aP(z; a) = P(z) = ,+o+

/J

(C - n)Pnzn n=o C/J - XZ X(Z)

(5.2.13)

Using P ( l ) = 1, we get n p n = c ( e o -p)

(5.2.14)

n=O

where Qo = P (all servers are busy). The results (5.2.13) and (5.2.14) are used in calculating expected values, which are discussed below. If L and L, represent the expected number in the system and in the queue, then in the case of c channels, it is easy t o show that C

L-L,

=

C+

. ,

For the system f l I W 3 , p = ha/(3p),

R @(a); a). Furthermore,

-

Hence

p
n=o C (n-c)Pn.

This relation, though intuitively obvious and easy to prove otherwise, is true irrespective of the nature of the distribution of interarrival times, service times, and size of arrival groups. To particularize this for the model M ~ ~ M weCget, using equation (5.2.14), L - L, = 6 a, 0 = X/p, which represents the average number of busy servers. From equation (5.2.13) one gets, by using equation (5.2.14)

The Waiting-Time (in Queue) Distribution

Let the queue discipline be FCFS by groups, and random within the groups. Let Vg be the waiting time of a random customer in an arriving group. In this section we find P(Vq > 0) and P(V, > t), t > 0,by a method which is instructive, though a bit laborious. There is, however, a much simpler method, discussed in Problems and Complements, for finding these two probabilities. To discuss P(Vq > 0), we reason as follows. If a group containing m customers arrives t o find the system in state n, 0 < n i c - 1, then for given m and n, the probability of immediate service for the customer under consideration is 1 if 1 < m < c - n and(c-n)/mifrn>c-n 1. Consequently, for all m and given n,

+

p(v,=~ln)=

c mum -=.a

c-n

c-n-

C-.=lZE%

+

m=s

m=c-n+~

~2

h'

Odnic-l

c-n

C (c-n-m)am

m=:

Li

where mam/# is the probability that the customer under consideration arrives in a group of size m. For details, see Chapter 3 (Miscellaneous Problems and Complements). Since the probability of V, > 0 is the complement of the probability of Vq = 0, the unconditional probability of V, > 0 is

Once Po, P I , . . . , PC-' are determined, L and consequently L, are determined. In particular, for the systemMX/M/l, c-n

a n=o

MULTICHANNEL BULK QUEUES

246

5.2

THE SYSTEM IWX/IW/c

The result (5.2.16) can be simplified. For details, see Problems and Complements. Let us now turn our attention t o P(Vq > t ) , t > 0. We show that This gives finally

To prove equation (5.2.17). first we need to discuss the relations between the limiting probabilities which may be obtained from equation (5.2.1). One can easily show that the limiting probabilities P,, in fact, satisfy the following relations: where we have used the important relation between the Edang distribution and the Poisson process. If Wq is the mean waiting time (in queue), then

wq = E(VqV,) =

" d l

-

d=l

To get equation (5.2.171, we proceed as follows. If a group containing m customers arrives to find the system in state n , c - m < n < c - 1 , then c - n are served immediately and m - c n are delayed. Of the m - c + n who are delayed, the customer under consideration must wait f o r d departures,d = 1 , 2 , . . . ,m - c n. On the other hand, if the group arrives t o find the system in state n , n 2 c , all the m customers are delayed. Thus, of these m customers who arrive when n = c k , k = 0, 1, 2, . . . , the customer under consideration must wait for d departures, d = k 1, k 2, . . . , k m. The conditional probability, given the state of the system n just before the arrival of the customer's group, of the customer waiting f o r d departures before his service commences is given by

+

+

+

+ +

CP P

which shows that Little's formula holds for the system Mx/M/c. One can easily find the higher moments of Vq. In particular, the variance of Vq is

+

[It may, however, be remarked here that the probability corresponding to d = O is P(V9 = Ojn), 0 < n < c - 1 , which has already been discussed and which will not be used further while discussing P(Vq > t).] Consequently, the probability that a random customer in an arrival group waits f o r d departures before beginning service is

where we have used equation (5.2.1 8). Since the service time is exponential with the same mean 1 / p for each of the c servers, the time until the first of the c servers becomes empty is exponentially distributed with mean equal t o l / c p . Thus the time the customer under consideration must wait for d departures from the system is the d-fold convolution of this and is the Erlang distribution with d degrees of freedom, yielding

Now consider the distribution of two different components of the waiting time Vq (in queue) of a randomly selected customer of an arrival group, namely, (1) waiting-time (in queue) distribution of the first member of the group, (2) distribution of waiting time (in queue) due t o the service times of the members of the group served before the selected customer. One first needs t o know the distribution of the number of departures from the queue the typical customer has t o wait for before his service commences. The consideration of these two components of waiting time leads to the generalization of the results discussed in Chapter 3 for the single-server case (see Miscellaneous Problems and Complements). We now proceed to discuss the two components of V,. In case (1) the distribution of the number of departures is Poisson, or the waiting-time distribution is Erlang, as before. Thus if V,, represents the waiting time (in queue) of the first member of an arrival group, one can easily see that if a group arrives to find d > 1 customers already in the queue, and the system in staten=c+d-l,d=1,2,3 ,..., "

P(v,,

>tln)=Jt

gd(u)du,

d = 1 , 2, . . .

This gives the unconditional distribution of V,, as

MULTICHANNEL BULK QUEUES

If Vq2 is the component of the waiting time (in queue) of the customer under consideration due to the queuing times of the members of his arrival group, then From this, or otherwise, one can easily see that E(Vq,) = C& (d/cp)Pd+,-, . The higher order moments may be obtained similarly. This completes the discussion of the distribution of Vq, . The discussion of case (2), however, is a bit more involved. For that we need to use the fact that the probability that a random customer has the jth position in his arrival group is

(see Miscellaneous Problems and Complements, Chapter 3). Let Di be the event that a customer, whose waiting-time distribution is required, experiences a delay due to the commencement of service of i > 0 members from within his arrival group before his service commences. Consequently, if a group arrives to find the system in state n ( 2 c), then P(Di In) = ri+l,i 2 0. In other words, the probability that the customer whose waiting-time distribution (in queue) is required is Ist, 2nd, 3rd, . . . , in his group is rl = P(Do In), r2 = P(D1 In), r3 = P(D2 In), . . . . On the other hand, if a group arrives to find the system in state n, 0 < n < c - I , then there is no delay for the first c - n customers in the group, and P(Do in) = Ciz:rj. However, if his position is c - n + i, i 2 I , then P(Di In) = rc-n+i,i 2 1. Hence we obtain the unconditional probabilities

=

C-1

c-n

n=O

j=l

2 r I p n + C P,,

n=c

C rj

Now define the "mean service position" or Ec(SP) as the expected number of customers from within a random customer's arrival group that enter service before the random customer. Then

The expected value of Vq2 is therefore

Numerical calculation is possible in all the above cases, but analytically explicit results can be achieved only for geometric distribution of the arrival group size. For more details, see Problems and Complements. Many of the results of this section are due to Abol'nikov(1967) or Kabak (1970). Hawkes (1965) obtained the distribution of waiting times of ordinary and priority customers in the priority queuing system ~ ~ 1 / G.I , G2/1, ~ 2 from which one can obtain the waiting time distribution for the system MX/G/1. Later Burke (1975) derived the waiting-time distribution for MX/G/l, and cited both correct and incorrect formulas for waiting time in the work of earlier authors. It is easy to show using Burke's results, that Little's formula holds for MX/G/l. In this connection, see Chapter 3 (Miscellaneous Problems and Complements). Those results of Kabak which were erroneously reported have been corrected by Cromie and Chaudhry (1975), a revised version of which is presented in this section. Abol'nikov and Yasnogoridskiy (1972) have also discussed the distribution for the number in the systems & f X / ~ / c / cand MX/M/c/~inite,but give the results in implicit form. For explicit results for certain particular cases of &fX/M/c/c, see Problems and Complements. Abol'nikov (1970) has transient solutions of MX/M/c. Cromie (1974) has discussed more elaborately the numerical aspect of the system MX/M/c (limiting case) for the three cases where the arrival group size has (1) constant input, (2) geometric input, and (3) positive Poisson distribution (left truncated). He has provided numerical values ,and curves for certain measures of efficiency such as L, and P(Vq >0) for certain combinations of values of the average group size 5, the traffic intensity p, and the number of channels c for the three cases mentioned above. In this connection, see also Cromie et al. (1979) who have simplified further some of the results of Cromie, and have given independent proofs for others. For numerical evaluations of the cumulative distributions of Vq and V,, , Grassmann (1 974) has shown that the following expression, which may be obtained either directly or by interchanging the summation in equation (5.2.19), for P(Vq > t ) converges faster than equation (5.2.19): %

The above results for the system MX/M/c have been extended to the system

E?/M/C. For details, see Holman et al. (1980) and Holman (1977).

MULTICHANNEL BULK QUEUES

250

5.2

THE SYSTEM M ~ I M / C

251

The Busy-Period Distribution

The distribution of a busy period for a multi-server queue may be defined in ere we initiate the discussion for the iMx/N/c queue when c = 2. The procedure, of course, runs on the same lines for the case of more than two servers. Two cases need to be distinguished, depending on whether a busy period is considered to be a period during which both servers are continuously busy, or a period during which the two servers are not simultaneously idle.

P Two Servers (at Least One Busy). The length of a busy period in this case may be defined as the interval of time from the instant of arrival of a unit that makes at least one of the servers busy (that is, the initial number, i = I ) to the subsequent instant when both servers become free for the first time. The p.d.f. for the distribution of a busy period is given by dPo(t)/dt.To compute dPo(t)/dt,we have the equations

p","(t)

=

- ( h + 2p)Pn(t) + X

n-1

m=l

ampn-,(t)

+ 2pPn+,( t ) ,

then

The inverse of zF is

Taking the inverse of equation (5.2.25) with the help of equation (5.2.17), we finally get the p.d.f. for the busy period starting with i customers in the f5rm

n >2

Define For the k l k / M / 2 queue with a, = 6,k, since b n , n ( k - l )= 1, b n j = 0, j # n ( k - 1 ) and i = I , we have the p.d.f. Taking the tion, we get

L.T.of the above equations, using Pi@) = 1 and the generating func. F(z;a) =

zi+' + w ( z - 2 ) P , ( a ) ( h + 2p a)z - 2p - hzA(z)'

+

This p.d.f. may be expressed in terms of a function I k ( x ) (defined in Chapter 1). Thus

The denominator of equation (5.2.24) has one zero zo inside the unit circle, lz I = 1. As the numerator of equation (5.2.24) must vanish a t 2 0 , we have t. if k = 1, that is, if arrivals follow a Poisson where r/2 = 2 ~ l ' ( ' + ~ ) pFurthermore, process,

To invert PI(a), we proceed as follows. Let p = h / 2 p , a = 2p/(X + 2p + c), w = A/(A 211 a), g(z) = zA(z), and put z m = f(z). Here m is an arbitrary integer, and f ( z ) is an analytic function whose Lagrange expansion (see Appendix A.6) is given by

+ +

I z=a

If we let

whereIm(r) %[;(I)

is a modified Bessel function of the first kind of index m.

2 Two Servers (Both Busy) The busy period In this case may be defined as the Interval of time from the instant of arrival of a unit that makes both servers busy (that is, z = 2 ) to the subsequent Instant when at least one of the servers becomes free for the first t ~ m e The . p d.f for the duration of a busy perlod is now given by dPl (t)/dt.We compute dP, (t)/dt from the following equations

MULTICHANNEL BULK QUEUES

252

PA'"^)

= -(A

+ 2p)Pn(t) + h

n-2

ampn-,(t) m=1

Define B(z;a) =

1

+ 2pPn+, (t),

n >3

P2(a)zn.

n=2

As in case (I), we now have P(z;a) =

zi+' (h

Each customer receives service for exactly b units of time. The queue discipline is FCFS for the batches, and the system is in statistical equilibrium. Let N(t) be the number of customers in the system at time t . Then the transition probability matrix (t.p.m.) of the process {N(t), t Z 0 ) is given by

- 2pz2F2(a)

+ 2 p + a)z - 2 p - hzA(z).'

The denominator of equation (5.2.27) is the same as that of equation (5.2.15). Therefore the zero inside Iz 1 = 1 and the inversion can be discussed as before. We give below the result.

As before, the results for the k-arrival and single-arrival cases when i = 2 are, respectively, 1: (r) - - - 4p7exp dP1(t) ( ( h + 2p)t) dt and

Although (N(t)) is non-Markovian, if we examine the sequence we see that it is Markovian. In fact (N;, n = 0, 1, 2, . . . ,) is a homogeneous Markov chain with one-step t.p.m. (Pij) = (P(N;+, = jjN;+= i)) given by

Since the input process is a compound Poisson process, the limiting distribution the number in the system encountered by an arbitrary arriving batch is the same as the limiting distribution Pi= lim,,,Pii(t), j = 0, 1, 2, . . . , of the process {N(t), t > 0 ). If Pj exists, so does Pjr = limn,,P (N,* = j IN$ = i}, j = 0, 1 , 2 , . . . , and they are equal. We therefore do not distinguish between probabilities such as P? and Pi, and will use Pj instead of Pj* in the sequel, although our interest is in the limiting distribution of the imbedded Markov chain (5.3.1). The Chapman-Kolmogorov equations for this distribution are

5 of

where r has been defined earlier.

5.3

THE SYSTEM M X I D I c

The Limiting Distribution of

N,*

where dn is the probability that not more than n customers are in the system and is thus given by n

dn =

In the system M ~ / D the / ~ input , scheme is the same as discussed in Sections 5.1 and 5.2. Once more, let X ( t ) be the number of customers that arrive during ] 0, t] . Then if nn(t) = P(X(t) = n), it can be shown (see Chapter 2) that the p.g.f. of .irn(t)is n(z, t ) = C s , nn(t) zn = exp [tP(z)] ,where P(z) = X(A(z) - 1) and A(z) = C& a m z m . Further, we shall write p i ~ ( ' ) ( 1 )= hCfmam and p2 = hCf m2am, that is, p i and y2 are the mean and the variance of the total number of arrivals per unit time. The probabilities nn(t) can be computed through the recurrence relation

C

P,.

(5.3.3)

m =O

Equations (5.3 2) may be paraphrased by writing P(n In system at end of an interval of length b) = P(queue empty at beginning of mterval) P(n customers arrive durlng interval) + CE", P(m customers In system at beginning of mterval) P(n - m + c customers arrive durmg mterval) Here we use the fact that during any interval of length equal to the constant service time b, all customers in service at the beg~nnmg of the mterval, and only those customers, complete service and leave the system. As

MULTICHANNEL BULK QUEUES

254

usual, we assume that p = p, blc < 1 so that the limiting distribution exists. Consequently, we need to solve the set of equations (5.3.2) for the limiting probabilities P,. To do this, we introduce the p.g.f. P(z) = Z,,Pnzn. Multiplying both sides of equation (5.3.2) by zn and summing over n, we have, after simplification,

where g(z) = Z,C=,Pnzn. Finally, the p.g.f. of the sequence {P,, n = 0, 1 , 2 , . . .)is given by

Since (P,)are probabilities, the function P(z) is regular in z < 1 and, therefore, the zeros of the denominator within and o n the unit circle must coincide with those of the numerator. Now for Izi = 1 6 , where 6 is sufficiently small. and positive, we have

Probability o f No Service Delay

The probability that a batch on arrival does not have t o wait for getting service started on its first member is equivalent to the probability dc-, that the batch finds at most c - 1 servers busy. In computing dc-, it is convenient t o introduce the generating function D ( z ) = C;==,dnzn.Since dn -dn-, = Pn, n = 1 , 2 , 3 , . . . , by equation (5.3.3) we get (1 - z ) D ( z ) = P ( z ) , and consequently by making use of equation (5.3.5) we have

The coefficient of zC-' in D ( z ) is the desired probability

+

=

exp [bpi 6

where we have used the fact that bpi

+ o (6)]

< c . Consequently, by RouchC's

,

whence In dc- = In ( c - p, b ) - ZF-' In ( 1 - zi).Using the generalized argument principle (see Appendix A.4), one can get an expression for In dc-, independent of the roots zi. To do this, let @(z)= ebo@)- - z C , and note that @(z)has simple zeros at z = z l , z 2 , . . . , z,-, , and no pole inside the circle D , iz I = 1 - E . Here E > 0 is chosen in such a way that all the roots zi, i = 1 , 2 , . . . , c - 1, lie inside D and zc(= 1) is outside D. Choose ~ ( z as) the principal branch of In (1 - z ) which is analytic inside and on D. The generalized argument principle yields

theorem, the

has exactly c roots within the region lz 1 < 1 + 6 . Let these roots be denoted by z l r z 2 , . . . , zc-, ,zc(= 1). It can be shown that these roots are all distinct. For note that zc = 1 is simple because 1 - z; e- b m c ) lim = bp, -c#O. ZC+l zc-1 To show that the remaining roots zi, i = 1 , 2 , . . . , c - I , are distinct, first show that these roots are within the unit circle, ) z I = 1. Suppose that lzil =1 for some i = 1, 2, . . . , c - 1 . Then from the equation giving the roots, lexp [bO(zijl I = 1, which in turn implies that the real part of b@(zi)must be zero, that is, Re[P(zi)] = 0. Thus one should have Re[P(zi)] = - Re[XZ:ak(l -z!)] = 0. Since all terms within the sum are nonnegative, one should have Re[l -zF] = 0, for all k, and therefore zi = 1. This is contrary to our assumption that the root z = 1 is simple. It then follows that lzil < 1, i = 1 , 2 , . . . ,c - 1. Since the numerator of equation (5.3.4) is a polynomial of degree c , it can be replaced by A(z - l)(z - z l ) . . . (z -z,-,), where the constant A is determined by P(l -) = 1 . Thus

It will now be shown that

It is easy to see that the principal branch of (1 -z) In [zC-' (1 - z)] is analytic in lzl< 1 - E , and consequently its integral on D is zero by Cauchy's integral theorem. Hence we get

Integration by parts gives

MULTICHANNEL BULK QUEUES

As a simple example, consider the case of single arrivals, in which case a t = 1, a, = 0 , r f l.Then Now we want to change the contour D to D , , jzl= 1 + 6 , where 6 > 0 is chosen in such a way that the only zeros of @(z) in lz I < 1 + 6 are 1, z l , z 2 , . . . ,z,..~. The integrand above has a pole z = 1 which is inside D,but outside D. Since the residue of the integrand above at the pole z = 1 is ln(c - bpi), we have

The Waiting-Time (in Queue) Distribution

To obtain the limiting distribution of Vq(t), define the probabilities where we have used the fact that the integral of [ln(l -z-l)]/(z1) on D I vanishes. This fact is easy t o establish. For as Izl-' < 1 on D l , the power series for In [ I - z-" converges uniformly on D l , and termwise integration is allowed, so that

which is seen t o be zero by calculating the residues at the poles z = 1 (order 1) and z = 0 (order n) and invoking the residue theorem (see Appendix A.3). Consequently, the value of J is as stated above. Once again as /z-Cebp(z)(< 1 on D l , by using the argument that the power series expansion for In 11 - z-'e bPe)] converges uniformly on D l . and termwise integration is permitted, we find

gm(t) = P (among the customers present at some time t o , at most m of them will remain in the system at timet+to,OO) and the p.g.f. G(z; t) = 2~=,g,(r)zm. Considering the state corresponding t o gmc+c-I (t), it can be seen that, at most, mc + c - 1 of the customers preceding the given batch will be in the system at time t later, and thus, at most c - 1 of them at time mb + t later. This is the condition for the waiting time (time until a first customer from the batch enters service) t o be less than mb t, that is,

+

P(Vq < mb + f) = gmc+c-1(t). Now in the limiting case we have the relation

which gives Now the integrand has poles at z = 0 (order nc) and at z = 1 (order 1). Clearly the residue at z = 1 is 1. The residue at z = 0 is the coefficient of z-' in the expansion of -ebnp(z)~-nc(1 z)-l as a power series in z, that is, it is the coefficient of z-' in ebnfl")z-(nc-~-')z-l . But the coefficient of znc-j-' in ebnP@),by definition, being nnc-j-l (nb), with j = 0, 1.2, . . . ,nc - 1, we have finally, by using the residue theorem, ' , r nc-1 1

which is the required result. Since the probability of n o delay = dc-,

:

By using equations (5.3.3), we have, as before,

(I --z)D(z)

= P(z).

(5.3.7)

Substituting the values of D(z) and P(z) in equation (5.3.6), we obtain

One can get the explicit probabilities, P(V, < bm + t), by first obtaining a Laurent series expansion of G(z, r) in the annulus 1 < lz I < IqI, where q is that root of zC exp [- bp(z)] = 1 which has the smallest modulus exceeding unity. It is easy t o see that such an q exists. For let us consider the functionxC exp [- bp(x)] I 5 f(x) (x = real part of z). Then as f(1 + 6 ) = 1 + c(1 - p)6 + o(6) > 0 (since p < 1) and f ( m ) = - 1, the function f(x) which is continuous must cross the x axis between 1 + 6 and -. This proves that there exists at least one real root of f(x) = 0, and consequently at least one root of zC expi- bp(z)] = 1, outside the circle lz I = 1 6 , where 6 is sufficiently small and >O. Select q such that q has the minimum modulus, that is, q is the next pole (in order of magnitude) of the expression

+

258

ULTlCHANNEL BULK QUEUES

for G(z; t ) , the first pole being at z = 1. nere we have used the fact that the numerator of G(z; t ) does not vanish at q for p < 1. For 1 < lz < I?], the absolute value of z-' exp [bp(z)]is less than 1. (Consider iz I = 1 + 6 and choose 6 in such a way that 1 + 6 < 1.) Expanding the denominator of equation (5.3.8) in powers of z-' exp [bp(z)],the exponential function in powers of z , and collecting like powers of z , we get for the annulus 1 < lzl< IqI

2, . . ., where P(uL+, - 5; = 1 ) = 1, for every n and 5; = 0. Since the interarrival times are constant and equal to 1, we have o; = n . The service times are independentIy exponentially distributed with mean 1/p. While in other models we have considered the number of customers in the system just before arrival instants, here we follow Cox and HinMey (1970) in an analysis of the imbedded Markov chain {N,} formed by considering the numbers of customers present immediately after (potential) arrival instants. Let the number of customers arriving at time n be A,, having a binomial distribution of index s and parameter p. Further, let N ( t ) be the number of customers present in the system at time t and let Nn = N(oA 0 ) . Also let Y , be the number of possible service completions during a unit interarrival interval from n t o n 1. Clearly the distribution of arrival groups is given by

+

where gn is the coefficient of zn in the polynomial (pib - c)~;:: [(z - zi)/(l z i ) ] .Since z = 1 is the only singularity of G(z;t ) in the circular region lz I < IqI (a simple pole with residue -I), G(z;t ) (z - I)-' is analytic in / zI < lql, and hence for lzl < / ? I , G(z;t ) + (z - I)-' = expansion (5.3.9).

+

+

where E(Ai) = sp, var(Ai) = spq, and

Therefore, for \z I < I , we must have

1

G(z;t ) = expansion (5.3.9) + 1 -z = expansion (5.3.9) +

rn

2 zn n=O

From this by picking the coefficient of zmC+C-', we get

Now because the distribution of service times is exponential, the distribution of Y,, given N, = I , is independent of the values of N,-, , Nn-,, . . . , and hence {N, ) forms a Markov chain. Let N , converge t o N in distribution as n + -. To find the limiting distribution of N which exists subject t o p = splw < 1 , we need first to find the transition probability matrix of N n . To get this, consider first the distribution of Y n ,given Nn = I. For this, we have e-CP(w)"m

The results of this section are due to Kuczura (1973),although we have corrected some minor errors in the original paper. The corresponding results for the single Poisson arrival case M/D/c have been discussed by Erlang (1920), Pollaczek (1930a, b) and more elegantly by Crommelin (1932, 1934), among others.

E SYSTEM DSIMIc The Limiting Distribution of N, In this section we consider a multiserver queuing system in which customers are scheduled to arrive in groups of exact size s at unit time intervals. There is an independent constant probability, q = l - p , that a customer scheduled t o arrive in fact does not d o so. In application, servers may be judges, customers cases. AII cases are scheduled t o start at the beginning of a day, and nonarrival corresponds t o a case being settled out of court shortly before the time set for its hearing. Suppose that s customers are scheduled t o arrive at each instant o;, n = 0 , 1,

qtm

P(N, - Y,

I,m>c

(1 - m)! '

= m IN, = I ) =

)

e

-e

m

,

(5.4.3) /, m < c

\

because in the first case c servers are working throughout the unit time period, whereas in the second case each of the I customers has independently a chance y of service completion where y = /; pe-Pf dt = 1 - e-p. This leaves the case m < c < I. For this it may be noted that the time x measured from n until the number of customers first falls from 1 t o c has the probability f(x)dx, where f ( x ) is the Erlang p.d.f. given by cp ( C ~ X ) ' - ~ - ' e-'pX (I-c-I)! . f(x) = Also, if x < 1 , the probability that in the time 1 - x , measured from x t o the end of the interval, further c - m services are completed is

):(

- e - ~ ~ l - xc) me-cbcl-x). 1-

MULTICHANNEL BULK QUEUES

260

Thus

P(N, - Y , = m IN,, = I ) =

The results of this section are due t o Cox and Hinktey (1970) who have considered approximations to the present model, numerical calculations, and further generalizations t o situations wherein not all customers are scheduled.

J ~ ( l - - e - P ( ~ - ~ ) ) ~ - ~m d< x c, < l . (5.4.4)

Besides, it may be noted that qtm = 0 if I < m. Now the entries Pij of the t.p.m. P may be obtained by using equations (5.4.1) and are given by

E SYSTEM MINIBIc The Transient Behavior

where we have used the independence of A,+, and (N,- Y,). With P determined from equations (5.4.1), (5.4.3), (5.4.4) (or its equivalent form - see Problems and Complements) and (5.4.5), the limiting distribution nj = limn,, P(N, = j ) can, in principle, be found from the relation- =-B, where rr is the row vector {rro,rrl, -7.3..

.I.

This section deals with a queuing system having c homogeneous servers, each with maximum capacity B. When one of the servers is available, then: 1

2

Measures of Delay

3

As an index of delay to customers, we may define the relative measure

which is the ratio of the expected number delayed to the expected number of arrivals. As in many practical situations, a customer delayed at time n stands a good chance of starting (and, possibly, even of finishing) service before n 1, whereas a delay beyond n + may be regarded as serious. An analogue of equation (5.4.6), calculated at n 4 instead of n , is a useful index and may be evaluated thus. Given N , = I > c, the probability that at time n there are m > c customers present,

+

+

4

+4

If number in queue > B > 1, then the first B customers enter service immediately If 0
Arrivals follow a Poisson distribution with mean rate X, batch service time is exponentially distributed with rate p, and FCFS queue discipline is observed. We first discuss the transient probabilities of the number of customers in the queue and the number of busy channels. Let N,(t) and Lb(t) be the number of customers in the queue and the number of busy servers at any time t , respectively. Let us define Pm,,(t) = P ( N J t ) = m , L d t ) = 2 ) .

It can be easily seen that these probabilities are nonzero for 0 G I < c, nz = 0 and I = c , 0 < m < -. The state equations for the system are:

Accordingly, the half-period analogue of equation (5.4.6) is

where bh = exp (-- lcp)

(4

--h!

Given p and p, feasible combinations of c and s may be found to satisfy tolerance and p , that is, the feasible number of appointments per unit time limits on cPo, for a given number of servers may be determined.

We solve this set of equations under the initial condition Po+,(0)= 1, that is, at time zero all servers are busy and no customer is waiting t o be served. Define P(z; r ) =

-

C pk,,@)zk 0

(5.5.5)

Cal tu ".

R

V

II

9" 1. tu

MULTICHANNEL BULK QUEUES

5.6

THE SYSTEM G I I M ~ I C

267

Hence combining the two cases, we see that the probability that the time spent by an arriving customer in the system is greater than t , is given by

P (system time > t ) -

=

-

I. 1

-Po,,

Now the expression for the probability that a customer joins the nth batch awaiting service and is still awaiting service at time t later is B n-1 kB - I (c"'

1 n2

] [

andn = Po.,

( k - I)@-'

r=o

r!

The probability that a customer arriving at any time will be delayed in the queue for a time greater than t , is obtained by summing a n d n over n. Thus

The joint distribution of the number of customers in the queue and the number of busy servers was found by Ghare (1968), but the various other distributions and results are due t o Cromie (1 974). * An elaborate discussion on the computational performance of the results presented in this section is available in Cromie (1974) and Cromie and Chaudhry (1976). Cromie has drawn curves for L, or X W,, tables of a root k of the characteristic equation (5.5.23) for various values of the parameters p and B, tables of state probabilities PO,,for various combinations of B, p and c, and so on.

E SYSTEM GIIMBI c where we have used equation (5.5.23) and p = h/c@. An alternative derivation of the p.d.f. of V q , following the procedure discussed in Section 5.6, is much simpler. For details, see that section. aiting-Time (in System) Distri

Since the p.d.f. of the time spent in the queue can be seen to be g ( t ) = Po,, the probability that a random customer enters service with his, her or its batch at a time lying between y and y dy after its arrival is given by g ( y ) d y . Further, if the batch starts service at time y , then the probability that the batch will complete service at a time between x and x dx after a random customer in the batch entered the queue is p e-@(X-Y)dx, x >y . Thus the probability that a customer who joins the queue at time t = 0 will complete service (with his or her batch) at a time . probability that lying between x and x dx later is J g g ( y ) d y p e - @ ( x - Y ) d xThe the customer under consideration will not yet have completed service at an elapsed . this does not time t after entering the queue is JTdx J g g ( y ) p e - @ ( x - y ) d yBut include the probability of the event that a customer enters service without waiting, which is C- I k

+

+

+

z

i=o

P0,i

= 1 -po,c

ri

The probability that such a customer has not completed service at a time t is just

e-pt .

The Limiting Distribution of N;

The M/M*/C queue in continuous time, a particular case of the system G I / M ~ / C , was discussed in the last section using the Erlangian procedure. In the present section we discuss G I / M ~ / cin full generality, but in discrete time, through the imbedded Markov chain technique. In G I / M ~ / Ccustomers , arrive at the sequence of epochs 0 = ah, a ; , . . . , a; such that the interarrival times ,,a: -a; > 0,n = 0,1, 2 , . . . , are i.i.d. r.v.'s with the common distribution function A(u) and mean I / h , 0 < h < m. The service times and the mode of servicing are as discussed in the last section, with the mean service time 1/p. It is convenient t o define the state of the system at epoch t asN(t), which now is taken t o mean the sum of the number of customers waiting in the queue and the number of groups of customers being served. Then N;; = N(a; - 0)is the state of the system just before the arrival of a customer at epoch a;. N-, = i may be taken to indicate ( I ) if i > c , that c servers are busy and i - c customers are waiting in the queue; and (2) if i < c , that i of the c servers are rendering service to i groups of customers, with no customers waiting in the queue. To get the one-step transition probability matrix P = [ P i j ] = [P(Nn+, = j IN-, = i)], i, j > 0, we use the notation introduced by Kendall (1953) in his study of the GI/M/C queue, which simplifies many things. We have the following notation: [ I lm; u ] is defined as the probability that I servers complete service during an interarrival period of duration u , given that rn servers were busy at the beginning of the period, where I < m < c. The probability that 2 independent exponential

.V1

.-X1

:

+4

a,

5 rr 0

2 a, +

a,

.-+

E"

8 8 t4

/-, U -

0

V

/I

5T On-

%

2 2

ULTlCHANMEL B U L K QUEUES

270 Table 1:

Matrix P of transition probabilities for ~ 0

1

2

1

1

~

3

~

1

~ 4

...

c - l

c

X =

where

{ x O , X I , X ~ ,

. . .I,

such that

-

C i=O

Ixil<m

x = xP.

(5.6.8)

It can be seen that the coefficients of the variables in the kth, k > c are the same. Thus we may write xk

=

+

+

+ ... ,

bO~k-l bl~B+h-lbZ~zB+k-l

+ 1 equation

k > f~1.

(5.6.9)

In order to solve the difference equation (5.6.9), let with

Xk

=

ok-c

,

k2c

+ 1.

Substituting equation (5.6.10) reduces equation (5.6.9) to

=

In order to prove the ergodic property, it is sufficient to prove that there exists a nonzero vector

~ ( d )

(5.6.1 I)

which is equation (5.6.3). It can now be shown, by branching process theory (see Appendix A.2) or otherwise, that equation (5.6.1 1) has a real root in ] 0, 1 [, i f f Bpc/h> 1 . It follows that this real root is the unique root of equation (5.6.11)

MULTICHANNEL BULK QUEUES

272

inside the unit circle, and hence that the ~ a r k o chain v is ergodic iff p = h/wB< 1. It can also be shown by substituting equation (5.6.10) in the (c 2)th equation (5.6.8) that x , = I so that equation (5.6.10) holds even for k = c. In order that equation (5.6.10) hold for all values of k >, 0 , x k = wk-C must satisfy the remaining first (c 1) equations of the set of equations (5.6.8). Substituting equation (5.6.10) into the equations from the second t o the (c 1)th we obtain c equations with c unknowns, x o , x l , x z , . . . , x e l . Since the determinant of the coefficients is [O 1 1] . [O 121 . . . [ 0lc - I ] . [Olc] f 0 , these equations are linearly independent, and consequently we find xi = pi, 0 < i < c. Since P is the t.p.m., the row sum is unity, so that the x = { p o ,p l , . . ., p c - l , 1 , w , w 2 ,. . .) obtained in this way will obviously satisfy the first equation of (5.6.8) where not only x f 0 , but also CIxil <m. Finally, we have t o prove equation (5.6.4) for p,, 0 < r < c , which may be done as follows. Define

+

+

+

where we have used xi = mi-',i > c . Differentiating equation (5.6.12) r times and letting z = 1, we have for r = 1 , 2 , . . . ,c - 1 that

where we have used equation (5.6.3), or

This is a set of first-order linear difference equations with variable coefficients, which can be solved (see Appendix A.9) by letting

C

U(z) =

C uj (z - 1)j

i=O

so that

j = 0 , 1 , 2,...,c

,

u.=_I ! dz'

'

Dividing both sides of equation (5.6.13) by C, and then summing over r from j to c , and using that Uc = x c = 1, we have

and let

Clearly, Uc = x c = 1 . This fact is used in the sequel. Now b y using equation (5.6.8), we can see that

j = 0 , 1 , 2, . . . , C - 1

By use of the identity

i = 0 , 1 , 2, . . . , c - I which can easily be proved by using the relation ( e - ~ v- e

-

- ~) uc j

dv d A ( u )

+

where k(i) = [(i - c B)+/B]when i >c , and [ x ] is the greatest integer contained in x . This, after simplification, gives the following integral equation:

and after some simplification, we obtain equation (5.6.5). Since

jo(1 - e-Gu + ze-pu) U ( l - e-pY + re-Pu) dA(u) m

U(z) =

+ -

(85'wi) 16

[J':pCecpvYB

i=o w

(I

(e-wY - e-pY

- e-pU + Z ~ - P ~ )&(u) ~+

+ ie-wu)c

by inversion we have

+1

5.7

MULTICHANNEL BULK QUEUES

274

(see Appendix A.10). This determines po, p l , . . . , ye-, , that is, equation (5.6.4). Thus we have completely determined the vector x, the normalization of which gives equation (5.6.1) with the normalization constant A given by equation (5.6.2). For an explicit form o f A , see Problems and Complements. An alternative procedure for getting the values of x,, r = 0, 1 , 2 , . . . ,c - I , which is based on solving the linear equations containing x,, is much simpler, and may be used for computational purposes. For details, see Problem 36.

OVERVIEW

275

The limiting behavior of {NJ for the system G1/MB/c when B = 1 has been discussed by Kendall (1953), of { N i } and { N ( t ) )by Takics (1962). The transient behavior of {N(t)) has been discussed by Wu Fang (1962) and of {N;} and ( N ( r ) ) by Xu Guang-Hui (1965).

aiting-Time (in Queue) Distribution

Suppose that the service discipline is FCFS, and let VP) be the waiting time in the queue of the n t h arrival. Let V'$) converge to Vq in distribution as n +m. Then the distribution of Vq is given by c-1

P(VP = 0) = A

x pi

(5.6.14)

i=O

p c exp I-- cpx (1 - wB)],

x > 0.

It is easy to see the result of the first equation of (5.6.14). To prove the second equation, we first recall k(i) = [(i - c f B)+/B] , which may be interpreted as an r.v. representing the number of groups awaiting service (including the arrival that has just joined). A customer joining the k(i)th batch must wait for k(i) service completions before his batch enters service. Thus under the condition that the system state is i 2 c, that is, k(j) > 0, the conditional density function for k(i) service completions is

and so the p.d.f. of waiting time in the queue is

The contributions of some researchers in multiserver bulk queues, whose work could not be included in the main text, for reasons given in other chapters, are mentioned here. In recent years, as we have stated earlier, Marcel Neuts has been advocating the use of matrix methods in queues. In the system GI~/M/C with bulk arrivals Neuts (1979) shows that the distribution of the number in the system is matrix-geometric, provided the group size cannot exceed a certain limit k. He also investigates the limiting distribution of the number in the system, both at a random epoch and immediately preceding an arrival epoch, and the limiting waitingtime distribution. Medhi and Borthakur (1972) discuss the steady-state distribution of the number in the system for the two-server q u e u e ~ / M k 3 B /using 2 Tauberian arguments. Later Medhi (1979) investigated the waiting-time distribution for the c-server queue M / M ~ * ~ /Numerical c. values of mean and variance are also discussed for the cases c = 1 and c = 2. Related work has been done by Neuts and Nadarajan (1980) on M / M ~ , ~ /by C algorithmic methods, and by Cosmetatos (1983) and by Sim and " / analytic c methods. Templeton (1983) o n M I ~ ~ ~ by In the queuing system M X / ~ j cwith bulk arrivals, Polyak (1968) investigates, among other things, the distributions of queue length and waiting time, and some inequalities. Other contributors who have done some work on bulk-arrival or bulkservice queues with a finite or infinite number of servers include Baily and Neuts (1981), Pearce (1965), Murari (1968), and Sharda (1969, 1970, 1981).

D COMPLEMENTS Section 5.1

The results of this section are due to Shyu (1960). Shyu (1964) has also given waiting-time distributions for random order of service and for last-come first-served. This author's name may be found with at least three spellings. Xu Guang-Hui, sii Kuang-Hui, and Shyu Kwang-Huei all occur in the English-language literature, and all refer to the same author. Roes (1966) has also discussed a system of the h;l/MY/c type using the theory of derived Markov chains due to Cohen (1969). Furthermore, the limiting behavior of ( N i )in Ek/Mr/c has been discussed by Love (1970) using the theory of semi-Markov processes which had previously been applied to M/Gr/l by Fabens (1961) and Fabens and Perera (1963). In both cases exactly r customers are always served together.

1

Using equation (5.1.9), show that P,(t) is given by

2

Using the transformation

MULTICHANNEL BULK QUEUES

276

PROBLEMS AND COMPLEMENTS

show that P(z; t ) of equation (5.1.9) asslmes the more convenient form

P ( Z ;t ) = g(v) exp where

[- J:

(1 -A(u))

x (t x

) dx

6

The result ( 1 ) is due to Sahbazov (1962) and equation ( 2 )is due t o Reynolds (1968). Continuation: limiting distribution of N ( t ) in MX/M/m with X(t) = h and t -+ -. Show from equation ( 1 ) or ( 2 ) of Problem 5 that

v = l - ( l - ~ ) e - ~ ~ , u=l-(1-z)e-'.LX.

P ( z ) = limP(z;t) = e r p t+-

Show further that in this case

Continuation. From Problem 2 derive the conditional mean number of busy servers for a given initial distribution {Pn(0)j and show that it is given by

3

4

7

Letting X(t) = X = constant, show from equation (5.1.9) that

P ( Z ;i) = g(u) exp

C

- -:j X ( U )du

[

This result confirms the existence of an equilibrium distribution independent of the initial state of the queue. Continuation. Show that P(z) of Problem 6 may be put in the form

where k = JA X ( u ) d u , and @(z)= k-' J ; X ( u ) du. Equation ( 4 ) shows that the limiting distribution of the number of busy servers is a compound Poisson distribution. Solution: To prove equation (4), we can write

Consider the following numerical illustration of the system discussed in Section 5.1. In the time interval [ 0 , TI let all the hypotheses assumed in that section be satisfied. To further simplify the analysis, assume that each of the arriving groups contains two customers so that a? = 1, am = 0 , rn f 2 , and that the arrival rate of the groups is represented by X ( t ) = eat where a is some arbitrary parameter, and that at the initial instant the system was empty. Show that under such assumptions the probability that the system is empty at time t is given by

and that the mean number of busy channels is given by

5

8

it follows that @(z)is the p.g.f. of the distribution pr = X,, /kr, r = 1 , 2 , 3 , . . . , whence equation ( 4 ) follows from equations (3) and (5). In Problem 6 , if X follows a geometric distribution with a, =am-' (1 -a), m > 1 , 0
Section 5.2

9

If a, = 6,,

show that in the limiting case Po is given by

Further, if the initial conditions are Pn(0) = Sni, then show that

where p = e-Pt, q = 1 - p number of busy channels is

Then using equation ( 2 ) , show that the mean

L ( t ) = E [N(t)IN(O) = i] = Further, if t

-+

-, then

277

h

( 1 - e-'lt) X ( 1 ) + N ( o ) ~ - ~ . i-1

X L = limL(t) = -X(l). t+i-1

a well-known result for the systemM/M/c. Hint: Use equations (5 2.12)

a?'

II

MULTICHANNEL BULK QUEUES

PROBLEMS AND COMPLEMENTS

P C

(iii) (a) un = an -----~ ( -PI' 1

where = @/a,and

Continuation.

If in Problem 12 we have c = 1, show that

This shows that the distribution of Pn has a probability mass concentration at n = 0 , and is otherwise a geometric distribution expect for a multiplicative constant p. Hence the truncated distribution P(N = n IN > 0), n = 1 , 2 , . . . , is

(0) p(v,

> t)

P

- ') - p)

+

= PC (1 -a)(l

exp [- cp(1- a ) ( l - p)tI ,

+

(d)

Observe that E(Vq) = E(Vql) E(Vq2),as it should. Also note that the truncated distribution P(Vql > t / Vql > 0) is exponential, as are the truncated distributions of Vq2, Vq . Show that the distribution of waiting time in the system (queue + service) for the first customer of a group, where g ( y ) = p.d.f. of 'V,, , is given by m

P(system time > t ) = [ l -P(Vql > O)] e-@

+

geometric with parameter ( p + a ( l - p)). The assumption of geometric group size is as restrictive as the exponential assumption for continuous time models. Nevertheless, both these distributions serve as useful first approximations, even in situations where the application of these distributions may seem unrealistic. Once these exact results are known, computations are easier. The computational results for other group size distributions may be compared against those for the geometric distribution. One may then see how the two differ for various parameters. For some work in this connection, see Cromie et al. (1979). If a, = amk in AfX/Af/c, show that the limiting probabilities Pn for k > c satisfy the following difference equations:

X

lXZt

Jy=o

g ( y ) dy

x pe-p(X-Y) dx.

where B = X/p. Solving this set recursively, show that

Hence show that if the group size distribution is geometric, thenP(system time > t ) is given by

where r ( - )is the gamma function. Show that the p.g.f. corresponding to equation (5.2.13) reduces to Use the arguments of Section 5.5. 12 Continuation of Problem 1 1. If in Problem 1l(a) a f 0, show that the results may be expressed in terms of gamma functions as follows: Hint:

B ANNEL BULK QUEUES

282

from which, by using P(1) = 1 (and the results on gamma functions discussed in Chapter I), find Po = c! r(B t 2)(1 - p ) / r ( e 4- c + I), where p = Xklcp < 1. Prove further that for the system M ~ / M / c where , k 2 c, the mean number of customers L in the system is given by

5 Numerical question. Let us consider the operation of receiving and checking luggage for passengers arriving at an airport by bus. Suppose that statistical analysis has demonstrated sufficient correspondence between the theoretical model and empirical data (exponential interarrival times and service times). Suppose further that two types of buses transport passengers t o the airport type 1 carrying 1 0 customers each and type 2 carrying 35 each. On the average, 8 buses arrive per hour so that A = 8 per hour, where the probability of an arrival of a bus of type 1 is 0.75 and that of type 2 is 0.25. Thus A(z) = 0 . 7 5 ~ " 0 . 2 5 ~Suppose ~ ~ . that c = 3 with p = 6 0 per hour for each server. Under these conditions it is easy t o verify that p = 13/18 < 1. Show that P,, n = 0, 1 , 2 , . . . , satisfy

+

11

1

PROBLEMS AND COMPLEMENTS

283

and after normalization [Kabak (1970).] Note that we could have obtained equation (6) by using equation (5.2.18). In applications of queuing theory to telephone systems, two concepts loss system and delay system - are frequently used. Systems which have been considered so far are delay systems. In a loss system, n o queue is permitted, so that in a loss system with c channels, no more than c customers can join the system. If c channels are already busy when a customer arrives, he leaves without service and is said to be lost to the system. Some probabilities concerning these two systems are discussed in the problems that follow. For va$ous other concepts in telephone congestion theory, we refer the reader t o Riordan (1 962). 17 Verify that equations (6), (7) and (8) of Problem 16 hold for n = 1 , 2 , 3 , . . . , u , where u = c refers to the loss system defined above (which may be denoted by MX/A4/c/c) and u = refers to the delay system discussed in Problem 16. Note that in the loss system Po is given by

-

[Kabak (1970).]

Prove that, on the average, the servicing apparatus will be free one-fourth of the operating time. Prove also that the p.g.f. of P,, n = 0 , 1, 2, . . . ,is (approximately) - 6z3 - 14z2 - 2052 + 225 P(z) = 10(z 36 32 - 942 90)

+

+

where we have used approximations in computing the coefficients. Show that L, = 39 customers. If c = 4, then show that L , = 12, but Po increases t o 0.37. 6 This problem is concerned with an alternative derivation of the probabilities Pn for the system I@/M/C without the use of a generating function. Solving recursively the set (5.2.1) for the first few Pn in terms of Po and noting that A , = 1, show that the probabilities Pn are connected by the relations

18 In the case of the system MX/Pd/c (limiting case), when a , = ?irnh, there are two interesting cases for which explicit results can be obtained. (a) Show that if a, = 6 &,, then

(b)

nt: (a)

(b)

where y(n) = X/p(n), p(n) = p min (n, c) andAj = Z", j a m . Letting G(0) = 1 and defining n-1 n = 1,2,. . . , (7) G(n> = y(n> An-k G(k), k=0

show that

If a, = b m k , where k 2 c , then

This may be proved on lines similar to the ones which lead to the derivation of equation (5.2.1 6). For this case the following method is more elegant. If a group of size k ( 2 c ) arrives t o find the state of the system n, O < n d c , then k -c n arrivals are blocked. However, if it arrives to find all the servers busy, then all the k arrivals are blocked. Consequently the conditional probability of V , > 0 , given that the state of the system is n, is

+

MULTICHANNEL BULK QUEUES

PROBLEMS AND COMPLEMENTS

Thus the unconditional probability is given by

Since the last expression within brackets is easily seen to be the average number of busy servers, it is equal to Xk/p and hence

P(Vq > 0 ) = (c)

k-c k

X

-+ -.

F

In general show that the result (5.2.16) can be simplified to

which is easy to evaluate numerically. Hint: This may be done recursively by taking c = 1, 2 , . . . and using equations (5.2.14) and (5.2.1) in the limiting case. In view of this value of P(Vq > 0). one may conclude that equation (5.2.19) holds even for t =0. For an independent proof of this and some other interesting results, numerical evaluation of certain measures of efficiency, and so on, see Cromie et al. (1979). 19 (a)

Show that the p.g.f. of the sequence {u,, n > 0 ) , defined by equation (5.2.22), is given by

where R ( z ) , the p.g.f. of the sequence {r,, n 2 I ) , is given by R ( z ) = z(1 - A(zjj/(l - 2 ) ~ .

(b)

By using U(z), verify equation (5.2.23), and in particular show that

E l (SP) = E2 (SP) =

25

Typically for the loss system Mk/M/c/c, one can see that theiirst two expressions together with Z,C=,Pn = 1 must hold. In the case of the loss system M k / M / c / cwith k > c , show that the solution is given by

+

where Po = c ! /l'IF=I [ ( h / p ) i] . In particular, show that the probability that a group on arrival finds the system full is h / ( h + cp) = 0/(8 c).

+

21 Continuation of Problem 20: the loss system M k / ~ / c / with c k >c. that in this svstem

Show

+

nt: Use arguments of Problem 18 and Z,C=,nPn = h c / ( A p), which can be obtained by using the first expression for Pn given in Problem 20. For k = 1, the loss system M/M/c/c is known as Erlang's loss system, and the delay system M/M/c/m is known as Erlang's delay system. These systems have been completely analyzed; see, for example, Parzen (1962).

22 The distribution of NO), the number in the system for M X / ~ / 2 ( p !1 J, ~ ) . Assume that an arriving customer who finds the system empty goes to the first server with rate p, and probability p and the second server with rate p2 and probability q , where p + q = 1. Define P, (1, 0 , tj, Pl ( 0 , 1, t ) as the probability that one unit is in the system and that it is with the first server or second server, respectively. Let PI ( t )= P, (1,O, t ) + P I ( 0 , 1, t ) , and let Pn(t) be the probability that n units are in the system at time t , n > 0 . Show that the L.T. of the p.g.f. forPn(t) is given by

= ~ ( ~ ~ - 1, ( 1 ) [Burke (1975)]

A y l )

a-

1

- -Po. 25 a

20 Continuation of Problem 16.

If a, = S ,h, then show that

where Pi(Oj = 1. Discuss the evaluation of the constants involved and also the cases (a) a, = Srl and (b) p1 = pl.

MULTICHANNEL BULK QUEUES

286

PROBLEMS AND COMPLEMENTS

287

of d,-, suggests the application of the generalized argument principle with ~ ( z as ) (1 -z)-' and @(z)as before. Then we have, say,

Section 5.3

23 Show that the constants P o , PI, . . .,PC-, in the numerator of P(z) given in equation (5.3.4) are uniquely determined. nt: Write equation (5.3.4) as

where D is the contour of the circle, lz I = 1 - E ,

E

> 0 . Now as

since the integrand has a simple pole at z = 0 , where Integration by parts yields

The condition P ( l -) = 1 gives c-1

Qi = c - p l b .

(9)

i =O

In addition,

The integrand has a pole at z = 1 (order 2). Its residue there is

Equations (9) and (10) determine the constants Qo, . . . , Qc-I uniquely if the determinant of the coefficients of Qo, . . . , Qc-, is nonzero, and that it is so can be easily seen. Finally Q's determine P's.

24 If the group size distribution a k , follows a geometric distribution k > 1 , O < q < 1,thenshowthat

qk-l

(1 - q),

2 e--j=nc

Consequently,

. .,

where D l is the contour of the circle, lzI = 1

+ 6 , S > 0. Now since

hbn

In d,-I

= -

n=l

by considering the expansion of In (1 - l/z) [since 1 l / z grating term by term, we can evaluate the integral

I1

< 1 on D l

and inte-

whcre Ln(x) is the Laguerre polynoniai, some details of which are given in Problem 9, Chapter 2. 25

or M ~ / D / Cin limiting equilibrium, show that the mean numbers in the system and in the queue are given, respectively, by

As discussed in Section 5.3, ~ z - ~ e ~I < o (1~on ) D l , In 1 - Z-C e bp(z)] has a power series expansion in z-Cebp(Z)which is uniformly convergent on D l , and hence term-by-term integration is permitted. Thus the integral becomes

and L, = L - p l b . One can eliminate the roots zi, i = 1 , 2,. . . , c - 1 , from the expressions for L and consequently for L,. The method used while discussing the evaulation

The integrand now has poles at z = 1 (order 2) and z = 0 (order nc) with residues n (bp, - c) and C&-,' (nc - k ) nk (nb), respectively. Finally combining all the results and using the residue theorem (see Appendix A.3), we have

MULTICHANNEL BULK QUEUES

This is easy for computation, but it may be written in a compact form as

where we have used

26 (a)

289

an average customer, he must (on the average) be in the middle of the group. As service times are constant, c customers are departing from the system at a time. (We assume that the first batch of m = k c customers finds an empty system.) Consequently the average queuing time of the customer under consideration arising from the delay due to the members of his group is kc-c b -.--(k-I)b 2 c 2 . Finally his total average queuing time is the sum of this expression and the expression calculated earlier, that is,

Continuation of Problem 24. The value of L, can be used to give W , , the average delay in queue. If the queue discipline is FCFS for the groups but random within the groups, then using Little's formula, we have

When groups arrive in batches of fixed size m, then a form which is easy for computation. Thus

PROBLEMS AND COMPLEMENTS

W, may be put in

If b is added to the above expression, we get the average time the customer will spend in the system. Also if k = 1, then W, = bp/2(1 -p), that is, the average queuing time of an average customer in MC/D/cis the same as the corresponding quantity inM/D/I, as it should be on the basis of intuition, but the average number in the queue in M l D l c will be c times the corresponding number inM/D/l. Section 5.4

27 Show that equation (5.4.4) can be put in the form

(b)

This form with b = 1 was used by Kuczura (1973) to do some numerical work, but he points out that the convergence of the series involved is slow in heavy traffic (when p is near 1). Therefore in the interest of speedy computations it may be necessary to solve for the roots of the denominator of equation (5.3.4). But as, in general, finding the many roots accurately is difficult, it might be better to find the root with minimum modulus and then apply some approximate procedure, such as one of those suggested in Chapter 1. The same remarks apply to the calculation of delay probability. Since the service times are fixed (= b ) and if groups arrive in batches of fixed size k > 1, then an interesting phenomenon occurs. One can observe that the average queuing time of the first customer of a group in Mh/D (with mean b)/l is the same as the average queuing time in a one-server queuing system with single Poisson arrivals and service time kb, MID (with mean kb)/l, both being equal to kbp/2(1 - p), p = hkb. The statement remains true if the queuing s y s t e m M k /(with ~ mean b)ll is replaced ~ mean b)/c, that is if the batch size is an integer multiple by M k C /(with of the number of channels c, that is, if m = kc. In this case we can get a simple expression for W,, the mean queuing time of the average customer of the arrival group. For as the customer under consideration is

-

e-c@C I-c C.1 c-m-1 m!

I-c-1

(-

1)j

C k=o C ]!(c-m-j)! . j=o

nt: Use the identity discussed in Chapter 1 to transform the integral into a summation. 28 Prove that if p = 1 in DS/M/c, then no = nl = n, = . . . = n,-, = 0. Section 5.5

29 Show that the root k of equation (5.5.23) which lies outside the u n ~ circle t is real and bounded above by m~n(p-l,1 + ( 3 p ) - I ) . 30 In the case B = 1 in M]MB/c, we may define PmVc= P,+,, m 2 0,c 2 0.In

MULTICHANNEL BULK QUEUES

290

the limiting case, show that the state probabilities for M/M/c may be deduced from M/MB/c and are given by

PROBLEMS AND COMPLEMENTS

Section 5.6

nt:

where

and p = X/cp. For independent proofs, see Saaty (1961) and Jackson and Henderson (1 966). In the limiting case of M / M B / c , if N , is an r.v. representing the number of customers in the queue (excluding those in service, if any), then show that the p.g.f. of Nq is given by

291

Use equation (5.6.5) and

36 There is an alternative procedure for finding the values of x o ,x l , . . . , xc-,. This is done by solving the second to ( c 1)th equations of the set of equations (5.6.8) recursively, starting with the value of xc = 1. In particular, show that

+

37 Let N i = ( N - - c)+, where N - is the number in the queue plus the number of busy servers in the limiting case. Then N i represents the number of customers in the queue. Show that the limiting distribution of NG is

var (N,) = E(&) 32 In the limiting case ofM/MB/c,show that the expected number of busy servers

Show also that E ( N ; ) = A w / ( l - u ) ~ . 38 The sytem IM/MB/l. In G/hfB/c,if A ( u ) = 1 - e-", so that the arrival rate is unity and c = 1, show that the vector x is { p o , 1, w , w 2 , . . .), where w is a root of the equation p~B+" ( p 1) w 1 = 0 and po = p. The normalization of the vector gives the probability distribution

+

33 From equation (5.5.27) show that the expectation of the time a customer spends waiting in the queue is given by

Note from Problem 3 1 that Little's formula

+

TAJ, = L q / h stands verified.

34 Define N t o be the number of batches awaiting service and let b, = P(H = n).

where Lq is defined in Problem 3 1. It may be noted that when B = 1, E(H) = Lq , as it should be.

Remark: It is interestmg to see that the same result is obtained for the system M/MB/l as in Section 5.5, although the processes cons~deredin the two sections are different. It may, however, be noted that the state of the system in the present section and in Section 5.5 is defined in the same manner, although the notation used is different. /~ c = 2 , B = I , and A ( u ) = 1 39 The system M/M/2. If in a G I / M ~system, e - U ,then show that the vector x 1s { p o , p l , 1, w , w 2 , . . . 3, where w = p 1/2p

MULTlCHANNEL BULK QUEUES

292

and p o = 1/2p2, p , = 1l p . T h e normaliz'ation of x gives Pl = (1 - p)/(l -I- p) and P; = 2pnP;, n 2 1. The same remark applies here as i n Problem 38. For a general proof o f this for the system MIMIC t h e reader is referred t o T a k l c s (1962).

40 (a)

Show that

ac.

W; =

(b)

(b)

j,*fvq(x)dx

Show t h a t W; m a y also b e obtained from

where

41 (a)

E(Vq) =

[x] is t h e largest integer less than o r equal t o x. G1/MB/1,show t h a t

F o r t h e single-server system

where Z(p) = Je-"" dA(u) and Zi(p(1 - w B ) ) = w. Show t h a t for A(u) = 1 - e-" t h e results in (a) reduce t o t h e results of Problem 38.

REFERENCES Abol'nikov, L.M. (1967). A multichannel queuing system with group arrival of demands. Eng. Cybern. 4,39-48. (1968). A nonstationary queuing problem for a system with an i n f d t e number of channels for a group arrival of requests. Problemy PeredaZi Informacii 4,99-102. English translation in Probl. Inform. Transm. (1968), 4,82-85. (1970). Transient regime in the system MX/M/n with nonordinary entering flow. Eng. Cybern. 5,881-885. a n d R.M. Yasnogoridskiy (1972). Investigation of many channel nonstationary Markov systems with non-ordinary input flow. Eng. Obern. 10,636-642. Burke, P.J. (1975). See Chapter 3. Cohen, J.W. (1969). The single server queue. North-Holland, Amsterdam. Cox, D.R., and D.V. Hinkley (1970). Some propexties of multi-server queues with appoint ments. J. R. Stat. Soc. Ser. A 133, 1-1 3.

-

-

REFERENCES

293

Cromie, M.V.(1974). Measures o f efficiency for the bulk-service queue M/MX/c/- and the bulk-arrival queue M ~ / M / C / - with computing algorithms, tables and charts, etc. M.Sc. thesis, Royal Military College of Canada, Kingston, Ontario. and M. L. Chaudhry (1975). Further discussion of the results for the queuing system MX/M/c. Presented at the joint national meeting of ORSA and TIMS, Las Vegas, November 17-19. (1976). Analytically explicit results for the queuing system M/MX/c with charts and tables for certain measures of efficiency. Oper. Res. Quart. 27, 733-745. and W.K. Grassmann (1979). Further results for the queuing system M X / ~ / cJ.. Oper. Res. Soc. 30, 755-763. Crommelin, C.D. (1932). Delay probability formulae when the holding times are constant. P.O. Elect. Eng. J. 25,41-50. (1934) Delay probability formu1ae.P.O. Elect. Eng. J. 26,266-274. Erlang, A.K. (1920). See The life and works o f A.K. Erlang, (1948) by BrocBmeyer, E., H.L. Halstr6m and A. Jensen, Copenhagen. Fabens, A.J. (1961). The solution of queuing and inventory models by semi-Markov processes. J. R. Stat. Soc. Ser. B 23, 113-127. and A.G.A.D. Perera (1963). A correction to Fabens (1961). J. R. Stat. Soc. Ser. B 25,455-456. Ghare, P.M. (1968). Multichannel queuing system with bulk service. Oper. Res. 16, 189-192. Grassmann. W.K. (1974). The steady state behaviour of the M/E& queue with state dependent arrival rates. INFOR 12, 163-173. Hawkes, A.G. (1965). Time-dependent solution of a priority queue with bulk arrival. Oper. Res. 3,586-595. Holman, D.F. (1977). Some problems in the theory o f bulk queues. M.Sc. thesis, Royal Military College of Canada, Kingston, Ontario. M.L. Chaudhry, and W.K. Grassmann (1980). Some results of the queuing system E~'$M/c. Naval Rex Logist. Quart. 27,217-222. Jackson, R.R.P., and J.C. Henderson (1966). The time-dependent solution t o the many server Poisson queue. Oper. Res. 14, 720-722. Jordan, C. (1965). Calculus offinite differences, 3rd edition. Chelsea, Neu York. Kabak, I.W. (1968). Blocking and delays in M(")/M/c bulk queuing systems. Oper. Res. 16, 830-840. (1970) Blocking and delays in M ( ~ ' / M / Cbulk arrival queuing systems. Manage. Sci, 17,112-115. Kendall, D.G. (1953). Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24, 338-354. Kuczura, A. (1973). Batch input to a multi-server queue with constant service times. Bell Syst. Techn. J. 52, 83-99. Love, R.F. (1970). Steady-state solution of the queuing system E,IM/s with batch service. Oper. Res. 18, 160-171. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. r Aufgabe der Wahrscheinlichkeitstheorie I. Math. 2. 32, 64Pollaczek, F. (1930a). ~ b e eine 100. r Aufgabe der Wahrscheinlichkeitstheorie 11. Math. 2. 32, 729(1930b). ~ b e eine 750. Reynolds, J.F. (1968). Some results for the bulkarrival infinite-server Poisson queue. Oper. Res. 16,186-189.

294

MULTICHANNEL BULK QUEUES

Riordan, J. (1962). Stochastic service systems. Wiley, New York. Roes, P.B.M. (1966). A many server bulk queue. Oper. Res. 14, 1037-1044. Saaty, T.L.(1961). Elements ofqueuing theory with applications. McGraw-Hill, New York. Sahbazov, A.A. (1962). Some problems of queuing theory (in Russian). Author's abstract of dissertation submitted for the degree of Candidate of Phys. Math. Sciences. Shanbhag, D.N. (1966). On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274279. Shyu Kwang-Heui (1960). On the queuing processes in the system GI/M/n with bulk-service. Acta Math. Sinica 10, 182-189. English translation in Chinese Math. (1960) 1, 196204. (1964). See Xu Guang-Hui (1964). Takics, L. (1962). Introduction to the theory of queues. Oxford Univ. Press. New York. Wu Fang (1962). On the queueing process GI/M/n. Sci. Sinica 11, 1169-1 182. Xu Guang-Nui (1964). The waiting time distribution for the queuing processes in the system GI/M/n with bulk-service. Acta Math, Sinica 14, 796-808. English translation in Chinese Math. (1965) 6,195-207. (1965). The transient behaviour of the queuing processes GI/M/n. Acra Math. Sinica. 15,91-120. English translation in Chinese Math. (1965) 6, 393-421.

-

Baily, D.E. and M.F. Neuts (1981). Alorithmic methods for multiserver queues with group arrivals and exponential senices. Eur. J. Oper. Res. 8,184-196. queue. To appear Cosmetatos, G.P. (1983). Closed-form equilibrium results for theM/&,,/n in EUY.J. Oper. Res. Medhi, J. (1979). Further results in a Poisson queue under a general bulk service rule. a h . Cen. Etud. Rech. Opir. 21, 183-189. a n d A. Borthakur (1972). On a two server Markovian queue with a general bulk senice rule. Cah. Cent. Etud. Rec. 0 p & 14, 151-158. Murari, K. (1968). An additional special channel, limited space queuing problem with service in batches of variable sue. Opeu. Res. 16, 83-90. Neuts, M.F. (1979). An algorithmic solution to the GI/M/c queue with group arrivals. Cah. Cen. Etud. Rech. Opir. 21,109-119. and R. Nadarajan (1982). A multiserver queue with thresholds for the acceptance of customers into service. Oper. Res. 30, 948-960. Peace, C. (1965). A queuing system with non-recurrent input and batch servicing. J. Appl. Prob. 2,442-448. Polyak, D.G. (1968). Some problems in the theory of multi-channel queuing systems with constant service time. Eng. Cybern. 2,86-95. Sharda (1969). A many server queuing problem with variable departures. Z. Angew. Math. Mech. 49,163-166. (1970). A discrete time limited space queuing problem with batch arrivals forming a Markov chain and service in M parallel channels in batches of variable size. Gzh. Cen. Etud. Rech. Opdr. 12, 178-194. (1981). A limited space correlated queuing problem with departures in batches of variable size. Cah. Cen. Etud. Rech. Opir. 23, 87-96. Sim, S.N. and J.G.C. Templeton (1983). Computational procedures for steadystate characteristics of unscheduled multi-carrier shuttle systems. To appear in Eur. J. Oper. Res.

This chapter seeks to give a unified discussion of several queuing systems by considering some common features of those systems. In particular, it deals with: 1

2

Relations among three different steady-state p.g.f.'s for the number in the system considered at different epochs: P-(z) is the p.g.f. just before an arrival epoch, P(z) the p.g.f. at a randomly chosen epoch, and P+(z) the p.g.f. just after a departure epoch. Formulas connecting these three p.g.f.'s are found for the queuing systems GIr/M/I and M/Gr/l with arrival or service in groups of exact size r. For systems in which all three p.g.f.3 P-(z), P(z), and P+(z) are known, we may wish to use any or all of them for a particular purpose. For example, an arriving customer may be interested mainly in the p.g.f. P-(z) at an arrival epoch and in the corresponding waiting-time distribution. In other systems we may know exact relations among P-(z), P(z), and P+(z) without knowing a formula giving any of the three p.g.f.'s in terms of the interarrival and service-time distributions. Either P-(z) or P+(z) may serve as an approximation t o P(z) in queuing systems in which P(z) is difficult t o obtain. For convenience, where the context makes it clear which p.g.f. is intended, we can use the term "queue size p.g.f." for any one of P-(z), P(z), and P+(z). Relations between bulk-arrival and unit-arrival systems, or between bulkservice and unit-service systems.

Several authors have considered some of the above properties. Cox and Smith (1967, ex. 2, 3, p. 166) state that the limiting distribution of queue size considered at an arbitrary instant for Mr/E/I is equivalent t o the limiting distribution of the number of phases considered at the corresponding instant for M/Er/I. [For explicit analytical details of this, see Gross and Harris (1 974).] A more general statement in the transient case (continuous time), the equivalence of the distribution of the number in the system M ~ / E /1 , and phases in the system M/Ex/l, where X is an r.v., has been discussed in Chapter 3. Brockwell (1963) discusses the relation of MP/EI/l to M/E,/I in the transient case (imbedded Markov chain). Some numerical work on a particular case of M r / E l / I , namely, for M/E2/1 and consequently for M2/EI/1, has been carried out by Kerridge (1966). Finch (1962) discusses the

RELATIONS AMONG QUEUl NG SYSTEMS

296

relation of M/Mr/l/finite to E,./M/l/finite in' the transient case (imbedded Markov chain). Khintchine (1932) proved that the limiting distributions of queue sizes considered at an arbitrary instant and just after a departure for M/G/l are the same. For some more general cases, the reader is referred to Takics (1 962). A unified approach in the limiting case for single-server bulk queues was made in Foster (1961,1964), Foster and Nyunt (19611, and Foster and Perera (1964, 1965). These authors and, more recently, Chaudhry (1978,1979), Chaudhry and Templeton (1981), and Holman et al. (1981) give results showing relations between different queuing systems and between different limiting solutions (random times, before arrivals, after departures) for the same queuing system. Since the results of the last four papers have already been discussed in Chapters 3 and 4, we consider here the results of the five papers cited above by Foster and his collaborators. We also try to derive some of the results of the earlier authors from the work of Foster, Nyunt, and Perera. It is shown, for example, that the limiting distribution of the number considered just before an arrival instant for the system GIr/E,/l is equivalent to the limiting distribution of the number of phases considered just before the corresponding arrival instant for the system GI/Er/l. In passing, the reader is reminded of the notational equivalence of E l ,E , and M. In what follows, we use El E 5 M indiscriminately.

6.7

THE SYSTEM

GIr/E/I

Oueue Size just before an Arrival lnstant

Define the probability

Py

=

n

lim P(N,- = j) +-

= lim P(N(a; - 0) = j). n

+-

It has been shown in Chapter 3 that the p.g.f. P-(z) of the limiting probabilities

4- is given by equation (3.2.3).

Queue Size just after a Departure Instant 5n

Let al, a 2 , . . . be the instants of departure of customers from the system. Define q(n) =

P(K = j )

= P(N(an

+ 0)

= j).

Since the customers arrive in groups of size r, Pi+(n) is nonzero only when j + n, the number of arrivals up to the instant of the nth departure, is a multiple of r, that is,

Equation (6.1 . l ) shows that the ordinary limit limn , , Pi+(n) does not exist. Let us then consider the CBsaro limit, ution of Queue Size

We start out by considering the system GIr/E/l. In Chapter 3 we discussed the p.g.f. P-(z) of the limiting probabilities of queue size considered just before an arrival instant. As in Chapter 3, batches of customers of fixed size r arrive at the sequence of instants 0 =a;, a; , . . . ,uL, . . . ,such that the interarrival times a;,, a; > 0 , n = 0, 1 , 2 , . . . , are i.i.d. r.v.'s with common d.f. A(u). The customers are served individually by a single server with an exponential service time distribution of mean 1/p. It will be convenient to take the mean of A(u) as klh so that when the interarrival times have the Eh distribution, l / h would represent the mean of each of the k phases. Further, define 7 as the ratio of mean service time to mean interarrival time so that 7 = h/k,u, and p = r7 = hrlkp. Assume that p < 1; this is a necessary condition for the existence of the limits to be considered next. In this section we use techniques used in renewal theory to derive steady-state distributions of N (the number of customers in the system at a random time), N(the number of customers just before an arrival), and N' (the number of customers just after a departure), all for the system GIr/E/l. A supplementary-variable derivation of the steady-state distribution of N, and related results not accessible by the techniques of this section, are given in Miscellaneous Problem 1.

Again by virtue of equation (6.1.1), one can see that this C6saro limit, if it exists, is also the CCsaro limit of the sequence

It is shown later that the ordinary limit of this new sequence exists. It then follows that the CCsaro limit (6.1.2) exists. Let us now write Pi' =

1

r

lirn Pi(sr -j ) s-+m

(6.1.3)

and define

For details of the existence of the Cksaro limit when an ordinary limit exists, we refer the reader to Parzen (1962).

300

RELATIONS AMONG OUEUlNG SYSTE

preceding instant of arrival, if the distribution of the time interval Y between these two instants of time is known. The limiting distribution of the eiapsed interarrival time Y then follows from renewal theory (see Chapter 2) and is given by

6.1

P(1-)

THE SYSTEM GIr/E/l

30 1

= 1 gives C and consequently the result (6.1.8)

The System E6/E1 /1

provided A(y) is a nonlattice distribution. Lct D* be the number of departures (real or virtual) during the time interval Y and define gj = P(D* = j) so that the p.g.f. of the sequence {gj) is G(z) = Cr=, gjzi. It then follows that

As an example, let us consider the system Ei/El / I [Problem 7(b) of Section 3.21. Here the interarrival time distribution is k-Erlang with

and mean k/h. In this case we have proved (see Section 3.2) that

*

and therefore where

7

= X/(kp) and y l ,y 2 , . . . , y, are the roots inside lz I = I of the equation

where *w

K(z) = ?i(b(l -z))

and

Z(a) =

j0

e-aU dA(u).

Now N and N- being the r.v.'s representing the limiting queue sizes at the arbitrary chosen instant and just before the preceding arrival instant of time, respectively, we have N = ( N - + r -I)*)* in distribution. Since Y and D* are independent of N-, we have, on taking the generating functions and proceeding as in Chapter 3, P(z) = Z'P-(z)G(z-I)

+

2 dA(l - z - ~ )

[In comparing results in this section with the corresponding results in Section 3.2, note that the mean interarrival time has been taken as l / h there and as k/X here. Thus T is the ratio of mean service time t o mean interarrival time in each case, and equation (6.1.18) is unambiguous, since it is given in terms of T and not X.] It can be shown (see Problems and Complements) that the roots of equation (6.1.18), and in particular the yl's, are all distinct. Consequently we may write equation (6.1.17), using the method of partial fractions, as

m=O

where dz = P(N-+r-D*

= -m).

Use equation (6.1.16) and then eliminate K(z-') by using equation (3.2.7) to get, on simplification,

[Recall from Section 3.2 that b, = P ( N - + r - D = -m), where D is the number of real or potential departures during a complete interarrival period.] Now since the left-hand side of this equation is a power series with nonnegative powers, so must be the right-hand side. Consequently all terms on the right-hand side with negative powers must cancel out, leaving at the most a constant C, say. Thus

where

Thus comparing the coefficients of z j on both sides of equation (6.1.19), we have

From equations (6.1.1 I), (6.1.13), and (6.1.20) we obtain

RELATIONS AMONG QUEUlNG SYSTEMS

GI/Erh

303

j = o

For convenience we suppose that the roots yl, 1 = 1 , 2 , . . . ,r , are all %tinct. If one or more of the roots are repeated, the analysis proceeds on similar lines. Now equation (6.2.4) can be rewritten, using partial fractions, as

11

Once we have discussed the relationships among Pd(z), P+(z), and P(z) for the system GIr/E/l, several other interesting properties follow. First we show that the number N - in the system GIr/E/l has the same distribution as the number N; of phases in the system GI/Er/l. This use of the notationN; in this section and in Section 6.2 is not to be confused with the earlier use of N i = N(ak - 0 ) to denote the number in the system just before the arrival epoch a;. Further properties of the system GIP/E/l are considered in Section 6.2 in connection with the relations established there between the systems GIr/E/l and GI/Er/l.

6.2

THE SYSTEM

It may be remarked here that Wishart's equation (in our notation) giving the roots { yl) is zr =B[r(l -z)/b] ,where b is the mean service time of a customer in GI/Er/l. This conforms to our equation (3.2.4) if and only if b = r/p, the mean service time of a group in GIr/E/I (see discussion later). We next show that the distribution of N; given in equation (6.2.1) is the same as the distribution of N - in GIr/El/I given in equation (3.2.3) by means of some algebraic manipulation of the roots { y l ) of the equation K(z) = zr. First consider equation (3.2.3), which is repeated here as equation (6.2.4):

and from equations (6.1.14), (6.1.15), and (6.1.21) we obtain

( 1 -8.

6.2

where

Now using equation (6.2.3), we can write, say,

THE SYSTEM GI/E,I1

The results for the system GI/Er/l can be derived from those for the system GIr/E/l. Wishart (1956) has shown that for the system GI/Er/l

Consequently from equation (6.2.5),

r

p i = P(N; = i) = K where

alyi, 1=1

i >, 0

(6.2.1) Comparing equations (6.2.6) and (6.2.1) and applying the normalizing condition gives A/K = 1. Hence PT for the system GIr/E/l is equivalent to p7 for the system GI/E,./I . We also have the following relations, which are often useful in problems:

-1

p i is the probability that there are i phases in the system (including those, if any, in service) just before an arrival instant, yl, I = 1 , 2 , . . . ,r, are the r roots (distinct or coincident) of the equation

1,

m

zr = ZC(I -z)]

=

exp [-pu(l

- z)] d ~ ( u )

(6.2.2)

[see Section 3.2, in particular, equation (3.2.4)] ,and [use P-(1) = 1 in equation (6.1.19) or otherwise].

304

ONG QUEUING SYSTEMS

For an independent derivation of p; Ode may see Wishart (1956). Once the limiting distribution { p ; ) of the number of phases in the system GI/Er/l is known, one can derive the limiting distributions of the number of customers in the system or in the queue, the limiting waiting-time distribution, measures of efficiency, and so on, (see Problems and Complements). The equivalence between customers in G I r / E l / l and phases in GI/Er/l holds even in the case of transient distributions of queue sizes, provided that the two systems start with the same initial conditions (number of customers in GIr/E1/l and phases in GI/Er/l). The transient distributions of queue sizes (number in the system GIr/EI/l and phases in the system GI/Er/l) have been independently discussed by rain and Rani (1971) and by Wu Fang (1960). Since both used the supplementary variable technique, their equations representing the two systems in continuous time can easily be seen t o be equivalent if the initial states (number in Glr/EI/I and phases in GI/Er/I) are the same. The above remarks hold also for the distributions of waiting times, busy periods, and so on, for the two systems. (For some explicit results o n waiting times, see Problems and Complements.) To see the equivalence in the case of busy periods, the reader is referred t o Takhcs (1962) or Conoily (1960) for the queuing system GIr/M/I, and to Prabhu (1965) for the queuing system GflEr/ I . In closing this section it may be remarked that the results considering instants just before arrival for the system GI/Er/l (or for GIP/E/I) can also be derived from those for the system Er/G/l (or for E/Gr/l). For such derivations the reader is referred to Prabhu (1965). In this case each system is called the dual of the other, for the reason that the distributions of interarrival and service times are interchanged.

We showed in Section 6.2 that the number of customers in the system GIr/E/l can be identified with the number of phases in a GI/Er/l system such that the service-time distribution of a group in GIr/E/l is the same as the service-time distribution of a customer in GI/Er/l. In other words, suppose that in the group arrival system GIr/E/l the groups retain their identity, in that a group is regarded as being present in the system until all its members have departed. The total service-time distribution of a group will have an Erlangian distribution E,. The resulting system, in which the groups are now t o be treated as the individual customers, is a GI/Er/l queue. Thus if uI , u z , . . . are the departure instants in GIr/E/l, then ur, u z r , . . . are the departure instants in GI/E,/l, the instants of arrivals being the same in both the systems. Now if E(t) represents the number of customers (not groups) in the system GIIE,/1(including the customer under service, if any) at time t , we have

6.3

RELATlONSHlP BETWEEN GI/Er/l

AND GIr/E/l

305

where [ x ] denotes the integral part of x . We shall consider the system length distributions of G1/Er/1 a t the three instants of time and their relations t o the corresponding distributions of GIr/E/l. Let us now define for GI/Er/l :

Case 1

From equation (6.3.1) we get at the instant t = UA - 0,

-+.,

Since the probabilities on the right-hand side of the above equations converge as n so do the ones on the left-hand side. Consequently q j = limn , ,q[(n) exists and

Let us define q-(z) = ZEo q j z i and put Ry = Pi and q-= Z/=, q; so that Ry and Tj- are the d.f.'s corresponding t o the two distributions involved. It is easy t o show that T i =Pi,T; = Rr-, . . . , Tj- = Rj;and

But in view of equation (6.3.31, the left-hand side of which is a power series, we may write, using complex variable theory (see Appendix A.3),

Here C i s a contour, traversed in the positive (counterclockwise) sense around the

306

RELATIONS AMONG QUEUING SYSTEMS

6.3

RELATlONSHlP BETWEEN

Gl/E,/l

AND Glr/E/l

origin, excluding the poles of P-(z)/(l - z ) add the point i = \/=I. Thus Therefore

Since the poles of the integrand within Care at wiz"', j = 1 , 2 , . . . , r, where wj is an rth root of unity, the residue at W ~ Z " is~

It is interesting to see that the p.g.f. q-(z) of the limiting queue size considered just before an arrival instant for the system M/E,./I agrees with the corresponding p.g.f. [which can be derived from equation (3.1.19)] which is considered at an arbitrary instant. Case 2

Therefore by the residue theorem (see Appendix A.3) we have

Again from equation (6.3.1) we get, at the instant t ,

where P-(.) is given by equation (3.2.3). Equation (6.3.4) expresses the p.g.f. of queue size for the system GZ/Er/l in terms of the corresponding p.g.f. for the system GZr/E/I. As an example, let us consider the case of the system M/Er/I in which the customers arrive singly. In this case, the mean arrival rate being h and the mean service time r/p, the traffic intensity p = b / p remains the same as in the case of Mr/E/l . Using equation (3.2.1 O), we have

On the basis of convergence of the probabilities [which do converge when p < 1 and A(u) is nonlattice] on the right-hand side, the probabilities on the left-hand side converge under the same restrictions, and consequently

exists, and we can write where we have used (wi)' = 1 . Therefore

By the arguments used in Case 1 , we can write the p.g.f.'s q ( z ) and P(z) as follows: It is possible to obtain a simpler expression for q-(z) by expressing the sum on the right-hand side of equation (6.3.5) as the sum of roots of a certain equation. Define y j = wiz"" [l p(1 - z)/r] SO that the sum on the right-hand side of equation (6.3.5) becomes Xi'=,( 1 -yj)-' = Z!=, xj, where we set xi = (1 -yj)-' . Now { y j ) ,j = 1 , . . . ,r, are the roots of the equation y r = z[1 + p ( l -z)/r] '. Setting x = (1 -y)-' or y = ( X- 1)/x transforms this equation to ( X- 1)" = xrz [ 1 p(l - z)/r]', which can be written as

+

+

For this last equation the sum of roots is easily found by taking the ratio of coefficients of x'-' and x', and is

Case 3

Once again from equation (6.3.1) we get, at the instant t = on,+ 0 ,

(

r

,

j>O.

6.4

THE SYSTEM E / G ~ A

ONG QUEUBNG SYSTEMS

308

where we have used

By virtue of equation (6.1 .I), this may be written as

qf(n)=P>(nr),

j=O,l,2

,....

But we have shown in Section 6.1 that the limits Pif = (Ilr) limn,exist. Consequently the limits 1 qf = lim q;(n) = r lim - Pjf(nr) n

n-t-

-+m

309

Pif(nr - j ) Also by equations (6.3.7) and (6.1.22) we obtain

Y

exist and q f = rFG, j = 0 , i , 2 , . . . . Now from equation (6.1.1 I ) , P: = (1b)P; and from equation (6.1.131,

Therefore

The p.g.f., in this case, is given by

Finally, by using equation (6.3.2), we have

q; = q i ,

that is,

q*(z) = q-(z).

(6.3.9)

Equation (6.3.9) shows that the distribution of the limiting queue size just before an arrival instant is equivalent to the one just after a departure instant for the system G I / E r / l .It may also be observed that q f , q j , and qj for GI/Er/l are identical with PjC,P;, and F j , respectively, for GIr/E/l if r = 1, as is expected. and get more explicit It is possible t o elaborate more on the system J!?~/J!?,./~ expressions by using the results of the example considered in Section 6.1. If GI = Ek in GI/Er/l with Ek having the distribution function

where (yl} are the roots within the unit circle, lz I = 1 , of equation (6.1.18), provided Er is taken such that the mean service time per customer is r/p. This is because a customer in GIr/E/l is identical with a phase in GI/Er/l (or equivalently a group in GIr/E/I isidentical withacustomer in GI/Er/l)if the rate of service of the customer in GIr/E/I is the same as the rate of change of phase in GI/Er/l (or equivalently the rate of service of a group in GIr/E/l is the same as the rate of service of the customer in GI/Er/l).However, if Er is taken in such a way that if its total mean is 1/p, that is, the mean of each service phase is l / r k , then in equation (6.1.18) we must change p t o rp so as t o get the desired equation giving the roots {yl)within the unit circle. It will be shown later that these distributions can also be derived by considering the system E / ~ , h / la, particular case of the system E/G'/I which we proceed t o discuss next.

and mean k/X, then by equations (6.3.9), (6.3.2), and (6.1.20) we obtain

E SYSTEM EIEIG~II The Limiting Distribution of Queue Size

This means that the corresponding generating functions are

q-(z) = q+(z)

In Section 6.1 we have discussed the distributions of the r.v.'s N-, N*, a n d N a n d the relationships among their p.g.f.'s for the system G I r / E / l . Since this section runs parallel to Section 6.1 and is also related to Section 4.3, some details will be left to the reader. Briefly, in E / G ' / ~ the customers arrive singly by a Poisson process with rate h and are served in batches of exact size k , the service-time d.f. B(v) for the batches being arbitrary with mean service time r/p =.I:vdB(v). We have chosen our units so that r/k is the mean service time. This choice will be

RELATIONS AMONG QUEUING SYSTEMS

310

convenient when we consider the special Lase ~ / E , k / lof E / G ~ /[see ~ equation (6.4.15)] . In E/l?,k/l with mean service time r / p the mean of each service phase will be 1/p. Define 7 1 = b / p , the ratio of mean service time to mean interarrival time, and take p, the traffic intensity, as p = r l / k = hY/kp. The limiting distributions to be considered exist if p < 1, which we shall assume to hold.

6.4.

THE SYSTEM E/G k / l

Pj = lim Pj(t). t--=

Define

For the existence of the limits (6.4.41, see Takics (1961).

Queue Size just before an Arrival Instant

Define the probability P,T(n) = P-(NG = j) = P(N(oL - 0 ) = j), j = 0 , 1 , 2 , . . . , where uk - 0 is the instant just before the nth arrival, uk being the instant of arrival of the nth customer. Since we are interested in the limiting distribution, the initial queue size is immaterial. However, for ease of analysis, one can take N(0) = 0 . Since the customers are served in groups of size k,Pj-(n) is nonzero only when n -j - 1, the number of customers departed from the system, is a multiple of k , that is,

Queue Size just after a Departure

Let

P; = P(N; = j ) = lim P(N(o, n 4-

+ 0 ) = j).

It has been shown in Chapter 4 that the p.g.f. of (P;) is given by equation (4.3.1). Relationships among the Functions P - ( z ) ,P + ( z )and , P(z)

Here we want to establish that Clearly, the ordinary limit of Py(n) does not exist for k consider the C6saro limit

> 1. Let

us therefore To prove the first equation of (6.4.5), we proceed as foIIows. Two cases arise according to whether j
N ( U ; ~ + -~0 + ) ~= j

By means of the relations (6.4.1) one can see that this CBsaro limit, if it exists, is also the Cksaro limit of the sequence

*

on

< okk+j+l* N(on +O) < j.

This implies

Py(nk+j+ 1) = P(o,

< U L , + ~ + ~=)

i

P&>.

(6.4.6)

i=O

However, if j

> k , say j = sk + m , where s > 0 and m < k , then

It will be shown later that the ordinary limit of this new sequence exists. It then follows that the CBsaro limit (6.4.2) exists. Let us then write

P- =

1 ks+-=

- lirn P;(sk

N ( ~ k ~ + ~ + ~ - O ) = i k + mi = , 0 , 1 , 2 , . . . ,s +

+j + 1)

a,


~ ~ +*~N(o, + , +0)

<j

which implies

and define

Queue Size a t an Arbitrary Instant

kt

and

The existence of the limits of the left-hand sides of equations (6.4.6) and (6.4.7) and consequently of equation (6.4.3) is to be proved on lines similar to the ones discussed in Section 6.1. Thus assuming that the limits exist, we have from equation (6.4.6), on proceeding to the limit when n -+ and using equation (6.4.31,

312

S AMONG QUEUING SYSTEMS

Similarly, from equation (6.4.7) we have

6.4

THE SYSTEM E/G k/l

313

we have

that is, Replacing j by j - k in equation (6.4.9) and subtracting, we get Now since P'(1) = 1, we have from equation (4.3.4), = 0 in view of the relation j - sk = m < k. The first where we have used Pp (, + equation of (6.4.5) now follows from equations (6.4.9) and (6.4.10) on taking generating functions. Foster has also discussed the first equation of (6.4.5) using heuristic arguments which make no assumptions about the input or service distributions. He states that the first equation of (6.4.5) should hold even for the more genera1 system GI/G'/I. To prove the second equation of (6.4.5), again two cases arise according to whether the arbitrary instant considered falls in an idle period (j < k - 1) or in a busy period (j> k). For the case j < k - 1 we argue as follows. The expected numbers of arrivals and departures during an arbitrary period of length T a r e hT and hTlk, respectively. But as the expected length of each service time is r / p , the expected time the server is busy is (hT/k)(r/p) = pT. Thus the probability that the server is busy at an arbitrary instant of time is p , and consequently the probability that he is idle is 1 - p . The server is idle when i = 0, 1 , 2 , . . . ,k - 1 customers are present in the system and gets busy as soon as k customers are present. The length of a period when the server is idle (idle period) is divided by the instants of arrivals into k - 1 intervals, each of expected length 1/X, and consequently the expected length of an idle period starting with i customers is (k - i)/h. The probability that an idle period selected arbitrarily starts with i customers is jointly proportional t o Pi+and t o the length of the idle period, namely, (k - i)/h. Now as P(there are j customers a t an arbitrary instant of time selected in the idle period / a n idle period starts with i customers) = 1/(k - i), j = i, i + 1, . . . ,k - 1, we have

Thus C/X = I l k , and therefore

For the case j > k , the arbitrary instant selected falls in a busy period. Let us then take Y as an r.v. representing the time between the start of service on the customer undergoing service and the instant of time considered above. Then reasoning as in Section 6.1, we get

P(y

< Y < y + dy)

=

PO < Y < y 4- dylserver is busy)P(server

is busy)

provided B ( y ) is a nonlattice distribution. Let A be an r.v. representing the number of arrivals during the time period Y, and define hi = P(A = j) so that the p.g.f. of the sequence (hj)is H ( z ) = ZGohjzi. It then follows that

and therefore

where C i s a normalizing constant. As the probability that the server is idle is

where k(z) =,;:L kjzi = 6 [ h ( l - z)] ,as defined in Section 4.3. Now N and N+ being the limiting queue sizes at the arbitrarily chosen instant

RELATIONS AMONG QUEUING SYSTEMS

314

THE SYSTEM E/G

and mean r/p. In this case we have (see Section 4.3)

and just after a departure respectively, we have

N = max(N+,k)+A,

6.4

N 2 k

where A is the number of arrivals during the period Y that is still going on at the arbitrary instant considered above. Making use of the convolution property and taking generating functions, we get

P+(z) = k ( l

-,qz - 1)

k-1

1 - zj) n (Z(zk-lkzj)/( (z)) - 1

(6.4.14)

I

.,

where z l ,z 2 , . . . ,zk are the k - 1 roots within the unit circle, jz 1 = 1 , of the equation zk = k ( z ) = [ I + ;rl(l -z)/r]-', where T, = Xr/p. This may be written as

E ( ~ N= ) E ( z ~ k~ ) ~) ~( ( ~ A~ ) , = [E(zN' IN'

2 k)P(N+ 2 k )

+ E(zk IN+ < k)P(N+ < k ) ]Ii(z).

But as

In view of equation (6.4.15), the denominator of equation (6.4.14) has k + r zeros, 1, z , , . . . ,~ k - z10,z20, ~ , . . . ,zro, the last r zeros being outside the unit circle. Consequently we can write the denominator of equation (6.4.14) as

we have where C is a constant to be determined. Replacing the denominator of equation (6.4.14) by this expression and normalizing, we get By equation (4.3.4) If we put zjo =

we can write P+(z)as

Using this and equation (6.4.12),

where j = 1,2, . . . ,r, are the roots outside the unit circle of equation (6.4.15). It can be shown that all the roots of equation (6.4.15) are distinct and hence that the l / e j are distinct. Consequently we can write equation (6.4.161, using the method of partial fractions, as

where Substituting this and H(z) in equation (6.4.13) finally gives the second equation of (6.4.5). As an example, let us consider the system E/E,h/l. In this case the service-time distribution is r-Erlang with

Note that since Pi(l) = 1,

RELATIONS AMONG QUEUING SYSTEMS

31 6

and equation (6.4.1 7) gives

Once P; is known, we have, using equation (6.4.51,

Thus for the system E/E,~/P the three probabilities Pi+,Pi,and PT are known in terms of the roots 1/q,E = 1 , 2 , . . . , r , outside the unit circle of equation (6.4.15).

6.6

RELATIONSHIP BETWEEN

Ek/G/l

AND E / G ~ / ~

317

equations (4.3.5) and (4.3.1) for the system E/G'/I and show that the result is equivalent to the one for E k / G / l given in Prabhu (1965). Furthermore, the joint distributions of duration of a busy period and the number served during a busy period, as well as the occupation-time distributions, will be the same. To see analytical equivalence in the former case, the reader is referred to TakAcs (1962) and to Prabhu (1965) who discuss independently the systems Af/Gk/1 and E k / G / l , respectively. For the distribution of occupation time for the queuing system Af/Gk/l, the reader is referred t o Problems and Complements in Chapter 4. However, no independent derivation of an analytical expression for the occupation-time distribution for the system Ek/G/l seems t o be available for comparison. The relations between the limiting distributions of queue sizes considered at three instants of time -just before an arrival instant, at an arbitrary ins@nt, and just after a departure instant - for the system E k / G / l , and the corresponding distributions for the system E / G ' / ~ , are considered next.

Ek/G/l AND E/Gk/l We have shown in Section 6.2 a relation between the two systems GI/E,/l and GIr/E/l. A similar relation exists between the two systems E / G k / l and E k / G / l if a group in the former system is identified with a customer in the latter, so that the interarrival time distribution for the group in E/Gk/l is the same as the one for the customer in E k / G / l . At each departure instant the queue for the system E / G ~ / Iis reduced by k , whereas the number of phases of E k / G / l is reduced by k. If the interarrival-time distribution for E/G'/~ is E with mean l / h (or for a group Ek with mean k/X), the interphase distribution for E k / G / l is E with mean 1/h (or the interarrival-time distribution for a customer is Ek with mean k/h). This makes the traffic intensity p equal to h/kp for both systems, since 11.~1is the mean service time for both systems. In E k / G / l a customer passes through k exponential phases (marked frontward as 1 , 2 , . . . , k), each of mean l / h , before actually entering the system. Consequently if Nl(t) represents the number of arrival phases completed by a customer at time t , then N1(u, 0) converges in distribution t o an r.v. N;. Hence N; represents the limiting number of arrival phases completed by the customer just after a departure. In view of this, the distribution of NT in Ek/G/I and that of N+ in the corresponding E/G'/I must be the same. As a check, one may easily see that the distribution of N: relating t o E k / G / l given in Prabhu (1965) is equivalent to the one given in equation (4.3.1). This statement remains true even in the transient case, provided the initial states (customers for E/G'/I and phases for E k / G / l ) are taken t o be the same. It can be shown (see Problem 4) that the waiting-time (in queue) distr~butionfor the system GI/E,/l is equivalent to the corresponding distribution for the first member of a group in GIr/E/l. Similarly for the systems E / G k / l and E k / G / l , the waiting time (in queue) of the last customer of a group in E / G k / l is identical with the corresponding distribution for the system E k / G / l . For an analytical verification of this statement, one may use

As in Section 6.3, we consider the relations between the limiting probability distributions of queue size in the systems Ek/G/I and E/G'/I. First recall from Section 6.5 that a group in E/G'/~ is identified with a customer in E k / G / l . Thus , instants of arrival in the if u;, u;, . . . are the instants of arrival in E / G ~ / ~the corresponding Ek/G/l are cr;, u i k , . . . since a customer in E k / G / l is considered to have arrived if and only if it has passed through all the arrival phases. A customer leaving the system E k / G / l reduces the number of phases in the system by k, which is equivalent to a group of size k leaving the system E/G'/~. The instants of departure for both systems E/G'/~ and E k / G / l are the same. Now if $(t) represents the number of customers (including the customers, if any, undergoing service) at t h e t for Ek/G/l, then we have

+

where [x] is the greatest integer contained in x. We shall discuss the three distributions of E k / G / l at the three instants of time and their relation t o the corresponding distributions of E / G k / l . Let us define

3

q;(n)

= P([(ukk

- 0)

= j).

RELATIONS AMONG QUEUING SYSTEMS

318

6.6

RELATlONSHlP BETWEEN Ek/G/l AND E / G ~ A

where w j is a kth root of unity, the residue at vj = wjzVk is

Case 1

From equation (6.6.1) we get at the instant t = a,

+ 0,

k-1

Therefore by the residue theorem (see Appendix A.3), we have Since the probabilities on the right-hand side of this equation converge as n -tm, p < 1 , so do the ones on the left-hand side. Thus we may write

qr so that Let us define q+(z) = ZTxoq f z j and put Rf = Z{=oP: and Tj+= and {Ti+} are the d.f.3 corresponding to the two distributions involved. It is {Rf easy to show that

where P+(.) is given by equation (4.3.1). Equation (6.6.5) expresses the p.g.f. - - of queue size for the system Ek/G/l in terms of the corresponding p.g.f. for the system E / G ~ / I . As an example, let us consider the special case Ek/E/I of the system Gk/G/l in which G = E. Using equation (6.4.1 7), which gives PC(z) for Ek/E/l, we have from equation (6.6.4) for the system Ek/E/I

T: = R i - , where E , E is the unique root inside / z I = 1 of the equation kzk+' - (k + p)z - p = 0, and p = h/kp is the traffic intensity for the system Ek/E/l, 1/X being the mean for each arrival phase. But as by equation (6.4.1 8), B1 = 1 - E, we have

and

by evaluation of the residue at the pole v = I/E outside C (see Problems and Complements). Equation (6.6.6) can also be derived alternatively by using equations (6.6.2) and (6.4.19). This procedure is discussed later when we consider the system Ek lE*/ 1.

Now reasoning as in Section 6.3, we can write

where Cis a contour around the origin excluding the poles of P+(z)/(l

- z). Thus

Case 2

Again from equation (6.6.1) we have

The probabilities on the right-hand side of this equation converge [Takdcs (1961)l when f +=, provided that p < 1 and B(v) is nonlattice. It follows that under the same restrictions qj = lim,,, qj(t), j = 0, 1,2, . . . ,exists, and Since the poles within C of the integrand are vj = wjzlik,

j = l , 2 , . . . ,k

RELATIONS AMONG QUEUING SYSTEMS

320

If we define q ( z ) = Z G o q j z l ,then proceeding as in Case 1 we can show that

6.6

RELATlONSHlP BETWEEN Ek/G/l AND

E/G~/~

32'0

From equations (6.4.20), (6.6.2), and (6.6.10) we get

In particular cases, however, it may be convenient t o use equation (6.6.8) rather than equation (6.6.9). (See example following Case 3.) which implies Once again from equation (6.6.1) we have qjT(n) = PG(nk)+Pj;,+,(nk)+ ...+P$+k-l(nk) = P$+k-l@k)

Also from equations (6.4.20) and (6.6.8)

since other terms vanish in view of equation (6.4.1). Now it follows from Section 6.4 that the limits ( I l k ) limn , , P;(nk j + 1 ) = Py exist, provided that p < 1. Consequently,

+

But by equation (6.4.1 O),

Therefore qf =:::c

Pj+k,;, which gives, by equation (6.6.21, qJ: = qif,

that is

q-(z) = qC(z)

(6.6.10)

where q-(z) = CJT=Oq1:zJ. This is the result corresponding t o equation (6.3.9) for thc system GIr/E/l and shows that, for the system E / G ~ /the ~ limiting , distribution of queue sizes just before an arrival instant is equivalent to the one just after a departure instant. It may also be observed that qif, q j , and q j for E k / G / l are identical with Pj+,Pi, and P;, respectively, for E/G'/I if k = 1, as is expected. In thiscase, as Pi,Pj,P i are all equal for k = 1, so are q f , qj, qf and, in fact, qif = qj = q:J = p I? = pJ. We have already discussed the system E k / E r / l . Now we give an alternative derivation of the same system. In the problems we consider a proof that the results for the system E k / E r / l derived by the two different methods are, in fact, identical.

If in the system E k / G / I , G Er (with mean r / p ) , then it is possible t o get qj:, qf, and qj explicitly in terms of the roots l / e j of equation (6.4.15).

where we have used equation (6.4.1 8). The p.g.f., in this case, is

The System E ~ / G ' / ~

We have shown in Section 6.6 that the two systems E / G ~ / Iand I E k / G / l behave ~. identically if we identify customers in Ek/G/I with groups in E / G ~ /Similarly, the results for the bulk-queuing system E ~ / G ' / Iin, which arrivals follow a k-Erlang distribution and the service consists of batches of fixed size I, may be obtained from the bulk-service queuing system E / G ~ / Iwhere , K = kl. In particular, if N is the number of customers in the queuing system E ~ / G ' / Ithen , P(N= n ) = 7$;F-Qj, where the probabilities Pj have been analyzed in Section 6.4. The queuing system E / G ~ / Idiscussed here may also be compared with an (s, S)

RELATIONS AMONG QUEUING SYSTEMS

322

inventory model which has attracted a great deal of attention in inventory controls. For details, see Fabens (1961).

6.7

EXPECTED BUSY AND I D L E PERIODS

In this section we obtain expected busy and idle periods for some bulk queuing systems by means of the alternating renewal process and its generalization, as discussed in Chapter 2. This method, in general, is more elegant for finding averages than that of finding these averages through their distributions. Of course finding the distributions gives more information than finding the averages only. Instead of giving these averages in the discussions of various bulk queuing systems in other chapters, we thought that it would be nice t o keep all these results together. This chapter was considered to be the best choice t o contain these results. As in Chapter 2, let T o ,T I , . . . and Td, T I , . . . be two sequences of r.v.'s representing busy and idle periods, respectively. Suppose that {T,) are i.i.d. as T , and that (TA} are i.i.d. as I , but that (T,) and { T ; ) are not necessarily mutually independent.

6.7

EXPECTED BUSY AND IDLE PERIODS

323

where P; is the probability that an arrival finds the system idle. Equation (6.7.3) is easy to prove otherwise; for E(I' 1 arriving group finds system idle) Pi E ( I ~ I arriving group finds system busy) (1 -PG) = E(I' / . . . idle) PG + 0 (1 = ~(1:).Although equation (6.7.3) is true for k = I , 2, 3, . . . , we need t o use it only for k = 1. If, however, the higher moments of I, are known, equation (6.7.3) gives higher moments of I. Higher moments of I, for the system GIr/iM/l are given in Miscellaneous Problem 1. Now for the system GIr/M/l, since E(I,) = (1 - p)/X by equation (6.7.21, E(I) is given by

+

- -Pa

where P; may be found either from Chapter 3 or from Miscellaneous Problem 1 and is given by

( y i } being the r roots within the unit circle of the equation

The System GI'IMII

Since for this system, in the limiting case, the proportions of times the server is busy and idle are p and 1 - p, respectively, we have, by equation (2.3.18),

To find E(I), we use the conservation principle that in the limiting case, mean input rate equals mean output rate. Thus in GIr/M/I treat a group as a single supercustomer so that the group is in the system as long as its last customer has not been served. Let I, be an r.v. representing the duration of the idle period between two successive groups (I, = 0 if service continues without interruption between two groups) and 111-1the mean service time. Then as E(I,) f r/y is the mean time between completion of service of successive groups, we must have

Recall that E(I,) is called the mean virtual idle period (since it includes the zero idle period) as opposed t o the actual idle period which excludes the zero value. If we define I as the actual idle period as observed by an arrival, then as we know (see Section 2.2) the moments of these r.v.'s are related through the equation

Using equation (6.7.4) in equation (6.7.1), we get

which is the expected busy period for the system GIr/M/l. The latter expression for E(T) has been obtained by using equations (6.1.6) and (3.2.3). The System GIIE,II

Recall the remarks made toward the end of Section 6.2 that if a group in GIrlM/l is identified with a customer in GI/Er/l so that the total mean service time of a group in GIr/M/l is the same as the mean service time of a customer in GflEr/l, the distribution of a busy period for the system GIr/M/l is identical with the distribution of a busy period for the system GI/Er/l. Consequently the mean busy period of the server for the system GI/Er/l is given by

where 111-1is the mean duration of each phase of service in GI/Er/l. In terms of the

324

RELATlONS AMONG QUEUING SYSTEMS

notation of Section 6.3, E ( T ) could as well be written as

The System N I X / ~ / l

For the s y s t e r n M X / ~ /since ~ , E(I) = 1/A, we have by using equation (6.7.1) E(T) = Z/(p - XT) = (pP;)-l, where P(X = x) = a, is the distribution of group size, Z = Cc C,=, xa,, 1/X is the mean interarrival time and 1/p is the mean service time. For more details on this, see Chapter 3. The second expression for E(T) has been obtained in the same way as equation (6.7.5) for GI'/M/I.

service-time distributions whose coefficient of variation is < 1, and has the advantage that numerical calculations are possible. The initial probabilities P,',i = 0,1 , 2 , . . . , k - I , may be found either as in Section 4.1 or as in Section 6.1. The computational work done by Holman (1977) may be extended and used to evaluate P:, i = 0, 1 , 2 , . . . , k - 1, and hence E(T) for various values of the parameters k, r, B, and p = h/Bp. The Systems M/GB/ll and NI/Gk/l

The systems M/GB/1 and M / G ~ are / ~ special cases of M / G ' * ~ / ~and , results for them may be obtained by putting k = 1 and B = k, respectively. We have

The System MIG k , B / l

This discussion should be read in conjunction with Section 4.1, where the notation is explained, and Miscellaneous Problem 8 in Chapter 4. For the s y s t e m M / ~ k 3 B / 1 the relation between the mean busy and idle periods [analogous to equation (6.7.1)] is k-l

For M / G ~ / I E(T) , may also be written as E(T) = p/h&-, , where we have used equation (6.4.12) and p = hlkp. In these special cases, some computational work has been done by Allan and Carignan (1978). The System E k / G / l

We first discuss the mean idle period E(I), which is easy to evaluate. Let N(t) be the state of the system at time t . N ( t ) enters the set of idle states, S {0,1,2, . . . , k - 11, at the termination of a service period. The conditional probability that N(t) enters state i, given that N(t) enters S, is therefore CP,',i = 0,1 , 2 , . . . ,k - 1, where 1 / C = C~Z;P,'. Now N ( t ) leaves S when k - i customers arrive, which happens in expected time (k - i)/h. Thus

First using equation (M21) of Chapter 4, and then equation (6.761, we finally get

For a generalization of this result, see Problem 16.

The derivation of the mean busy period of Ek/G/l from IM/Gk/l is analogous to that of GI/E,/l from GIr/M/l. Recall the remarks made in Section 6.5 that if a group in IM/Gk/l is identified with a customer in E k / G / l so that the mean interarrival time for a group in&f/Gk/l is the same as the one for a customer in Ek/G/l, the mean busy period for the system Ek/G/l is the same as that for the system lM/Gk/l. Thus the mean busy period of the server for the system E k / G / l is given by

where p = X/kp = Xr/p and 1/X is the mean of each input phase so that the total mean input rate is X/k = X'. In terms of the notation of Section 6.6, E(T) could as well be written as E(T) = 1/1-14; This tallies, as it should, exactly with the result obtained earlier for GI/Er/l. It may, however, be noted that in the system Ek/G/l, l / p is the mean service time whereas r/p is the mean service time in the system GI/Er/l.

The System M / E , ~ , ~ I ~

]If we take the r-Erlang distribution of service times, we have the system I M / E , ~ * ~ / I . This system is a good approximation to the more general case discussed above, for

Shanthikumar and Chandra (1980) discuss, among other things, relations between different limiting probabilities for general multiserver bulk-arrival queuing systems

326

RELATIONS AMONG QUEUING SYSTEMS

of which special cases are M X / ~ / cand GIK/M/c. For some other similar results relating to nonbulk queues, the reader is referred to references contained in Shanthikumar and Chandra, and to the recent work of Franken et al. (1981) and Miyazawa (1977), among others. In reading these publications, one should keep in mind that some authors describe arrival-point (departure-point) and random-point probabilities as arising from customer and time processes, respectively.

PROBLEMS AND COMPLEMENTS

3

n

2

Using renewal-theoretic arguments, Foster and Perera (1965) have proved, for , Pj(t) exists, provided that p < 1 and the system GIr/EI 11, that Pj = lim, , A(u) is nonlattice, as is expected on the basis of our experience gained through the previous chapters. Show that the mean queue size L (mean number in the system considered at an arbitrary instant) for the system GIr/E/l is given by

< t)

W; is

Hint :

4

and deduce that

Hint: use ~-(')(1),which may be obtained from Chapter 3 [equation (3.2.3)] and equation (6.1.8). The result (1) has also been independently obtained by Conolly (1960) for the system GIr/EI/I considered at an arbitrary instant by using the supplementary-variable technique. For details, see Miscellaneous Problem 1. Observe that by using P-(')(l) and equation (6.1.6) one can also obtain the mean queue size just after a departure instant. It may be observed by looking at L, and W , (Problem 9 of Section 3.2) that Little's formula L, = & W q , where X, = Xr, for the system GIr/M/l is satisfied. The formula is true even for the more general system G I ~ / M /[for ~ details see Easton (1981)], as it should be. It is true without any constraints on the input distribution, the service-time distribution, or the group-size distribution, if the customer whose mean waiting time is required is chosen at random from an arrival group. For the verification of Little's formula for M ~ / G /see ~ ,Miscellaneous Problem 4(a) in Chapter 3. It should be pointed out that in both G I ~ / M / Iand MX/G/l, the queue size L , is calculated as a steadystate average over all instants of time, whereas W i (or W, in MX/G/l) is calculated at arrival instants.

+-

where K i s defined in Section 6.2, and deduce that the mean waiting time given by

Sections 6.1, 6.2,and 6.3 1

If v:) is an r.v. representing the waiting time (in queue) of the nth arrival, then show that the limiting waiting-time distribution function for the system GI/Er/l is given by

WG(t) = lim P(V?)

PROBLEMS AND COMPLEMENTS

327

5

Use equation (6.2.1) and the other relations found while discussing the system GZ/Er/ 1. Continuation of Problem 3. Show that the waiting-time (in queue) distribution obtained in Problem 3 for the system GI/Er/I is really equivalent to the corresponding distribution for the first member of a group in GIr/E/I . Hint: From equation (3.2.1 1) we have

where C2is defined in Section 6.1 and a: is the parameter of the transform. Also recall that Zf=, Cl = K, Cl = Kcq. Get the result by inverting or by taking the L.-S.T. of equation (2). It has been assumed that the roots { y r )are distinct. As a consequence of Problems 3 and 4, it follows that the waiting-time distributions for GIr/El/l and GI/Er/l are identical. These remarks apply equally well to both systems if they are considered at arbitrary instants of time. (a) The system GI/Er/l. Show that the limiting distribution of the number $- in the system GI/Er/l considered just before an arrival instant is given by

RELATIONS AMONG QUEUING SVSTE

328

PROBLEMS AND COMPLEMENTS

where

(c)

329

Show that for the system M/Er/I , p i = 1 Brit. Equation (6.2.1) gives

xcul K = A n (I-,,) r

r

= K

The cumulative distribution of 5- is given by

=

=

1=1

?=1

.

- p.

,

i I

?

, Ldp-

\?

where yl are the roots of the equation Z ~ - ~ ( Zf~. .-. + ~ z)-7 and the p g f . of $- by

6

)

(a)

Show that equation (3) can also be obtained by using equation (6.3.1 1). Show that the mean queue size L- (the mean number in the system) is given by

= 0.

Substitute z = 1 - y and then 1 - yl are the roots of the equation in y, and so on. The system GIT/E1/1 : the waiting-time distribution. Let V,(t) be the virtual waiting time (in queue) at the instant t . Show that the limiting distribution for the first member of a group Wq, ( x ) = lim P(V,(t) t -+-

< x)

has the L.-S.T. given by

and deduce that the mean number L, in the queue is

(b)

int: To discuss this problem, first note that the relation between the number I of phases and the number 5- of customers in the system GI/E,/ I is given by

I = r(5-

Show that the mean waiting time for the first member of a group in (a) is given by

7

For the system M r / E 111 show that

8

Hint: Use equation (3.2.10) or equation (3.1.19). Intuitively, the equivalence of P-(z) and P(z) may be argued on the basis of Poisson (random) input. Such an equivalence has already been observed in Chapter 3 (see Problem ll), even in the more general case when the arrival group follows an arbitrary distribution. But here, of course, we have given the proof in the case when the arrival group size is fixed. Show that all the roots of the equation

- I) fS

where S is the phase of service of the customer undergoing service at the instant of the last arrival. In addition, the system is uniquely defined if it is supposed that whenever $- = 0, I = 0. Now for n > 0,

and so on. Also note that equation (6.3.1) gives are distinct. Proof: Differentiating equation (4), dividing one equation by the other, and

"

&A

RELATIONS AMONG QUEUING SYSTEMS

330

PROBLEMS A N D COMPLEMENTS

setting rr = p, leads to the equation .z = (pk 4- r)/(k -t- r). Substituting this value of z in equation (4) gives the condition for the roots to be repeated, which is

Let us now see if this equation holds for the traffic intensity p < 1. Putting < e < 1, in equation (S), we get (1 - E )=~(1 - ce), where c = k / ( k f r). But this is impossible, unless c = 0 . Thus the roots of equation (4) are distinct if p < 1. Since the roots can be computed only up to a certain degree of accuracy, they may become computationally inseparable as p -t 1. I t appears from our computational experience that when p < 1, the roots will be computationally distinct so long as the degree of the equation to be solved is kept small. Muller's algorithm [see Conte and de Boor (1972)l may be used if the roots are repeated. If GI El in GI/Er/I , then show that p = 1 -E, 0

9

q-(z) = q+(z) = q(z) =

(l-~w-z) . 1 -z{l ?(I -z)Y

+

12

and so on. One particular case of E / E , ~1/is E/E:/ 1, when r = 1. Show that for this case

where E is the unique root inside the unit circle of the equation

13

This shows that for the system El / E r / l ,the distributions of $-, $+, and $ are identical. This is true even for the more general system M/G/1;see Section 6.4 f o r k = 1. Hint: Use equations (6.3.8) and (6.3.9) and the procedure discussed in the example of Section 6.3. See also comments after equation (6.3.6). We can also discuss this problem by using the example of Section 6.3 and Problem 7.

Evaluate the integral given in equation (6.6.6). Proof:

Sections 6.4.6.5,and 6.6

10

11

since the neglected part has no poles inside C. Now transform the integral by making the substitution v = l / V . Then the integrand has the pole V = E inside C. The residue at V = E being ( 1 - ek)/(l - ekz), we have by the residue theorem (see Appendix A.3)

Show that the roots of equation (6.4.15),

are all distinct. Hint: Proceed as in Problem 8. Show how to get the results of equation (6.4.20). Hint:

14

(a)

Show that when r = 1 in E k / E r / l ,that is, for the system E k / E / l ,

and the corresponding p.g.f.'s are

332

RELATIONS AMONG QUEUlNG SYSTEMS

PROBLEMS AND COMPLEMENTS

333

ej" =

$.

But by equations (I 1) and (12)

(b)

where E, E is the root inside the unit circle of equation (6). Hint: Use equations (6.6.11), (6.6.13a), (6.6.13b), and (6.4.18). Show that for the system Ek/E/l q (z) = 1 - p

5

+ pzq-(z).

It may be observed that this relation between q(z) and q-(z) is true even for the more general system GI/E/l, as may be seen from equation (6.1 3 ) . We have discussed queue- size distributions {q;), {qj}, and { q j }for Ek/Er/l by discussing two different systems E;/E/l and ~ / ~ , h /Now l . we 'show that equation (6.3.10) is, in fact, identical with equation (6.6.11) and equation (6.3.12) with equations (6.6.13a) and (6.6.13b). To do this, we first need to establish relations between the roots y and I/E of equations (6.1.18) and (6.4.1 5). Since the traffic intensity for both systems is p = Xrlkp, we rewrite equations (6.1.18) and (6.4.15) in terms of p. Thus

gives the roots yj, J = 1 , 2 , . . . ,P , within the unit circle, whereas

which shows that yj is a root of equation (7) and also l yjl < 1 by equation (12). Relation (10) gives a one-to-one correspondence between the r roots l / e j outside the unit circle of equation (8) and the r roots yj in$de the unit circle of equation (7). Now from equation (1 0)

By subtracting we get

By dividing equation (13) by equation (14) we get

gives the roots l/q, j = 1 , 2, . . . , r , outside the unit circle. Since l/ej is a root of equation (81,

Using the definitions of C and B (Sections 6.2 and 6.4) we have

Now if we define

which implies Bl G-Y:-r = -.

then it follows that

(15) 1-€2 1-71 Since both equations (6.4.11) and (6.6.13b) give the value of q,, we have on comparing

and by equation (91,

Recall that

RELATIONS AMONG QUEUlNG SYSTEMS

MlSCELLANEOUS PROBLEMS AND COMPLEMENTS

=

(a)

(b)

Show by using equations (1 2), (1 51, and (1 7) that equation (6.3.1 0) is identical with equation (6.6.1 1). Show by using equations (12), (13), (IS), (16), and (17) that equation (6.3.1 2) is identical with equations (6.6.13a) and (6.6.13b).

Section 6.7

16

Equation (6.7.8), derived above for M / G ~ , ~is/ valid ~ , for G I ~ / G ~ , as~ / ~ , indicated by the following heuristic argument: E(T) = E(service time) * E(number of groups served in a busy period) since service time and service group size are independent. Hence E(T) =

E(service time) P(the end of a service is the end of a busy period)

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

2 b-,,, I, [a(y)(py)me-YY/m!]dy,

m=o

In this problem we discuss the queuing system GIr/iM/l already considered in Section 6.1. The analysis used here runs parallel to Section 2.2. For notation and other terminology, see Sections 6.1,2.2, and 3.2. In addition to rediscovering some of the results of Sections 6.1 and 3.1, we discover some new results which are difficult to obtain using other techniques. Briefly, GIr/M/l is a single-server queuing system in which groups of size r arrive. Interarrival-time distribution is arbitrary with mean l / h , and servicetime distribution is exponential with rate p.

n

>r+l

(M2a)

together with

P: = 0,

n

< r.

Wb)

It may be noted, as indicated, that equation (M2b) is true for n < r, for the number in the system after an arrival cannot be less than r. Equation (MI) corresponds to the case n = r. Write in words equations (MI) and (M2), which completely describe the system GIr/M/I. Show further that the solution of the set given by equations (MI) and (M2) is given by

where

and yj, i = 1 , 2 , . . . ,r, are the r roots inside the unit circle of the equation a(p(i

1

335

where

- z))

= zr

(M5)

In writing the solution (M3), it has been assumed that the roots of equation (M5) inside the unit circle are simple, as they are at least when GI = Ek. For details, see Problem 8 of this chapter. Hint: Proceed, in general, as in Section 2.2 for GI/M/l. Replacing Pi by zn,we get equation (M5). Now corresponding to equation (2.2.33), we have

The Limiting Distribution of the Number of Customers in the System

Show that the equations for GIr/M/l corresponding to equations (2.2.30) and (2.2.31a) of the system GI/M/1 are

where

Pz = P(n in system just after arrival in the limiting case)

where the r constants {Cj) which are independent of n are to be determined by requiring equation (M6) to satisfy equations (Ml) and (M2b). Hence

336

RELATIONS AMONG QUEUING SYSTE

MISCELLANEOUS PROBLEMS A N D COMPLEMENTS

337

where the summations are over i = 1 t o i = r , a convention which will be adopted in all problems in this section unless otherwise stated. Solving the set (M7), one can show that

One way t o get equation (M8) is to solve the set (M7) as simultaneous equations. Another way is to first consider the identity (M9), which is easily proved by considering partial fractions of xS [F(x)]-\

2

3

Then use equation (M9) with x = 1 and s = r - 1 and show that equation (M8) satisfies the first equation of (M7). Similarly, use equation (M9) with x = 0 , 1 < s < r - 1 , to show that equation (M8) satisfies the second equation of (M7). This, in turn, shows that equation (M8) is the solution of equations W7). One more point which needs attention is that equation (M3) is true for n > 1, for its right-hand side automatically vanishes when 1 < n < r - 1 , as it should. This can easily be seen by using the identity (M9). Continuation: {Pi).,the distribution of the number in the system just before an arrival epoch. (a) Show that P i is given by

Hint: Let P i = KP;+,(O), where K is a normalizing constant, and proceed as we did in getting equation (2.2.35). ( b ) Show that the p.g.f. of P i is the same as the one obtained in Chapter 3. int: Use the identity (M9). (c) The waiting-time distribution for the first customer and a random customer of an arrival group may be obtained by using P i . For details, see Chapter 3 and problems therein. Continuation: { p n } ,the distribution of the number in the system at a random epoch. Show that the distribution {P,,}is given by

4

and then P, = JomP,(y) d y , and so on. (iii) Po may be obtained either by using (1 -xr)/(l - x ) = Z;t:\ xk , the identity (M9), and 1 -Po = Z r = l P,; or independently by J: Po(y) d y , where Po@) is given by equation (M13) (see later). (a) Continuation: the virtual idle-period and the actual idle-period distributions for the queuing system GIr/M/l.Adapting the arguments used in the discussion of the virtual idle-period distribution for the queuing system GIIMII in Section 2.2, show that the distribution of I,, the virtual idle period, for GIr/M/I is given by

Hint:

(b)

Use

where Pn(y) is given in Hint (ii) t o Problem 3. Note that equation (M13) for the bulk-arrival system GIr/M/l remains the same as for the singlearrival system GI/M/l. Continuation: moments of I,. Show that the first two moments of I , are given by

where, if need be, one can replace F ( ' ) ( I ) / F ( I )in terms of the roots ) Ci(1 - yi)-' , and a, = 1," u2a(u)du, 2 = 1 , 2 , 3 , given by F ( ' ) ( I ) / F ( I= . . . , with a, l/h. The moments of I may be obtained through the relation

Hint:

Hint: (i) Adapt the procedure used to get equations (2.2.42) and (2.2.43). (ii) First note that P,(y) here is the same as that given in equation (2.2.40),

Take the L.T. of equation (M12), which is

1 e-"tdFr,(t), and so where we have used equation (M5) and &(a) = ; on. Now adapt the procedure used in problems on GI/M/l discussed in Chapter 2 and the identity (M9) and its derivatives.

338

5

RELATIONS AMONG QUEUING SYSTEMS

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

The system G I ~ / G / I . The distributiohs of waiting times and their moments for the queuing systems GI~/M/I and M ~ / G have / ~ been discussed in Chapter 3. The discussion of these distributions for the more general case GIX/G/1 poses problems. It is, however, possible to obtain some relations between expected values by completely elementary arguments, even far the most general case G I ~ / G / ~This . problem, and the next one which deals with G I / G ~ / ~are , devoted t o these relations. (a) Show that for the queuing system G I ~ / G /in~ the steady state, p = ?Zip < 1, the expected waiting time (in queue) of a random customer of an arrival group is given by

and since V A , and Un+, are independent, a2 var(S) = o: f af, = Z U ; ~+ L + a; !J2

where we have used equation (MI 5) on the right-hand side. Now one can use Problem 9(b) of Chapter 2 to get W i , the first part of Wi.To get the second part of W, given in equation (M14), one may proceed as follows. Since the probability that the customer selected randomly in his batch is positioned j t h is ai/Z, where 1 is the upper bound of the size of a group, the mean wait (in queue) of this customer during service of other customers in the same batch is given by

cI=~

where

w,,

= expected wait (in queue) of the first customer of a group

h ' ( c r f , + ~ a ~ + u ~ / ~ ' _(I2) ) W) 2 w - P)

-

X = an r.v. representing the size of a group such that

0;

a,=P(X=x),

x = 1 , 2 ,...,

a

m2 = E(X')

= E(X),

(b)

which can be simplified t o get the desired result. The sum of the two parts gives the result (M14). The system G I / G ~ / I . Everything else in this problem is the same as in (a), except that service is now in batches of fixed size k , and arrivals are by singlets. Show that in this case (when p = Xlkp < I ) the mean wait (in queue) of an arrival is given by

= variance of service time of a group

- zo:,

a: =

m2 -

w;

+ 0: /p2 ~

2

= variance of group size.

Other quantities are as defined in Problem 9(b) of Chapter 2. Hint: Since the procedure used in this problem is similar t o the one used for a special case in Problem 9 of Chapter 2, only a brief discussion is given here. Let Vr,, be the service time of the r t h customer of the (12 1)th arrival group (since the first arrival group is at a; = 0; see definition in Section 2.2) and define

+

Now if v:) is the waiting time (in queue) of the first customer served in the (n 1)th group, then = ( v p &)+ (M 16)

+

vi:+"

+

where Sn = V:+, - Un+, and U,+, is the time between the arrival of the (n 1)th and the (n 2)th groups. In steady state as n + =,

+

+

= Wik

k-l +2h

where Wik is the mean wait (in queue) of the customer who is the last (kth) arrival in his group. Hint: Let U,,,, 0 G r < k, be the interarrival time between the (r + 1)th and the (r + 2)th arrivals of the nth group t o be served, and let v,$) be the wait (in queue) of the last (kth) customer of the nth group. Then if we define U,* = 2::: Up,. ,we have

v$+l) =

(v$'

+ vn - u;>+

(M18) where Vn is the service time of the entire nth group. Now since equation (M18) is similar to equation (M16), one can find, in the steady state, Wik = E(Vqk). The second term on the right-hand side of equation (M17) is the expected wait of a randcm customer before his service group is completed and made available for service by the arrival of its kth customer. The rth customer in a service group must wait for k - r arrivals, and the mean interarrival time is 1/X. Therefore the mean wait of a random customer for completion of his service group is k-r

k-1 - = -.

k-" r=l

2X

RELATIONS AMONG QUEUING SYSTEMS

340

(c)

Continuation of (b): t h e system A4/Gh/1. S h o w t h a t if in (b) then Little's formula f o r A4/Gk 11 is satisfied.

GI=M,

T o s h o w that Little's formula L, = XW, f o r the queuing system Hint: & f / G k / l is satisfied, w e first need t o evaluate L,. Let N, be a n r.v. representing t h e number in queue at a random e p o c h in t h e steady-state case. T h e n a s

where

N is t h e n u m b e r in t h e system a t a random epoch,

where

P(z) is

given in equation (6.4.5). N o w f r o m equations (6.5.5) a n d

(M 191,

k-1 L = L++----

2

a n d from

@,'(a)

h I.I

= L, f -

(M20)

given in equation (4.3.5),

T h e use of equations (M17), (M20), a n d (M21) gives the desired result. Most o f t h e results o f this problem are d u e to Marshall (1968).

Allan, S. E. A , , and P. L. B. Carignan (1978). Queueing theory - A derivation o f the length o f the queue for the model Ek lE,ll . . . . Engineering Report, Royal Military College of Canada, Kingston, Ont. Brockwell, P. J . (1963). The transient behaviour of a single server queue with bulk arrivals. J. Aust. Math. Soc. 3, 241 -248. Chaudhry, M. L. (1978). Marriage between the supplementary variable technique and the Imbedded Markov chain technique - I. Traizsactions of the 8th Prague Conference on Information Theory, Staristical Decision Functions, Random Processes, vol. A, 133-141. / ~ its ramifications. Naval Res. Logist. Quart. (1979). The queueing system M ~ / G and 26,667-674. and J . 6 . C. Templeton (1981). The queueing system M / G ~ / and I its ramificat~ons. Eur. J. Oper. Res. 6, 57-61. Conolly, B. W. (1960). Queueing at a single serving point with group arrival. J. R. Stat. Soc., Ser. B 22, 285-298. Conte, S. D., and C. de Boor (1972). See Chapter 1. Cox, D. R., and W. L. Smith (1967). Queues. Methuen and Co., London. Easton, 6. (1981). Some problems in the theory o f single-server b u k queues. M.Sc. thesis, Royal M~litaryCollege of Canada, Kingston, Ont.

REFERENCES

Fabens, A. J. (1961). See Chapter 4. Fang, Wu (1960). See Wu Fang. Finch, P. D. (1962). On the transient behaviour of a queueing system with bulk service and finite capacity. Ann. Math. Stat. 33, 973-985. Foster, F. 6. (1961). Queues with batch arrivals I. Acta Math. Acad. Sci. Hung. 12, 1-10. (1964). Batched queueing processes. Oper. Res. 12,441-449. -, and K. M. Nyunt (1961). Queues with batch departures I. Ann. Math. Stat. 32, 1324-1332. ----, and A . G. A. D. Perera (1964). Queues with batch departures 11. Ann. Math. Stat. 35,1147-1156. -, and ____ (1965). Queues with batch arrivals 11. Acta Math. Acad. Sci. Hung. 16,275-287. Gross, D., and C. M. Harris (1974). Fundamentals o f queueing theory. Wiley, New York. Holman, D. F. (19771. Some problems in the theory o f queues. MSc. thesis, aoyal Military College of Canada, Kingston, Ont. ----, M. L. Chaudhry, and A. Ghosal(1981). See Chapter 4. Jain, J. L., and R. G. Rani (1971). The transient behaviour of the queueing system GI/M/l: (-/FCFS) with arrivals in batches. Ekonom.-Mat. Obzor. 7, 299-304. Kerridge, D. (1966). A numerical method for the solution of queueing problems. New J. Stat. Oper. Res. 2, 3-1 3. Khintchine, A. Y. (1932). Mathematical theory of a stationary queue (in Russian). Mat. Sb. 39,73 -84. Marshall, K. T. (1968). Bounds for some generalizations of the GI/G/1 queue. Oper. Res. 16, 841-848. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Prabhu, N. U. (1965). Queues and inventories - A study o f their basic stochastic processes Wiley, New York. Takacs, L. (1961). Transient behavior of singleserver queueing processes with Erlang input. Trans. Amer. Math. Soc. 100, 1-28. ---- (1 962). Introduction to the theory o f queues. Oxford Univ. Press, New York. Wishart, D. M. G. (1956). A queueing system with X' service-time distribution. Ann. Math. Stat. 27, 768-779. Wu Fang (1960). Some results about the queueing system G I / E k / l . Chinese Math. 1,205-216. .

ADDITIONAL REFERENCES

Franken, P., D. Ktinig, U. Amdt, and V. Schmidt (1981). Queues and point processes. Akademie-Verlag, Berlin. Miyazawa, M. (1977). Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349-357. Shanthikumar, J. G., and M. J. Chandra (1980). Application of level crossing analysis to discrete state processes in queueing systems. Working paper #80-010, Dept. of Ind. Eng. and Oper. Res., Syracuse University, Syracuse, N.Y.

A.3

TRANSFORM INVERSION AND RESIDUE CALCULUS

343

To show that f(z) has a root q , 0 < 7) < 1, we must show that f(0) and f(1) have opposite signs. It is easy t o see that this is so, since f(1 - 6) = 1 --KC1)(1) f terms containing 6 and its higher powers. The converse may be seen to be true by retracing the steps backward. The uniqueness of the root follows by RouchB's theorem.

AND RESIDUE CALCULUS Proofs of the results and theorems stated here without proof may be found in many books; some references are given for the convenience of the reader.

Pn some situations the solution of a given problem is easily obtained if we differentiate, with respect t o a parameter, a function which is given in the form of a definite integral. Of course, one way is t o perform first the integration and then differentiate with respect to the requisite parameter. It is often easier and may be necessary, however, t o differentiate without actually performing the required integration first. In such situations we may use the following elegant procedure due t o Leibnitz. Let the definite integral g(u) be given by

where h(. , .) is a function of x, u and a(-) and b(-) are functions of u only. Then the derivative of this integral with respect t o u is given by

There are several inversion methods, such as series expansion or the partial fractions method, for inverting the various transforms. One such method for the inversion ; f,zn (which is a p.g.f. if the f, are probabilities) of the z transform F(z) =,,Z involves evaluating the following complex integral: s

where C is a closed contour around the origin and is traversed in the positive (counterclockwise) sense. If F(z) is a rational function, one can also use a modified form of the above formula. Let F(z) = F1(z)/F2 (z), where F1(z) and F2(z) are polynomials. Suppose that the degree of Fl(z) is less than that of F2(z), or if not, consider the rational function R(z)JF2(z) where R(z) is the remainder when Fl(z) is divided by F2(z), and therefore R(z) is of lower degree than F2(z). There is another procedure if F(z) is of the form zk.F1(z)/F2(z) where k is a positive integer and the degree of F, (z) is less than that of F2(z). In such cases one can either use the transformation z = I/w in order t o evaluate equation (I), or evaluate equation (1) by omitting the factor zk and then use an elementary transform t o get the final result. For details of the latter procedure, see Kleinrock (1975), and for details of the discussion on complex rational functions, see Green (1939). Now if F(z) =Fl(z)/F2(z) and the above assumptions are made, then clearly F(z) -+ 0 as z -, and by using this argument one can show that the inversion formula ( I ) modifies t o -+

wherever the operations on the functions are defined.

A.2

DAM GESS

If K(z) = C y kjzi is a p.g.f., and ko > 0, 0 1 [Prabhu (1965), Feller (1968), and Parzen (1962)l . Proof: This lemma may be proved by several methods. The following geometrical proof seems t o be elegant and simple and hence is given here. Since K(z) - z has a zero at z = I , we define f(z) = (K(z) -z)/(l -z). Note that we have defined f(z) in such a way that f(0) = k, > 0.

where C1 is a closed contour large enough so as t o include all the poles of F(z) but exclude the point z = 0. It may be noted that the degree of the denominator in the integrand in equation (2) is at least 2 higher than that of the numerator. Furthermore equation (2) holds whenever F(z) 0 as z -+-,but we use it only for the rational functions. The contour integration is easily performed by making use of the residue theorem, which may be stated as follows. Residue Theorem: Let C be a closed contour within and on which a function F(z) is analytic except for a finite number of isolated singular points z, ,z 2 , . . . ,z, interior to C. If K 1 , K 2 , . . . ,K, denote the residues of F(z) at these points, then -+

APPENDIX A

For our purpose, equation (3) may be taken as a definition of residue. A function F(z) of a complex variable z is said to be analytic (regular, holomorphic) in a region R of the complex z plane if it is single-valued and differentiable at every point of R. A point of the complex z plane at which a function ceases to be analytic is called a singular point. A function which has no singular point, finite or infinite, in the whole complex plane, must be a constant. Furthermore, a singular point of an analytic function F(z) is called an isolated singular point if it has a neighborhood which does not contain further singular points of F(z). An isolated singular point of f(z) at z = a f 00 is a pole of order m , m = 1,2, . . . ,if and only if (z -a)mf(z) but not (z f(z) is analytic at z = a . If F(z) has a pole of order nz 2 1 at z = a , then the residue of F(z) at z = a is given by 1 Res F(z) = ------z=a

A pole of order 1 is called a simple pole. For a simple pole equation (4) simplifies

A.7

THE RIEMANN-STIELTJES INTEGRAL

345

If f(z) and g(z) are functions of z, analytic inside and on a closed contour C, and if If(z>l< Ig(z)l on C, then g(z) and g(z) f(z) have the same number of zeros inside C [Titchmarsh (1 939)j .

+

If f(z) and g(z) are functions of z, analytic on and inside a contour C surrounding a point a , and if w is such that

is satisfied at all points z on the perimeter C, then the equation

regarded as an equation in $, has exactly one root in the interior of C. Further any function f(g) of analytic on and inside C can be expanded as a power series in w by the formula

Res F(z) = lim ((2 - a)F(z)). z=a

z -+a

If F(z) = F,(z)/F2(z), where Fl(z) and F2(z) are both analytic at z = a , and

F1(a) # 0, F2(a) = 0, and ~$')(a)# 0, then F(z) has a simple pole at z = a. Equation

[Whittaker and Watson (192711.

(5) then reduces to

The method can be extended to poles of order greater than 1. For.details, see Churchill (1 960).

A.4

GENERALIZED ARGUMENT PRINCIPLE

Suppose that $(z) is a function of a complex variable z which is analytic within and on the contour C. Suppose also that @(z)is another function analytic within and on C except at a finite number of poles. If cuk, k = 1 , 2 , . . . , are the zeros of order (multiplicity) rk and B h , k = 1, 2 , . . . , are the poles of order (multiplicity) sk of the function @(z) within C [@(z) must not have poles or zeros on C], then the generalized argument principle states that

The proof is based on the residue theorem [Whittaker and Watson (1927) and Ahlfors (1 966)] .

The Riemann-Stieltjes Integral

Integrals may be viewed essentially as sums, since they are limits of approximating sums. Readers may recall the definition and the usefulness of the Riemann integral which we study in elementary calculus. For some applications an extension of the definition of the Riemann integral, known as the Riemann-Stieltjes (or Stieltjes) integral, provides a substantial simplification in notation. It also affords a useful tool in fields such as physics in which mass distributions are partly continuous and partly discrete, or the theory of probability in which r.v.'s are discrete, continuous, or mixed. Before defining the Riemann-Stieltjes integral, we discuss below certain concepts of elementary calculus. A function f(x) is said to be nonincreasing (either decreasing or stationary) in the closed interval [a, b ] if and only if f(x2) X I , where x l and x 2 is any pair E [a, b ] , and is called strictly decreasing if and only if f(xS) x l . Similarly, f(x) is said to be nondecreasing (increasing or stationary) or strictly increasing according to whether f(x2) >f(xl) orf(x2) >f(xl) for xz > X I .

APPENDIX A

346

A function which is either nonincreasing ot nondecreasing is called monotonic. A function f(x) is called bounded on [a, b3 if and only if m
By a partition of the interval [a, b ] we mean a finite set {xo, x , , . . . , x,) of points such that For any partition of the closed interval [a, b ] define

A function f(x) is said to be of bounded variation [see Apostol(1974) for this and some of the other statements given here] if and only if there exists a number M such that n

C /Mil < M for all partitions

A.7

THE RIEMANN-STIELTJES INTEGRAL

347

integral) is that either 1 g(x) is continuous and f(x) is of bounded variation, ox 2 f(x) is continuous andg(x) is of bounded variation.

Although we have defined the Riemann-Stieltjes integral for real-valued functions, the definition easily extends t o complex-valued functions. For details, see Apostol (1974). Alternatively, one may define the Riemann-Stieltjes integral of g(x) with respect to a function f(x) by considering the least upper and the greatest lower bounds of g(x). For this we refer the reader t o Rudin (1964) or Apostol (1 974). One can easily see that if f(x) = bx, where b is a constant, the RiemannStieltjes integral reduces to the usual Riemann integral. Whereas the Riemann integral represents the area under g(x), the Riemann-Stieltjes integral does not do so, except when f(x) = x , in which case the two integrals coincide. To give such a geometrical interpretation t o the Riemann-Stieltjes integral, the sum may be considered as a weighted sum, weighted by the values that f(x) takes in [a, b] . As an application of the Riemann-Stieltjes integral t o the theory of probability, let us consider fx(x) t o be a density function, discrete, continuous, or mixed. Its (cumulative) distribution function or d.f. may now be written as

of the interval [a, b ] . Alternatively, f(x) is of bounded variation if and only if it can be written in the form with where the functions fl(x) and f2(x) are nondecreasing and bounded on [a, b ] . Also, if the function f(x) has a finite number of relative maxima and minima, a finite number of discontinuities, and is bounded in ] a , b [, then it is of bounded variation in ] a , b [. Now we are ready t o define the Riemann-Stieltjes integral. Let g(x) andf(x) be real-valued functions on [a, b ] . Further let ci be a point in the ith interval of a partition of the interval [a, b ] , that is, let

If Fx(x) possesses a derivative, then

with

Form the sum

The limit of this sum if it exists as AxLJXm + 0 and n -+ is called the RiemannStieltjes integral of g(x) with respect to f(x) and is denoted by

which is the usual way of writing the distribution function. The above discussion follows immediately from one of the properties of the Riemann-Stieltjes integral (see property 1 below). The Riemann-Stieltjes integral, when it exists, has a number of properties, some but not all of which are analogous to those of the Riemann integral, as may be seen from the properties stated below:

1

A sufficient condition for the existence of the limit of the sum (and hence of the

If f(x) possesses a continuous derivative on [a, b ] ,then

APPENDIX A

2

A.8

SOME THEOREMS ON LABLACE-STIELTJES TRANSFORM

where the integral on the right is in the Riemann sense. If a , and a2 are constants, then Laplace-Stieltjes Transform

I

An Abelian Theorem.

If for some real number /3 > 0,

and 3

If g is a constant in [a, b ] ,then exists for Re a! 2 0, then

Jabgdf = g [ f ( b ) - f ( a ) l . 4

Iff is a constant in [a, b J , then

2

In this book we have used the special case of this theorem when 6= 0. A Tauberian Theorem. If F(t) is increasing or stationary and

exists for Re cu > 0, and if there is some number /3 > 0 such that

6

Ifg is continuous at a and b , and f is the constant r in ] a , b [ ,then

Iim d f ( a ) = C cu-0

j a g d f = g(a)[r -f(a)l + g ( b ) t f ( b ) -rl 7

then

.

From integration by parts, one can see that

[Widder (1 946)] . Generating Function

The Laplace-Stieltjes Transform

1

The definition of the Laplace-Stieltjes transform is analogous to the definition of the Laplace transform. Whereas the latter corresponds to the Riemann integral, the former corresponds to the Riemann-Stieltjes integral. Thus the function g(cu), defined by

If limn , , a, = a and CE=, anzn converges for 1% I < 1 ,

lim 241-

2

provided the integral on the right exists for some range of values of a , is called the Laplace-Stieltjes transform of g(x).

Abel's Theorem. then

[Titchmarsh (1 939)j . Tauber's Theorem. If

and

[

M

(I

-2)

C

n=o

anzn

!

= a

APPENDIX A

A.10

INVERSION OF A DISCRETE TRANSFORM

lim n(an - a ~ - , ) = 0 , n+-

then lim a, = a

ittaker and Watson (1927)j.

STA

We give below outlines of the solution procedure of two types of difference equations which have frequently been used in the theory of queues. They are:

B

Linear difference equations with constant coefficients. First-order linear difference equations with variable coefficients.

For more details on these and other difference equations we refer the reader to the publications mentioned below.

Consider the following first-order difference equation with variable coefficients: f,-a&,-,

An equation of the type

A linear combination of all solutions t o the homogeneous equation, plus A particular solution to the nonhomogeneous equation.

The solution to the homogeneous part of the equation proceeds along the following lines. Letting f, = C y X in the homogeneous equation leads t o a,cyx+n

+ a l ~ y x + n -+ l . . . + a , - l ~ y X ' l +a,CyX = 0

and a,yn + a , -yn-l f

= b,,

x = 1 , 2, . . . , n .

The general solution of this set of equations is

where the ai are known constants, fi are unknown functions t o be determined, and b, is a given function of x , is called a nonhomogeneous linear difference equation of order n. If b, = 0, for all x, then it is called the homogeneous linear difference equation with constant coefficients. A general solution to the above nonhomogeneous equation consists of two parts:

2

+

hinear First-Order Difference Equation with Variable Coefficients

Linear Difference Equation wit Constant Coefficients

1

+

+

A.

2

+

If a root, say y I , is repeated m 1 times, then the term C1y f in the solution is replaced by (Clo C l l x C I 2 x 2+ . . . C1, x + Cl,xm) y f . Since the complex roots occur in pairs, if need be, two terms corresponding to complex conjugate roots y = a +- ib may be combined into 1 y lX (c cos x0 d sin x 0 ) (0 = arc tan bla). The constants Cij,c, d, . . . must be chosen so that the solution satisfies the boundary or initial conditions on the f,. So far we have discussed the solution procedure for the homogeneous difference equation. Several special methods are used to find a particular solution of the nonhomogeneous difference equation. One such method is the method of undetermined coefficients, another one is the z-transform method, still another one is by appropriately guessing from the form of the function f,. Since the particular solutions have not been used in the text, we omit their discussion and refer the interested reader t o the publications mentioned below.

+

n-+m

$

.

.++,-,y + a ,

=:

0.

This last equation in y, being an nth-degree equation, gives n roots (real or complex, distinct or coincident). As a consequence, assuming that the roots are distinct, the general solution of the homogeneous part is written as

where

The details of the solution are given in the text [Milne-Thomson (1933), Jordan (1965),and Hildebrand (1965)l.

lSCRETE TRANSFORM Let us suppose that an r.v. X assumes only nonnegative integer values. Let Ph = P(X = k), k = 0 , 1 , 2 , . . . ,N . The mth binomial moment of X is defined by

If the moments b , determine (Pk uniquely, then we have

APPENDIX A

Proof:

Putting the values of b , in the right-hand side of the second equation, we

since

REFERENCES Ahlfors, L. V. (1966). Complex Analysis, 2nd ed. McGraw-Hill, New York. Apostol, T. M. (1974). Mathematical analysis, 2nd ed. Addison-Wesley, Reading, MA. Churchill, R. V. (1960). Complex variables and applications, 2nd ed. McGraw-Hill, New York. Feller, W. (1968). An introduction t o probability theory and its applications, vol. 1 , 3rd ed. Wiley, New York. Green, S. L. (1939). The theory and use of the complex variable. Pitman, London. Hardy, G. H. (1949). Divergent series. Oxford Univ. Press, Oxford. Hildebrand, F. B. (1965). Methods o f applied rnnthematics. 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. Jordan, C. (1965). Calculus o f finite differences, 3rd ed. Chelsea, New York. Kleinrock, L. (1975). Queueing systems, vol. 1 . Wiey, New York. Milne-Thomson, L. M. (1933). The calculus offinite differences. Macmillan, London. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Prabhu, N. U. (1965). Queuesand inventories. Wiley, New York. Rudin, W. (1964). Principles of mathematical analysis, 2nd ed. McGraw-Hill, New York. Titchmarsh, E. C. (1939). Theory of functions, 2nd ed. Oxford Univ. Press, London. Whittaker, E. T., and G. N. Watson (1927). A course o f modern analysis, 4th ed. Cambridge Univ. Press, London. Widder, D. V. (1946). The Laplace transform. Princeton University Press, Princeton, NJ.

As one might expect, the study of queues in which both arrival and service take place in batches is not easy. These bulk-arrival, bulk-service queues have all the complexity of the bulk queues discussed in Chapters 3 to 6. TheSe queues also have some additional complications due to the necessity of breaking up arrival batches in order to construct service batches. Most of the literature we have found on these queues uses techniques which are now familiar from their use in Chapters 3 and 4 for queues with bulk arrival or bulk service. It is sometimes useful to use two or more techniques for a single model, for example, the Erlang method for the arrivals and the imbedded Markov chain method for the services. Dagsvik has presented analytical methods for bulk-arrival, bulk-service queues which are very general, but mathematically and computationally very difficult. Since we found no general method for bulk-arrival, bulk-service queues that we could present at the mathematical level of this book, we have omitted these queues from our discussion. We provide below, however, a list of publications for readers interested in such systems.

REFERENCES Atfa, A. S. (1982). Time-inhomogeneous bulkserver queue in discrete time: A transportation type problem. Oper. Res. 30,650-658. Bagchi, T. P. (1971). Contributions to the theory o f bulk queues. Ph.D. Thesis, University of Toronto, Toronto, Ont. Bagchi, T. P., and J . G. C. Templeton (1972). Numerical methods in Markov chains and bulk queues. Lecture Notes in Economics and Mathematical Systems, 72, Springer-Verlag, New York. and (1973a). Finite waiting space bulk queueing systems. J. Eng. Math. 7,313-317. -and G / l , K bulk queueing system. J. Appl. (1973b). A note on the M ~ / Y Prob. 10,901-906. and (1974). Some finite waiting space bulk queueing systems, in Mathematical methods in queueing theory. Lecture Notes in Economics and Mathematical Systems, 98, Springer-Verlag, New York, 133-137. Bhat, U. N. (1964a). Imbedded Markov chain analysis of single server bulk queues. J. Aust. Math SOC.4,244-263.

354

APPENDIX B

(1964103. On the busy period of a single server bulk queue with a modified service mechanism. Calcutta Stat. Assoc. Bull. 13, 163-171. Borovkov, A. A. (1965). Some Limit theorems in the theory of mass service. Theory Prob. Appl. 10,375-400. Borthakur, A. (1975). On busy period of a bulk queueing system with a general rule for bulk servicc. Opsearch 12,40-46. . - and J. Medhi (1974). A queueing system with arrival and service in batches of variable size. Cah. Cen. thud. Rech. OpQr. 16, 117-126. Chaudhry, M. L.,and J. G. C. Templeton (1972). The theory of bulkarrival bulk-service queues. Opsearch 9, 103-121. Chiamsiri, S., and M. S. Leonard (1981). A diffusion approximation for bulk queues.Manage. Sci. 27,1188-1199. Cohen, J. W. (1980). The single server queue, 2nd rev. ed. North-Holland, Amsterdam. Dagsvik, J. (1975a). The general bulk queue as a matrix factorization problem of the WienerHopf type. Part I. Adv. Appl. Bob. 7,636-646. (1975b). The general bulk queue as a matrix factorization problem of the WienerIIopf type. Part II.Adv. Appl. h o b . 7,647-655. Delbrouck, L. E. N. (1970). A feedback queueing system with batch arrivals, bulk service and queue dependent service time. J. Assoc. Comput. Mach. 17, 314-323. Gaur, R. S . (1972). A limited queueing problem with any number of arrivals and departures. Rev. Fr. Autom. Inf. Rech. Opkr. 6, 87-93. (1973). An intermittent M I ( ~ ) / G ( ~ ) /system I with multiphased capacity of the service channel. Rev. Fr. Autonz. Inf Rech. OpQr.7,97-106. Goldman, A. J. (1968). Fractional container-loads and topological groups. Oper. Res. 16, 1218-1221. Hirasawa, K. (1971). Numerical solutions of bulk queues via imbedded Markov chain. Elec. Eng. Jpn. 91,127-136. Keilson, J. (1962). The general bulk queue as a Hilbert problem. J. R. Stat. Soc. Ser. B 24, 344-358. Kinney, J. R. (1962). A transient discrete time queue with finite storage. Ann. Math. Stat. 33, 130-136. Lambotte, J. P. (1968). Processus semi-markoviens et files d'attente, Cah. Cen. h i d . Rech. Opkr. 10, 21-31. LeGall, P. (1962). Les systenzes avec ou sans aitente et les processus stochastiques, vol. 1. Dunod, Paris. Loris-Teghem, 9. (1966a). Un systkme d'attente B arrivbes et services en groupes d'effectif albatoire. Cah. Cen. h i d . Rech. Opkr. 8, 179-191. (1966b). Systbmes d'attente A plusieurs guichets et B arrivbes en groupes d'effectif albatoire. Cah. Cen. E'tud. Rech. Opkr. 8, 98-1 11. (1966~).Condition nbcessaire d'ergodisme pour un processus stochastique lib B un systbme d'attente B arrivbes et services en groupes d'effectif albatoire. Bull. Acad. R. Belg. 52, 382-389. Madan, K. C. (1976). Interrupted service queueing with arrivals and departures in batches of variable size and genera1 repair time distribution. Math. Operationsforsch. Stat. 7, 139-149. Miller, R. G. (1959). A contribution to the theory of bulk queues. J. R. Stat. Soc. Ser. B 21, 320-337. Murari, K. (1972). A queueing problem with arrival and service in batches of variable size. Metrika 19, 27-35.

REFERENCES

355

Powell, W. B. (1981). Stochastic delays in transportation terminals - New results in theory and application o f bulk queues. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA. Prabhu, N. U. (1965). Queues and inventories - A study o f their basic stochastic processes. Wiley, New York. Rana, R. K. (1968). On a certain type of bulk queueing problem with general service time distribution. 2. Angew. Math. Mech. 48,495-497. Rao, S. S. (1965). Bulk queues with arbitrary arrivals and exponential service time distributions. Def Sci. J. 15, 1-9. Schellhaas, V. H. (1971). Ein Algorithmus fiir das zeitabhHngige Verhalten eingebetteter Markov-Ketten bei den Wartesystemen M ( ~ ) / G und ~ / ~G I ~ / M ( ~ ) /Unternehmens~. forschung 15,229-239. Sharda (1968). A queueing problem with intermittently available server and arrivals and departures in batches of variable size. 2. Angew. Math. Mech. 48,471-476. (1970). A discrete time limited space queueing problem with baich arrivals forming a Markovian chain and service in M parallel channels in batches of variable size. Cah. Cen. ktud. Rech. Opkr. 12,178 -1 94. --- (1979). Apriority queueing problem with intermittently available phase type service. Cah. Cen. ktud. Rech. Opdr. 21,191-205. Suzuki, T., and Y. Yoshida (1971). Inequalities for many-server queue and other queues. J. Oper. Res. Soc. Jpn. 13, 59-77. Syski, R. (1960). Introduction t o congestion theory in telephone systems. Oliver and Boyd, London. Teghem, J., J. Loris-Teghem, and J. P. Lambotte (1 969). Modkles d kttente MIGII et GI/M/I & arrivdes et services en groupes. Lecture Notes in Operations Research and Mathematical Economics, 8, Springer-Verlag, New York. Watanabe, M., H. Miyahara,and T. Hasegawa (1977). An analysis of passenger queues at stations in series. Proceedings o f the 7th International Symposium on Transportation and Traffic Theory, Japan, 717-742. .

GLOSSARY O F SYMBOLS

357

Probability density functidn of service time, b(t) = dB(r)/dt nth binomial moment of a random variable X,

where Pk = P(X = k) Laplace transform of b(t), b(a) = J," e-nfb(t) d t Expected (mean or average) service time, = j t dB(t) = 1 / p Constant Square of the coefficient of variation of arrival group size,

Following is a list of frequently occurring mathematical symbols together with their definitions. Symbols which occur infrequently, in isolated sections or problems, are not always included. I t is possible that the symbols defined below may sometimes have other definitions in isolated portions of the text. English symbols precede Greek symbols. Symbols from each alphabet are given in their own alphabetical order. Nonliteral symbols are listed after both the English and the Greek symbols. A?(~)/B? b/clM

= 0,"/(q2

Square of the coefficient of variation of service,time, = (pub)' 1 Number of servers in multiserver queuing system 2 Constant 3 As superscript, denotes complementary distribution function, FC(x) = 1 - F ( x ) Probability that customer enters in the rth phase in a modified Erlangian distribution Probability generating function of (c,) Expected value (mean or average) of c, = Zrc, Same as d.f. 1 Arbitrary constant 2 deterministic (constant) interarrival-time or service-time distribution Distribution function Expected value

The above notation represents a c-server queue with:

A,(t) = interarrival time distribution with arrival rate depending on t , n (if t , n in A,(t) are missing, it means arrival rate is constant) X, = arrival group size distribution with group size probability depending o n n (if n in X, is missing, it means group size probability is independent of n ) = service time distribution with service rate depending o n n (if n is missing B, in B,, it means service rate is constant) a b

M

= quorum for service group = capacity for service group = storage capacity (if last descriptor is missing, it is assumed t o be infinite)

Most of the models are covered by the above notation. The only apparent exception t o the above notation is the priority queuing system model discussed in Section 3.6. The notation for that model is confined t o Section 3.6 and is not used elsewhere. A(t)

A (z) a (t) ?(a) am -a a B(f)

Cumulative distribution function of interarrival (renewal) time U such that P ( U < t) = A ( t ) Probability generating function of (am ) Probability density function of interarrival time, a([) = dA (t)/dt Laplace transform of a ( t ) , ?(a) = e-fffa(t) d t P(arriva1 group size = rn) Expected (mean or average) value of arrival group size, = E ma, Expected interarrival time, t dA(t) Cumulative distribution function of service time V such that P ( V G t ) = B(t)

jy

=I

GCD GI

Exponential distribution Erlang-k, k fixed, distribution for interarrival or service time Erlang-J, J a random variable, distribution for interarrival or service time Exponential function, exp (x) = ex First-come, firstserved queue discipline P ( X G x ) . The notation Fx(x) may be replaced by F(x) if it is clear whatzandom variable X is under consideration; similarly fx(x) and fx(a) may be replaced by f(x) and f(a), respectively Complementary distribution function, F$(x) = 1 - Fx(x) d F x ( x )/dx E I ~ -=~j eJ - = d ~ ~ ( x ) General probability distribution of service times; independence is usually assumed Greatest common divisor General independent probability distribution of interarrival times

APPENDIX C

Actual ~dle-timerandom variable Number of phases in system Virtual (including zero idle time) idle-time random variable If and only if Independent identically distributed random variables Probability generating function of {kj) Probability of j arrivals (departures) during a service (interarrival) period E ( N ) , steady-state expected (mean or average) number in system at random epoch E(Nq), steadyatate expected number in queue at random epoch ~(fl), steady-state expected number in system at (immediately after) departure epoch E ( N - ) , steady-state expected number in system at (immediately before) arrival epoch Laplace transform Limit of a continuous function f ( t ) as t * Natural logarithmic function or logarithm t o the base e Eaplace-Stieltjes transform 1 Exponential interarrival- or service-time distribution 2 Capacity of waiting room Number in system at time t + steady-state number in system at a random epoch (This definition is not valid in Section 5.4.) Number in queue at time t + steady-state number in queue a t a random epoch Number in system at time t = a, i- 0 +-steadystate number in system immediately after a departure epoch Number in system at time t = o; - 0 + steady-state number in system immediately before an arrival epoch [ f(x)/g(x)] = K, f(x) = O [g(x)] as x + a if and only if Iim,

1 2

Iv iff i.i.d. r.v.'s l((z) kj

L.T. limr - f ( t ) In (-1 E.-S.T.

A4

-+

,,

IKl<-

Pij Pi, j

p,(t) Pz, Pi,P,,P,

f(x) = o [g(x)l a s x -t a if and only if lim, [ f(x)/g(x)] = O Bold face denotes a matrix of one-step transition probabilities, = [Pij or a vector One-step transition probability of going from state i t o state j 1 Steady-state joint probability for Erlang models of i customers (or phases) in system together with the customer being in the jth service (or arrival) phase 2 Steady-state joint probability of i in queue and j servers busy in multiserver queuing system Probability that n customers are in system at time t Steadystate probability of n in system immediately after an arrival epoch, after a departure epoch, before an arrival epoch,

GLOSSARY OF SYMBOLS

a t a random epoch, in that order Probability generating function of (P,(t)) Laplace transform of P(z, t) Steady-state joint probability generating function Laplace transform of P(z, x ; t) Steady-state number of phases in system at (immediately before) an arrivai epoch Probability density function Probability generating function Probability mass function Probability generating function of (P,), (Pi),(Pi),If':), in that order Real part of a complex variable oc Random variable Steady-state difference between service time and interarrivai time, S = V - U Interarrival time between arrivals at 0:-, and 0:. Service time of kth customer Time spent in system in the steady-state case Time spent in queue by the first customer of a group in the steady-state case (subscript 1 is often omitted in case of single arrivals) Time spent in queue by a random customer of an arrival group in the steady-state case Cumulative distribution function of the virtual [actual] waiting time in queue of the random customer of an arrival group Expected virtual [actual] waiting time in queue of a random customer of an arrival group Expected virtual [actual] waiting time in queue of the first customer of an arrival group (subscript 1 is not used in case of single arrivals) Probability density function of virtual [actual] waiting time in queue of a random customer Probability density function of virtual [actual] waiting time of the first customer t o be served in an arrival group (This definition is not valid in Section 3.6.) Laplace-Stieltjes transform of the distribution function of the virtual [actual] waiting time in queue of a random customer Laplace-Stieltjes transform of the distribution function of the virtual [actual] waiting time in queue of the first customer t o be served in an arrival group (This definition is not valid in Section 3.6.) z-transform variable

APPENDIX C

Greek Symbols

Parameter of the Laplace or Laplace-Stieltjes transform Infinitesimal interval of time Dirac delta function 6 (x) = 0 for all x other than x = 0, is such that :J 6 (x) d x = 1 ; as a consequence e-ax 6 (x) dx =1

A Kronecker symbol, =

(b, =

= otherwise Conditional service rate, ~ ( x ) b (x)/(l - B ( x ) ) Expected (mean or average) arrival rate independent of t , = I/a Expected (mean or average) arrival rate at time t Effective input rate Expected (mean or average) service rate per server, = l l b

bl

0)

. b2 .

'

bk (product notation)

Traffic intensity, utilization factor, or congestion index Epochs (instants) of departures (arrivals) ordered by the temporal parameter n An epoch immediately after a departure (before an arrival) Variance of arrival group size Variance of service time Variance of a random variable X Time; sometimes T is used in dimensionless form as a ratio of mean service time t o mean interarrival time, particularly in Chapter 6

Nonliteral Symbols

[.I

1

Greatest integer contained in argument, [XI = n , n n + l

2

(,I-

Simple brackets 3 Matrix notation min (0, .),as in (X)- = min (0, X )

=

(*I+

(

>

0,

XGO

x>o

max (0, .),as in (x)+= max (0, X )

=

*

(x. [o.

X, Usually denotes convolution Superscript parenthesis:

XO

<x <

GLOSSARY OF SYMBOLS

361

1 Order of differentiation 2 Order of convolution Bar over symbol indicates transform of a function "given," as in P(A IB) = Prob(event A given event B) Divisiori, as in a/b = a f b Approximately equal t o Set membership Such that Implies Implies and is implied by

Denotes the number of combinations of n objects taken r at a time, ! / ( ! ( r)!),

nr

Less than or equal t o Greater than or equal t o An interval (simply connected subset of the real line) open o n the left but closed on the right, ( x : c < x [. , . [, [. , .I, and 1. , .[ are interpreted similarly One of the definitions of the exponential function


Authors referred to in the text by year generally are given here. Abol'nikov, L. M., 240, 249, 282, 292 Abramov, A. Kh., 203, 234 Abramowitz, M., 14, 17, 18, 20, 32, 90, 104 Adiri, I., 175 Ahlfors, L. V., 344. 352 Aiagaraja, K., 203, 234 Alfa, A. S., 353 Allan, S. E. A,, 325, 340 Apostol, T. M., 347, 352 Arndt, U., 326, 341 Badii, L., 14, 33 Bagchi, T.P., 353 Bahadur, R. R., 125, 175 Bahary, E. S., 203, 234 Bailey, N . T. J., 20, 32, 107, 125, 175, 185, 186, 203, 209, 232 Baiiy, D. E., 275, 294 Barber, B., 121-122, 156, 175 Bartlett, M. S., 40, 104 Beightler, C. S., 3, 32 Bhalaik, H. S., 151, 177 Bharucha-Reid, A. T., 46, 104 Bhat, U. N., 84, 104, 150, 151, 177, 178, 187, 202, 226, 232, 234, 353, 354 Blake, F., 1, 32 Bloemena, A. R., 185, 232 Borovkov, A. A,, 354 Borthakur, A., 203, 226, 228, 232, 233, 234, 275, 294, 354 Brockmeyer, E., 8, 32, 107, 175 Brockwell, P. J., 127, 175, 295, 340 Bunday, B. D., 32 Burington, R. S., 222, 232 Burke, P. J., 93, 104, 172, 175, 249, 284, 292 Cadzow, J. A., 3, 32 Carignan, P. L. B., 325, 340 Chakravarthy, S., 151, 178 Champernowne, D . G., 125, 175 Chandra, M. J., 325, 341

Chang, W., 202, 234 Chaudhry, M. L., 16, 33, 151, 172, 176, 177, 185, 202, 226, 230, 233, 224, 249, 267, 281, 284, 293, 296, 340, 341, 354 Chiamsiri. S., 354 Chung, K. L., 46, 104 Churchill. R. V.. 344, 352 Cinlar, E., 40, 84, 104 Clarke, A. B., 125, 176 Cobham, A., 141, 176 Cohen, 3. W., 71, 84, 104, 114, 176, 274, 292, 354 Conolly, B. W., 71, 84, 104, 121, 156, 158, 176, 304, 326, 340 Conte, S. D., 15, 32, 330. 340 Cooper, R. B., 71, 105 Copson, E. T ., 20, 33 Cosmetatos, G. P., 275, 294 Cox, D. R., 40, 57. 73, 105, 259, 261, 292, 295, 340 Crabill, T. B., 129, 176 Crane, M. A., 202, 234 Craven, B. D., 203, 235 Cromie, M. V., 16, 33. 249, 267, 281, 284, 293 Crommelin, C. D., 153, 176, 258, 293 Dafermos, S. C.. 71, 105 Dagsvik, J., 354 Dave, H. B., 203, 235 Deb, R. K., 203, 235 de Boor, C., 15, 32, 330, 340 Delbrouck, L. E. N.. 354 Dick, R. S., 190,233 Doob, J. L., 41, 105 Douglas, J. B., 16. 33 Downton, F., 186,205, 208, 233 Dynkin, E. B., 42, 105 Easton, G., 230, 233, 326, 340 Eilon, S., 172, 176

Erdklyi. A,. 14. 33 Erlang, A. K., 153, 176, 258, 293 Ernst. M., 134

Holman. D. F.. 223, 226, 233, 249, 293, 296, 341 Ivnitskiy, V. A.. 133, 176, 203, 235

Fabens. A. J., 192, 233, 274, 293, 322, 341 Fainberg,M.A., 138, 151, 176,177 Fang Wu, see Wu Fang Feller, W., 1, 16, 23, 33, 46, 54, 73, 78, 105, 342, 352 Finch. P. D., 57, 105, 168, 176, 202, 209, 216, 233, 295, 341 Forsyth, A. R.. 20, 33 Foster, F. 6 . . 121, 151, 176, 177, 216, 233, 296, 326. 341 Frank. H.. 202, 203, 235 Frankel. E., 185, 234 Franken, P., 82. 105, 326, 341 Fujisawa, T., 232, 233 Fukuta, J . , 202, 235 Gani, J.. 40. 105 Garber. H. N., 134, 176 Gaur. R. S., 354 Gaver, D. P., 46, 105, 107, 113, 116, 176 Georganas, N. D.. 203, 235 Ghare, P. M., 267, 293 Ghosal. A., 180. 209, 226, 233, 296. 341 Gnedenko, B, V., 9, 33 Goldman, A. J.. 354 Goyal, J . K., 151, 177, 203, 235 Grassmann, W. K., 16, 33, 249, 281, 284, 293 Green, S . L., 343, 352 Griffin, W. C., 222. 233 Gross, D.. 45, 54, 105, 164, 176, 209, 233, 295, 341 Gupta, S . K.. 111, 151, 176, 177 Halstrbm, H. L.. 8, 32. 107, 175 Handa, J. M.. 203, 235 Hardy. G . H.. 350, 352 Harris, C. M., 45. 54, 105, 132, 164, 176, 203, 209, 233. 235. 295, 341 Hasegawa, T., 355 Hawkes, A. G., 150, 176. 249. 293 Henderson, J . C., 290, 293 Henderson, W.. 71, 105 Henrici, P.. 15, 33 Hildebrand. F. B., 351, 352 Hille. E.. 85, 105 Hinkley, D. V., 259, 261, 292 Hirasawa, K., 354

~ a c k s o n R. , R. P., 134, 164, 165, 176, 290, 293 Jam, J . L., 151, 177, 202, 235, 304, 341 Jaiswal, N. K., 46, 57, 63, 105, 150, 176, 185, 186, 207, 209, 233 Jensen. A,. 8, 32, 107, 175 Jewell, W. S., 172, 176 Jordan, C., 262, 293, 351, 352 Jury, E. I., 3, 33 Kabak, I. W., 249, 283, 285, 293 Kambo, N. S., 151, 177 Karlin, S., 40, 54, 73, 79, 105 Kashyap, B. R. K., 203, 235 Kaufman, H., 14, 33 Keilson, J.. 151. 177, 354 Kemeny, J . G., 46, 105 Kendall, M. G., 14, 33, 87, 105 Kerridge, D., 151, 177, 295, 341 Khintchine, A. Y., 62, 105, 207, 233, 296, 341 Kingman, J . F. C.. 2, 33, 84, 105 Kinney, J . R., 354 Kleinrock, L., 14, 33, 39, 40, 54, 55, 72, 84. 105, 151, 177, 343, 352 Kolesar, P., 203, 234 Kolmogorov, A. N., 9, 33 Kosten, L., 57, 105 Kotiah, T. C. T., 203, 235 Krakowski, M., 72, 106, 172, 176 Kuczura, A,, 258, 288, 293 Laha, R. G., 13, 33 Lambotte, J . P., 354, 355 Ledermann, W., 111, 125, 176 Lee, A. M., 202, 234 Le Gall, P., 354 Leonard, M. S.. 354 Levy, P., 41, 106 Lindley, D. V., 54, 57, 106, 168, 176 Lippman. J . A , , 151, 177 Little. J . D. C., 176 Lo&ve,M., 2, 33 Loris-Teghem, J., 354, 355 Love, R. F., 274, 293 Lukacs, E., 13. 33 Luke, Y. L., 20, 33 Lwin, T., 209, 233

McMahon, G. B., 203, 236 Madan, K. C., 354 Magnus, W., 14. 33 Markin, P. G., 203, 235 Marshall, K. T., 340, 341 Maxwell, W. L., 172, 176 May, D. C., 222, 232 Medhi, J., 226, 228, 233, 275, 294, 354 Meisling, T., 71, 106, 193, 197, 233 Mercer, A., 203, 233 Miller, H. D., 40, 105, 150 Miller, R. G., 354 Milne-Thomson, L. M., 351, 352 Mitten, L. G., 3, 32 Miyahara, H., 355 Miyazawa, M., 326, 341 Mohan, C., 203. 235 Mohanty, S. G., 151, 177, 203, 235 Moore, S. C., 151, 178 Morse, P. M., 114, 125, 134, 176, 216, 234 Muntz, R. R., 151, 177 Murao, H., 203, 235 Murao, Y., 202, 209, 234, 235 Murari, K., 151, 178, 203, 236, 275, 294, 354 Nadarajan, R., 275, 294 Nair, S. S., 202, 228, 234, 235 Nakamura, G., 151, 178, 202, 203, 235 Narasimham, G. V. L., 151, 178 Natarajan, R., 197, 234 Nemhauser, 6 . L., 3, 32 Neuts, M. F., 13, 33, 71, 73, 78, 105. 106, 150, 151, 178, 202, 223, 228, 234, 235, 275, 294 Neveu, J., 2, 33 Newell, G. F., 84, 106 Nickols, D. G., 134, 164, 176 Novaes, A., 185, 234 Nyunt, K. M., 216, 233, 296, 341 Oberhettinger, F., 14, 33 Palm, C., 52, 106 Parzen, E., 13, 23, 33, 40, 45, 106, 285, 293. 297, 341, 342, 352 Pearce, C., 275, 294 Pegden, C. D., 125, 177 Perera, A. G. A. D., 192, 233, 274, 293, 296, 34 1 Piaggio, H. T. H., 20, 33 Pike, M. C., 122, 177 Pollaczek, F., 258, 293

Polya, G., 3, 33 Polyak, D. G., 275, 294 Powell, W. B., 355 Prabhu, N. U.,40, 54, 71, 73, 84, 106, 164, 177, 180, 211, 234, 304, 316, 317, 341, 342, 352, 355 Rainville, E. D., 90, 106 Ramaswami, V., 151, 178 Rana, R. K., 355 Rani, R. G., 304, 341 Rao, S. S., 355 Regis, R. C., 57, 106 Restrepo, R. A,, 114, 177 Reuter. 6 . E. H., 111, 125, 176 Reynolds, J. F., 277, 293 Riordan, J., 283, 294 Roberts, G . E., 14, 33 Rodemich, E., 151, 177 Roes, P. B. M., 274, 294 Romanovsky, V. I., 48, 106Rosenlund, S. I., 151, 178 Rosenshine, M., 125, 177 Ross, S. M., 116, 151, 177 Rudin, W., 347, 352 Runnenburg, J. Th., 202, 235 Russell, A., 15, 33 Saaty, T. L., 164, 177, 290, 294 Sack, R. A., 111, 125, 177 Sahbazov, A. A,, 172, 177, 277, 294 Schellhaas, V. H., 355 Schmidt. V., 325, 341 Seal, H . L., 40, 106 Seneta, E., 48, 106 Serfozo, R. F., 203, 235 Shanbhag, D. N., 122, 151, 177, 178, 240, 294 Shanthikumar, J . 6..325, 341 Sharda, 151, 178, 203, 235, 275, 294, 355 Sharma, S. D., 151, 178 Shyu Kwang-Huei, 274, 294. See also Xu Guang-Hui Sim, S. H., 275, 294 Singh. V. P., 190. 234 Smith, W. L., 54, 78, 79, 106, 165, 177, 180, 234, 295. 340 Snell, J . L.. 46, 105 Soriano, A., 151, 172. 177, 178 Stegun, I. A., 14, 17, 18, 20, 32, 90, 104 Steinijans, V. W.. 96. 106 Stidham, S., 172, 177, 180, 234 Stuart, A., 14, 33, 87, 105

Stuart, I. M., 203, 236 Suzuki, T., 122, 177. 355 Syski, R.. 355

Van Dantzig. D., 185, 234 Van Hoorn, M. H., 151, 178 Watanabe, M , 355 Watson. 6 N 344, 345, 350, 352 Waugh, W A O'N , 203, 235 We~ss,H J , 203, 236 Whittaker, E T , 344, 345, 350, 352 W~dder,D V , 349, 352 Wtshart, D M G , 46, 71, 106, 196, 234, 302. 304, 341 Wu Fang, 275, 294, 304, 341

.

TakBcs, L.,40. 63, 64, 73, 78, 84. 106, 11 1. 122, 127, 150, 177, 178. 192, 224,225, 234. 275. 292, 294, 296, 304, 311, 317, 319. 341 Taylor. W. M., 40, 54, 73, 79, 105 Taylor, R. 6 . . 15, 33 Taylor. S. J., 2. 33 Teghem. J., 355 Templeton, J. 6 . C., 185. 202, 226, 233, 234, 275, 294, 296, 340, 353, 354 Thompson, J. W., 203, 235 Titchmarsh, E. C., 345, 349, 352 Tricomi. F. G.. 14. 33 Tsvirkyn, A. D., 203, 234

Xu Guana-Hut. 275, 294. See also Shyu Kwang-Huei Yao, D. D. W., 151, 178 Yasnogoridskiy, R. M., 249, 292 Yoshida, U.,355

The presence of subscripts and superscripts makes it difficult to arrange the queuing models given below in an unambiguous alphabetical order. Recall that in queuing models the exponential distribution of interarrival time or servicc time may be denoted by M, E or El interchangeably. Alternating renewal process, 82-84 Applications: b~ologicalscience, 23. 32 bus, multichannel bulk-arrival queue, 282 bus, renewal process, 80 car ferry, 179, 202-203 computers, 35, 216-219 computing systems, 202 dams, 180, 21 1 equipment failure, 82 guided tours, 179 multichannel bulk queues, 237 processor-sharing, 151 taxis and customers. 203 telephone systems, 283 toll gate, 37 traffic, 23, 179 trains, 179 transit system, 218-221 transportation networks, 202 see also Examples Autocorrelation. 86 Bulk-arrival, bulk-service queues, 353-355 Bulk-arrival queues. 107-178 Bulk-service queues, 179-236 Busy and idle periods, expected, 322-325 E d G / l , 325 G I / E J l , 323 GIr/M/I, 322 M / E $ ~ / ~324-325 , M / G ~ / I and M / G ~ / ~325 , M / G ~ ' ~ /324 ~, MX/G/1, 324

Characteristic equation, 67 Characteristic funct~on,13 Collective marks, method of, 185 Coefficient of variation, of deterministic r.v., 28 of Erlang r.v., 28 of exponential r.v., 28 of hyperexponential r.v., 28 Concentrated service, 202 Conservation principle, 71-72 Convolution, 4-8. 24 Cumulant generating function, 13

Dr/M/l, 156 DB/M/c, 258-261 delay measure, 260 limiting behavior of N,, 258-260 Differential equations: partial, 20, 31-32 solution by Lagrange's method, 20 Differentiation, definite integral, 342 Dirac delta function, 114 Dirichlet series. 13 Discrete time analysis, 71. See also M/GB/1, imbedded Markov chain Distribution: binomial, 22, 125 busy period, 39 chi-square, 8 deterministic, 30 Erlang: generalized, 24, 133 modified, 22, 30, 108

SUBJECT INDEX

ordinary, 7-8, 22, 25, 133, 139, 184, 228, 229. 324 exponential, 7, 9, 25 gamma, 8 geometric, 23, 113, 278 hyperexponential, 9, 27-28 idle period. 39 negative binomia!, 22 Poisson, 11, 23, 35, 75 regular, 30 renewal, 74-75 system length, 38 truncated. continuous, 9-10, 29 truncated, discrete, 10-11, 23, 29 waiting time, 38 see also Life Double-ended queue, 203 E / G ~ / ~309-316 , queue size N at arbitrary instant, 310 queue size N- before arrival, 310 queue size N' after departure, 31 1 relations among p.g.f.'s of N-, N, N', 31 1-316 see also M / G ~ / ~ E d E d 1, 308-309, 320-321, 332-334 EkG/l, 316-317 expected busy and idle periods, 325 E d G / l and E/Gk/1, 317-321 EdG1/1, 321-322 EVEI/I, 155-156, 301-302 E~/E,/ 1: generalized Erlang input, 133-139, 161-166 limiting behavior of Nft), 133-137 L, Lq, 137-139 ordinary k-Erlang input, 139 special cases: E d M / l , 165 Et/M/I, 163-165, 331-332 E;/D/~, 139 E?/D/I, 162 Ef/E./l, 162 MX/D/l, 168 M X / E d l , 162, 166 E , / M ~ . ~ / I230 , Erlang: A. K. Erlang. 7 delay system, 285

distribution, 8, 23, 228, 229 generalized, 24, 133 modified, 22, 30, 108 ordinary, 7-8, 22, 25, 133, 139, 184, 228, 229, 324 loss system, 285 Examples: airport luggage handling, 282 computer core storage system, 216 equipment failure, 82 G I / G / l , 53 GI/M/c, 50 imbedded Markov chains, 50 M/G/l, 50, 54-65 p.d.f. of largest r.v., 88 p.d.f. of smallest r.v., 89 Poisson process branching, 97 supplementary variable technique, 54-63 transit system, 219 see also Applications Function, 16-20 autocorrelation, 86 Bessel, 17-18 characteristic, 13 conditional rate of failure, 27 correlation coefficient, 86 Dirac delta, 114 gamma, 16-17 generating, see Generating functions hazard, 26-27 hypergeometric, generalized, 18-19 hypergeometric, ordinary, 19 intensity, 27 mean value, 75, 90 modified Bessel, 18 renewal, 75-78, 90, 91 Generalized argument principle, 344 Generating functions, 13-14 cumulant, 14 moment, 13 probability, see Probability generating function GE0M/GB/1, 193-198, 211-212 constant service time, 194 geometric service time, 196 limiting behavior of Ni,193-198 waiting time (in queue), 198

SUBJECT INDEX

GI/D/c and GI/Dc/l. 202 G I / E d l , 302-304 expected busy and idle periods, 323-324 waiting time (in queue), 327 G I / W l and GIr/M/l, 304-309, 327-329 GI/G/l. 53-54, 102-104 with delay for first customer in busy period. 56-57 Lindley's results for, 54 G I / G ~ / ~339 , GI/M/l, 65-71, 100-102, 155, 202 characteristic (operator) equation for, 67 number N- in system before arrival, 67-68 number N in system at random epoch, 68-69 virtual and actual idle time for, 69-71, 102 waiting time (in queue), 68 G I / M ~ / I , 292 GI/M'/C, 267-275, 291-292 limiting behavior of N', 267-274 waiting time (in queue), 274 GI/M'/~, 202 GIr/E/l, 296-302 expected busy and idle periods, 322-323 idle period, 337 queue size N at arbitrary instant, 298, 326, 336-337 queue size N' before arrival, 297 queue size N' after departure, 297 relations among p.g.f.'s of N-, N, N', 298-301 see also GIr/M/I GIr/M/l, 116-122, 156-158, 334-338 limiting behavior of Ni, 116-121, 297 limiting behavior of Ni, 297 limiting behavior of N(t), 298, 336-337 limiting number in system after arrival, 334-336 waiting time (in queue), 121, 156, 329 see also GIr/E/l GI'/G/I, 168-169, 338-339 G I ~ / G " ~ / I334 , GI'/M/I, 72, 158-159 GI'/M/C, 275 Increments: independent, 42 stationary, 42

Inversion: of discrete transform, 351 of other transforms, 14 Kolmogorov equations, 109 Laplace-Stieltjes transform (L-S.T.), 23, 345 Leibnitz's rule, 124 Length-biased sampling, 92. 174 Life: current, 81 excess, 80 joint distribution of past and residual, 93-94 past, 81, 91, 171 remaining, 80 residual, 80, 91 total, 91 Linear difference equations, 350-351 Little's formula, 165. 166, 172, 208, 247 M/E& . . . Ed1, 231 M/Ef/l, 184-185 M/E!/I, 314-316 M/G/l, 54-65, 217 busy period, 63-65 expected busy and idle periods, 83-84 idle period, 63-65 relation between p.g.f.'s of N, Nt,207 waiting time (in queue), 62 waiting time, Lindley's approach, 54-56 functional equation approach, 99 supplementary variable approach, 98 M/GB/1, 180-186, 203-209, 216-219 expected busy and idle periods, 325 imbedded Markov chain, 183, 203-206 limiting behavior of N,(t), 180-185 special cases: M/D*/I, 205 with variable capacity, 208 waiting time (in queue), 185-186 M/EB/I, 204, 207, 215 M/E@/I, 204. See also Ej/E,/I M / E ~ I 207 , M/G/l, 206-207, 217, 219. See also M/G/1 M/G2/1, 217 M/MB/l, 217 M/G'/I/M, 186-190, 209

SUBJECT lNDEX limiting behavior of N,. 186-190 special cases: M/D/l/M. 209 M / E d l / M , 210 M/M/I/M, 209-210, 216 M / G I~, 202 M/Gk/l, 191-193, 223-225, 340 expected busy and idle >eriods, 325 limiting behavior of NL and v?'. 191-193 waiting time (in queue), 192-193 see also E/Gk/l M / G " ~ / I . 222-228 with balking, 190 expected busy and idle periods, 324 special cases: M/E?/I, 324-325 W M B / l , 291-292 M/M'/I, 225, 331 h4/MksB/I , 225-23 1 M/Mk."/ 1, 226 M&?/I. 202 Iblk/M/c/c, 285 M/M'/I/MB. 198-202, 212-216 limiting behavior of NA,201-202 transient behavior of N',, 198-201 M/MS/c, 261-267, 289-290 limiting behavior of N,(t), 264-265, 290 transient behavior of N,(t), 261-264 waiting time (in queue), 265-266, 290 waiting time (in system), 266-267 M / M ~ . ~ / 275 C. M/Mk.."/c, 275 Mr/M/l, 116, 120-127, 329 limiting behavior of Ni, 125-126 moments, 160 number of batches served in a busy period, 126 transient behavior of Ni, 122-125 waiting time (in queue), 126 M'/D/c, 252-258, 286-289 limiting behavior of N,*, 252-254, 286-288 probability of no service delay, 255-257, 286 waiting time (in queue), 257-258, 288-289 M'/EI/I, 111, 152, 154 M'/EJ/I, 107-116, 153, 155 limiting behavior of N(t), 112 L, L,, W, 115-1 16

special cases: M/M/l, 100, 113, 115 M X / ~ / l ,163 M k / E d l , 113 M ' / E ~ I , 112-113, 115-116, 151. 154, 162, 166 M ' / E I / ~ , 111, 113, 115, 152, 154 transient behavior of N(t), 107 waiting time (in queue), 114-115 M X / ~ / l 72, , 130, 167-175 expected busy and idle periods, 324 by imbedded Markov chain, 174 by supplementary variable technique, 172 waiting time (in queue), 167-169, 249 with warm-up time, 167-172 M'/G,/I, 127-133, 161 limiting behavior of N',,127-129 modifications, 130 M:/G,/~, 133 M x / ~ ~ , / 1151 . M:,' MZ2/G1, G d 1 , 140-150, 166-167, 249 limiting behavior of joint distribution, 145 transient behavior of joint distribution, 140 waiting time (in queue), 146 non-priority customer, 148 priority customer, 146 MX/M/c, 240-252, 277-285 busy period (when c = 2), 250-252 limiting behavior of N(t), 243-245 special cases: M/M/c, 277, 289-292 Mk/M/c, 278, 281, 283 MX/M/2(p1,, ~ 2 3 285 , transient behavior of N(t), 240-243 waiting time (in queue), 245-249 MX/M/c/c, 283 MX/M/-, 24 1, 277 MX(t)/Wm, 237-240, 275-277 transient behavior of N(t), 237-240 M'/SM/I, 150 Markov chains, 41 aperiodic state of, 49 ergodic state of, 50 homogeneous, 48-52 irreducible, 48-49 periodic state of, 48-49 recurrent state of, 48

SUBJECT INDEX

stochastic matrices of, 48 transient (non-recurrent) state of. 48 Memoryless property of exponential, 37, 85, 93 Multichannel bulk queues, 237-294 Occupation time, 39, 224, 230-231 Order statistics. 86 Partial differential equations, 20, 31-32 Past life. 81, 91, 171 Phases, 8, 22, 36, 45, 108, 11 1, 133, 134 Poilaczek-Khintchine transform formula, 56 Priority queues, 140-150, 166-167 non-priority customer, 148 priority customer, 146 Probability generating function (p.g.f.). 2-3 multivariate, 5 Process, stochastic, 40-45 special cases: alternating renewal, 82 arrival, 34 bulk-arrival Poisson, 42-44, 89-90 completion time. 140 compound, 98 compound Poisson, 42, 98, 127 counting, 74 delayed renewal, 73 difference of two Poisson, 97 equilibrium renewal, 74, 96 general renewal, 73 Markov, 41 modified renewal, 73 non-Markovian, 45 ordinary renewal, 73. 75, 91, 96 Poisson, 35, 75, 84-88, 91 Poisson, by branching a given Poisson, 97 regenerative, 52, 91 stationary renewal, 74 sum of Poisson processes, 97 time-homogeneous Poisson, 42-45, 84,

Queue discipline: FCFS, 36, 39, 53, 141 head-of-the-line priority, 141 LCFS, 36

priority, 140 Quorum, 179 Random customer. 93 Random events, 88 Random variables (r.v.'s), addition of, 4. 27. 30 difference of, 24 independently and identically distributed (i.i.d.r.v.'s), 4 largest, 88 lattice. 2-3 least, 89 mean of, 3, 12 moments of. 3, 12 non-lattice, 2-3 sum of continuous i.i.d.. 6-8 sum of discrete i.i.d., 4-6 variance of, 3, 12 (X-Y)', 25-26 Regeneration points, 52 Renewal theory, 72-84 density, 76 equation, 76, 90 function, 75, 78 process, 72 Residual life, 79-80 paradox of, 80-81 Residue calculus, 101, 343 Riemann-Stieltjes integral. 345-348 Service mechanism, 36 Strong law of large numbers, 8 Techniques: imbedded Markov chain, 46-52 integral equation, 53-54 modified Erlangian, 46 phase, 45-46 supplementary variable, 57-62 Theorems: Abelian, 349 Abel's, 349 Blackwell's, 78-79, 94, 96 elementary renewal, 78 key renewal, 79, 80, 95, 96 Lagrange's, 345 Rouchk's. 345