Mathematical theory of connecting networks and telephone traffic (Mathematics in science and engineering series;vol.17)

Mathematical Theory of Connecting Networks and Telephone Traffic

MATHEMATICS IN SCIENCE A N D E N G I N E E R I N G A Series of M o n o g r a p h s a n d Textbooks

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TRACY Y . THOMAS. Concepts from Tensor Analysis and Differential Geometry. Second Edition. 1965 TRACY Y . THOMAS. Plastic Flow and Fracture in Solids. 1961 RUTHERFORD ARIS. The Optimal Design of Chemical Reactors: A Study in Dynamic Programming. 1961 JOSEPH LA SALLE and SOLOMON LEFSCHETZ.Stability by Liapunov’s Direct Method with Applications. 1961 GEORGE LEITMANN(ed.) . Optimization Techniques: with Applications to Aerospace Systems. 1962 RICHARDBELLMANand KENNETHL. COOKE. Differential-Difference Equations. 1963 FRANK A. HAIGHT.Mathematical Theories of Traffic Flow. 1963 F. V. ATKINSON. Discrete and Continuous Boundary Problems. 1964 A. JEFFREY and T. TANIUTI. Non-Linear Wave Propagation: with Applications to Physics and Magnetohydrodynamics. 1964 JULIUS TOU.Optimum Design of Digital Control Systems. 1963 HARLEYFLANDERS. Differential Forms: with Applications to the Physical Sciences. 1963 SANFORD M. ROBERTS.Dynamic Programming in Chemical Engineering and Process Control. 1964 SOLOMON LEFSCHETZ.Stability of Nonlinear Control Systems. 1965 Systems and Simulation. 1965 DIMITRISN. CHORAFAS. A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems. 1965 MARSHALL C. PEASE,111. Methods of Matrix Algebra. 1965 V. E. B E N E ~Mathematical . Theory of Connecting Networks and Telephone Traffic. 1965

In preparation WILLIAMF. AMES.Nonlinear Partial Differential Equations in Engineering A. HALANAY. Differential Equations: Stability, Oscillations, Time Lags R. E. MURPHY.Adaptive Processes in Economic Systems DIMITRISN. CHORAFAS. Control Systems Functions and Programming Approaches J. ACZEL.Functional Equations A. A. FEL’DBAUM. Fundamentals of the Theory of Optimal Control Systems DAVIDSWORDER. Synthesis of Optimal, Discrete Time, and Adaptive Control Systems (tentative). S. E. DREYFUS. Dynamic Programming and the Calculus of Variations.

MATHEMATICAL THEORY OF

CONNECTING NETWORKS AND

TELEPHONE TRAFFIC V. E. Bend BELL TELEPHONE LABORATORIES, INCORPORATED

MURRAY HILL, NEW JERSEY

ACADEMIC PRESS New York and London

COPYRIGHT 0 1965,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published by

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I J B R A R Y OF CONGRESS CATALOG C A R D

NUMBER: 65-21 156

PRlNTED I N THE UNITED STATES OF AMERICA

To John Riordan

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Preface Applications of the mathematical theory of probability in modern technology range from sampling methods in quality control to noise calculations for radio receivers. It is not widely known, though, that Some of the earliest practical successes of the theory of stochastic processes were scored ten to twenty years before that theory received its elegant foundations in measure theory at the hands of A. N. Kolmogorov in 1933. Foremost among these early successes was the application of the theory of probability to the engineering of telephone systems, and, especially, the formulation and use by A. K. Erlang of the concept of statistical equilibrium. T h e early applications of probability to telephony were followed by three important developments:

( I ) T h e journal literature saw a great proliferation of probabilistic treatments of very specific problems, usually involving small, isolated portions of telephone systems. From 1918 to 1939 these were concerned mainly with loss in gradings and with delays in multiserver queues. After the advent of crossbar switching and common control in 1938, a new class of difficult problems was added: calculation of loss due to mismatching of links. (2) With the identification, during the period 1939 to 1948, of Erlang’s concept of statistical equilibrium with the invariance of a measure for a Markov stochastic process, the early results were imbedded tightly and elegantly in the modern theory of stochastic processes. (3) A realization that the performance of a telephone exchange is determined in large part by the structure of its connecting network led, after 1950, to increased interest in the purely combinatorial properties of such networks, without reference to stochastic models for exchange operation. I n spite of the very large amount of research on telephone traffic and similar topics that has been conducted over the last 50 years, the theoretical methods in use today for the design and engineering of telephone systems often are not even u p to the theoretical standards set by A. K. Erlang in 1909-1918. This apparent paradox results from the fact that the early telephone systems were simple in structure and operation, and thus amenable to Erlang’s method. Modern systems, based on the philosophy of common control that seems best vii

...

Vlll

PREFACE

to utilize switching equipment, are large and complex ; they require taking a global, rather than a piecewise, view of the system, and so they pose much bigger problems to which the statistical equilibrium method is hard to apply. As a result, engineers faced with immediately pressing tasks of system evaluation and connecting network design have abandoned Erlang’s method in favor of admittedly less attractive, less sophisticated, and less accurate approaches for getting out some numbers. Some of these “counsels of desperation” agree well, and others poorly, with experiment or with Erlang’s method, but in the absence of demonstrably superior practical alternatives they continue to be used. Useful theoretical methods superior to those in current use must be based on a general mathematical understanding of the structure and properties of telephone systems, followed by full use of the now widely available modern tools of close approximation and rapid computation. Thus, a half-century after Erlang’s productive years, the time appears ripe for a study of telephone traffic that is systematic rather than encyclopedic, one that takes the widest possible view of the subject, and tries to expose the structural features that affect the performance of traffic systems. I t is intended that this book be the beginning of such a systematic study. Only a beginning is possible herein, because in a volume of modest size it has been necessary to forego completeness. Indeed, attention has been restricted to three kinds of problems of traffic theory: ( 1 ) Combinatorial problems of network design, ( 2 ) Probabilistic problems of traffic analysis, comprising (a) statistical problems of traffic measurement and (b) analytical problems of calculation of the grade of service, ( 3 ) Variational problems of routing traffic in networks.

Most of the text is concerned with problems (1) and ( 2 ) , but many topics are noticeable because they are absent. Thus, e.g., problems concerning delays in telephone systems, although mentioned in Chapter 1 , are not considered in later chapters. Some mathematical formulations of routing are given, and some simple problems solved in the text, but no adequate theory of routing exists as yet. T h e combinatorial aspects of the subject have been emphasized, both for their own interest and for their relevance to the probabilistic problems.

PREFACE

ix

Only a token attempt to describe the literature is made, and the reader interested in it is referred to R. Syski’s treatise (“Introduction to Congestion Theory in Telephone Systems.” Oliver & Boyd, London, 1960). T h e pursuit of modern telephone traffic theory requires an acquaintance with probability theory, stochastic processes (especially the theory of Markov processes), and combinatoric, in addition to set theory, analysis, and algebra. Since it is not possible to abstract or review these prerequisite topics without writing a much longer book, some guidance is due the reader in the form of references, in which he can look up the necessary theoretical background. These are listed at the end of the book. With the exception of Chapter 4, this book is based entirely on research papers on telephone systems written by the author at Bell Telephone Laboratories during the 10 years prior to 1964. T h e material of Chapter 4, on nonblocking networks, is drawn from a research paper of C. Clos; it ties in well with the combinatorial emphasis of the book, and is included with his kind permission and encouragement. All the papers represented in the book appeared in the Bell System Technical Journal. T h e author’s gratitude is extended to his many colleagues at Bell Telephone Laboratories. Their readings of first drafts, critical comments, helpful suggestions, and continuing interest greatly advanced the research that is reported here. T h e extensive secretarial and drafting work involved in preparation of the manuscript was provided by Bell Telephone Laboratories.

March 196.5 Murray Hill,N e w Jersey

V. E. BENES

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Contents vii

PREFACE Chapter 1

Heuristic Remarks and Mathematical Problems Regarding the Theory of Connecting Systems 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Introduction Summary of Chapter 1 Historical Sketch Critique General Properties of Connecting Systems Performance of Switching Systems Desiderata Mathematical Models Fundamental Difficulties and Questions T h e Merits of Microscopic States From Details to Structure The Relevance of Combinatorial and Structural Properties: Examples Combinatorial, Probabilistic, and Variational Problems A Packing Problem A Problem of Traffic Circulation in a Telephone Exchange An Optimal Routing Problem References

1 3 3 5 6 13 15 17 18 19

20

24 28 29 35 38 51

Chapter 2

Algebraic and Topological Properties of Connecting Networks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction Summary of Chapter 2 The Structure and Condition of a Connecting Network Graphical Depiction of Network Structure and Condition Network States The State Diagram Some Numerical Functions Assignments Three Topologies Some Definitions and Problems Rearrangeable Networks Networks Nonblocking in the Wide Sense Networks Nonblocking in the Strict Sense Glossary for Chapter 2 References

xi

54 55 56 57 58 62 64 68 69 71 75 71 19 80 80

CONTENTS

Chapter 3

Rearrangeable Networks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Introduction Summary of Chapter 3 The Slepian-Duguid Theorem The Number of Calls That Must Be Moved: Paull’s Theorem Some Formal Preliminaries The Number of Calls That Must Be Moved: New Results Summary of Sections 8-15 Stages and Link Patterns Group Theory Formulation T h e Generation of Complexes by Stages An Example Some Definitions Preliminary Results Generating the Permutation Group Construction of a Class of Rearrangeable Networks Summary of Sections 17-21 The Combinatorial Power of a Network Preliminaries Construction of the Basic Partial Ordering Cost Is Nearly Isotone on T(C,) Principal Results of Optimization References

82 84 86 88 89 90 96 98 100 102 103 105 106 109 113 119 121 125 127 128 130 135

Chapter 4

Strictly Nonblocking Networks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction Square Array Three-Stage Strictly Nonblocking Connecting Network Principle Involved Five-Stage Network Seven-Stage Network General Multistage Switching Network Most Favorable Size of Input and Output Switches in the Three-Stage Network Most Favorable Switch Sizes in the Five-Stage Network Search for the Smallest N for a Given n for the Three-Stage Network Cases in the Three-Stage Network Where N = r(mod n) Search for the Minimum Number of Crosspoints between N = 23 and N = 160 Search for the Minimum Number of Crosspoints for N = 240 Rectangular Array

136 137 137 139 140 141 142 144 146 147 147

150 150 152

CONTENTS

15. 16. 17. 18. 19.

N Inputs and M Outputs in the Three-Stage Array Triangular Network One-way Incoming, One-way Outgoing, and Two-way Trunks Comparison with Existing Networks Conclusion Reference

...

Xlll

152 153 155 155 157 158

Chapter 5

A Sufficient Set of Statistics for a Simple Telephone Exchange Model 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13.

Theoretical Problems and Methods of Traffic Measurement Description of the Mathematical Model Discussion of the Model Summary of Notations for Chapter 5 The Average Traffic Maximum Conditional Likelihood Estimators Practical Estimators Suggested by Maximizing the Likelihood L , Defined in Section 11 Other Estimators The Joint Distribution of the Sufficient Statistics The Distributions of 2 and M Proof that (n,A , H , 2 )Is Sufficient Unconditional Maximum Likelihood Estimators The Joint Distribution of xT , n, A , H , and Z References

159 165 167 168 169 170 171 172 173 177 183 184 185 186

Chapter 6

The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Discussion, Summary, and Conclusions The Covariance Function The Covariance in Terms of the Recovery Function The Variance of the Number of Paths in Service The Variance of Time Averages Derivation of the Covariance Approximation to the Dominant Characteristic Value References

188 190 202 208 209 212 215 219 221

xiv

CONTENTS

Chapter 7

A CCThermodynamic” Theory of Traffic 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Introduction Preliminaries Summary of Chapter 7 and Discussion Equilibrium The Maximum Entropy Principle The Canonical Distribution Properties of the Partition Function A Reversible Markov Process for Which the Canonical Distribution Invariant Analogy with Statistical Mechanics Discussion and Critique A Markov Model Based on Terminal-Pair Behavior The Approach to Equilibrium Covariance of Functions of zt Applications to Sampling Error A Generalization References

223 224 225 230 232 234 237

IS

24 1 244 246 250 252 251 261 262 264

Chapter 8

Markov Processes Representing Traffic in Connecting Networks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction Preliminary Remarks and Definitions Summary of Chapter 8 Probability Operation Transition Rates Markov Processes Probability of Blocking A Basic Formula Solution of the Equations of Statistical Equilibrium Stationary Probability Measures for Ergodic Markov Processes Expansion of the Stationary Vector p in Powers of A Expansion of the Probability of Blocking in Powers of A Combinatorial Interpretation and Calculation of the Constants {cm(x,o), ~ E Sm ,2 0) 15. Calculation of c,(x, A) References

266 267 269 271 272 273 214 276 28 1 285 290 295 300

Suggested Reading

314

AUTHORINDEX SUBJECT INDEX

315 317

303 309 313

CHAPTER

1

Heuristic Remarks and Mathematical Problems Regarding the Theory of Connecting Systems

A connecting system consists of a set of terminals, a control unit for processing call information, and a connecting network. Together, these three elements provide communication, e.g., supply telephone service among some customers. In this chapter we present a comprehensive view of the theory of connecting systems, an appraisal of its current status, and some suggestions for further progress. The existing probabilistic theory is reviewed and criticized. T h e basic features of connecting systems, such as structure, random behavior, complexity, and performance, are discussed in a nontechnical way, and the chief difficulties that beset the construction of a theory of traffic in large systems are described. It is then pointed out that despite their great complexity, connecting systems have a definite structure which can be very useful in analyzing their performance. A natural division of the subject into combinatorial, probabilistic, and variational problems is drawn, and is illustrated by discussing a simple problem of each type in detail.

$1. Introduction Mass communication long ago spread beyond the manual central office and assumed a nationwide character; it is presently becoming 1

2

1.

HEURISTIC REMARKS A N D MATHEMATICAL PROBLEMS

world-wide in extent. Many of the world’s telephones already form the terminals of one enormous switching system. T h e scale, cost, and importance of the system make imperative a comprehensive theoretical understanding of such global systems. Nevertheless, a lack of knowledge about the combinatorial and probabilistic properties of large switching systems is still a major lacuna in the art of mass communication. I t is a fact of experience that each time a’new switching system is planned, its designers ask once again some of the perennial unanswered questions about connecting network design and system operation: How does one compute the probabilities of loss and of delay? What method of routing is best? What features make some networks more efficient than others ? Etc. T h e present chapter is an informal discussion of problems in the theory of traffic flow and congestion in connecting systems(cal1ed traffic theory, or congestion theory, for short). Th e comments to be made are prefatory, tutorial, and illustrative. They are intended as background for later chapters, of a more technical nature, in which topics tpuched on in the present one are considered in greater depth and detail. Together, the chapters are an attempt to describe a comprehensive point of view towards the subject of connecting systems. I believe that this point of view will be useful in constructing a general theory of connecting networks and switching systems. What follows is then in part a prospectus for research. My concern in this chapter is with some of the physical bases and principal problems, with the fundamentals and difficulties, of the subject. I wish to emphasize some important properties and distinctions on which a systematic approach may be based. I am making a plea for a much more general, abstract, and systematic approach to large-scale congestion problems than has been envisaged heretofore. Naturally, it is impossible to explore all the consequences of such a comprehensive approach in one work; I pretend to have solved only a few of the basic problems of the theory, and I am saying “Look, perhaps these observations will help provide a general approach.” Examples and simple problems appear in the text as illustrations of the principal points made. For tutorial purposes, I have chosen particularly simple and clear illustrations, which may seem trivial to cognoscenti of traffic theory. Nevertheless, it has been my experience in talking with engineers that the comprehensive view here presented

3.

HISTORICAL SKETCH

3

is sufficiently new to warrant clear, simple examples. More complex problems do not belong in an introductory portion; they are to appear in later chapters.

$2. Summary of Chapter 1 I n Section 3 we give a historial sketch of traffic theory, which is followed by a critique of existing theories in Section 4. T h e general properties of switching systems are discussed in Section 5. T h e performance of switching systems and desiderata for a theory of congestion are considered in Section 6 and Section 7, respectively. Sections 5 to 7 are heuristic and nonmathematical in character. Mathematical models are considered in a general way in Section 8, while Section 9 concerns itself with some of the basic difficulties and questions that arise in constructing a theory of traffic in a large-scale system. I n Sections 10 and 11 we show that, despite their great complexity, connecting systems actually have a definite structure which can be very useful in analyzing their performance. This usefulness is exemplified by four specific instances in Section 12. I n Section 13 we make a general division of the subject into combinatorial, probabilistic, and variational problems. T h e remaining sections, Sections 14 to 16, are devoted to illustrating this division by working out a simple problem of each type in full detail.

$3. Historical Sketch We shall not attempt to canvass systematically the literature of congestion theory. For the interested reader, the best single theoretical reference on the theory of probability in connecting systems is undoubtedly the treatise of R. Syski (1); the historical development of the subject has been described in papers by L. Kosten (2) and R. I. Wilkinson (3). Nevertheless, we include a brief account of previous work in order to substantiate our critique (Section 4) of present theories of traffic in connecting systems. T he first contributions to traffic theory appeared almost simultaneously in Europe and in the United States, during the early years of the 20th century. I n America, G. T. Blood of the American

4

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

Telephone and Telegraph Company had observed as early as 1898 a close agreement between the terms of a binomial expansion and the results of observations on the distribution of busy calls.* I n 1903, M. C. Rorty used the normal approximation to the binomial distribution in a theoretical attack on trunking problems, and in 1908 E. C. Molina improved Rorty’s work by his ( 4 ) (or Poisson’s) approximation to the binomial distribution. I n Europe, the Danish mathematician A. K. Erlang, from 1909 to 1918, laid the foundations of the first dynamic theory of telephone traffic, which is in general use today. Perhaps influenced by statistical mechanics, Erlang introduced the notion of statistical equilibrium, and used it as a theoretical basis for deriving his now well-known loss and delay formulas. An account of Erlang’s work is given by Jensen (5). From 1918 to 1939 traffic theory developed in many directions that are (on retrospect) closely allied to specific problems that arose in the design of the automatic telephone systems that were coming into use, and in related queueing systems. We mention only a few topics: T. Engset (6) introduced the notion of a finite number of sources of traffic, G. F. O’Dell (7) published a classical paper on gradings, C. D. Crommelin (8) studied constant holding-time delay systems with many servers, E. C. Molina ( 9 ) made contributions to trunking theory. F. Pollaczek (10) and A. I. Khinchin (11) studied the queue with one server, and derived the delay distribution that bears their linked names. Pollaczek has also solved single-handedly many other difficult loss and delay problems. All these important contributions are concerned with congestion in specific parts of connecting systems. During this period, T. C. Fry (12) wrote the first systematic and comprehensive book on applied probability; this book devoted a chapter to telephone traffic, and appeared in 1928. Between 1939 and 1948 there developed an increasing awareness (among workers in traffic theory) that the mathematical bases of traffic theory were closely related to the modern theory of stochasitc processes initiated by Kolmogorov (13) in 1933. In particular, Erlang’s idea of statistical equilibrium was identified with the stationary measure of a Markov process (or more generally with a semigroup of transition probability operators). Also, C. Palm (14)

* Blood’s unrecorded work was reported by E. C. Molina and described by R. I. Wilkinson (3).

4.

CRITIQUE

5

stressed the importance of recurrent processes, and W. Feller (15) that of birth-and-death processes, to traffic theory. However, particular problems continued to form the bulk of the new literature. Palm (14) made a penetrating theoretical analysis of traffic fluctuations, and L. Kosten studied such topics as retrials for lost calls (16) and error in measurements of loss probability (17). T h e introduction of crossbar switching and common control of connecting networks in 1938 (18) was accompanied by a new kind of problem: calculating the loss due to mismatching of available links (rather than to unavailability of trunks). T h e first comprehensive treatment of loss in such systems was given by C. Jacobaeus (19); his theory is adequate for practical purposes, but is based on assumed a priori distributions for the state of the system. Fortet and Canceill (20) have also made contributions to this topic in the spirit of Jacobaeus’ approach. Another method for the same problems, based only on the possible paths for a call, has been developed (independently) by C. Y. Lee (21) and P. Le Gall (22). T h e statistical equilibrium approach to congestion in crossbar systems is rendered extremely arduous by the large number of possible states. T h e difficulties in this method have been faced with some success by K. Lundvist (23) and A. Elldin (24). However, no practically feasible approach exists at present that simultaneously includes both the concept of statistical equilibrium and the structure of the connecting network. A fortiori, no approach exists that also includes the effect of the common control equipment that places calls in the network.

$4, Critique I n comparison with the highly sophisticated communications systems that are being built, the models and assumptions on which theoretical studies are based are often crude and fragmentary, almost more indicative of our ignorance than of the properties of systems. It may be argued that such a harsh appraisal of the condition of traffic theory is unjustified, and is disproved by the practical successes of current engineering methods. However, it is not the efficacy of these methods, but their theoretical basis and scope, that we are questioning. Who knows to what extent present systems are “overdesigned” ? T o be sure, measures of performance, loss and delay formulas, and

6

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

routing methods are in daily use. Still, only in very special cases have they been investigated, let alone analyzed and understood in the full context of the system to which they are applied. Although the published literature on telephone traffic alone is vast, and many models and problems have been considered, the existing theories tend to be incomplete and oversimplified, applicable to at most a small portion of a system. Useful comprehensive models are needed; to date, only individual pieces of systems have been treated with theoretical justice. As R. Syski remarks ( I , p. 611) “At the present stage of development ... the theoretical analysis of the [telephone] exchange as a whole has not been attempted.” T h e general theory of switching systems now consists of some apparently unrelated theorems, hundreds of models and formulas for simple parts of systems, and much practical lore associated with specific systems. I t will stay in this condition until sufficient theoretical underpinning is provided to unify the subject. We believe that this sad “state of the theory” is due largely to these three factors:

(i) T h e large scale, and consequent inherent difficulty of the problems. (ii) T h e absence of a widely accepted framework of concepts in which problems could be couched and solved. (iii) T h e lack of emphasis on and success with the combinatorial aspects of the problems. More generally, many of the basic mathematical properties of connecting networks and switching systems have either never been studied, or, if studied, have not been digested, advertised, and disseminated for engineering use. As a result, the design and complexity of systems has consistently run ahead of the analysis of their performance.

$5. General Properties of Connecting Systems \Ve start by discussing some universal properties of connecting systems from the point of view of congestion, without reference to definite mathematical models for their operation. Specifically, we describe, in a nontechnical way, (i) the general nature and outstanding features of connecting systems, (ii) the principal kinds of congestion

5.

GENERAL PROPERTIES OF C ONNECTI N G SYSTEMS

7

that interest engineers, and (iii) some of the difficulties and desiderata in both the theory and practice of large-scale switching. No mathematical abstractions are used at first. Some observations made may seem obvious or trivial; nevertheless, they are necessary for the general understanding that we desire. O n these observations, we shall base a systematic division of the theory into three kinds of problems, combinatorial, probabilistic, and variational. By a connecting system we shall mean a physical communication system consisting of (i) a set of terminals, (ii) control units which process requests for connection (usually between pairs of terminals), and (iii) a connecting network through which the connections are effected. T h e system is to be conceived as operating in the following manner: (1) calls (or requests for connection) between pairs of idle terminals arise; (2) requests are processed by a control unit, and desired connections are completed, if possible, in the connecting network; (3) calls exist in the network until communication ends; (4) terminals return to an idle condition when a call terminates. (Naturally, the arising requests may “defect” at any point during the process of connection.) T h e gross structure of a connecting system is depicted in Fig. 1.

FIG. 1. Connecting system.

Most modern connecting systems follow this basic pattern. Particularly important examples are telephone central offices, toll centers, telegraph networks, teletypewriter systems, and the many military communications systems. All the examples cited share three important properties. These are (i) great combinatorial complexity, (ii) definite geometrical or other

8

1. HEURISTIC REMARKS

AND MATHEMATICAL PROBLEMS

structure, and (iii) randomness of many of the events in the operating system. I t is obvious that many connecting systems are highly complicated. Both the control unit and the connecting network contain thousands of parts which may (together) assume millions of combinations. That is, the system can be in any one of millions of possible “states.” These numbers are increased when several switching centers are considered together as a unit, as in toll switching. Our purpose in calling attention to this complexity is to suggest that it calls for theoretical methods that, like those of statistical mechanics, are especially designed to distill important facts from masses of detail. It is less often realized, however, that this complexity is accompanied by definite mathematical structure and is frequently alleviated by many symmetries. T h e control unit and the connecting network always have a specific combinatorial, geometric, and topological character, on which the perfarmance of the system closely depends. By imputing randomness to the systems of interest we do not imply that their operation is unpredictable; we mean only that the best way of describing this operation is by use of probability theory. It is not practical, even though it might be possible in principle, to predict the operation of a switching system by means of differential equations in the way that the flight of a rocket is predicted. However, differential equations have been used for many years to describe, not the motion of an actual system, but the changes in the likelihoods or probabilities of its possible states. Such equations govern the flow or change of probabilities and averages associated with the system, not the detailed time behavior of the system itself. I t is in this weaker sense of assigning likelihood to various events that we can predict the behavior of switching systems, a fact first emphasized by A. K. Erlang’s pioneering work on telephone traffic (5). For instance, certain features (such as average loads offered and carried) of telephone traffic that are predictable in this weaker sense form the basis on which toll trunking routes are engineered. We now turn to examples of the structure of connecting networks and of control units. T h e basic features of the connecting network for the No. 5 crossbar system are shown in a simplified form in Fig. 2. T h e network has two sides, one for subscribers’ lines and the other for trunks. Small squares represent rectangular crossbar switches, capable of connecting any inlet terminal to any outlet terminal. These switches are arranged in groups called frames, either line link frames,

5.

9

GENERAL PROPERTIES OF CONNECTING SYSTEMS

4 KS

FIG. 2. Basic No. 5 crossbar network.

for subscribers’ lines, or (on the other side) trunk link frames for trunks. Frames are indicated in Fig. 2 by large dashed squares enclosing four small squares; dots indicate repetition. T h e pattern of links which interconnect the switches is shown by solid lines between small squares. At most one link connects any pair of switches. As a second example of a connecting network, consider the threestage Clos network (25)depicted in Fig. 3. T h e interpretation of this figure is the same as that of Fig. 2; small squares stand for crossbar switches, and lines between them represent links. Each call can be put into the network in m ways, one for each of the m switches in the middle column. This network has the property that if m 2 2n - 1, it is nonblocking. A control unit consists of parts that are arranged in a manner reflecting their function, and are determined by the operations necessary to establish a connection, and by the philosophy of design and the technology that are basic to the system. T o establish a connection, the control unit must do some or all of the following: (i) identify the calling party or terminal, (ii) find out who the called party is, and (iii) complete the connection. Three examples will be considered, in order of increasing complexity and modernity.

10

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

rxr

nxm

FIG. 3.

mxn

Clos three-stage network.

A simple example of the structure of a control unit is given in Fig. 4. T h e unit consists of a dial-tone marker, which assigns and connects available idle registers to subscribers for dialing. T h e dialed digits remain in the register until a completing marker (one of possibly several) removes them and uses them to complete the call. T h e calls, or requests for connection, may be thought of as arriving from the left, and proceeding through the diagram from left to right. There may be a delay in obtaining dial tone, a delay in securing the services of a completing marker, or a circuit-busy delay (or rejection) in the network. It should be observed that the switching equipment necessary for connecting subscribers to registers, or registers to completing markers, is left out of account in this model. A second example is obtained from the first by inserting a buffer memory between the registers and the markers, as shown in Fig. 5. (One can argue that registers are expensive special-purpose units and should not be used for storing call information when cheap memory REGISTERS

--I

REOUESTS FOR SERVICE

T, MARKcn

,

COMPLETING MARKERS

I

El El I l l

TO CONNECTING NETWORK

/

DIALED DIGITS STAY I N REGISTER UNTIL A COMPLETING MARKER IS AVAILABLE

FIG.4. Simple control unit.

5.

GENERAL PROPERTIES OF C O N N E C T I N G SYSTEMS

11

COMPLETING MARKERS

REGISTERS

BUFFER MEMORY

El

TO -CONNECTING NETWORK

CALL INFORMATION MOVED TO BUFFER MEMORY AS SOON AS POSSIBLE, TO FREE REGISTER

FIG. 5.

Control unit with buffer memory.

is available.) When dialing is finished, the call information is forthwith transferred to the buffer memory, there to wait for a completing marker without preempting a register. T h e markers and registers are now effectively isolated, so that delays in completing calls do not cause delays in obtaining dial tone. Again, traffic is viewed as moving from left to right. T h e high speeds possible with electronic circuits have led to new configurations and problems (for control units and networks) which have not yet received much attention in congestion theory. Although it performs the same functions, the control unit of a modern electronic central office usually has an organization differing from that of the examples of Figs. 4 and 5 , which are characteristic of electromechanical systems. Four principal reasons for this contrast are: (i) T h e electronic office relies heavily on a large digital memory to aid in processing calls and (in time-division systems) to keep track of calls in progress; electromechanical systems, on the other hand, are based largely on “wired-in” memory. (ii) I n the electronic office, processing a given call usually requires several consultations of the digital memory; thus, the flow of traffic in the control unit is reentrant and not unidirectional as in Figs. 4 and 5 . (iii) T h e speed of electronic components often makes it possible to perform only one operation at a time; thus, a single unit may be (alternately) part of a dial-tone marker, part of a register, part of a completing marker, etc., depending on the details of organization of the control unit. (iv) T h e replacement of “wired-in” memory, whose stored information is immediately available, by an electronic memory which

12

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

has to be consulted, creates problems analogous to the problem of connecting completing markers to registers in the No. 5 crossbar system: special access units are needed. Subunits of the control unit, such as dial-tone markers, completing markers, senders, etc., must take turns in using the access circuit to the digital memory. Figure 6 depicts a (hypothetical) control unit for an electronic switching system built entirely around a memory that stores all information on the current status of calls. T h e control unit consists of various special-pupose units such as a sender, a receiver, a completing marker, a dial-tone marker, and registers. Each of the listed units can operate independently of and simultaneously with the others; however, they compete for (take turns at, possibly with priorities) the access circuit to the memory. Each unit depends on the memory to give it a new assignment, to file the results of the last one, or both. Every operation of a special-purpose unit requires access to the memory, either to obtain data from it, or to file data in it, or both. T h e memory contains several classes of calls: those waiting for dial tone,

v - RECEIVER

HANGUPS I

MEMORY

REQUESTS FOR

SERVICE

-

-

COMPLETING MARKER OR NETWORK CONTROL

I

ACCESS CIRCUIT

,

DIAL TONE MARKER

LB REGISTERS

FIG. 6 . Block diagram of electronic control unit.

TO CONNECTING NETWORK

6.

PERFORMANCE OF SWITCHING SYSTEMS

13

those waiting for a completing marker, those actually in progress in the connecting network, etc.

$6. Performance of Switching Systems I n general, the gross or average features of switching systems are both more accurately predictable‘and more economically important than the specific details. T h e average load carried by a trunk group is usually more easily predicted than the condition of a particular trunk; and the “all trunks busy” condition of the group is of greater concern to the telephone administration than the busy condition of a single trunk. From the point of view of economics and traffic engineering, only certain average features of the behavior of a system (used as measures of performance) are important. These few quantities of interest depend on the multitude of details of “fine structure” in the control unit and the connecting network. Although the intricate details give rise to the important averages, the details themselves are of relatively little interest. I n the rest of this chapter, we shall repeatedly contrast the few average quantities that are of engineering interest with the many millions of detailed features and properties (of connecting systems) on which the averages are based. T h e central problem in the theory of connecting systems is to understand how the interesting quantities arise from the details, and to calculate them. We shall start our discussion of the contrasting roles of averaged features and details by considering some of the different kinds of congestion that interest engineers, and in addition some associated measures for the performance of systems. Congestion is said to occur in a connecting system when a requested connection cannot be completed immediately. By “immediately” we mean, of course, not “instantaneously,” but “as fast as control equipment, assumed available, can do its work.” T h e time it takes to complete a call contributes to congestion only if it keeps other calls from being completed at the normal rate. T h a t a call cannot be completed immediately (in this sense) may be due to facts of three kinds: (i) certain necessary units of switching equipment (like trunks or markers) are all busy; (ii) there are available units, but they occur in an unusable combination, or “fail to match”; (iii) congestion has occurred previously, and other requests are awaiting completion.

14

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

I n telephone traffic theory, requests for connections that encounter congestion are traditionally termed lost calls. This terminology is used whether the request is refused (and never completed), or merely delayed (and completed later). Switching systems differ in the disposition of lost calls, i.e., in what is done with requests that encounter congestion. There are in theory two principal ways of disposing of lost calls. I n the first way, termed “lost calls cleared,” the request is denied and leaves the system; this way of dealing with lost calls naturally gives rise to the proportion ofrequests denied, or the probability of blocking or loss, as a measure of performance. T h e second way of disposing of lost calls is termed “lost calls delayed,” and consists in delaying the request until equipment becomes available for completing the connection; associated with this is the probability of delay in excess of a specified time t , as a measure of performance. O n the simplified account of the last paragraph we must impose at least two qualifications. First, whether a request suffers blocking or delay (or both!) may depend on the condition of the system at times shortly after the request is made; second, the completion of a request usually involves a sequence of steps, any one of which may expose the request to delay or loss. For example, a request may encounter delay in obtaining dial tone, delay in securing the services of a completing marker, and delay or blocking in the attempted completion of the desired connection through the connecting network. We conclude this section by briefly considering what general features of connecting systems are particularly relevant to their performance as measured (for example) by probabilities of blocking or delay, or by average loads carried, offered, or both. Now, the features of a connecting system that are relevant to performance are conveniently distinguished according to whether they are features of the control or of the network. This distinction is fundamental because the performance of the control is largely determined by the speed and number of the various subunits comprising it, while the performance of the network is largely dependent on what combinations of calls can be in progress simultaneously. T h e control unit is basically a data processing system: I t collects information about desired connections, digests it, makes routing decisions, and issues orders for completing requested calls in the connecting network. Its capacity is measured, e.g., by the number of customers who can be dialing simultaneously, or by the number of calls which are being completed in the network at the same time.

7.

DESIDERATA

15

Its performance is described by the probability distributions of delay before receipt of dial tone, and of delay after completion of dialing until the desired connection is completed. For a simple model of a control unit (such as depicted in Fig. 4), the features pertinent to performance are: (i) the calling rate, (ii) the number of registers for dialing, and (iii) the speed and number of completing markers. I n the case of the prototype electronic control unit (depicted in Fig. 6) some additional features appear: (iv) the speed of the access circuit to the memory; (v) the order of priority of the functions being performed, the discipline of access to various services, and the competition for access among marker, dial tone marker, sender, etc. ; (vi) the presence of reentrant traffic (every call must “use” the access circuit at least twice); and (vii) the number and arrangement of the various functions that are going on simultaneously. T h e connecting network, in contrast to the control unit, determines what calls can be in progress, rather than how fast they can be put up. Its configuration determines what combinations of terminals can be connected simultaneously together. For example, if rn 3 n, the Clos network of Fig. 3 has the property of rearrangeability: any preassigned set of calls can be simultaneously connected. T h e No. 5 network of Fig. 2 does not have this property: the number of calls between a line link frame and a trunk link frame is limited by the number of links between those two frames. Such combinatorial properties of the structure of the connecting network play a determining role in estimating the cost and the performance (probability of blocking) of the network. If the structure is too simple, very few calls can be in progress at a given time and blocking is high; if it is extensive and complex, it may indeed provide for many large groups of simultaneous calls in progress, and so a low probability of blocking, but the network itself may be prohibitively expensive to build and to control.

$7. Desiderata Our discussion of the three prominent features of switching systems -(i) great complexity, (ii) definite structure, and (iii) randomnesshas exposed or suggested some of the problems and desiderata that a theory of congestion in large-scale systems must (respectively) encounter and supply. Specific statements of requirements and tasks are now given.

16

1.

HEURISTIC

REMARKS AND MATHEMATICAL PROBLEMS

General desiderata can be obtained by examining the purpose served by a theory of congestion. T h e function of such a theory is twofold: it is (i) to describe the operation of switching systems, and (ii) to predict the performance of systems. More specifically, the descriptive function (i) is to provide a theoretical framework into which any system can be fitted, and which permits one to evaluate the performance of the system, e.g., to compute the chance of loss, to estimate a sampling error, or to prove a network nonblocking. T h e predictive function (ii) has logically the same structure as (i), but emphasizes the use of theory to make future capital out of past experience, to extrapolate behavior, and thus to guide engineering practice. More specific tasks than these appear when we list some of the activities comprised by the theory and practice of traffic engineering. A possible list is as follows: (i) Describing and analyzing mathematical models. (ii) Computing measures of performance for specific models. (iii) Studying the accuracy of traffic measurements, the effects of transients, and problems explicitly involving random behavior in time. (iv) Comparing networks, control systems, methods of routing, etc. (v) Using traffic data to verify empirically the assumptions of theories. (vi) Making predictions and estimates for engineering use. On the basis of this list, and of our previous discussions of complexity, randomness, gross features, and details, we can say that a satisfactory theory of congestion must meet the following requirements: (i) It must be sufficiently general to apply to any system. (ii) It must yield computational procedures for system evaluation and prediction of performance, based on masses of detail. These procedures must be at once feasible and sufficiently accurate, and if approximations are made, their effect must be analyzable. (iii) It must encompass all the three basic elements simultaneously, viz., the random traffic, the control unit, and the connecting network.

8.

MATHEMATICAL MODELS

17

$8. Mathematical Models We shall now consider what mathematical structures are appropriate theoretical descriptions of operating connecting systems. T h e discussion will provide an intuitive picture of an operating system, and will help to motivate a natural division of our subject into combinatorial, probabilistic, and variational problems. By a state we shall mean a partial or complete description of the condition (of the system under study) in point of (i) busy or idle network links, crosspoints, and terminals and (ii) idle or busy control units or parts thereof. Complete, highly detailed descriptions correspond to fine-grained states specified by the condition of every crosspoint, link, or other unit in the system, in absolute detail. Incomplete descriptions correspond to coarse-grained states, or to equivalence classes of fine-grained states. During operation, the connecting system can pass through any permitted sequence of its states. Each time a new call arises, or some phase of the processing of a call by the control unit is finished, or a call ends, the system changes its fine-grained state. These changes do not usually occur at predetermined epochs of time, nor in any prescribed sequence; they take place more or less at random. At any particular time, it is likely that some terminals, links, and parts of the control unit are idle, that various requested calls are being processed, and that certain calls are in progress in the connecting network. T h e last paragraph suggests the following intuitive account of an operating switching system: it is a kind of dynamical system that describes a random trajectory in a set of states. Such an intuitive notion can be made mathematically precise in many ways. Any one precise version is a mathematical model for the operation of the switching system. I n constructing such a model, it is neither necessary nor desirable always to use the most detailed (the fine-grained, or microscopic) states; often a partial description in terms of coarsegrained states suffices and is less difficult to study. Indeed, in building a model it is to some extent possible to choose the set of states to suit special purposes. One can, for instance, control the amount of information included in the state so as to strike a balance between excessive detail and insufficient attention to relevant factors. I t is Possible to make the notion of state more or less complete so as to achieve certain (desired) mathematical properties (such as the Markov

18

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

property, or a suitable combinatorial structure) which simplify the analysis of the random trajectory. Finally, one can add supplementary variables analogous to counter readings or cumulative measurements and obtain their statistical properties. (Cf. Chapter 5 . ) T h e abstract entity appropriate for describing the random behavior of a switching system is a stochastic process, For our present heuristic purposes, we can define a stochastic process as follows: by a possible history of the system we mean a piecewise constant function of time taking values in the chosen set of states; a stochastic process is then a collection l2 of possible histories of the system in time, with the property that many (presumably interesting) subsets A of l2 have numerical probabilities Pr{A} associated with them. T h e probability Pr{A) of the set A of possible histories is interpreted as the chance or likelihood that the actual history of the system be one of the histories from the set A . Models of this kind furnish information because desired quantities can be calculated from the basic probabilities Pr{A}.

$9. Fundamental Difficulties and Questions T h e systematic use of mathematical models (such as stochastic processes) in congestion theory and engineering has been largely limited to small pieces of systems like single-server queues, groups of trunks with full access, etc. More complex models of systems involving connecting networks have hardly been touched by theory. This limitation has been due almost entirely to the large number of states such models require, and to the complex structure of the transitions (changes of state) that can occur. I n short, the essential characteristics (of large-scale connecting systems) themselves generate the basic difficulties of the theory. I n most congestion problems, it is easy enough to construct (say) a Markov process that is a probabilistic model of the system of interest. But it is difficult, because of the large number of states and the complexity of the structure, to obtain either analytic results or fast, reliable simulation procedures. This circumstance has been a major obstacle to progress in the congestion theory of large systems. One of its consequences has been that, in some cases, models known to be poor representations of systems have been used merely because they were mathematically amenable, and no other tractable models were available. Even overlooking such extremes, it is fair to state that,

10.

THE MERITS OF MICROSCOPIC STATES

19

to date, problems of analysis and computation have limited the amount of detail embodied in the notion of state for models of switching systems. Every effort has been made to keep the number of states in models small, and their complexity low. Having exposed some basic properties of and theoretical problems arising from congestion in connecting systems, let us acknowledge that an operating, large-scale connecting system cannot be done full theoretical justice except by a stochastic model with an astronomical number of states and a very complicated structure of possible transitions. At this point, let us try to take a synoptic view of the subject, and ask some general questions whose discussion might indicate new approaches and emphases. Let us, in the current idiom, lean back in our chairs, make a(n) (agonizing ?)reappraisal, and draw ourselves the “big picture.”* T h e following three questions seem (to this writer) to be pertinent, and are taken u p in the next sections: (i) What is the value of mathematical models that have a very detailed notion of state ? (ii) Is it possible to make explicit theoretical use of the very properties of connecting systems that appear to be most troublesome ? How can the two principal difficulties (large number of states, complex structure of changes) be turned into positive advantages ? (iii) What features of connecting systems are especially relevant to the mathematical analysis of system operation ? We do not pretend to provide iron-clad answers to these questions. We try to give a helpful discussion of relevant matters, illustrated by examples.

$10. The Merits of Microscopic States We have raised the question: T o what extent can detailed probabilistic models of the minutiae of operating switching systems (i.e., models with “microscopic” states) improve our understanding of these

* Supplying those clichts whose substitution leaves the content of this last sentence invariant is left as an exercise for the reader.

20

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

systems and, therefore, our ability to engineer them ? Against the value of such detailed models it can be argued that for engineering purposes only certain performance data are of interest, and that the detailed model produces a vast amount of information with no apparent practical method for reducing this information to probabilities of delay or blocking. Since the usefulness of mathematical models depends entirely on the desired information they can be forced to yield, it is not reasonable to dismiss detailed models a priori. For, in truth, few if any such models have been considered, and it has not been shown that they are useless in the sense that no practical method for extracting useful quantities from these models exists. T o be sure, the congestion engineer is not as concerned with the minutiae themselves as with their effect en masse. But he has to base his conclusions and recommendations in some way on the total effects of a large number of individually trivial events. Hence, at some point in his procedure, he must take account of the large number of states and the complex structure of possible transitions of his system. Traffic engineering practice is based on (relatively few) probabilities and averages, such as average loads, deviations about them, and blocking or delay probabilities. Any reliable theoretical estimate of these averages must be based on the combinatorial and probabilistic properties of a theoretical model (stochastic process) for system operation. At worst, an approach or model that provides detailed information might yield a much needed check point for the methods that are in current engineering use, and so increase the engineer’s understanding of and confidence in these methods. However, there is a much more general, positive sense in which attention to the details of connecting systems can contribute to theoretical progress. This is taken up in the next section.

$11. From Details t o Structure T h e prospect of solving (say) statistical equilibrium equations for models with a very detailed notion of state is discouraging indeed, although it has been faced, notably by Elldin (24) in Sweden. (Cf. also Chapter 8.) Nevertheless, a sanguine and useful approach (along this line) to connecting systems can be obtained by a shift of emphasis from “details” to “structure.” We have emphasized that describing

11.

FROM DETAILS TO STRUCTURE

21

an operating connecting system means keeping track of numerous details, none of which is interesting in itself. We have said that the operation of such a system could be pictured as a trajectory in a very complicated set of states. We now claim that the inclusion of enough details (in the notion of state for a model) gives the set of possible states a dejinite structure that is useful because it makes possible or simplifies the analysis of the probabilistic model. Whatever may be the value of detailed probabilistic knowledge for the immediate problems of engineering, such knowledge is useful if not essential in theoretical studies. By using a highly detailed, “microscopic” description for the state of the system, it is possible to exploit the extensive mathematical structure (properties) that such a set of states naturally has. I n fact, the combinatorial properties and geometrical structure of the set of states are two of the very few weapons available for attacking large-scale problems of traffic theory. I believe that in the past these properties and this structure have not been sufficiently exploited. They can only be put to use by a systematic application of “microscopic” states. T h e three basic properties of switching systems discussed in Section 5 were (i) extreme combinatorial complexity, (ii) definite geometrical structure, and (iii) randomness. T h e preceding paragraphs of this section can be related systematically to these properties, and elaborated into a sort of program: Instead of throwing up our hands at (i) in trying to do justice to (iii), we should realize that a detailed notion of state allows us to turn (ii) to our advantage in studying (iii). Let us then disregard the fact that there are many states, and analyze the structure of possible changes of state, to see how to capitalize on it. For, indeed, the possible microscopic states of a particular connecting system are not arbitrary. They are rigidly determined by the combinatorial and topological properties of the connecting network and by the organization of the control unit. Such a set of possible states has a mathematical structure of its own, and this structure is relevant to the performance of the system, and to any stochastic process that represents its operation. It can be seen quite generally that when a switching system changes its microscopic state, it can only go to a new state chosen from among a few “neighbors” of the state it is leaving. These neighbors comprise the states that can be reached from the given state by starting a new call, ending an existing call, or completing some operation in the

22

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

control unit. I n a large system, a state may have many such neighbors, but they will be few in comparison with the total number of microscopic states. A striking and useful example of how details give rise to structure can be obtained by considering the possible states of a connecting network. These states can be arranged in a pattern as follows: At the bottom of the pattern we put the zero or ground state in which no calls are in progress; above this state, in a horizontal row, we place all the states which consist of exactly one call; continuing in this way, we stack up level after level of states, the kth level L,c consisting of all the states with k calls in progress. We now construct a graph by drawing lines between states that differ from each other by exactly one call. (Such states, needless to say, are always in successive levels of our diagram.) This graph we call the state-diagram. I t is a natural (and standard) representation of the of the states: When x and y are states, partial ordering

<

X


means that y can be obtained from x by adding zero or more calls to x, or alternately, that x can be got from y by removing zero or more calls. T h e importance of this state-diagram lies in two facts: (i) T h e state diagram gives a geometrical representation of the possible states of the system. T h e myriad choking “details” of the connecting network have been converted into a vast geometrical structure with special properties. T h e operating system describes a trajectory through the state diagram, moving between levels as calls begin and end. (ii) Any stochastic process describing the operation of the connecting network is a point moving randomly on the state diagram. T h e motion is only between adjacent levels. New calls put into the network correspond to jumps to the next higher level; hangups correspond to jumps to the next lower level. As a simple example, we consider the possible states of a single 2-by-2 switch. These consist of (i) the zero state, (ii) the four ways of having one call up, and (iii) the two ways of having two calls up. These states are depicted in Fig. 7. Figure 8 shows the states of a 2-by-3 switch.

11.

FROM DETAILS T O STRUCTURE

FIG. 7. States of a 2-by-2 switch.

FIG. 8. States of a 2-by-3 switch.

23

24

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

$12. The Relevance of Combinatorial and Structural Properties: Examples I n this section we elaborate, by discussing examples, our theme that the combinatorial and structural properties of connecting systems are of the greatest import (i) to their performance, and (ii) to the analysis of mathematical models of their operation. T h e organization of the control unit and the configuration of the connecting network largely determine the possible microscopic states of the system. Let us see what effects these features can have on problems of system analysis. Example 1. Any connecting system has a “zero” or ground state in which all terminals and links are idle, no calls are being processed by the control unit, and the connecting network is empty. T h e existence of this zero state is a structural property common to all switching systems. This zero state seems most uninteresting. Nevertheless, many probabilistic models (for switching system operation) have the property that if the equilibrium probability of the zero state is known, then that of any other state can be determined in a simple way. Several specific examples of this phenomenon are worked out later in this chapter, so none will be given here. (See Sections 15 and 16.) Example 2. T h e relevance of combinatorial properties of the connecting network to the calculation of probabilities can be vividly illustrated by reference to Clos’ work on nonblocking networks (25). T h e blocking probability of a connecting network is the fraction of attempted calls that cannot be completed because no path for the call exists in the current state of the network. Until Clos’ article appeared it was not generally known that, no matter what probabilistic model was used, an exact calculation of blocking probability for a Clos network 2n - 1 (see Fig. 3) would yield the value zero!* with m

>

Example 3. Consider the class of connecting networks that have the property that, in any state of the network, two idle terminals (forming an inlet-outlet pair) can be connected in at most one way.

* Zero,

not zero factorial, which equals unity!

12.

COMBINATORIAL AND STRUCTURAL PROPERTIES

25

For each member of this class of networks we construct a Markov stochastic process to represent its operation under random traffic, as follows: I n any state, if an inlet-outlet pair is idle, the conditional probability is Ah o(h) that it request connection in the next interval h, as h -+ 0; also, an existing call terminates in the next interval h with a probability h o(h), as h + 0; requests that encounter blocking are denied and do not change the state of the system (lost calls cleared). If X is a finite set, let I X 1 be its cardinality, i.e., the number of elements of X , and let S be the set of all states of the network under discussion. For x in S , define

+ +

A, B,,

Ix1

=

set of states accessible from x by adding a call

=

set of states accessible from x by removing a call

number of calls in progress in state s L, = set of states with /2 calls in progress. =

Note that I B , , = 1 x ;. Let p x be the stationary or equilibrium probability that the system is in state x. By reference to Fig. 9, it can be seen that the statistical equilibrium equations for our probabilistic model are

FIG.9.

A . state x, and the sets A , , B , in the state diagram.

26

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

Since in any state an idle pair can be connected in at most one way, no routing decisions need to be made, and the solution of this equation (regardless of the network configuration!) is given by

p, p,'

= pox',',

=

1+

x f 0,

c; h ' Y '

YES

Y>O

where 0 is the zero state. We have therefore shown that the simple combinatorial property, that a call can be put up in at most one way, implies that the stationary probabilities of the Markov process we defined are of a simple geometric type. Note the important role played by the zero state, as discussed in Example 1. Example 4. T h e Markov stochastic processes of the previous example can be used to illustrate another important point. There are many switching system models for which quantities of interest (such as the probability of blocking) can be given rigorously, without approximations, by a formula in which the distinction between system combinatorics and random customer behavior appears explicitly. I n Example 3, the state probabilities {p, , x E S> are completely determined by the quantities

( L , I,

k 20,

i.e., by the number of states with k calls in progress, for k 0. For these models we can express the blocking probability as a function of the traffic parameter h and of 1 L, 1, k 3 0. T h e numbers 1 L , I represent purely combinatorial properties of the network. T h e blocking probability b can be calculated as follows: b is the fraction of attempted calls that are unsuccessful, so that 1 -b

total rate of successful attempts

= _____________,

total rate of attempts

I n equilibrium, the total rate of successful attempts must equal the total rate of hangups. T h e total rate of hangups is

p, XES

1 x I = mean number of calls in progress

12.

COMBINATORIAL A N D STRUCTURAL PROPERTIES

27

(because the mean holding time is used as the unit of time). Let T be the number of terminals offering traffic. Since an idle inlet-outlet pair calls at a rate A, the attempt rate in a state x is X (number of idle pairs in a state x)

=h

T he total rate of attempts is then

Hence, h = l -

xeS

x Cp. XES

(

2I 2

-

I

1

where [T/2]is the greatest integer less than or equal to T/2. This formula exhibits the blocking probability as a rational function of the calling rate X per idle pair and as a bilinear function of the combinatorial constants {I L, I, k 2 O}. T h e degree of the denominator in X is one more than that of the numerator, so b 3 1 as X 3 co; also note that

This limit is greater than zero if there are calls that cannot be put up in any way. Finally, we observe that if the network is nonblocking, then

28

1.

and so b

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

=

0, as it should, if we interpret

as zero.

$13. Combinatorial, Probabilistic, and Variational Problems T h e preceding discussions have established that the ingredients going into a mathematical model of a connecting system are of two kinds. On one hand are the combinatorial and structural properties and, on the other, the probabilistic features of traffic. We emphasize the distinction between these aspects, and claim that by carefully drawing it, we can extend the general understanding of connecting systems, unify or modify existing theoretical methods, and obtain new engineering results. Our discussion also suggests that, to study stochastic processes that represent operating connecting systems, it is essential to have an extensive theory of the combinatorial and topological nature of the microscopic states of such systems. I n any specific model of a connecting system, one can distinguish the combinatorial from the stochastic features. However, it is also of interest to compare models of systems in an effort to determine optimal systems. These facts suggest a useful though imprecise division of the entire subject (of connecting system models) into three broad classes of problems. I n order of priority, these are: (i) Combinatorial problems. (ii) Probabilistic problems. (iii) Variational problems. This order of priority arises in a natural way: One needs to study combinatorial problems in order to calculate probabilities; one needs both combinatorial and stochastic information in order to design optimal systems. T h e tripartite division just made provides a rational basis for organizing research effort. Since so many of our pronouncements have been generalities, we devote the remainder of the chapter to

14.

A PACKING PROBLEM

29

illustrating carefully each of the three divisions (combinatorial, probabilistic, variational) by working out and discussing in detail a very simple (yes, a trivial) problem from each division. These problems have been chosen for their tutorial value rather than their realism or usefulness. I n discussing them, we place emphasis on furthering insight rather than solving practical problems, on exposing principles rather than providing engineering data.

$14. A Packing Problem It has long been suspected (and, in some cases, verified experimentally) that routing calls through a connecting network “in the right way” can yield considerable improvements in performance. T h is procedure of routing the calls through the network is called “packing” (the calls), and the method used to choose routes is called a “packing rule.” T h e use of the word “packing” in this context was surely suggested by an analogy with packing objects in a container. However, the existence and description of packing rules that demonstrably improve performance (e.g., by minimizing the chance of blocking) are topics about which very little is known. What, then, is the “right way” to route calls ? I t has been argued heuristically that it is better to route a call through the most heavily loaded part of the network that will still take the call. Appealing and simple as this rule is, nothing is known about it. We know of no published proof of either its optimality or its preferability over some other rule. T h e rule will be proven optimal for an example in Section 16. T h e question naturally arises, though, whether for a given network in which blocking can occur there exists a packing rule so cunning that by following it all blocking is avoided. Then, use of the rule makes the network nonblocking. Such a network may be termed nonblocking in the wide sense, while a network none of whose states has any blocked calls may be termed nonblocking in the strict sense. T h e existence of such a rule is a purely combinatorial property of the network, and so serves as an example of the first type of problem described in Section 13. Unfortunately practically useful connecting networks that are nonblocking in the wide sense are yet to be found. Since we are primarily interested in exemplifying principles, we shall be content with discussing an impractical network that is

30

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

nonblocking in the wide sense. T h e example to be given was suggested by E. F. Moore.* Let us first consider the three-stage connecting network depicted in Fig. 10. All switches in the middle column are 2 by 2, and there are 2n - 1 of them, therefore, by a result of C. Clos (25), the network 2x2

2n -1

FIG. 10. Three-stage nonblocking connecting network (Clos type).

is nonblocking. Suppose that we adopt the rule that an empty middle switch is not to be used unless there is no partially filled middle switch that will take the call. I n other words, do not use a fresh middle switch unless you have to! I n general, this rule is not quite the same as the one exhorting use of the heavily loaded switches wherever possible, because it only tells us what to avoid, but it is in the same spirit. I n the case to be considered, however, a middle switch is either empty, half-full, or full; hence the two rules coincide. We shall show that if this rule is used, then no more than [3n/2] middle switches are ever used, where [x] is the greatest integer less than or equal to x . Th u s the rest, about one-quarter of the middle switches, could be removed, and no blocking would result if the rule were used. I t can be verified by examples that if there are only [3n/2] middle switches, then calls can be blocked if the rule is violated. Thus, the network of Fig. 11 is not nonblocking in the strict sense, but is nonblocking in the wide sense. A state x of a connecting network is called reachable (under a rule p )

* Private communication.

14.

A PACKING PROBLEM

31

2x2

FIG. 11. Three-stage network which is nonblocking if proper routing is used.

if using the rule p to make routing decisions does not prevent the system from reaching x from the zero state. We set S(x)

=

number of middle switches in use in state x.

Let us use the diagram of Fig. 12 as a canonical representation for a 2-by-2 middle switch. T h e numbers at the left [top] indicate to which outer switch on the left [right] the numbered link connects. T h e seven possible states of a middle switch are depicted in Fig. 13 and are indexed therein by letters a, b, ...,g. A state x may then be represented (to within renaming switches and terminals) by giving seven integers a(x),b(x), ...,g(x) where u(x) = number of

middle switches of type a when network

is in state x

g(x)

=

number of middle switches of type g when network is in state x.

‘33 t

2

2

FIG. 12. Representation of a 2-by-2 middle switch.

32

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

MIDDLE SWITCH STATE

+

TYPE -

CALLS

a

NONE

= CLOSED

CROSSPOINT

FIG. 13. Seven possible states of a middle switch.

It is clear that for any state x

+ b ( x ) + ... + g ( x ) = 2n - 1, b ( r ) + c ( x ) + ... +g(x) = S ( x ) .

) . ( a

14.

A PACKING

33

PROBLEM

Theorem 1.1.*

Let p denote the rule: Do not use an empty middle switch unless necessary. Let x be a state of the network of Fig. 10, reachable under p. Then for n 2 2

<

(1)

~ ( J c ) [3n/21,

+ 4.4 +f(.) < n, 4.)+ 44 + g(4 < qJC)

(2)

72.

Proof. Each reachable state is reachable in a certain minimum number of steps. T h e theorem is true if x consists of one call and is reachable from the zero state in one step. As an hypothesis of induction, assume that the theorem is true for all states reachable in k steps or fewer. All changes in the state are either hangups, or new calls of the following kinds:

Type 1.

(1, 1) (2, 2) ( 2 , 1) ( I , 2) Type 2.

( I , 1)

-

a(y)+ u ( y ) - 1, and one of b(y) c(y)-. c(y) d(y) d(y) e( y)+ e(y> kY)

+1 +1

+1

+1

with

4 y ) = 0, with b(y) = 0, with e(y) = 0, with d ( y ) = 0.

(preferred by p )

-

n(y) remains

f ( r ) - j ( y ) + 1, (2,2) f ( y ) f ( Y ) 1, (2, 1) g(y)-.g(y) + 1, (1, 2) g ( y )- g ( y ) 4-1,

+

fixed and one of

-

c(y)WY) 4Y)d(Y)

-

c(y) - 1

with

b(Y) - 1 e(y) - 1 d(y) - 1

with

with with

> 0, b(Y) > 0 , e(y) > 0 , 4 Y ) > 0. C(Y)

All states, reachable or not, satisfy the inequalities b(Y)

+ e(r) + f ( A+ d Y ) G

b(Y) 4Y)

+ 4 Y ) + f ( Y ) + d Y ) G n, + 4 Y ) + f ( Y ) + g ( y ) G n.

439 + 4Y)+ f ( Y ) + A Y ) G 72,

* T h e notation “Theorem a.6” refers to the bth theorem of Chapter a. Similarly u i t h definitions, remarks, corrollaries, etc. Formulas are simply numbered sequentially in each chapter, without prefixes.

34

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

T h e alternative preferred by p changes neither the value of S( . ) nor the truth of (2) of the theorem. Consider a state x first reachable in k f 1 steps. If x is first reachable by a hangup or by putting up a call of T y p e 2, then (1) and (2) are true of x. Suppose then that x is first reachable in k 1 steps only by putting u p a call of Type 1. Without loss of generality we can consider only the case where the new call is a (1, 1) call; the other three cases are symmetric. Let y be a state from which x is thus first reachable. Since the avoided alternative is used, we have c ( y ) = 0.

+

Since a (1, 1) call is possible in state y , we must have b(Y)

b(Y)

+ 4 Y ) +f'(r)+ g ( v ) < + 4 Y ) +m+ d Y ) G n

-

1, 1,

and from the induction hypothesis

4y) Hence,

W(y)

+ 4 Y ) + g(Y) G n.

+4Y)+

or, since c(y) = 0,

+ g(Y)>

+.f(Y)

372

-

2

3n S ( y ) < - - 1. 2

+

However, S(x) = S ( y ) 1, so S(x) also holds of x consider that b(Y)

< [3n/2]. T o

show that (2)

+ 4 Y ) + f ( Y ) + d Y ) G n - 1, c(y) = 0.

It follows that b(Y)

+

C(Y)

+f(Y) G n

-

1.

However, since x is obtained from y by putting up a (1, 1) call of Type 1, we have

+

e(x) = 4 Y ) ,

b(x) = b ( y ) 1, 4.) = C(Y) = 0,

f(.)

=f(.y),

0)= 4Y)V

gw

=d Y ) .

Hence, (2) of Theorem 1.1 is true of x. This proves the result.

15.

TRAFFIC CIRCULATION I N A TELEPHONE EXCHANGE

35

$15. A Problem of Traffic Circulation in a Telephone Exchange We shall describe and analyze a simple stochastic model for the operation of the control unit of a switching system. T h e connecting network is assumed to be nonblocking and is left out of account. T o set u p a telephone call in a modern electromechanical automatic exchange usually involves a sequence of steps that are (traditionally and functionally) divided into two groups. T h e first group consists in collecting in a register the dialed digits of the called terminal. T h e second group, performed by a machine called a marker, consists in actually finding a path through the connecting network for the desired call, or otherwise disposing of the request for service. For even if a path to the called terminal be found, this terminal may already be busy. I n the exchange, enough registers and markers must be provided to give customers a prescribed grade of service. For engineering purposes, then, it is desirable to know the probability that r registers and m markers are busy. Let us assume that the exchange serves N customers, and that there are R registers and 111 markers. *4ll calls are assumed to go to terminals outside the exchange. We may think of each customer’s line as being in one of a number of conditions, and moving from one condition to another. It makes no difference whether we ascribe these “conditions” to the line itself, or to a fictitious single customer if several people use the line. A given line may be idle (i.e., not in use); at some point in time it may request a connection, i.e., the customer picks u p the receiver and starts waiting f o r dial tone; after obtaining a register he spends a certain amount of time dialing; he then waits f o r a marker to complete his call (freeing the register meanwhile); upon obtaining a marker, he must wait until the marker completes the connection; at this point he begins his conversation; at the end of his conversation his line becomes idle again. One may now ask, what is the distribution of the N customers among these various conditions ? Clearly, if not enough markers are provided there will be a tendency for the customers to collect in the “waiting for a marker” condition; a lack of registers will make the customers collect in the “waiting for dial tone” condition. T o obtain a simple probabilistic model for the “circulation” of

36

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

customers, we assume that the probability that an idle customer starts a call in the next interval of time of length h is Ah + o(h), the chance that a dialing customer completes his dialing in the next o(h),the chance that a busy marker finishes the call interval h is Sh it is working on is p h + o(h), and the probability that a conversation ends is h o(h), all as h -+ 0. T h e probability of more than one such event in h is o(h) as h -+ 0. These assumptions are in turn consequences of assuming that the time a customer stays idle, the time a customer takes to dial, the time a marker takes to complete a call, and the holding time (conversation length) are all mutually independent random variables, each with a negative exponential distribution, and the respective means A-l, 6-l, p-l, and unity. T h e number h is the calling rate per idle customer, 6 and p are the average rates of dialing and call completion by a marker (respectively), and time is measured in units of mean holding time, so that the hangup rate per call in progress is unity. T h e assumption that the marker operation times are exponentially distributed is not realistic, but we make it here in the interest of obtaining a global model whose statistical equilibrium equations can be solved in a simple way. This restrictive assumption could be avoided at the cost of complicating the mathematics. T h e important features of our model are depicted in Fig. 14; the labeled arrows indicate the rates of motion for various transitions. T h e state of the system is adequately described by giving the

+

+

R M A X @,r-A) WAITING FOR DIAL T O N E

i IDLE

I

-

,YMMIN(m,M) PROGRESS

FIG. 14.

REGISTERS

El

M A X (0,m-M) WAITING FOR A MARKER

Diagram of a telephone system.

15.

TRAFFIC CIRCULATION IN A TELEPHONE EXCHANGE

37

number i of idle customers, the number Y of customers that are dialing or waiting for dial tone, the number m that are being serviced by a marker or are waiting for a marker, and the number c of calls in progress. Actually, any three of these numbers suffice, since for physically meaningful states

i

+ + m + c = N. Y

Let pi,mc by the equilibrium (or stationary) probability of the state (i, Y, m , c). T h e “statistical equilibrium” equation is, with suitable

conventions at the boundaries,

This equation states that the average rate at which a state is left equals the average rate at which it is reached from other states. We observe that the flow of calls in the exchange is in a sense cyclic; in making a call, each customer passes through four stages: idle, dialing, marker, conversation, then back to idle, in that order. This fact yields a way of solving the equation. Each side of the equilibrium equation has four terms, one for each of the four stages of a call. We shall find a way of assigning to each term on the left a corresponding equal term on the right which will cancel it. T h e solution of the equation for (i, Y, m, c) f ( N ,0, 0, 0) is proportional to

f~max (1, j / ~ ) max (1, j ni

Iv!

j=o

/ ~ )

j=O

A W p

-firm =

T h e constant of proportionality is the probability of the “zero” state PNOOO =

(1

4-

2

i+r+m+c=N i,r,m,e>O i i N

-1 -fiymc)

,

obtained from the normalization condition for probabilities. T h e algebraic character of the solution is closely analogous to the actual

38

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

pattern of circulating traffic in Fig. 14, for the easiest way of showing that fir,,rcis actually a solution of the statistical equilibrium equations is to make the following correspondence between terms on opposite sides of the equations: XiPtrmc

s min P

(y1

&Jtrrnc

-

-

(c

+ ')P(z-l)rm(c+lv

+

1)P(t+l)(7-l)md

min ( m , M)Ptrmc 8 min ( y CP,,,,

P

+ 1,

min (m

+

mt(r+l)(m-l)c?

'7

~~lPtr(m+l)(e-l) *

I t can be seen that each term on the left cancels the corresponding one on the right when fir,,,, is substituted. Each term represents the (total) rate of occurrence on one of the four kinds of possible event: request for service, completion of dialing, completion of a call, and hangup. I n the life history of a given call, these events occur in the natural cyclic order given. Events associated with corresponding (i.e., canceling) terms are next to each other in this cyclic order.

$16. An Optimal Routing Problem Our final example is a variational problem involving both combinatoric and probability. We shall exhibit some particular answers to the following question: If requested connections can be put up in a connecting network by several different routes, leading to different states, which routes should be chosen so as to minimize the probability of blocking? This question poses a variational problem in which many possible methods of operating a connecting network of given structure are compared, rather than one in which different network structures are compared. We shall consider this question for a connecting network that is of little practical significance because it is obviously wasteful of crosspoints. Its virtues, however, are that it is perhaps the simplest network for which our question can be asked, and that it clearly exhibits the principles and arguments involved, so that these can be understood. T h e network is shown in Fig. 15, the squares standing for square 2-by-2 switches. T h e possible states of this network are determined by all the ways in which four or fewer inlets on the left can be connected pairwise to as

-

16.

SWITCH 2x2

FIG. 15.

AN OPTIMAL ROUTING PROBLEM

=

39

x X

= CROSSPOINl

A simple network in which optimal routing is studied.

many outlets on the right, no inlet being connected to more than one outlet, and vice versa. These possible states are depicted in a natural arrangement in Fig. 16; states that differ only by permutations of customers or switches have been identified in order to simplify the diagram. Th at is, there is essentially only one way to put up a single call, there are four ways of having two calls up, two ways each of having three and four calls up. These “ways” have been arranged in rows according to the number of calls in progress, and lines have been drawn between states that differ from each other by only the removal or addition of exactly one call. For ease of reference, let us number the states in the (partly arbitrary) way indicated in Fig. 16; insofar as possible, we have used small numbers for states with small numbers of calls. T h e set of possible states of our example then consists of (essentially) ten different configurations of calls in the basic network of Fig. 15. T h e state diagram, with each state identified now only by its number within a small circle, is schematized in Fig. 17. Also indicated in this schema are two important sets of quantities associated with the states. T o the left of each state is the number of idle inlet-outlet pairs, and to the right of each state is the number of idle inlet-outlet pairs that can actually be connected, i.e., that are not blocked. Only in the state numbered 4 are there any blocked calls. I t is to be noticed that state 4 realizes essentially the same assignment of inlets to outlets as state 2, which has no blocked calls. T h e difference

40

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

I

0 0 0

o n n

LO

0

FIG. 16. (Reduced) state diagram for the network shown in Fig. 15.

between the two is that in state 2 all the traffic passes through one middle switch, leaving the other entirely free for any call that may arise. Clearly, then, this difference illustrates the “packing rule” that one should always put through a call using the most heavily loaded part of the network that will still accept the call. (If state 4 were always rearranged to 2, there would be no blocking.) T h e question naturally arises, therefore, whether this packing rule is in any sense optimal for our particular example. We shall prove that it is, in two senses. I t is clear from an inspection of the state diagram that only in state 1 is there ever a choice of route, and that this choice is always between states 2 and 4. From the fact that state 4 is the only state with any blocked calls, it is intuitively reasonable to expect that

16.

41

AN OPTIMAL ROUTING PROBLEM

3

r4

FIG. 17. Schema of

47

state

4

diagram.

the probability of blocking is the least if the “bad” state 4 is avoided as much as possible, i.e., if from state 1 we always pass to either 2, 3, or 5, and visit 4 only when we have to, via a hangup from state 6. T h e next task is to choose a probabilistic model for the operating network; this will be done in the simplest possible way. We postulate that, in any state of the system, the probability that a given idle inlet-outlet pair request connection in the next interval of time h is Ah o(h), the chance that an existing connection cease is h o(h), and the chance that more than one event (new call or hangup) occur in h is o(h), as h -+ 0. T h e number h is the calling rate per idle pair, and time is measured in units of mean holding time, so the “hangup” rate is unity. New calls that are not blocked are instantly connected, with some specific choice of route, while blocked calls are lost and do

+

+

42

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

not affect the state of the system, their terminals remaining in the idle condition. T o complete the probabilistic description of the behavior of the system, it remains to specify how routes are chosen. I n our example, this amounts to specifying whether, for certain calls arising in state 1, the route leading to state 2 or that leading to 4 is chosen. At first we shall consider only those methods of choice that are independent of time, i.e., the choice is made in the same way each time. T h e methods of choice over which we shall take an optimum may be parametrized as follows: each time a choice is to be made between going to state 4 and state 2, a coin is tossed with a probability 01 of coming up heads. If a head comes up we choose state 4; if a tail, we choose state 2; the toss of the coin is independent of previous tosses and of the history of the system. T h e parameter 01 may take on any value in the interval 0 01 1; the value 01 = 0 corresponds to choosing state 2 every time; a value of 01 = 1 corresponds to choosing state 4 every time; a value of 01 intermediate between 0 and 1 means that 4 is chosen over 2 a fraction a: of the time. Introducing a natural terminology (from the theory of games), we may say that a choice of 01 represents apolicy or strategy for making routing decisions; a value 0 or 1 of 01 represents a pure strategy, in which the route is specified by a rigid rule, and there is no randomization; an intermediate value of 01 represents a mixed strategy. A choice of 01 determines a matrix Q = Q(a, A) of transition rates (Fig. 18) among states of the system, and so a Markov stochastic process taking values on those states. As a measure of performance we shall use the fraction b of requests for connection that encounter blocking, defined as follows: Let b(t) be the number of blocked calls occurring in the interval (0, t]; and let a ( t ) be the number of attempts for service occurring in (0, t ] ;then

< <

I t can be shown that this limit exists and is constant with probability one, so b is well defined. (Cf. Chapter 8.) T h e number b = b(01, A) can be calculated from the matrix Q as follows: If (i, i = 0, ..., 9) is a state, let p(i) be the number of blocked idle pairs in state i, and let y ( i ) be the number of calls in existence in state i. T h e stationary state probabilities { p i ,i = 0, ..., 9} exist

16.

43

AN OPTIMAL ROUTING PROBLEM

FIG. 18. Schema of state diagram showing transition rates.

and are the unique solution of the matrix-vector equation Qp Then b is given by

5

=

PzP(i>

c;Pd4 9

z=o

i=o

-

Y(i)I2

where the inner product ( p , x) is Z:-opix,

.

=

0.

44

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

We may therefore formally state our variational problem for this example as follows: to find that 01 in the interval 0 01 1 for which the ratio

< <

subject to the conditions

QP

=

0,

$pi

=

1.

i=O

I t is natural to expect that in choosing an optimum routing method in the example above there is no point in randomizing, i.e., using a mixed strategy with 01 unequal to either 0 or 1. T h a t this is so is not obvious from our mathematical statement of the problem, arid requires proof. We shall demonstrate a more general result:

Let x and y be vectors of 10 dimensions, with y Theorem 1.2. nonnegative and not identically zero,

16.

AN OPTIMAL ROUTING PROBLEM

45

These are the standard “statistical equilibrium” equations for the probabilistic model we have assumed. They can be solved by successively eliminating every pi except p , and obtaining a solution of the form Pi = f A , i # 0. T h e value of p , is then determined by the normalization condition ~ 9 = =~ 1pas~ 1

Po

=

Z=1

Thefi are of course functions of X and a:. We shall prove that they are linear functions of the parameter 01. We first eliminate p , and note that f , = 16h. Since the relations (iii)-(iv) contain the variables { p i ,i = 2, 3, 4, 51 only on the left, these variables may be eliminated entirely from (ii), and from (vii)-(x). But substitution for these variables in (vii) and (viii) in terms of (iii)-(vi) introduces a: and p , only in inhomogeneous terms. Hence, f6 and f , are linear in a , and so all {fi, i = 1, ..., 9} are linear in 01. Clearly, we have

+

C:=lfi cancel out, and so it because the normalizations terms 1 follows that ( p , x),’(p,y ) is a bilinear function of 01, i.e., it has the form

where A , , A,, B , , and B , are constants. Now d

-g(.)

da:

&(A,

+

= ____-

(A,

%%

+ BPI~ -

+

H I 4

which is of the same sign as its numerator. T h u s g’(a) is either always nonpositive or nonnegative, and so any extremum ofg(n) in 0 01 1 is assumed at the boundary, either for 01 = 0 or a: = 1. Since the solution p of Qp = 0 is known to have all strictly positive components for all a: in the unit interval, we have A , B2a:= ( p , y) > 0.

< <

+

46

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

It follows in particular that the minimum of blocking probability b is achieved for 01 = 0 or 01 = 1. It is unthinkable that visiting a blocking state (state 4) more frequently should decrease b, so we conjecture (and shall shortly prove that) 01 should be zero rather than one. Before doing this, though, let us observe that there is only one blocking state (viz., 4), and that the blocking probability b can be written as 2P45 h= 1%

+ 9P1 + 4 C P Z + P s + P , i=2

These facts and our intuition suggest that b should be a monotone increasing function of f 4 = rP,4. This conjecture is correct, and provides an easy way of showing that CY = 0 gives the least blocking probability. Let us prove it. From (i) and (ii) we find that

5 i='L

pi = 8h(9h + l ) p o - 2Apo = 72X2po,

whence b=-

16

2f4

+ 144h + 288X2 +f 6 + f 7

'

From (vii)-(x) we find that

= 96X3p,

-

$hp4.

Therefore b=

16 + I44h

2.f4

+ 556P + 192h3 - $hf4

'

16.

47

AN OPTIMAL ROUTING PROBLEM

This is of the form

2x n

-

cx

where a and c are strictly positive constants. Now 2x

d dx a

-

-

cx

-

2 n

-

2cx cx + ( a

2u

-CX)~

> 0.

( u - CX)Z

Hence, h is a monotone increasing function of f4 . I t follows that b is a minimum iff4 is a minimum. T o prove that the blocking probability b is a minimum for 01 = 0, it remains to calculate p4 from the equilibrium equations. By eliminating all the equilibrium probabilities except p , and p , , we find 1

p6

=

h+3 ( 1

p7=h+3(

+

8X2(l - a)16hpo 8X'%16hp0 240, 7 h 2 +-2h+2 4X216hp0 4hp6 + 4h+2

+

+

+

8h2(1- a)16hpo

4 x + r

+ APT)

+

(16h)'Apo 4hp7 4h+2

We have purposely not simplified the terms so that their origin can be verified. From these two equations we find that

where X=A+3-----

2x3

> 0.

x

h+l

2x 2h+1

+ + 7~ + 3 + 3x + 1 5 ~ 2

2x2

-

+

2 ~ 3 2x2 w3+4x+3 2x3

+ u"

+ +3

2 ~ 3 4~

1.

48

HEURISTIC REMARKS A N D MATHEMATICAL PROBLEMS

T h e coefficient of

oi

in f6 is 1 +------- h h+3----

2x

+1

This is positive, because

x

I f ______3-3 -__ 2x 3h 1 ~

+

i

X - L l 2X”+5h+3 =l--L-(---2x I 2h2 4h 3

+ +

+

1

However, Hence, W e shall now consider the problem of optimal routing in our (trivial) network from a different point of view. Instead of minimizing the ratio of unsuccessful attempts to attempts, let us simply minimize the average number of unsuccessful attempts in any finite number of events, counting changes of state and unsuccessful attempts as events. I n our example, the only choice is between states 2 and 4, when a particular call requests connection in state 1. By a policy, let us mean a function p( ) on the nonnegative integers taking the values 0 and 1. Let x, be the state of the network after n events, n >, 0. We say that the system is operated according to policy p ( ) if, for each n >, 0, given that x, = 1 and a choice occurs, the system moves to state 2 if and only if p(n) = 1, state 4

if and only if p(n) = 0.

Now our intuitive feeling is that going to state 2 is preferable over going to state 4 under all circumstances. At the cost of anticipating results to be proven, let us partially order all the possible policies by the definition: If p( * ) and q( . ) are policies, then

p 2 q if and only if p(n) 2 q(n)

* Read “ p > q”

as

“ p is no worse than

q”!

for all

n

2 0.-

16.

49

AN OPTIMAL ROUTING PROBLEM

T h e shift transformation T of policiesp( * ) is defined by the condition

+ I),

Tp(n) = P(.

n

2-0.

It is evident that p 3 q implies Tp >, Tq. Let E,,,(x) define

I

E 0,

and

number of unsuccessful attempts after n events E,,,(x) = E starting from state .r if the system is operated according to policy p( .)

Let S be the set of states (0, 1, ..., 9}. We shall prove Theorem 7.3.

If p >, q, then f o r all n 3 1 and x

<

En.,(X)

E

S

~fl.&).

As a preliminary result (not without its own interest) we shall need the Lemma 7 . 7 .

For n >, 1 and any policy p( ) E,.,(4)

=

1nax T E S Efl,&)*

This says that starting in the (sole blocking) state 4 is always the worst way to start, no matter how long we run the system. Proof. For n = 1 and x f 4, E,,,(x) = 0 since no unsuccessful attempt can occur in any state except 4.However,

<

so the lemma is true for n 1. Assume as an hypothesis of induction that it is true for n k. Now for x # 4,Ek+l,Jl(x)is a convex combination of values of E,,,,,( . ), so clearly, for x # 4,

<

< y$:

J%,&)

E,,T,(Y) = E7.,T17(4)*

However, elementary probability arguments establish that for each policy s( . ) Ek+i,s(4)

=

+

E R , , ( ~ ) Pr{x,

so the lemma is proven.

=4

I Xo

=

~)EI,T’,(~),

1.

50

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

-

For any policy s( )

Proof of Theorem 1.3.

if x # 4,

El.l,(x) = 0

Hence,

for all x s S.

El,,(x) = E,,,jx)

Assume as an hypothesis of induction that p 3 q implies Ea,&)

<

< En,a('4

for all x and all n k. Now for x # 4 or 1 and any policy s( Ek+l,s(x)is a convex combination of values of

*

),

E k , Ta(.).

For x

=

4, we have for any policy E k t *.s(4)

=W

2h

h

s(

-)

+ convex combination of

where the coefficients of the convex combination are transition probabilities independent of the policy s( * ) , and

I

first event is a start in state 4 i i' blocked attempt

- - -2h -=~r\ 24hi

Hence, p 3 q and x # 1 implies

<

Ek+l,P(X)

For x

=

Eki1,4(X)'

1 and any policy s( - ) we have

Ek+l,s(l) =

4h

{41)Ek,Ts(2)

+ [1

--

41)I%,Ts(4)}

+ 1 + 9 h convex combination of EkSTs(-), where the coefficients of the convex combination are independent of . ), and

s(

4h ___ 1 +9h

= Pr

1

/first event requires start in state I (routing decision

I.

51

REFERENCES

Suppose now that p 3 q. It is sufficient to show that

+

+

G mEk,TQ(2)

~ ( i ) ~ ~ , ~ ~[1(2)

-

~wI-G~J~).

If p( I )

= q( l), this follows from the hypothesis of induction. T h e only other possibility is that p ( 1) = 1 and q( 1) = 0. By Lemma 1.1 and the hypothesis of induction we find EI,.TD(2)

<Ek.Td4) < E,,Ta(4)-

This proves Theorem 1.3. T h e result at once shows that the policy

p = 1 is optimal in the sense that it minimizes lim sup n-lEn,fl(x). n +io

REFERENCES

1. R. Syski, “Introduction to Congestion Theory in Telephone Systems.” Oliver & Boyd, London, 1960. 2. L. Kosten, The Historical Development of the Theory of Probability in Telephone Traffic Engineering in Europe, Teleteknik (English Edition) 1, 32-40 (1957). 3. R. I. Wilkinson, The Beginnings of Switching Theory in the United States, Teleteknik (English Edition) 1, 14-31 (1957). 4 . E. C. Molina, Computation Formula for the Probability of an Event Happening at Least c Times in N Trials, Am. Math. Monthly 20, 19&193 (1913). 5. A. Jensen, An Elucidation of A. K. Erlang’s Statistical Works Through the Theory of Stochastic Processes in the Life and Works of A. K. Erlang, Trans. Danish Acad. Sci. No. 2, 23-100 (1948). 6. T. Engset, Die Wahrscheinlichkeitsrechnung zur Bestimmung der Wahleranzahl in Automatischen Fernsprechamtern, E.T.Z., 31, 304-306 (1918). 7. G. F. O’Dell, An Outline of the Trunking Aspect of Automatic Telephony, J . Inst. Elec. Engrs. 65, 185-222 (1927). 8. C. D. Crommelin, Delay Probability Formulae, P.O. Elec. Engrs. J., 26, 266274 (1933-1934). 9. E. C. Molina, Application of the Theory of Probability to Telephone Trunking Problems, Bell System Tech. J . 6 , 461-494 (1927). 10. F. Pollaczek, Uber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. 2. 32, 64100, 729-750 (1930). 11. A. I. Khinchin, Matematicheskaya Teoriya Statsionarnoi Ocheredi, Matematicheskii Sbornik 39, 73-84 (1932). 12. T . C. Fry, “Probability and Its Engineering Uses.” Van Nostrand, New York, 1928. 13. A. N. Kolmogorov, “The Foundations of Probability,” 2nd ed. Chelsea, New York, 1956.

52

1.

HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS

14. C. Palm, Intensitatsschwankungen im Fernsprechverkehr, Ericsson Technics 44, 1-189 (1943). 15. W. Feller, On the Theory of Stochastic Processes, with Particular Reference to Applications, Proc. 1st Berkeley S y m p . M a t h . Stat. and Prob., 1949 pp. 403-432. 16. L. Kosten, On the Influence of Repeated Calls in the Theory of Probabilities of Blocking, D e Ingenieur 59, 1-25 (1 947). 17. L. Kosten, J. R. Manning, and F. Garwood, On the Accuracy of Measurements of Probabilities of Loss in Telephone Systems, J . Roy. S t a t . SOC.B11, 54-67 (1949). 18. F. J. Scudder and J. N. Reynolds, Crossbar Dial Telephone SwitchingSystem, Bell System Tech. J. 18, 76-118 (1939). 19. C. Jacobaeus, A Study on Congestion in Link Systems, Ericsson Technics 51, 1-68 (1950). 20. R. Fortet and B. Canceill, Probabilite de Perte en SClection ConjuguCe, Teleteknik 1, 41-55 (1957). 21. C. Y. Lee, Analysis of Switching Networks, Bell System Tech. J . 34, 1287-1315 (1955). 22. P. Le Gall, Methode de Calcul de 1’Encombrement dans les Systhmes TelCphoniques Automatiques B Marquage, Ann. Tilicom. 12, 374-386 (1957). 23. K. Lundkvist, Method of Computing the Grade of Service in a Selection Stage Composed of Primary and Secondary Switches, Ericsson R e v . No. 1, 11-17 (1948). 24. A. Elldin, Applications of Equations of State in the Theory of Telephone Traffic, Thesis, Stockholm, 1957. 25. C. Clos, A Study of Non-Blocking Switching Networks, Bell System Tech. J . 32, 406-424 (1953).

CHAPTER

2

Algebraic and Topological Properties of Connecting Networks

A connecting network is an arrangement of switches and transmission links allowing a certain set of terminals to be connected together in various combinations, usually by disjoint chains (paths): e.g., a central office, toll center, or military communications system. Some of the basic combinatorial properties of connecting networks are studied in the present chapter. Three of these properties are defined informally as follows: A network is rearrangeable if, given any set of calls in progress and any pair of idle terminals, the calls can be reassigned new routes (if necessary) so as to make it possible to connect the idle pair. A state of a network is a blocking state if some pair of idle terminals cannot be connected. A network is nonblocking in the wide sense if by suitably choosing routes for new calls it is possible to avoid all the blocking states and still satisfy all demands for connection as they arise, without rearranging existing calls. Finally, a network is nonblocking in the strict sense if it has no blocking states. A distance between states can be defined as the number of calls one would have to add or remove to change one state into the other. This distance defines a topology on the set of states. Also, the states can be partially ordered by inclusion in a natural way. This partial ordering and its dual define two more topologies for the set of states. The three topologies so obtained are used to characterize (i.e., give necessary and sufficient conditions for the truth of) the three properties of rearrangeability, nonblocking in the wide sense, and nonblocking in the strict sense. Each of these three properties represents a degree of abundance of nonblocking states ; the mathematical concept used to express these degrees is the topological notion of denseness.

53

54

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

$1. Introduction Any large communication system contains a connecting network, an arrangement of switches and transmission links through which certain terminals can be connected together in many combinations, usually by many different possible routes through the network. Examples of connecting networks can be found in telephone central offices, toll centers, and military communications systems. T h e connections in progress in a connecting network do not arise usually in a predetermined temporal sequence; instead, requests for connection (new calls) and terminations of connection (hangups) occur more or less “at random.” For this reason it is customary to use the performance of a connecting network when subjected to random traffic as a figure of merit. One precise measure of this performance is the fraction of requested connections that cannot be completed in a given (long) time interval, or the probability of blocking. I n a telephone connecting network this probability measures to some extent the grade of service given to the customers. T h e performance of a connecting network for a given traffic level is determined largely by its configuration or structure. This configuration may be described by stating what terminals or transmission links have a switch placed between them and can be connected together by closing the switch. T h e configuration of a connecting network determines what groups of terminals can be connected together simultaneously. Any one set of permissible connections may be called a state of the network. Quantities such as the number of combinations of terminals that can be connected, and the number of states in which a given combination is connected, clearly are indicative of both the performance and the cost of the system. If these numbers are small the performance may be poor and the cost low; if large, the performance may be unnecessarily good and the cost prohibitive. T h e numbers are among the purely combinatorial and topological properties of the connecting network. For example, in a telephone exchange, the network configuration determines what pairs of terminals can be simultaneously connected by disjoint paths, that is, what calls can be in progress. If this configuration is too simple, only a few pairs of terminals can have calls in progress between them at the same time. If the configuration is extensive and complicated it may provide for many large groups of

2.

SUMMARY OF CHAPTER

2

55

simultaneous calls in progress, but the network itself may be expensive to build and difficult to control. T o design connecting networks with confidence, then, it is desirable to have an adequate general understanding of their combinatorial and topological properties. A discussion, in part heuristic and tutorial, of connecting systems and of some associated mathematical problems has been given in Chapter 1, based on a paper ( I ) by the author. There a division of the topic into combinatorial, probabilistic, and variational problems was drawn, and it was argued that the elements of this division have a natural order of priority: one must know the combinatorial properties of a system in order to calculate its probabilistic properties, i.e., its performance in the face of random traffic; and one must know both the combinatorial and the probabilistic properties of systems in order to compare them and to select optimal ones. I n this chapter we shall be concerned exclusively with those combinatorial and topological properties of a general connecting network that seem to be most relevant to its performance.

$2. Summary of Chapter 2 Some of the basic combinatorial properties of connecting network are studied in the present work. Three of these properties, rearrangeability, nonblocking in the wide sense, and nonblocking in the strict sense, can be defined informally as follows: For brevity, define an idle pair to be a pair of idle terminals consisting of an inlet and an outlet. A network is rearrangeable if, given any set of calls in progress, and any idle pair, the existing calls can be assigned new routes (if necessary) so as to make it possible to connect the idle pair. A state of a network is a blocking state if some idle pair cannot be connected. A network is nonblocking in the wide sense if by suitably choosing routes for new calls it is possible to avoid all the blocking states and still satisfy all demands for connection as they arise, without having to rearrange existing calls. Finally, a network is nonblocking in the strict sense if it has no blocking states. A distance between states of a connecting network can be defined as the number of pairs of terminals that are connected in one state and not in the other. This distance defines a topology on the set of states. Also, the states can be partially ordered by inclusion in a natural way. This partial ordering and its dual define two more topologies

56

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

for the set of states. T h e three topologies so obtained are used to characterize (i.e., give necessary and sufficient conditions for the truth of) the three properties of rearrangeability, nonblocking in the wide sense, and nonblocking in the strict sense. Each of these three properties represents a degree of abundance of states in which calls are not blocked; the mathematical concept used to express these degrees is the topological notion of denseness. Studies of some particular connecting networks that are rearrangeable appear in another paper (2),and in Chapter 3 .

93. The Structure and Condition of a Connecting Network

I n discussing connecting networks, we shall abstract from the many possible technological realizations and actual designs of connecting networks, and shall consider only certain relevant features on which we can base a useful and sufficiently general mathematical theory. Most real telephone switching networks consist of pairs of wires for talking paths and electromechanical switches for crosspoints; in certain experimental systems the talking paths are pulse code modulation channels, and the crosspoints are time-division gates made of transistors. However, any attempt to formulate some general properties of connecting networks must be independent of the network configuration chosen, and of the technology used to build the network, for a particular real system. A theory must apply equally well to Strowger switches, crossbar switches, gas-diode switches, and timedivision switches. Unless it is independent of technology, a theory of connecting networks is limited in scope and may have missed the heart of the problem. We therefore use some of the terminology of switching engineers but understand it to refer to defined mathematical idealizations of switches, gates, crosspoints, transmission links, etc., rather than to the physical entities themselves. We distinguish between switching networks used for communication and those used for control functions and logical transformations, like relay nets. Our concern is with networks of the former kind, and we call these connecting networks. A communications switching network or connecting network consists of three kinds of entities: (i) wires or other transmission media along which communication may take place; (ii) terminals to which the

4.

GRAPHICAL DEPICTION OF NETWORK

57

wires are attached; and (iii) crosspoints or switches which can be used to connect the terminals, and hence the wires, together in various combinations. Each crosspoint can connect together exactly one pair of terminals, and it has two conditions: in the “on” or closed condition the two terminals are connected and communication can pass from one to the other; in the “off” or open condition the terminals are disconnected, and no information can pass through. From the point of view of switching, two terminals connected together by a wire are essentially one terminal, albeit a spatially extended one. We therefore regard terminals as identical if they are wired together; in mathematical terms, we identify terminals under the equivalence relation of “being wired together.” Henceforth, then, our considerations will leave wires out of account, and will be based only on the notions of terminal and crosspoint. By the configuration or structure of a connecting network, we mean a specification of the terminals between which individual crosspoints have been placed. By the condition of a connecting network, we mean a specification of the closed and open crosspoints. I n most cases of interest the structure is invariant in time, while the condition changes in a random way. We shall assume that at most one crosspoint is placed between distinct terminals, and that no crosspoint is placed from a terminal to itself.

54. GraphicaI Depiction of Network Structure and Condition A simple device can illustrate the four notions we have introduced

so far. I n Fig. l(a) the nodes (points) represent terminals, and the branches (lines) labeled xi,i = 1, ..., 6, represent crosspoints placed between the terminals. T h e resulting graph represents the structure of a network. If we interpret the labels xi as binary variables specifying

the condition of the (respective) crosspoints, with 0 meaning “open” and 1 meaning “closed,” then an assignment of values 0 or 1 to { x ~ ..., , x6} represents a possible condition of the network, illustrated in Fig. l(b). We are purposely avoiding the term “network state’’ here in order to assign it a useful precise meaning in the next section. We have illustrated the use of a labeledgraph as a general representation for (simultaneously) the structure and condition of a connecting network. This representation is useful because it identifies the

58

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

2,

\

\

- r

x2 x3 z4 x5

=

=

= = =

re =

1

0 1

0

0 0

(4 FIG. I .

(a) Representation of structure; structure and condition.

(b) (b) simultaneous representation of

structure and condition of the network with a definite mathematical entity. It will become apparent that simple properties of this mathematical representation have great theoretical and practical relevance to congestion problems. I n general, the labeled graph g representing* the structure and condition of an arbitrary connecting network is constructed as follows: (i) Nodes (points) of g correspond to terminals of the network. (ii) Branches (lines, or edges) of g correspond to crosspoints of the network. (iii) Open crosspoints are labeled 0. (iv) Closed crosspoints are labeled 1. Two terminals are connected in g if g contains a chain of closed crosspoints from one terminal to the other.

$5. Network States Let G be a graph representing the structure of a switching network, and let V be the set of all labeled graphs g (labeled “versions” of G) obtained by assigning 0 or 1 to each line of G. There are several reasons why not every element g of V represents a physically meaningful state of the network.

* A glossary of mathematical notations appears at the end of this chapter, Section 14.

5.

NETWORK STATES

59

I n most switching systems there is an explicit functional distinction between terminals which are used only to connect other terminals together, and those between which desired connections arise, and which are never used to connect other terminals together. Terminals of the former kind we shall call links, because of their intermediary nature, and those of the latter kind, inlets and/or outlets. Desired connections always arise between two or more inlets or outlets. If more than two are involved, the connection is termed a “conference” call. Usually, though, the connections are disjoint chains of closed crosspoints, assuring private conversations between inlets and outlets by pairs only; we restrict attention to these. I n terms of our graphical representation of the structure and condition of a switching network, the distinctions made above impose restrictions on the elements of V which represent realistic conditions of a network having the structure of G. T h e restriction on the assignment of the labels 0 or 1 listed above are (perhaps the most important) among many which are imposed by the functional and operational features of a real switching system. I n general, a real connecting network specifies (or uses) only a subset of the set V of all possible labeled versions of the graph G that represents the structure of the network being studied. We have therefore avoided calling elements of V “states of the network” because not all members of V can reasonably represent the condition of an actual network. We now attempt to characterize those subsets S of V which can represent real networks. Each such subset S will be called a class of network states. T h e Boolean operations of join u and meet n (union and intersection, respectively) are definable for elements x,y of Y in an obvious way: I/-element having a 1 wherever either x or y has a 1, and 0 elsewhere,

x u y = the

x ny

= the

V-element having a 1 wherever both x and y have a 1, and 0 elsewhere.

T h e complement x’ and the difference x - y can be defined analogously. I n view of this it is natural to inquire whether these Boolean operations can be used to characterize subsets S of V which are classes of network states. If the elements x,y of V belong to such a subset (class of network states) S , it is not necessarily true that x u y , nor that x n y , belongs to

60

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

S. I n the case of x u y , there may be links and crosspoints used in both x and y , and so x u y may violate the requirement of privacy. Even if x n y = 0 there may still be inlets used in both x and y , so that x u y would lead to undesirable paths of extreme length. I n the case of x n y , there may be so little in common to x and y that x ny reduces to a single closed crosspoint between two links (i.e., not between an inlet and an outlet). Th u s the Boolean operations do not yield a useful way of describing S. T h e preceding remarks suggest that since any connection is a chain, none of whose terminals and crosspoints occurs in another connection, the labels 0 and 1 are really superfluous, aIthough they served a tutorial purpose heretofore. Th at is, in describing the possible subsets S of network states, we can (and should) take advantage of inherent physical restrictions, and conveniently replace our representation* x E V of the structure and condition of a network by a corresponding set of disjoint chains, since each physically meaningful element x from I/ is equivalent to such a set. A formal development of this suggestion follows. Let T be the set of terminals of a connecting network. T h e graph G representing the structure of the network is a subset G of the product

I’ x T

= {(u,a):

u E T,

E

T}

with the properties (u, v ) E

G

(u, u )

if and only if (v, u) E G

is never in G

and the interpretation (u, v ) E G

if and only if

(u, a ) is

an edge of graph G

if and only if nodes u and v are adjacent in the graph G

if and only if there is a crosspoint between

terminals u and c.

* ‘‘Y

E

V” means that x is an element of the set

V.

5.

61

NETWORK STATES

A chain p of length n between terminals u and v is a sequence of i n) such that elements (ziE T , 0

< <

zo = 11,

zi# zi

(zi, zi+,)E G

z, = 21,

for i # j ,

for i

= 0,

..., n - 1.

Two chains p , and p , are called disjoint if they have no nodes (terminals E T ) in common; in this case we write symbolically p , n p , = 8, with 8 = null set. We shall henceforth assume that the set T of terminals has been (functionally) decomposed into three sets: T

=IVQVL,

where I is a set of inlets, Q a set of outlets, and L is the set of links. It is possible that I = Q or that I n Q = empty set, or that some intermediate condition obtains. However, we shall insist that ( I u Q) n L be null, i.e., that no link be an inlet or an outlet. T h e set C of connections consists of all chains p = {ziE T , i = 0, ..., n(p)}such that xo E 1, ziE L ,

Z A P ) E Q,

zo

# ZAP)

for i # 0 or n(p).

Each element p of C represents a possible connection from an inlet to an outlet through the network whose structure is represented by the graph G. Elements of the set S of network states will be defined as subsets x of C, x C C, consisting entirely of disjoint chains, that is, such that

p , ,p , E x

implies

p , np 2 = 8.

Two subsets x and y of C are called compatible if

p,

E x,

p, Ey

implies

p , np z

= 8.

T he connections that comprise compatible states can all be put UP simultaneously without interfering with each other or violating the requirement of privacy.

62

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

T h e functional and physical restrictions imposed by real networks determine (in any particular system) a subset E of C consisting of (what we shall call) the elementary states, or single connections that can actually be used. For example, chains in C that double back and are wastefully circuitous may be excluded from E. Given such a subset E of elementary states, we can define a class of network states S, associated with E, in a natural way as follows: S is the smallest class of subsets of E containing all unit subsets of E, and closed under formation of arbitrary intersections (meets) and unions (joins) of compatible subsets of E. That is, S is the smallest class of E-subsets such that

pEE

implies ( p ) E S, implies x ny E S , S and p , E x , p , E Y implies p , n p , = 8, then XU~ES.

x,y

if

x,y E S

E

We henceforth use “S” as a generic notation for a class of network states defined as above. T h e word “network” will refer to a graph G representing structure, choices I and Q of inlets and outlets, respectively, and a choice E of elementary states. T h e choice of G, I , Q, and E uniquely determines a class S of network states according to the definition given previously. T h e quadruple (G, I , Q, E ) will be called a network v. I t is easily verified that a class S of network states is partially Moreover, any two elements x, y of S have a ordered by inclusion, unique intersection (meet) consisting of just those connections common to both x and y , and S itself has a unique least element included in every other element, viz., the ground state in which no calls are in progress. However, since only infima exist, and since there may be many maximal elements in the partial ordering, S is not a lattice, in general (3).

<.

$6. The State Diagram

<

T h e partial ordering of S has a special nature that allows us to arrange the network states x E S in a particularly intuitive and useful pattern. T h e following conventions and definitions will be helpful in discussing this pattern.

6.

63

THE STATE DIAGRAM

If K is any set, we use the notation 1 K 1 to mean the number of elements of K . E.g., if x is a network state,

1x I

==

the number of calls in progress in state x.

T h e sets L, are defined by the conditions L,={xES:

IxI=k),

k=0,1,

...,

that is, L, is the set of all network states consisting of exactly k connections. Lo is a unit set containing just the zero state. T h e sets L, are a partition of S corresponding to the equivalence relation of “having the same number of calls in progress,” T o obtain our pattern for arranging network states we start with the zero or ground state in which no calls are in progress: this is the empty set (of chains). Above this zero element, in a horizontal row, we place all the states consisting of a single connection, i.e., all the elements of E . Continuing in this way, we put the set Lk+lof states consisting of (k 1) disjoint chains (i.e., k 1 calls) in a horizontal row above the set L, of states with k disjoint chains (i.e., k calls in progress). We call L,
+

+

Y-XEE, i e., if and only if y results from x by putting u p one more call. T h e resulting graph can be termed the state diagram D of the network v described by the quadruple (G, $,Q, E ) . T h e state diagram D is a natural and standard representation of the partial ordering of S. T h e history of the connecting network when operating can be thought of as a trajectory on D. We shall use the network depicted in Fig. 2 to illustrate the state diagram D. For practical purposes this network is wasteful of crosspoints, but it makes a suitably simple example of the partial ordering of the states. T h e network has four inlets and four outlets, and no inlet is an outlet. T h e squares in Fig. 2 represent 2-by-2 switches, as indicated.

64

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

SET I OF INLETS

I TL -

LINKS

SWITCH

=

2x2

X

n

SET OF OUTLETS

= CROSSPOINT

FIG. 2. Illustrative three-stage connecting network.

T h e possible states of this network are determined by all the ways in which four or fewer inlets can be connected pairwise to as many outlets on the right, no inlet being connected to more than one outlet, and vice versa. These possible states have been depicted in a natural arrangement in Fig. 3, which shows a reduced state diagram in which states that differ only by permutations of inlets, outlets, or switches have been identified. There is essentially only one way of putting in a single call; there are four ways of putting in two calls; and there are two ways each of putting in three and four calls. T h e states have been arranged in levels according to the number of calls in progress. In each state only links actually in use are shown, and the different notations on the links indicate the routing.

57. Some Numerical Functions T h e finite set S of network states is partially ordered by inclusion, which we shall denote by A chain in S is a subset X of S which is simply ordered by (the restriction to X of) that is, for any two elements x, y E X , we have either x >, y or y 3 x. Such a chain is not to be confused with the “chains” on the graph G that are elements of states x E S. T he dimension or height 1 x j of a state is the maximum < xd = x that have x for greatest “length” d of chains 0 < x1 < element. (This usage is consistent with the previous definition of I I.)

<.

<;

9

7.

SOME NUMERICAL FUNCTIONS

65

L4

Ll

LO

FIG. 3.

(Reduced) state diagram for the network shown in Fig. 2.

T h e dimension 1 x 1 of a state x is the number of Remark 2.1. busy pairs, or the number of calls in progress, in the state x.

A state x is said to cover another state y if and only if x > y , and there are no z E S such that x > z > y . T h e state x is then “immediately above” y . I t is apparent that x covers y if and only if x > y and I x I = 1 y I I. I n fact, the construction of the partial ordering of S arranges the states according to levels, each level being the (equivalence) class of all states having the same dimension. I n determining dimension one need only consider chains that are “maximal” or “connected” in the sense that xi covers xiPl for all i. Also, it can be Seen that the partial ordering of S satisfies the Jordan-Dedekind

+

<

66

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

chain condition: all connected chains between fixed end points have the same length. T h e present section will be devoted to various relationships between numerical functions defined on S , counting or “enumeration” problems, etc., based largely on the dimension function and the chain condition. T h e Mobius function ,LL(* ) of the partially ordered system (S, <) is defined recursively by p(0) = 1,

p(x) =

-

2p(y)

if

x

> 0,

Y<X

where 0 denotes the zero or ground state in which no calls are up: T h e Mobius function has the following two important properties. (i) Let f( ) be any function defined on S , and let F ( x ) = CJ(Y). Y<X

Then f( . ) and F( . ) are related by the Mobius inversion formula

f(4 = 2 P(Y)F(x - Y ) . 1J < X

Here x - y denotes the state obtained from x by removing all the calls of state y; this makes sense, since y < x. [See Weisner (4.1 (ii) Let X(x, n) be the number of chains of length n that can be interpolated between 0 and x.P. Hall ( 5 ) has shown that -p(x)

= h(x, 1) - A(.%, 2)

+ ..-.

By the Jordan-Dedekind chain condition, all the chains from 0 to x have the same length, viz., 1 x I. Hence for x > 0 p(x) = (-l)iX’A(x,

I x 1).

For simplicity of notation set

I *% I)

= .i(x> = number

of way of “climbing,, from 0 to x.

7.

SOME NUMERICAL F U N C T I O N S

67

Also, we introduce the following sets: A,

= {y:

y covers x}

coversy}

Bx = { y :

x

L,

I x I = a}.

= {x:

These have the following respective intuitive meanings: A, is the set of states immediately above x, i.e., obtainable from x by adding one more call; B, is the set of states immediately below x, i.e., obtainable from x by removing one call; L , is the nth level, the set of all states having n calls up. T h e cardinality of a finite set X is designated by 1 X 1. Remark 2.2.

1 B, 1

Remark 2.3.

For each x E S

= j

x I for each x E S. Clearly, x covers exactly

i x 1 states, each obtainable from x by removing one call. ri(4 =

c

VEB,

ri(Y)*

Indeed, every state y covered by x gives rise to exactly ~ ( yclimbing ) paths from 0 that reach x via y . Remark 2.4. For x E L , , ~ ( x )has the constant value n! This is obvious intuitively, since there are n! orders in which the n calls of x E L , could be put up. More formally, the result is true for x = 0; assume it true for y then, by the previous results, rlw =

Remark 2.5.

c

YEB,

rl(Y)

=

I Bx I

=

n!.

*

( n - I)!

T h e Mobius function p( * ) is given by p(x) = ( - 1 ) i q =

(-1).n!

x I)! for

XEL,.

Theorem 2.7.

Proof. T h e segments in the partial ordering passing upward from elements y EL,-^ are just those that pass from some x EL, to L,-l,

68

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

and by Remark 2.2, each x E L , has exactly 1 x I( =n) such segments. Therefore, n * l L l =

2

YEL,-l

!'%I?

and the sum on the right is exactly divisible by n. We define C , as the total number of chains (of length n) from 0 into L, , i.e., to some state of L, . Remark 2.6. cn =

2 dx) 2 =

XELn

V"L,-1

77(Y). I A ,

I.

It can be seen that x E L , has ~(x)chains climbing to it from 0; for x,y E L , , x # y , these chains are distinct since their highest elements are unequal. This proves the first identity. Also each chain climbing to L,, from 0 must pass through some unique y E Ln-, . Each y E Lnp1 has ~ ( ychains ) of length n - 1 reaching it from 0, and each such chain can then be completed to reach L, in 1 A, 1 ways. I t follows also that

$8. Assignments By an assignment we shall mean any one-to-one map u( . ) of a subset of I into Q. An assignment is to be interpreted as a specification of what inlets are to be connected to what outlets, without regard to the possible routes that these connections might take through the network. If I n Q is non-null, we restrict assignments so as to satisfy u(u) # U. Let x be a network state consisting of chains p , ,p , , ..., p , with n = n ( x ) > 0 and each pi a chain between uiE I and vi E a. We say that x realizes the assignment u( * ) if and only if

< I x 1) < < I x I}

(i) the domain of a( * ) is (zi, 1 ,< i (ii) the range of a( * ) is (vi , 1 i i [ x 1. (iii) a(uJ = v i , 1

< <

An assignment is realizable if some network state realizes it; a state realizes exactly one assignment; the zero state realizes the null assign-

9.

69

THREE TOPOLOGIES

ment. A maximal assignment is one that has either domain I or range

52. T h e set of all assignments is denoted by A , and that of all maximal

assignments by A. Two terminals, u E I and v E Q, are connected in state x if and only if some chain p E x is a chain between u and v, i.e., if and only if ( p } realizes the (unit) assignment {(u,

We define the function y( . ) from S into (the set of) subsets of

I x Q by the condition

y ( x ) = ((u, v) € 1 x

are connected in x}.

Q : u and

Formally, then, y ( x ) is the assignment realized by state x ; heuristically, we may think of y ( x ) as the set of calls that are in progress in state x. T h e set of unit assignments, that is, of c = {(u, v)}

such that

( u , v) € 1x

Q,

will be denoted by U , and a unit assignment C E U will be referred to informally as a call. If a = a( ) E A is an assignment, we use the notation Y-W

-

for the inverse image of a( . ) under y( ), i.e., the set of (equivalent) states y such that y ( y ) = a. In a similar vein, if X is a set of states, we define y ( X ) = { u E A j a = y(x)

for some x EX},

that is, y ( X ) is the set of assignments realized by members of X .

$9. Three Topologies Two network states x and y are equivalent, written x only if they realize the same assignment, i.e.,

-

y , if and

Y(X) = Y(Y)-

Intuitively, equivalent but nonidentical states correspond to different ways of putting up the same set of calls.

70

2.

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

A pseudometric (6, p. 118) on S can be defined by the formula d(x, Y ) = I y ( x ) dY(Y)

( 9

x, Y

E

s,

where d denotes the symmetric difference of sets, and [ . I cardinality, as before. I n plain words, the distance d ( x ,y ) between x and y is the number of pairs ( u , v ) G I x Q that are either connected in x and not connected in y , or connected in y and not connected in x. Clearly, d(x, 0)

and also

=

1 x I,

d(x,y) = 0

O

=

zero state,

if and only if

x -y,.

T h u s d( * ) only identifies states up to equivalence. T h e function d( *, . ) is obviously symmetric, and the triangle inequality is a consequence of the set inclusion a,

(XA Y ) c (XLl 2 ) u (k’ d Z). T h e pseudometric d( *, . ) can be used to define a topology for S in a standard way [see Kelley (6, p. 118 et seq)]. T h e closure of a set X in the d-topology consists of all states equivalent to members of X , and is denoted by X d . For each subset X of S , we define its <-closure X by the condition

3 ={yE

S 1 3’

<x

for some x E X I .

T h e operation on sets so defined satisfies the Kuratowski closure axioms [cf. Kelley (6, p. 43)]: 0 1 8 -

X_C&

&=& XUY uy __ = x and so defines a closure topology for S . T h e set & consists of all states that are “below” some member of X in the state-diagram D, i.e., can be reached from a member of X by removing calls. In a similar way, we define the >-closure X of a set X s S as x = ( y ~ ~ I y > xforsome

EX).

10.

SOME DEFINITIONS AND PROBLEMS

71

T h e converse of a partial ordering relation is also a partial ordering, called its dual. Hence the mapping X + is also a closure operation, defining a third topology on S.

x

$10. Some Definitions and Problems An inlet or outlet is idle in a network state x if it belongs to neither the range nor the domain of the assignment y(x) realized by x. An idle pair of the state x is an element (u,v) of I x such that both u and v are idle in x. A call c = ( ( u , u ) } is new in x if (u,v) is an idle pair of x. We shall now define what is meant by a blocked call. Let x E S realize the assignment y(x) and let c be a new call in x, i.e., let

u

c = {(?.I, v)) E

be a unit assignment such that (u,v) is an idle pair of x. T h e new call c is blocked in x if there is no state y > x such that Y ( Y ) = Y ( 4 u c-

A state x is a blocking state if some call is blocked in x. T h e state x is called nonblocking if and only if, for every idle pair (u,v) of x, the call c = {(?I, a ) )

is not blocked in x, i.e., there is a y larger assignment y ( x ) u c, so that Y ( Y ) = Y(4

y > x.

E

"

S above x which realizes the

{(I!,

4)

T he set of nonblocking states is designated by the symbol B'. A state that realizes a maximal assignment has no idle pairs, and is (trivially) nonblocking. I n plain terms, a nonblocking state x is one in which any idle inlet u can be connected to any idle outlet v without disturbing the calls that are already present; in this case there is a path r , disjoint from all paths p E x, between u and v, and x

u {Y}

E

s,

i.e., use of this path results in a network state.

2.

72

ALGEBRAIC AND TOPOLOGICAL PROPERTIES

A network v = (G, I , Q, E ) will be called nonblocking in the strict sense if and only if every state is nonblocking, i.e., B' = S. Such networks have been discovered and studied extensively by C. Clos. [See Clos, (7) Kharkevich, (8) and Chapter 4.1 A network that is nonblocking in this strong sense has the property that no matter in what state it is, any idle pair can be connected (in a way that results in a legitimate network state). I n most switching networks there may be several or many ways of connecting an idle pair, i.e., putting up a new call, in a given state, all of which lead to legitimate network states. Thus, even if the set S of network states contains blocking states, it is conceivable that by making the right choices of paths for connections one might avoid all the blocking states and still satisfy all demands for connection as they arise, without disturbing calls already present. That is, there may exist a rule for choosing paths which, if followed, confines the trajectory of the system to nonblocking states (without refusing any demands for connection by idle pairs). We next discuss what is meant by a rule. If a call c = {(u, v)} is blocked in a state x it cannot be put up without disturbing existing calls of x, and there is no question of using a rule. Also, if x is a maximal state, no new calls can be put up, and a rule is unnecessary. But if a call c can be put up in one or more ways in the state x, then there is at least one y > x such that y(x) = y(x) u c. I n such a case some method of specifying permitted or prohibited new states could be used in order to improve performance. A rule p( *, * ) for a network v is a mapping of the Cartesian product [S - y-l(A)] x

u

into subsets of S , with the properties: if x E S and c = ((u, c)} E U with (u, v) an idle pair of x (so that c is a new call in x), then 0

s p(x, c) c y-'(y(x) u c);

if x is maximal, or if (u, v) is not idle, p(x, c) is defined (arbitrarily) as the null set. If for some call c not up in x we have Y

E P(X9

4,

we say that the transition (between states) x -+y is permitted by P(

-9

*

>.

10.

SOME DEFINITIONS AND PROBLEMS

73

We say informally that a state x is reachable under a rule p( *, * ) if there is some sequence of changes of state, consisting of either hangups or transitions permitted by p( ), and leading from the zero state to x. More precisely, we define the notion 0

,

-

x is reachable under p(*, -) in n steps

recursively, as follows:

) in zero steps. (i) T h e zero state is reachable under p( (ii) If x is reachable under p( -, * ) in n steps, and for some call c E U , y(x) = y ( y ) u c, then y is reachable under p( *, * ) in (n 1) steps. (iii) If x is reachable under p( * ) in n steps, and for some call c E U , c is new in x and y E p(x, c), then y is reachable under p( -, . ) in ( n 1) steps. a,

+

a,

+

A state is reachable under p( ., ) if it is reachable under p( *, * ) in n steps, for some n >, 0. T h e set of states that are reachable under p( *, * ) will be denoted by R,, . A network v = (G, I , Q, E ) will be called nonblocking in the wide sense if and only if there is a rule p( * ) for v under which no blocking state is reachable, i.e., a,

R,

c B’.

In words, we may say that a network is nonblocking in the wide sense if there is a rule, depending on the states, and on the connections that are requested, such that if the rule is used (starting from the zero state) no blocking state is ever reached, and hence no request for connection by an idle pair (of a state that can be reached) need ever be refused. In making this definition, we think of the system as starting (empty) at the zero state; in any state x that it reaches, any idle pair of x may demand connection; it must always be possible to make this connection without disturbing existing calls, and reach a (nonblocking) state y one level higher, y EL,^^+^; at any instant an existing call may terminate, and the system move to a state of An example of such a network was given by the author ( I ) , and in Chapter 1, Section 14. Finally, we consider a still weaker property of networks than the first two defined, namely, the possibility of satisfying a demand for

74

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

connection by rearranging (if necessary) the existing calls in such a way that the desired call can then be accommodated. Let x be a network state realizing the assignment y(x). We call x rearrangeable if and only if for every idle pair (u, v) of x there is a y E S, possibly depending on (u, a) and x, which realizes the larger assignment y(x) u { ( u , v)}, i.e.,

Alternately x is rearrangeable if for every call c new in x there is a a y E S such that

This definition is the same as that of a nonblocking state except that the condition x < y is omitted. That is, to realize the larger assignment y(x) u c it may be necessary to reroute existing calls to x which is not comparable to x, and which has a give a new state z path r , disjoint from p E z , between u and a. T h e state y may then be taken to be z u { r } . A network v is called rearrangeable if its states x E S are rearrangeable. With these definitions laid down, we can formulate several problems of the combinatorial theory of connecting networks:

-

(i) Can general characterizations of the properties of being rearrangeable, and of being nonblocking (strict or wide sense) be given ? (ii) What relationships exist among the concepts we have defined ? (iii) What specijic networks are rearrangeable, or nonblocking (strict or wide sense) ?

To attack problem (i) we make the following observations: T h e three properties of interest represent different degrees of abundance of states of v in which calls are not blocked. T h e relative abundance or density of such states throughout S determines which (if any) of the three properties v has. T h e heuristic concept of abundance suggests the topological one of denseness, and the possibility of characterizing the three properties in terms of denseness. This idea is developed in the remaining sections; it leads to answers to problems (i) and (ii) above. Specific rearrangeable and strictly nonblocking networks are considered in Chapters 3 and 4, respectively.

11.

REARRANGEABLE NETWORKS

75

$11. Rearrangeable Networks Let X be a subclass of the class S of network states. We say that X is su&ient if y ( X ) = A, i.e., if every assignment is realized by some state of X . We make two comments: Remark 2.7. If A^ E y ( X ) , then X is sufficient. This can be seen as follows: Every assignment is a subset of some maximal assignment, and so belongs to the <-closure 8 of X . For the same reason we have Remark 2.8.

T h e following properties of a network v are equivalent:

(i) v is rearrangeable. (ii) Some sufficient class exists. (iii) T h e range of y ( * )includes A. I t is convenient to approach the study of rearrangeable networks by taking the point of view of a particular pair of customers, i.e., of a particular inlet-outlet pair (u,a ) E I x a. Such a pair corresponds to a unit assignment or call c = { ( ? I , v)} E

u,

any realization of which is among the states of E, the set of elementary states. For each call c E U we define Ic B,

= {x

E

= {x E

S : c is new in x, i.e., (u,v) is idle in x}, S: c is blocked in x}.

It can be verified that B, CI,, ~1

=

for

n( B J ,

CE

U,

C€U

s

-

y+(A)

=

u

I,.

C€U

We call a network v rearrangeable for the unit assignment or call c if and only if for every x E I , there is a y E S - I , which realizes the larger assignment y(x) u c = ~ ( y )I n. words, this condition states that for any state in which the pair (u,u ) is idle there is a (possibly

76

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

rearranged) state in which all the same calls are up, and in addition u is connected to v . It is easy to see that Y is rearrangeable if and only if it is rearrangeable for all calls c E U. Let X , Y be arbitrary subsets of S. I n accord with a standard definition (6, p. 49), X is said to be dense in Y in the d-metric if Y is included in the d-closure of X , i.e., YG

Xd.

Now in a pseudometric space the closure of a set X is the set of all points that are at distance zero from X , when the distance of a pointy from a set X is defined as inf d(x, y ) .

XEX

-

Hence the closure of X is the set of all y such that for some x E X , d(x,y ) = 0, or equivalently, x y. That is, the d-closure of X is the set of all states that are equivalent to a member of X : Xd={y~SIy-x

forsome X E X } .

These observations lead to the following result: Theorem 2.2.

v

is rearrangeable

(Bc)'is d-dense in I , ,

if and only if for each

c E U.

Let v be rearrangeable; let c E U ; and pick x in I , . T h e n Proof. there exists y E S such that

AY) = Y ( 4 " c, and so there exists a z Obviously then

E

E

A

y

y-l(c) such that x -

< y and x

-

y

-

Z.

x E (&)',

and since x is equivalent to y - z we have x

E

((BCW

Since x was an arbitrary member of I , , we have proved I, L Conversely, assume that the condition in the theorem holds, and pick

12.

NETWORKS NONBLOCKING I N THE WIDE SENSE

-

77

any c E U , and x E I, . Then x y for some y in (B,)', so that c is not blocked in y . Th u s u is rearrangeable for all C E U , and so is rearrangeable. A simimar argument yields the weaker and simpler result:

If B' is d-dense in S , then v is rearrangeable.In this Remark 2.9. case, since S E given a state x there is always an equivalent nonblocking state y , with y

-

X,

y

E

B'.

Hence rearrangements can be made uniformly in the calls new to x.

$12. Networks Nonblocking in the W i d e Sense We now turn to the characterization of networks for which there is a rule for routing calls that allows the operating system to avoid blocking states entirely. T h e case in which the network is actually nonblocking in the strict sense, so that any rule will do, is excluded here as trivial. T h e point is to use a network with blocking states, but to manage to avoid them by clever routing. T h e following general criterion of a useful rule p(., .) suggests itself: p(*, .) should make as many blocking states as possible unreachable, consistent with satisfying requests for connection by unblocked new calls. T o exhibit, in an intuitive way, all the relationships that obtain, it is convenient to introduce an additional concept: A class X of network states is preservable ( b y new calls) if and only if for any x E X and any call c that is new to x and not blocked in x, there is a state y E X such that y >x and y ( y ) = y(x) u c. That is, if an idle pair (u, v ) of x corresponds to a call c = {(u,v ) } that is not blocked in x, then some state y E X realizes y(x) u ((u, v)}, a ndy is above x in the state-diagram, y > x.I n words, X i s preservable if any call that can be put up at all in a state of X can be put u p salva staying in X , that is, in such a way that the system stays in X . A &-closed class is always preservable (by new calls). We make Remark 2.10.

sufficient.

If X is preservable, 0 E X, and X

E

B', then X is

78

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

It is then possible to start at the zero state, and call by call realize any maximal assignment salva staying in X . We now state 2.3. v is nonblocking in the wide sense i f and only i f there exists a nonempty subset X of states such that

Theorem

(i) X is preservable. (ii) X c B‘. (iii) X is <-closed, i.e., X

=

X.

Proof. Let (i)-(iii) hold for some subset X,and define a rule p(., -) by the condition that if c E U is new to x , then p(r, c ) = y - l ( y ( x ) u c ) n X .

Use of p ( - , -) is tantamount to requiring that any call must be put u p so as to lead to a state of X . By (i) and (ii), this can always be done. Since X is ,<-closed, hangups preserve membership in X ; since X is nonempty it contains the zero state. Hence all states reachable under p ( . , belongs to X n B’ and a)

R,

c B’,

so that v is nonblocking in the wide sense. Conversely, if v is nonblocking in the wide sense, then some rule p ( - , is such that no blocking state belongs to R, . Set X = R, . Then X is <-closed, because any state below a reachable state is reachable by hangups. Also X G B’, because p(., avoids all blocking states. Finally, X must be preservable since one can “preserve” X simply by using only state-transitions permitted by p(., .), i.e., by putting up unblocked new calls so as to lead only to states vouchsafed by P(., -1. We recall that for x E S, a)

a)

A,

y covers x}, = { y : y u z for some z E E}, = (set of states immediately above x}. ={y:

T h e property of preservability (of a set X of states) will now be given a topological characterization in terms of denseness, in the following result:

13.

79

NETWORKS NONBLOCKING I N THE STRICT SENSE

A nonempty subset X of S is preseservable

Theorem 2.4.

if and only

;f for every x E X , A, n X is dense in A, in the sense of the d-metric;

i.e., x

EX

implies A,

s (A,n X ) d .

Take x E X and y E A,, so that y is “immediately above” or y covers x. Then there is a call c new in x such that

Proof. X,

rfr) = A x ) “ c,

and so if A, n X is dense in A,, there is a z E A, n X which is equivalent t o y . Since z covers x, it follows that the call c new to x can be connected in state x so as to give rise to a state of X . That is, we have Z€X

y ( z ) = y ( x ) u c.

Since c was an arbitrary new call of x E X , the set X is preservable, if the condition of Theorem 2.4 is true. Conversely, let X be preservable, and take x E X and y E A, . Then there exists a call c not blocked in x with y ( y ) = y(x) u c. But since X is preservable, and c is not blocked in x,there is a z in A, n X such that y ( z ) = y(x) u c, that is, z y . Hence y is equivalent to an element of A, n X . Since y was arbitrary it follows that for x E X ,

-

A, c ( A , n X)d,. Remark 2.11. T h e sets {A, , x E X } in the condition of Theorem 2.4 may be replaced by the “x-cones”

{Y

IY > X I .

$13. Networks Nonblocking in the Strict Sense A network that is nonblocking in the strict sense has no blocking states whatever. A simple characterization of this property is given by Theorem 2.5.

is a subset X

of

v is nonblocking in the strict sense if and only B‘ such that

(i) X is sujicient (ii) X is d-closed, i.e., X

=

Xd.

if

there

80

2.

ALGEBRAIC A N D TOPOLOGICAL PROPERTIES

Proof. If v has the property, then X = B', and we may take X = S. Conversely, if (i) and (ii) obtain, take any x E S; since X is sufficient, there exists y E X for which y ( x ) = ~ ( y )i.e., , x m y . But X = X d , so x G X , and hence x E B'. By Remark 2.10, the condition (i) that X be sufficient can be replaced by the condition that X be preservable and nonempty.

$14. Glossary for Chapter 2 G g

V S

T P 0

I 9

L C E V

Q

0

1x1

Lk D

An arbitrary graph A copy of G with each edge labeled 0 or 1 T h e set of all labeled versions g of G T h e set of network states (typical members x, y , z ) The set of terminals (nodes of G) A typical path (chain) on G Null set T h e set of inlets T h e set of outlets The set of links The set of all connections (paths from I to Q) The set of elementary states An arbitrary network, specified by choosing G, I, Q, and E Partial ordering of V or S by inclusion Zero state Number of elements of the set X Set of states with exactly k calls up State diagram (Hasse figure of < on S)

p(.) Mobius function of < h(x, n) Number of chains of length n from 0 to x

wx, I x

I)

Set of states directly above x Set of states directly below x

c.

XSL,

An assignment (any 1 - 1 map of subset of I into Q) Set of maximal assignments Set of unit assignments or calls A call, or typical member of U The assignment realized by state x Equivalence of states

8 -

X Xd B p(., .)

R,

<-closure of X >-closure of X d-ciosure of X Set of states in which some call is blocked A rule for operating a network T h e set of states reachable under p(., .)

REFERENCES 1 . V. E. BeneQ,Heuristic Remarks and Mathematical Problems Regarding the Theory of Switching Systems, Bell System Tech. J. 41, 1201-1247 (1962). 2. V. E. BeneH, On Rearrangeable Three-Stage Connecting Networks, Bell System Tech. J. 41, 1481-1492 (1962).

REFERENCES

81

G. Birkhoff, Lattice Theory, A m . Math. SOC.Colloq. Publ. (rev. ed.), XXV (1948). L. Weisner, Trans. A m . Math. SOC.38, 474-484 (1955). P. Hall, Quart. J. Math. 7 , 134-151 (1936). J. L. Kelley, “General Topology.” Van Nostrand, New York, 1955. C. Clos, A Study of Non-Blocking Switching Networks, Bell System Tech. J. 32, 406-424 (1953). 8. A. D. Kharkevich, Multi-Stage Construction of Switching Systems (in Russian), Doklady Akad. Nauk S.S.S.R. 112, 1043-1046 (1957).

3. 4. 5. 6. 7.

CHAPTER

3

Rearrangeable Networks

In the interest of providing good telephone service with efficient connecting networks, it is essential to have at hand a knowledge of some of the combinatorial properties of such networks. One of these properties is rearrangeability a connecting network is rearrangeable if its permitted states realize every assignment of inlets to outlets, or alternatively, if given any state x of the network, any inlet idle in x, and any outlet idle in x, there is a way of assigning new routes (if necessary) to the calls in progress in x so that the idle inlet can be connected to the idle outlet. New and known results about rearrangeable networks built of stages of square crossbar switches are collected, systematized, and extended. A natural approach to their study using group theory is described, and the synthesis problem of selecting (from a given class) a rearrangeable network with a minimum number of crosspoints is solved.

$1. Introduction Most communications systems contain a connecting network as a basic functional unit. A connecting network is an arrangement of switches and transmission links through which certain terminals can be connected together in many combinations. T h e calls in progress in a connecting network do not usually arise in a predetermined time sequence. Requests for connection (new calls) 82

1. INTRODUCTION

83

and terminations of connection (hangups) occur more or less at random. For this reason the performance of a connecting network when subjected to random traffic is used as a figure of merit. This performance is measured, for example, by the fraction of requested connections that cannot be completed, or the probability of blocking. T h e performance of a connecting network for a given level of offered traffic is determined largely by its conjiguration or structure. This structure may be described by stating what terminals have a switch placed between them, and can be connected together by closing the switch. T h e structure of a connecting network determines what combinations of terminals can be connected together simultaneously. If this structure is too simple, only a few calls can be in progress at the same time; if the structure is extensive and complex, it may indeed provide for many large groups of simultaneous calls in progress, but the network itself may be expensive to build and difficult to control. T h e structure of a connecting network also gives rise to various purely combinatorial properties that areuseful in assessing performance. For example, C. Clos ( I ) has exhibited a whole class of connecting networks that are nonblocking: no matter in what state the network may be, it is always possible to connect together a n idle pair of terminals without disturbing calls already in progress. We call such a network nonblocking in the strict sense, because it has no blocking states whatever. If a connecting network does have blocking states, it is nevertheless possible that by suitably choosing routes for new calls one can confine the trajectory of the operating system to nonblocking states. T h a t is, there may exist a rule whose use in putting up new calls results in avoiding all the blocking states, so that the system is effectively nonblocking. T h e rule only affects new calls that could be put into the network in more than one way; no call already in progress is to be disturbed. Connecting networks for which such a rule exists we call nonblocking in the wide sense. I n telephone practice, the procedure of routing the calls through the network is called “packing” (the calls), and the method used to choose the routes is called a “packing rule.” T h e use of the word “packing” in this context was undoubtedly suggested by a natural analogy with packing objects in a container. Virtually nothing rigorous is known about the effect of packing rules on network performance. Finally, a connecting network may or may not have the property

84

3.

REARRANGEABLE N E T W O R K S

of being rearrangeable: given any set of calls in progress and any pair of idle terminals, the existing calls can be reassigned new routes (if necessary) so as to make it possible to connect the idle pair. These three combinatorial properties of connecting networks have been given general topological characterizations in Chapter 2. However, while it is clear how the properties relate to the performance of a connecting network, it has also become apparent that they are also relevant to its sound economical design. T h e reasons for this relevance are described in the next paragraph. T h e impact of recent discoveries and developments in electronics has been felt in the telephone switching field. Evidence of this is the fact that many laboratories throughout the world have research and development programs for arriving at economical electronic connecting systems. I n many of these experimental systems, the role of the connecting network becomes much more important than in presentday commercial telephone systems. I n particular, the modern common control equipment is less expensive, and the modern crosspoint (which assumes some of the control functions) is more expensive, than its (respective) counterpart in contemporary relay systems. T h e requirements on the connecting network of such an electronic system are that the number of its crosspoints be kept at a minimum and yet that it be able to permit establishing as many simultaneous connections as possible. These are opposing requirements, and an economical system must, of necessity, be based on a compromise. I n the search for this compromise, convenient starting points are provided (i) by rearrangeable networks, in which a blocked call can always be unblocked by assigning new routes to the calls in progress, and (ii) by strictly nonblocking networks, in which it is always possible to connect an idle inlet to an idle outlet, regardless of the state of the network. Rearrangeable networks are considered in the present chapter, and strictly nonblocking networks in Chapter 4.

$2. Summary of Chapter 3 A rearrangeable connecting network is one whose permitted states realize every assignment of inlets to outlets, that is, one in which it is possible to rearrange existing calls so as to put in any new call. Figure 1 shows a typical member of an interesting and useful class of connecting networks that has been suggested and studied by C.

2.

SUMMARY OF CHAPTER

INLET SWITCHES

85

3 mxn

OUTLET SWITCHES

I1 Y

FIG. 1 .

Three-stage Clos network v(m, n, r ) .

Clos ( I ) . We refer to this class as that of three-stuge Clos networks. Such a network consists of two symmetrical outside stages of rectangular switches, with an inner stage of square switches. It is completely determined by the integer parameters m, n, r that give the switch dimensions. I n one of the few outstanding contributions to the theory of connecting networks, Clos ( I ) showed that for m 2 2n - 1 the network is nonblocking in the strict sense. T h e network defined by the parameters m, n, r will be denoted by v(m, n, r ) . First, the following two known results about rearranging three-stage Clos connecting networks are discussed (2): (i) T h e Slepian-Duguid theorem, which states that the network v(m, n, r ) is rearrangeable if and only if m >, n. (ii) T h e theorem of M. C. Paull, which states that if m = n = r , then at most n - 1 existing calls need be moved in v(n, n, n ) in order to connect an idle terminal pair. A new result proven is the generalization of Paull’s bound in (ii) for any m, n, r with m 2 n to r - 1. T h e Slepian-Duguid theorem is proved in Section 3 by an inductive method due to Duguid (3)depending on the combinatorial theorem of P. Hall on distinct representatives of subsets. We discuss Paull’s theorem in Section 4, but defer our simple proof of it to Section 6, which presents simple inductive proofs of various bounds on the number of calls that must be moved. All the proofs to be given depend on a “canonical reduction“ procedure that consists in removing a middle switch from the network and reducing the parameters m and n by unity. Next, a natural algebraic and combinatorial approach to the study

86

3.

REARRANGEABLE NETWORKS

of rearrangeable networks is described, with attention centered principally on two-sided networks built of stages of square crossbar switches, each stage having N inlets and N outlets. T h e approach is based in part on the elementary theory of permutation groups. T h e principal problem posed (and partly answered) is this: What connecting networks built of stages are rearrangeable ? Sufficient conditions, including all previously known results, are formulated and exemplified ( 4 ) . Finally, in the effort to provide adequate telephone service with efficient networks it is of interest to be able to select rearrangeable networks (from suitable classes) having a minimum number of crosspoints. This problem is fully resolved for the class of connecting networks built of stages of identical square switches arranged symmetrically around a center stage: roughly, the optimal network should have as many stages as possible, with switches that are as small as possible, the largest switches being in the center stage; the cost (in crosspoints per inlet) of an optimal network of N inlets and N outlets is nearly twice the sum of the prime divisors of N , while the number of its stages is 2x - 1 where x is the number of prime divisors of N , in each case counted according to their multiplicity. By using a large number of stages, these designs achieve a far greater combinatorial efficiency than has been attained heretofore (5).

$3. The Slepian-Duguid Theorem T h e first part of the present chapter is devoted to studying the property of rearrangeability for three-stage Clos networks. We shall be particularly concerned first with the possibility of rearranging calls, and later with the number of calls that must be moved. Strictly nonblocking Clos networks will be considered in Chapter 4, in detailed exposition based on Clos’ own definitive study (1) of them. T h e first result is due essentially to D. Slepian (6),and is Theorem 3.1. (Slepian-Duguid).

with m >, n is rearrangeable.

Eaery three-stage Clos network

Proof. T h e proof to be given is due to A. M. Duguid (3).* Slepian’s proof was stated for the case m = n = r , but actually gave * I n a private communication from J. H. Dbjean, the author has learned that Theorem3.1 wasalso proved by J. LeCorre in an unpublished memorandum dated 1959.

3.

THE SLEPIAN-DUGUID

THEOREM

87

an explicit procedure for rearranging the existing calls so that the additional desired call could be put up. He showed for this case that at most 2n - 2 calls need be disturbed. This bound was subsequently improved to n - 1 by M. C. Paul1 (7). (See Section 4.) Duguid’s proof depends on a combinatorial theorem of P. Hall (8), which has recently come into prominence in studies of maximal flows in networks. [See D . Gale (9).] Hall’s Theorem. Let A be any set, and let A , , A , , ..., A,. be any r subsets of A. A necessary and sufficient condition that there exist a set of distinct representatives a , , ..., a, of A , , ..., A , , i.e., elements a , , ..., a, of A such that a, E a,

i

A,

# a3

for

=

1, ...,r

j # i,

< <

is that f o r each k in the range 1 k r , the union of any k of the sets A , , ..., A, have at least k elements. T h e condition given is obviously necessary. T h e interest of the theorem, and our application of it, concern the sufficiency. We proceed now to the proof of Theorem 3.1. I t is obviously sufficient to consider only the case m = n. Let the inlets to the network be denoted by u l , ..., u , , where N = nr, and let the outlets be denoted by v l , ..., v, . I t is sufficient to prove that every maximal assignment of inlets to outlets can be realized by a state of the network. Here “maximal” means that each inlet is to be connected to exactly one outlet, and vice versa. Such a maximal assignment is obviously equivalent to a permutation on Nobjects. We let {i+n(i),i = 1, ..., N} be such a permutation; also we denote the j t h inlet switch by I j and th e j t h outlet switch by Oj . It is convenient to think of Ii as the set of i for which ui is on the j t h inlet switch, and of Oj as the set of i for which v i is on t h e j t h outlet switch. Let K be the set of integers {I, 2, ..., n}. We define the subsets { K i ,i = 1, ..., n } of K by the condition K,

= { j : n(m)E

0, for some m € I , } .

Now let Ii(,), ..., li(k) be any k of the inlet switches, and set T

u k

=

j=1

K,,,,.

88

3.

REARRANGEABLE NETWORKS

Suppose that there are t distinct elements in T . Then all the kr inlets in the set

u L, t

3=1

are assigned by T ( . ) to outlets from t of the outlet switches, that is, to outlets from a set of tr outlets. But two distinct inlets are not assigned to one outlet, so t 3 k. Th u s any union of k sets among the Kicontains at least k elements. Hence by Hall’s theorem there is a set of distinct representatives ( k ( i ) , i = 1 , ..., n} with k(i) E K , , i = 1, ...)n k(i) # k(j) for i + j . Since K contains n elements, it follows that {i + k ( i ) , i = 1, ..., n} is a permutation. However, the interpretation of the fact that k ( i ) E Ki is that ~ ( mE )OIL(?) for some rn E I , . I n other words, to every inlet switch Ii there corresponds a unique outlet switch such that T ( * ) maps some inlet on Ii into some outlet on O,,i, . That is, there is a subassignment of T ( . ) that involves exactly one terminal on every inlet and outlet switch. I t is evident that such a subassignment can always be satisfied on a single middle switch (Fig. l), say that numbered 1. If this subassignment is completed, that one switch is filled to capacity, and the rest of the network is essentially v(m - 1, n - 1, Y), i.e., that of Fig. 1 with the parameters m,n reduced by unity. T h e theorem is clearly true for m = n = 1. As an hypothesis of induction assume that it is true for a given value of m - 1 (=n - 1). T h e argument given above proves that it is then also true for m ( = n ) , for the induction hypothesis implies that the remainder of the assignment T ( - ) that was not put up on the first switch is satisfiable in the subnetwork, i.e., essentially in v ( m - 1, n - 1, r ) . Hence T ( * ) is realizable, and the theorem follows by induction on n.

$4. The Number of Calls That Must Be Moved: Paull’s Theorem In view of the result of Slepian and Duguid that every three-stage Clos network with m >, n is rearrangeable, it is natural to ask, for a

5.

SOME FORMAL PRELIMINARIES

89

given state x of such a network, how many calls of x need actually be changed to new routes in order to put in a given call between idle terminals. Slepian’s original procedure was for the case m = n = r , and gave the upper bound 2n - 2 (uniformly for all states) to the number of calls that need be disturbed. T h a t is, he showed that if m = n = r, then at most 2n - 2 calls need be rearranged. By a similar but more complicated method, M. C. Paul1 (7) halved this bound, proving

Let v(n, n, n) be a three-stage Clos network with Theorem 3.2. m = n = r. Let x be an arbitrary state of this network. The number of calls in progress in x that need be rerouted in order to connect an idle pair of terminals is at most n - 1. There exist states which achieve this bound. Since Paull’s proof was involved, we have looked for and found simpler ways of proving and extending his result. In Section 6 we give a simple inductive proof; the argument to be given, of course, also provides a proof of the Slepian-Duguid theorem not depending on the Hall combinatorial result used in Section 3.

$5. Some Formal Preliminaries I n order to state and prove the rest of our results, it is useful, and indeed necessary, to introduce a systematic notation. Such a notation has been described and used in Chapter 2 and in a paper (10) by the author; the notation to be used is a consistent extension of this. T h e set of inlets of a network is denoted by I , and that of outlets Q. The set of possible states of a connecting network is dented by S. For a three-stage Clos network, S consists of all the ways of connecting a set of inlets to as many outlets by disjoint chains (paths) through an inlet switch, a middle switch, and an outlet switch. (See Fig. I .) States of the network may then be thought of as sets of such chains. Variables X, Y , x, ..., at the end of the alphabet, range over states from S. A terminal pair ( u , u ) E I x Q (with u an inlet and u an outlet) is called idle in state x if neither u nor u is an endpoint of a chain belonging to x. A call c is a unit subset c = {(u, n ) } C I x Q; c is new in a state x if ( u , u ) is idle in x. Th e assignment y(x) realized by x is the union of all calls c = {(u, u ) } such that x contains a chain from u to u. If a is

90

3.

REARRANGEBALE NETWORKS

an assignment, y-l(u) is the set of all states realizing a. T h e cardinality of a set X i s denoted by 1 X I. T h e states x E S are partially ordered by in a natural way. inclusion A distance between states can be defined as

<

S(x, Y ) = I x

= the

y

I!

number of calls that would have to be added, removed, or rerouted to change x into y,

where A is symmetric difference. T h e distance of a state x from a set X of states us defined in the usual way as S(x, X ) = min S(x,y). YEX

A call c new in a state x is blocked in x if there is no state y > x such that y ( y ) = y ( x ) u c. For a three-stage Clos network v(m, n, r ) with m >, n we define vr(m, n, r )

=

c

max

new in x

S(x, y-l(y (x ) u c)) - 1

= the maximum

number of calls that must be rerouted in order to put up a call c new in x.

We also set

d m ,n, 4

=

T.,Xd

m ,n, 4.

I n this last definition, it is assumed that S is the set of states determined by the parameters m, n, r in Fig. 1. I n the notation introduced above, the Slepian-Duguid theorem guarantees that for m > n and c new in x y-'(y(x)

u c) # 8,

and Paull's theorem may be cast as stating that q(n, n, n ) = n - 1.

$6. The Number of Calls That Must Be Moved: N e w Results We now present some new methods for studying the number of calls that must be moved; these yield extensions of results of D. Slepian (6) and M. C. Paul1 ( 7 ) .

6. Theorem 3.3.

4 2 , 2,r )

NEW RESULTS

91

< 2r - 2.

Proof. Suppose that a blocked new call between input switch I , , and output switch 0, is to be put in when the network is in a state x. Consider any sequence c l , ..., ck of existing calls of x with the properties

(i) Either c, is on I , , c1 and c2 are the same outlet switch, ..., ci and ci+, are on the same outlet switch, i odd, i < k ci and ci+l are on the same inlet switch, i even, i < k, or c1 is on 0, , c1 and c2 are on the same inlet switch, ..., ci and ci+, are on the same inlet switch, i odd, i < k ci and ci+l are on the same outlet switch, i even, i < k. (ii) ck is the only call on some outer switch. Since neither I , nor 0, is full, the largest k for which such a sequence exists is 2r - 2. T h e reader can verify that a possible strategy for rearranging existing calls of x so as to put in an Il - 0, call is to take each call of the sequence c,, ..., ck and reverse its route, i.e., make it go through the middle switch other than the one it presently uses. Th u s for all x ~ ~ (2,2I ),

< 2r

-

2.

Let x be a state of v(m, n, r ) , and let M be a particular middle switch. A canonical reduction of x with respect to M will consist of (i) removing M , (ii) on each outer switch that has a call routed via M , removing the link, crosspoints, and terminals associated with that call, (iii) on each outer switch that has an idle link to M , removing the link, the crosspoints associated therewith, and one arbitrarily chosen idle terminal.

It is easily seen that a canonical reduction of a state x of v(m, n, r ) leads to a state of u(m - 1, n - 1, r). Theorem 3.4.

rp(n, n, r )

< 2r - 2.

Proof. By Theorem 3.3, the result holds for n = 2, so assume it for a given value of n - 1 2 2, and try to rearrange a given state x of v(n, n, r ) so as to put in a new blocked call from Il to 0, .

92

3.

REARRANGEABLE NETWORKS

Case 1. There is a middle switch M with both an I , and an 0, call on it. Perform a canonical reduction of the state x with respect to M . This yields a state of v(n - 1, n - 1, r ) , for which the result holds.

Case 2. No middle switch has both an I , and an 0, call on it. Since the call to be put in is blocked, it must be true that #(idle links out of I*) + #(idle links out of 0,) = YZ

and hence

max{#(idle out of 11),#(idle out of 0,))> 1.

Suppose that #(idle out of I J > 1. There is a middle switch M with an idle link to I , , and a busy link to 0, . Perform a canonical reduction of x with respect to M , yielding a state of v(n - 1, n - 1, r ) in which each of Il , 0, still has an idle terminal. A refinement of this method suggested by M. C. Paul1 will halve the last two bounds. We prove Theorem 3.5.

rp(2,2,

Y)


-

1,

(r

3 2).

Proof. T h e result is true for r = 2, since is that case the network has only one blocking state (see Fig. 2), and both blocked calls can be unblocked by changing the route of one ( = r - 1) existing call. Let us assume as an hypothesis of induction that the theorem holds for some value of r - 1 3 2, and in 4 2 , 2, r ) attempt to put u p a blocked new call c between input switch I , and output switch 0,. Since c is new and blocked, there must be an idle and a busy link on both of Il and 0, , and each of the busy links must pass through a different middle switch. Let c1 be the call on Il, and c, be the call

2 FIG. 2.

Network with only one blocking state ( r =

2).

6.

93

NEW RESULTS

on 0, . We may suppose without loss of generality that c, is a call from Ilto 0, , while c, is a call from I , to 0, . Case 1. I, has only one call on it, viz., c, . Move c, to the other middle switch (see Fig. 3). INPUT

---

I

OUTPUT

M

0

c2

/

/

/

. FIG. 3. I, with one call,

cz

.

I, has two calls on it. Remove both c, and c, , so that I , and 0, become empty. Consider now the state x of the subnetwork of parameter r - 1 obtained by removing I , and 0, and reducing the dimension of the two square middle switches by unity to r - 1. Each of I , and 0, has at least one idle terminal in x, since c, and c, were removed. Hence by the hypothesis of induction the subnetwork can be rearranged so as to put in a call from I , to 0, while disturbing at most r = 2 existing calls. If the I , - 0, path thus provided is via M , then c1 and c, can be replaced as in Fig. 4. This leaves a path for the new I , - 0, call c via M , , and shows that it was never necessary to move c, , and that hence at most r - 1 calls were disturbed. If the I , - 0, path provided by rearranging the subnetwork is via M , , then c, and c, can be replaced as in Fig. 5 . This leaves a path for c via M , , and shows that c, did not really have to be moved, so that at most r - 1 calls were disturbed. Theorem 3.6. ~ ( nn,, r ) < r - 1. Proof. T h e result is true for n = 2. Assume that the theorem is true for a given value of n - 1 3 2, and seek to rearrange a state x of u ( n , n, r ) so as to put in a new call blocked in x between I , and 0, . Case 2.

3.

94

REARRANGEABLE NETWORKS

INPUT

OUTPUT

I

0

FIG.4.

Calls c1 and c2 over path via M ,

OUTPUT

INPUT

I

0

0

FIG. 5.

Calls c1 and c2 over path via M z .

T h e theorem follows by induction on n by distinguishing two cases as in Theorem 3.4, and using a canonical reduction of x. Theorem 3.7.

For m - 1 3 n, P h n, 4


-

1, n, 4.

Proof. This is almost obvious. Remove any middle switch M of v(m, n, r ) and make all terminals on which there were calls routed via M idle. This gives a state of v(m - 1, n, r ) ; in this state the desired call can be put in by rearranging at most g)(m - 1, n, r ) existing calls. Now replace M and the calls that were routed through it.

6.

95

NEW RESULTS

M. C. Paull (7) has conjectured that if Y v(m, n, n)

< 2n

-

=

n, then

1 - m.

This bound agrees with Theorem 3.2 if m = n, and with Clos’ results on nonblocking networks if m = 2n - 1. Paull has proved the result for m = 2n - 2. However, no proof of the full conjecture has been found. I t is tempting to try the stronger conjecture that p(m, n, r )

< 2n - 1

-

rn

for any m, n, and r . This can be disproved by the counterexample shown in Fig. 6. There is no way of connecting Il to 0, without moving a call on one of I , , 0,. However, all possible alternative routes for these calls are pre-empted, so at least two calls must be moved.

OUTPUT

INPUT

I

0

U

U

FIG. 6. Network showing that I , and O5 cannot b e connected without moving a call o n one of Il , 0,.

96

3.

REARRANGEABLE NETWORKS

$7. Summary of Sections 8-15 Figures 7 and 8 show the structure of two connecting networks built out of square crossbar switches, with each switch capable of connecting any subset of its inlets to an equinumerous subset of its outlets in any desired one-one combination. T h e network of Fig. 7 is often found in modern telephone central offices; we may call it the No. 5 crossbar network. It is not rearrangeable. T h e network of Fig. 8 is rearrangeable, but so far it has not found extensive practical use.

J KS

FIG. 7. Structure of No. 5 crossbar network.

I n the second portion of this chapter we describe a natural algebraic and combinatorial approach to the study of rearrangeability ( 4 ) . For the most part we restrict attention to two-sided connecting networks that are built of stages of crossbar switches, and have the same number N of inlets as outlets. T h e approach is based in part on the elementary theory of permutation groups. T h e way the connection with group theory arises can be summarized as follows: A maximal state of the network is one in which no additional calls can be com-

7.

SUMMARY OF SECTIONS

8-15

97

FIG.8. Rearrangeable Clos network.

pleted in the network; suppose that both the inlets and the outlets are numbered in an arbitrary way from 1 to N ; each maximal state realizes some submap of a permutation on (1, ..., N ) ; the network is rearrangeable if and only if the whole group of all permutations of (1, ..., N } is generated in this way by the maximal states of the network. Details are worked out in the main body of the text. It is not possible to explore in one chapter all the possible uses of group theory in the study of connecting networks. Indeed, we shall restrict ourselves to formulating the fundamental problem of rearrangeable networks in terms of complexes of permutations, and to giving a partial answer. One of the difficulties with the approach is that it always seems to be easier to obtain results about groups by the few available methods known for rearrangeable networks, than aice versa. T h e last part of the present chapter is concerned with the problem of synthesizing a rearrangeable network (for N inlets and outlets), subject to certain structural conditions and to the condition that it have a minimum number of crosspoints (5). I n Section 8 we define a precise general notion of a “stage” of switching in a connecting network, and, after describing how the networks which will be of interest are built out of stages by joining them together by patterns of links, we pose two problems: first, to discover what networks built in this way are rearrangeable; and second, to synthesize optimal rearrangeable networks of given size, optimal in the sense of having fewest crosspoints (among those in some class of networks having practical interest). Section 9 is devoted to giving a formulation of the first problem (discovering rearrangeable networks) in terms of partitions and

98

3.

REARRANGEABLE NETWORKS

permutation groups, using the notion of stage. I n Section 10 we discuss how stages generate complexes (in the group theory sense, i.e., sets of group elements). I t is shown that a stage can generate a subgroup only if it contains a substage made of square switches, a result that indicates to some extent the “best possible” nature of stages made of square switches. A known example, discussed in Section 11, indicates how a particular symmetric group S is generated by a rearrangeable network with three stages, in the form

with cpl, cp3 in a subgroup H and v2 in a certain subgroup q--’Hy conjugate to H . Definitions and preliminary results appear in Sections 12 and 13, respectively. Sections 14 and 15 are devoted to proving two “rearrangeability” theorems for connecting networks built of stages of square switches. T h e first theorem gives sufficient conditions under which a set of stages of square switches connected by link patterns will give rise to a rearrangeable network. T h e second theorem indicates a simple way of describing link patterns and stages that satisfy the hypotheses of the first theorem, and so yield many specific rearrangeable networks, generalizations of those given by Paul1 (7).

$8. Stages and Link Patterns T h e switches in Figs. 7 and 8 are arranged in columns which we shall call stages, the switches in these stages being identical. Two adjacent stages are connected by a pattern of links or junctors. Along with the switches, the link patterns are responsible for the distributive characteristics of the network. They afford an inlet ways of reaching many outlets. Obviously, each outlet on a switch in a given stage is some inlet of the next stage, if there is one. Suppose that the N inlets are numbered in an arbitrary way, and that the N outlets are also numbered in an arbitrary way, both from 1 to N . Then it is clear that each link pattern, and each permitted way of closing the largest possible number of crosspoints in a stage, viz., N , can be viewed abstractly as a permutation on (1, ..., N}. Both the networks in Figs. 7

8.

99

STAGES AND LINK PATTERNS

and 8 have the property that all maximal* states have the same number N of calls in progress, and any such maximal state realizes a permutation which is a product of certain of the permutations represented

by the link patterns and the stages. I t will be convenient to generalize the usual notion of a “stage” of switching in a connecting network. By a stage [of switching] we shall mean a connecting network constructed as follows: with I the set of inlets, and Q that of outlets, we choose an arbitrary subset Y of I x 52, and we place a crosspoint between all and only those inlets u E I and outlets v E 52 such that ( u , v) E 9’.We shall also speak of Y itself as the “stage.” Thus we make

A stage is a subset off x 52. This terminology is easily seen to be an extension of the usual one, according to which, e.g., a column of swiches in Fig. 7 forms a stage, the network having four stages separated (or joined) by three link patterns. Actually, a link pattern may be associated with one or the other (but usually not both) of the stages it connects, to define a new stage; we do not usually do this. Definition 3.7.

Definition 3.8. A stage 9’ is made of square switches there is a partition 17 of (1, ..., N} such that

Y =

u

if

and only

if

(A X A )

AEIZ

A substage 9’’of Y is a subset of 9’. Except in the trivial case in which 9’ is actually a square N-by-N switch (i.e., Y = I x Q), a stage 9’ will not by itself give rise to a rearrangeable network. Still, it is known that several stages joined end to end by suitable link patterns can together give rise to such a network, e.g., that of Fig. 8. We can thus formulate two fundamental questions about connecting networks built out of stages: Definition 3.3.

(i) What stages and link patterns can be used to construct a rearrangeable network ? (ii) What stages, and how many of them, should be used to construct a rearrangeable network that has a minimum number of crosspoints (switches) for a given number of terminals on a side ? -___

* I.e.,

states in which no additional calls can be completed.

100

3.

REARRANGEABLE NETWORKS

Question (i) is studied in Sections 8-15, while question (ii) is treated in Sections 16-21.

$9. Group Theory Formulation We adopt some notational conventions from group theory to simplify our presentation. Let G be a group. I t is customary to speak of a subset K 5 G as a comptex. If x E G, then xK denotes the set of products x y with y E K , K x denotes the set of products yx with y E K . Similarly, if K , and K , are complexes, K,K, denotes the set of products y z with y E K , and z E K , . A group G of permutations is called imprimitive [Hall (IZ, p. 64)] if the objects acted on by the permutations of G can be partitioned into mutually disjoint sets, called the sets of imprimitivity, such that every y E G either permutes the elements of a set among themselves, or carries that set onto another. That is, there is a nontrivial partition f7 of the set X of objects acted on such that y E G and A E f7 imply y ( A ) E Ill. We shall extend this terminology as follows: Definition 3.4. G is called strictly imprimitive ;f it is imprimitive, and each set A of imprimitivity is carried into itself by elements of G , i.e., there is a nontrivialpartition I7 of X such that A E Ii’ implies T ( A )= A for a l l y E G, so that y E G is “nonmixing” on f7. Consider a stage of switching that has N inlets and N outlets. I t is evident that such a stage provides ways of connecting some of the inlets to some of the outlets. If the stage contains enough crosspoints it can be used to connect every inlet to some outlet in a one-to-one fashion, i.e., with no inlet connected to more than one outlet and vice versa. With the inlets and outlets both numbered 1, 2, ..., N , such a setting of the switches corresponds to a permutation on { 1, ..., N } . Indeed, there may be many ways of doing this, differing in what inlets are connected to what outlets, that is, corresponding to different permutations.

Definition 3.5. A stage Y generates the permutation y if there is a setting of N switches of Y which connects each inlet to one and only one out let in such a way that i is connected to T ( i ) , i = 1, ..., N , that is if (i, P(9) €97.

9. Definition 3.6.

denoted by P(S”).

101

GROUP THEORY FORMULATION

The set of permutations generated by a stage 9 is

Definition 3.7. A network (with N inlets and N outlets) generates a permutation cp if there is a setting of the switches in the network that connects, by mutually disjoint paths, each inlet to one and only one outlet in such a way that i is connected to cp(i),i = 1, ..., N . If two stages S”, , Y zare connected by a link pattern corresponding to a permutation c p z , then the permutations that they generate together are those of the form

v ~ F ~ T~~E P~( , Y, , ) ,

i

=

1 or 3.

If a network consists of two stages Yspl , 9, joined by a link pattern corresponding to a permutation p, then it can be seen that it generates exactly the permutations in the set P(Yl)@(Y,).

A network of s stages, Yi , i = 1, ..., s, with a link pattern corresponding to c p i , i = 1, ..., s - 1, between the ith and the (i 1)th stages, generates the complex

+

91P192

**.

FS-lYS

to refer to or indicate such a network. It is now possible to formulate a group-theoretic approach to the analysis and synthesis of rearrangeable connecting networks made of stages of switching joined by link patterns. Consider such a network, generating the complex P(91)Fl... PS-lP(9J.

T h e factors cpiP(YiTl),i = 1, ..., s - 1, occuring herein are themselves again just complexes. Thus, given any product of complexes

102

3.

REARRANGEABLE NETWORKS

we seek to know whether the product is the whole symmetric group, and whether the factor complexes K i can be written in the form

where 9 is a permutation and Y is a stage. I n this general form the problem is largely unsolved; however, some special cases are worked out in the sequel.

$10. The Generation of Complexes by Stages We start with this elementary result: Let M be a complex (i.e., a set) of permutations. Remark 3.1. Define a stage Y by Y

= {(i, j ) :

~ ( i=)j

for some g, E MI

Then P ( Y ) 2 M , and no smaller stage has this property. I n many cases of practical importance, such as shown in Figs. 7 and 8, the stages are made of square switches, and it is clear that a stage Y (with N inlets and N outlets) is capable of effecting certain special permutations on X = (1, ..., N } , and of course, all submaps thereof. Indeed, for each switch there are numbers m and n with m < n such I)! perthat the switch is capable of performing all the ( n - m k n among themmutations of the numbers k in the range m selves. Since no inlet [outlet] is on more than one switch, these permutations form a subgroup of the symmetric group S ( X ) of all permutations on (1, ..., N } . This subgroup has a property which might be described intuitively by saying that there exist sets on which the subgroup elements can mix “strongly,” but which they keep separate. I t is apparent, indeed, that the subgroup generated by a stage made of square switches is strictly imprimitive, the sets of imprimitivity being just the elements of the partition I7 of ( I , ..., N } according to what switch an inlet [outlet] is on. This situation might also be described by saying that a permutation 9 from the subgroup is nonmixing on 17. Our second observation is

< <

+

11.

AN EXAMPLE

103

Remark 3.2. Let H be a strictly imprimitive group of permutations on X = {I, ..., N } , with sets of imprimitivity forming the partition I7. Let Y be the smallest stage with P ( Y ) 2 H . Then

Y = U A X A , AEI?

i.e., Y is made of square switches. T h e main result of this section states that a stage can generate a subgroup only if it contains a substage made of square switches. This suggests that stages made of square switches necessarily arise in the generation of the symmetric group by products of complexes some of which are subgroups.

Let Y be a stage, and let P( 9) contain a subgroup H of S ( X ) . Then there is a substage 9of 9 'which is made of square switches.

Theorem 3.8.

Proof. Define a relation 92 on {I, ..., N } by the condition that i 9 j if and only if j = p ( i ) for some p E H . Since H is a subgroup, it must contain the identity permutation, i.e., igi for all i = 1, ..., N . Let i,,j, k be numbers in (1, ..., N } such t h a t j = p ( i ) and k = #(j)for some permutations p, # E H . Th en # y E H and k = # y ( i ) , that is, i92k; hence 92 is transitive. Finally, if j = y ( i ) with y E H , we have i = q-'(j) with 9-l E H , since H i s a group. Hence 92 is an equivalence relation, and there is a partition 17 such that

9 = (J ( A x A). AE I 7

Since i92j obviously implies ( i , j ) E 9, we have

9 E 9. 9 is clearly a substage of Y made of square switches.

$11. An Example As is well known, elementary group theory contains many results that allow one to write a group as a product of complexes. These results often involve a subgroup of the group in question. We shall quote an elementary result of this kind and interpret it in terms of a network that is known to be rearrangeable.

104

3.

REARRANGEABLE NETWORKS

Let G be a group, and let H I and H , be subgroups of G, not necessarily distinct. A double coset is a complex of the form HlTH,,

PEG.

I t is a known result (ZZ) that two double cosets are either identical or disjoint. Thus there is at least one complex M with the properties

u

H1PHZ

=

W M

G

if y #

H,yH, n Hlt,hH2 = 9

t,h,

with

q ,t,h E M .

I n particular G

= H,ICIH,,

and we have factored G into a product of three complexes, two of which are subgroups. Now suppose that G is actually S ( X ) , the symmetric group of all permutations on X = (1, ...,N ) , and that m and n are positive integers such that mn = N . Let Il be a partition of X = (1, ..., N } into rn sets of n elements each, and let H be the largest strictly imprimitive subgroup of S ( X ) whose sets of imprimitivity form II. Also let y be a self-inverse permutation, and IIqa partition, such that A E 17,B E If, imply*

I y(Aj A B 1

= 1.

Let K be the largest strictly imprimitive subgroup of S ( X ) whose sets of imprimitivity form Ifq. By Remark 3.2, Section 10, H and K can each be generated by stages of square switches. Returning to the earlier discussion leading to the factorization G = H1h4H2, we let H I = H , = H. Now it can be seen that the complex HYKVH

is generated by a network of the form shown in Fig. 8. By the SlepianDuguid theorem [Theorem 3.1, or see Bene5 (2),p. 1484, Theorem 11 this network is rearrangeable, so that HqKqH

*iX

=

S(X).

] denotes the number of elements of a set X

12.

105

SOME DEFINITIONS

Since q = q-l, the complex yKg, is itself actually the subgroup q-lKq conjugate to K . T h u s for G = S ( X ) and H , = H , = H , the factor M in S(X)= HMH can be chosen to be q-lKp.

$12. Some Definitions T h e number of elements of a set A is denoted 1 A 1. Let X , Y be arbitrary finite sets with I X 1 = 1 Y 1, let B be a subset of Y , let Ill , IT2 be partitions of X , Y , respectively, and let g, be a one-to-one map of Y onto X . Let 0 be the null set.

X : q-'(x)

E B}

Definition 3.8.

g,(B) = (x

Definition 3.9.

y hits 17, from I3 i f and only i f A

E

E

17, implies

y ( B ) n A # 8.

if and only if B E 17, implies

Definition 3.10. q hits 17, from B.

q covers Ill .from 17,

Definition 3.11.

~(17,) is the partition of X induced by q acting on

elements of 17, , i.e.,

dG)= {dB): Definition 3.12.

.g,

B

E

4.).

is the restriction of g, to B.

Let A be a subset of X .

is the partition of A induced by Ill , i.e.,

Definition 3.13.

= {C n A :

g, B-covers Ill f r o m 17,

Definition 3.14. ,,€&I

from

Bfl2

C E ITl].

if

and only

if Bp covers

'

Let nu, IIl be partitions of X . Then Ill > no Definition 3.15. (read pi-one refines pi-zero) if and only if every set in II0 is a union of sets in II1 , and II1 # .

106

3.

REARRANGEABLE NETWORKS

$13. Preliminary Results Let X and Y be any sets with 1 X 1 = 1 ' I < co, let ( A , , ..., A,) and I7, = ( B , , ..., B,?&) be partitions of X and Y , respectively, and suppose that for k = 1, ..., n the union of any k elements of 17, has more elements than the union of any k - 1 elements of 17,. Then Lemma 3.1.

fl,

=

(i) m >, n

(ii) For each one-to-one map f of X onto Y there exists a set of n distinct integers k(1), ..., k(n) with 1 k(i) m, i = 1, ..., n, and i = 1, ...,n. f(AJ n Bk(i)$. 0 ,

<

Proof.

<

Since 17, and I7, are partitions, and 1 X j = 1 Y

,=1

1,

3=1

If m were less than n, then the union of m B's has as many elements as the union of n A's, for m < n; this contradicts the hypothesis. Let K , = { j : f(Z)

Also let

E

B, for some 1 E Ail,

i

= 1,

..., n.

, ..., A{,,, be any k elements of 17,,1 < k

< n, and set

All of the

elements of

are mapped by f into of II, has

1 T I sets

of I7, . Since no union of k - 1 sets

13.

107

PRELIMINARY RESULTS

elements, it follows that I T 1 >, k. Th u s the union of any h of the sets {Ki, i = 1, ..., n>has at least K members. Hence by P. Hall’s theorem ( 8 )there is a set of n distinct representatives k ( l ) , ..., k(n) with k(i)

E

K,

K(i) # k ( j )

i

=

1, ...)71

i #j.

for

But clearly k ( i ) E Kiif and only if J’(l) E Bk[,)

I E Ai,

for some

that is, if and only if f ( A J n B k ( , ) f 0.

Let II1,17, be partitions of sets X , Y , respectively, Lemma 3.2. with the properties 1 X j = I Y I and C, , C, E Dlu 17, implies

I C, 1

=

Then for every one-to-one map cp of X onto Y there is a set D s X such that 9 hits 17,from D and 9-l hits II1from ~(0). Proof. We observe that 1 fl1 1 = j 17,/, and that the conditions of Lemma 3.1 are satisfied, with m = n. For each onto map ‘p there is a set D E X with 1 D 1 = I l7,1, such that

A

E

D, implies

DnA # R

(ql hits Dlfrom y ( D ) )

B E D2 implies v(D) n B # R

(91 hits

17, from 0)

Let X be any set and let U1 and 17, be partitions of X such that A, B E Ill 112 implies I A I = 1 B 1. Then for every permutation ‘p on X there is a partition 17, of X such that (i)91 covers 17,from fl,, (ii)~ - covers 1 Illfrom ~(fl,), Lemma 3.3.

Let cp be a permutation on X . T h e hypothesis implies Proof. that the union of any k elements of has more elements than the

n,

108

3.

REARRANGEABLE N E T W O R K S

union of any k - 1 elements of 17,,for k = 1, ..., 1 17,I. Hence by Lemma 3.2, with X = Y and m = n, there is a set D , E X such that y hits 17, from L), y-l

hits IT, from y(D,).

Now we consider the sets X , partitioned by XJ7,

and

=

X

YlII2,

-

D , and Y ,

=

Y

-

y(D1)

respectively

and we apply Lemma 3.2 to X,q,i.e., to the restriction of y to X - D,. This gives a new set D, L X such that again y hits 112 from D ,

y-l hits 17, from y(D,).

We proceed in this manner till X and Y are exhausted, and set = {D,, ..., DyL},where n = 1 A I for all A E 17,n 17,. It is clear that 17,has the stated properties. Let q be a permutation on X , and let 17,,17, be partitions of X .

If q covers 17, from 17,, and B l B ( = (IIlI,thenA~17,irnpZies~A =1 117,I.

Lemma 3.4.

E

Il, implies

Proof. Since q covers 17,from 17,, then A E fl1and B E 17, imply that there is an x E B with x E B with y-l(x) E A. T h u s y-l(x) E A for at least I 17, 1 distinct values of x, and so I A 1 3 1 17, 1. Since B E 17, implies 1 B 1 = 1 17, I, it follows that 1 I7, 1 divides 1 X I and

Clearly,

Since there are

1 IT, I sets in Ill each with at least 1 II, I elements,

14.

109

GENERATING THE PERMUTATION GROUP

If any A E Ill had more than 1 112 I elements the sum would exceed j X j, which cannot be. Thus A E Ill implies I A I = I 112 I.

$14. Generating the Permutation Group I n this section we exhibit a sufficient condition on permutations ..., ps-l and stages Y , , ..., Y , under which the complex

yl,

... ? s - l p ( p s )

P(pl)(Pl

is actually the whole symmetric group, and the corresponding network (obtained by linking Yiand Yi+lby q i , i = 1, ..., s - 1) is rearrangeable. I n order to focus on the mathematical character of the results, on their purely formal aspects divorced from physical considerations having to do with switches, etc., we phrase the conditions on the y's and the 9 " s purposely in a quite abstract way. Consequently, the practical implications and applications of the result may be unclear and require discussion. This discussion is given after the theorem has been stated, and is followed by its proof. Theorem 3.9. Let s > 3 be an odd integer, let y l ,..., qs-l be permutations on X = (1, ..., N } , let I l k , k = 1, ..., s, and Ilk, k = 1, ..., i ( s - l), be partitions of X, and let

y'. =

1

i

w>

?;I pk

..* (f&)(lW

k=O

k

=

1,

k = &(s ?;(s+l)(nS--k) {X>

k

..., &S

- 1)

+ l), ..., s

-

= S.

Suppose that (i) I f s > 3, then IIk < Ilk+1, k = 1, ..., *(s - 3). (ii) Ill(s+l) = Ii't(s-1) (iii) For k = 1, ..., &(s - l), and every B E Y k - I , y k B-covers Ilk f r o m y k ( Y k ) . (iv) For k = ;(s l), ..., s - 1, and every B E Yk+l, B-covers Ii'lL.rl .from p;'(Yk). (v) If A E Ilk and B E Y k - l u Y k + ! ( , + l ) then I B f l k 1 = I Bns--x.+l 1 = i A j, k = 1, ..., i (-~1).

+

1,

110

3.

REARRANGEABLE NETWORKS

Let HI<, k = 1, ..., s be the largest strictly imprimitive subgroup of S ( X ) whose sets of imprimitivity are exactly the elements of Ill; (i.e., A E 17, implies E H k } = S ( A ) ) .Then the complex K deJined by K

has the property K rearrangeable.

=

=

ffl%H,

..-

HS-fP3.

,Hs

S ( X ) ,and any network generating this complex is

T h e theorem given above does not provide any new designs of rearrangeable networks that are not already implicit in the work of M. C. Paull (7) and D. Slepian ( 6 ) ;thus no new principle is involved. Rather, in formulating the result, we have tried to state a generalized, purely combinatory form of these previous results. T h e theorem exhibits this generalization, first as providing a way of generating the symmetric group in a fixed number of multiplications of certain restricted group elements, and second as based on certain purely abstract properties of some partitions and permutations. As in A. M. Duguid’s proof of the Slepian-Duguid theorem (3), the basic combinatory theorem of P. Hall on distinct representatives of subsets is used repeatedly. This means (roughly) that the proof proceeds by showing that an arbitrary permutation (to be realized in the network) can be decomposed into submaps each of which can be realized in a disjoint part of the network, thereby not interfering with the realization of the other submaps. A significant departure from the work of Paull (7) is that we try to obtain rearrangeability directly from conditions that are stated for the network as a whole, as well as by building it up from rearrangeable subnetworks. T h e following intuitive guides may be useful in understanding Theorem 3.9. T h e permutations y l , ..., ySpl are of course intended to be those corresponding to the link patterns between the stages of a network. T h e partition 17, corresponds to the assignment of the terminals entering the kth stage to various square switches, all u E A for A E I l k being on the same switch. T h e partitions Llk are used in defining the submaps mentioned above. T h e “covering” properties (iii) and (iv) of the y k in Theorem 3.9 ensure (roughly) that the y k are sufficiently mixing or distributive to be able to generate all permutations in the restricted ways permitted in the definition of K . They are generalizations of the property, exhibited in Fig. 2, that every middle switch is connected to every side switch by a link. T h e property (v), finally, implies that various

14.

GENERATING THE PERMUTATION GROUP

111

sets of switches all have the same cardinality; this ensures (again, roughly) that if a crosspoint is not being used for a connection between one inlet-outlet pair, then it can be used for a connection between some other pair. Proof of Theorem 3.9.

there is only one

Ilk,

We use induction on odd s >, 3. If s = 3, viz., 17l. Let y be a permutation; we show that ‘p E

Hl’plH2V2H3 *

T he argument to be given is constructive, in that we do not use proof by contradiction, but actually give a kind of recipe for finding three permutations yi E Hi , i = 1, 2, 3, with

T o prove the theorem for s = 3, it is enough, for i = 1,2, 3 to exhibit a partition IT(i)and to define yi on A E 17(i),i.e., to give a7i ,

Condition (v) for k

=

A

E

n(q,

1 [=&(s

-

i

=

1,2,3.

1) here] tells us that for A

E17l

However, the “middle stage” condition (ii) states that 17, = 17l. Hence j Ill i 1 IT2 1 = N . Since [condition (iii) now] y;’ covers 17, from yl(Y1) = yly;l(n1) = ITz, it follows that B E 17, implies 1 B 1 3 1 17,j. If for some B E 17, it was true that I B 1 < 1 17, 1, then

-

which is impossible. Th u s 1 B I = I 112 1 for B E Ill 1 I n exactly the same way, using condition (iv), we find that C E 17, implies j C j = 1 II2 1. Therefore B, C€IT1 u 17, implies I B j = 1 C 1. Returning now to the chosen permutation y , we apply Lemma 3.3 to conclude that there exists a partition 17, of X such that y covers 17, from I?,, ‘p-l covers 17, from y(17,), 1 17,l = 1 A 1 for A E 17, n 17,, and I B 1 = I II11 = 1 113 I for B E 17,. Hence also

117,r

=

IW*

112

3.

REARRANGEABLE NETWORKS

Let p : IIPt)17,be any map of 17,onto 17, . T h e desired partitions n(i), i = 1, 2, 3 will be taken to be n(1)= P l ( m n ( 2 ) = {p(D): D

E

17,) = IT,

W 3 ) = n,, and the desired permutations q i , i these properties: for D E 17, D

173

71:

Fl(CL(4

172:

P)2173(L))

=

++

1, 2, 3 , are defined so as to have

P)pl(CL(D))

* P(D) P)L1?FIP)(D)*

That this can be done (uniquely, indeed) can be seen as follows: Let D E 17,,and p ( D ) = B E IT, . Since y-l covers 113 from y(17,), and y covers 17, from 1?,, it must be true that (1) y-l hits U3from y ( D ) (2) y hits ITl from D. But at the same time, by conditions (iii) and (iv), and the fact that 1 7 l = 17,,yl covers II1from 17,and y;' covers I7, from I7,,and so

(3) y1 hits LIlfrom B (4)yp' hits 17,from B. Thus if u E D n A and A E 113, there is a unique u E A such that E B, and we take q3(u) = u. Similarly, if x = y ( u ) E C and C E ITl, there is a unique w E C such that y;l(w) E B , and we take rll(w)= x. Finally, define 9, so that q 2 ( y 2 ( v )= ) y:'(w). Since p(.) is onto, each D E 17, deals with a unique B = p ( D ) E 17,, and the definition of q i , i = 1, 2, 3 can be made for each D and its associated B , independently of the others. I t is apparent that y,(u)

71P)172Y273(W = P ) ( W ,

or

D

E

17, ,

P) = 171P)1172T2173 *

Since q i for i

=

1, 2, 3 is onto, and is a subset of

u

EEIl*

E x E ,

15. it follows that rli

A CLASS OF REARRANGEABLE NETWORKS E

113

Hi,i = 1, 2, 3, and thus that K

=

S(X).

We now assume, as an hypothesis of induction, that the theorem is true for a given odd s - 2 3 3, and that we are given permutations q k , and partitions {n,} and { I l k } , satisfying the conditions of the theorem. No loss of generality is sustained if it is assumed that each A E 112 is invariant under q 2 , ..., qSp2 . This invariance can always be achieved by redefining the F ~ without , loss of properties (iii) to (v). I t can now be seen that for k = 2, ..., s - 2 the restrictions AKlk,

with

Ank,

AEW,

satisfy all the conditions on I l k ,nk(respectively) used in Theorem 3.9. Hence by the hypothesis of induction, for each A e n 2 , the restriction of the complex

H2%

.**

ve--$Ha--l

to A generates S ( A ) .T h e argument used for the case s = 3 can now here) playing the role be used to complete the induction, !?'2(=!?'s-1 of n2.

$15. Construction of a Class of Rearrangeable Networks

We consider a network v built of an odd number s v = .4p,v,

... v s - 1 Y s

> 3 of

stages,

7

satisfying the symmetry conditions

with each stage ,4ak made of identical square switches. T h e pfcwill be chosen in the following way: Order the switches of each stage; to define pk for a given 1 k i ( s - 1) take the first switch of Y , , say with n outlets and n a divisor of N , and connect these outlets one to each of the first n switches of .4pk+l; go on to the second switch of 9, and connect its n outlets one to each of the next n switches of Yk+l;

< <

114

3.

REARRANGEABLE NETWORKS

when all the switches of Ykflhave one link on the inlet side, start again with the first switch; proceed cyclically in this way till all the are assigned. (See Fig. 9.) outlets of 9,

FIG. 9.

Link assignment.

We shall show that a network v constructed in this way is always rearrangeable. Theorem 3.10. Let s >, 3 be an odd integer. Let nk , k = 1, *(s I), be any positive integers such that

+

+( S+l)

rl[ n i = N k=l

and

n, 3 2.

...,

15.

+

For each k = 1, ..., %(s l), Zet II, be the partition of X into the Nln,c sets of the f o r m A k i ==

115

A CLASS OF REARRANGEABLE NETWORKS

{ t : (i - 1 ) n k


< ink},

i

=

( 1 , ..., N )

I , ..., N / n k .

=

Let pk , k = 1, ..., *(s - 1) be permutations with the property that n = t (mod N / n k t l ) i f and only if Vk(.)

Ak+l,t

= O, **.) ( N / n k + l )

9

-

Define y k = ~ ~ - 2 , for

&(s

-


1)

< s - 1.

Let 9, , k = 1, ..., s, be the stages made of square switches defined by Y , =Ys--k+l, k

u

Y k =

=

1,

..., &(s

- 1)

A X A

AEng

and v be the network constructed by putting a link pattern corresponding to q k , k = 1, ..., s - 1 between stages 9,and Y k + l .Then v is rearrangeable. Proof.

It is readily seen that for k P(9,)

= the

=

1, ..., s

largest strictly imprimitive subgroup

with sets of imprimitivity I7,, = H,

,

in the notation of Theorem 3.9. Th u s to prove the theorem by appeal to Theorem 3.9 it is enough to exhibit suitable partitions IP, k = 1, ..., i ( s - l), and to show that these, together with q l ,..., ps-l, satisfy the conditions of Theorem 3.9. For k = 1, ..., ~1 ( s 1)) set

ITk = class of all { j : (i - I)N/n, ... n , <j

< iN/nl ... n k , i

I t is evident that the n k

Ilk

= 1,

...)121712

nk}.

are successively finer partitions, i.e., that

< I7k.1,

k

=

1, ..., $(s

-

3).

3.

116

REARRANGEABLE NETWORKS

Also, since 17i(s-1) consists of the n1n2 ... n,,,-,, sets

< k%+(s+l),

( j : (k - l)nbcs+l,< j

k

=

1,

...,~2

nb(s-l)l

it can be seen that nt(S-1)

~

:

{A+(S+lLi

1

< i < Nh+(S+l)h

and hence that the middle stage condition (ii) in Theorem 3.9,

n&(s+l) = m-1),

is satisfied. The remainder of the proof, in which the requisite covering properties of the y k are demonstrated, is based on some auxiliary results. Lemma 3.5. For k = 2, following identity holds

..., &s

-

< i < N/nknk+l the

l), and 1

and the sets on the right are disjoint f o r dzfferent i. Proof.

Since T ' ; ~ ( A ~= + {n: ~ . ~ n)

-

t (mod N/nk+l), 1


the union on the right in the lemma is the set of all n that are in,. Consider =t(mod N/nk+l) for some t with (i - 1)nk < t such an n, with say

<

+ t,

n=-- IN nk+l

Then n =nk-

with 0


< n k , and so

IN

nknk+l

t i-l<-
0

<1<

+ (i - l)n, +

nk+l*

u,

n E Ak.(lN/n,n,+,+i-l)

or

n

E

AKSt with t = (i - 1) (mod N/nknk+l).

15.

117

A CLASS OF REARRANGEABLE NETWORKS

Since the representation of n in terms in I and u is unique, the lemma follows. T h e practical or physical import of the lemma is this: I n any stage k + 1, I k < *(s - I), the ith block of nk switches is connected by the link pattern vk to exactly those nk+l switches (in the kth stage) whose number t = (i - 1) (mod N/n,n,+,).

<

For 1

Definition 3.16.

Bik =

u t

IAk,t: t

< i < n, and 2 < k < m = i ( s + 1)

= Y (mod n1n2 *.. nk-J

< 11283 where n2n3 Lemma

3.6.

nk..l is taken = 1 ;f k For 1

=

(i - 1)

for some Y with

r **'

nk-1

2.

< i < n, and 2 < k

<m

Bik = v i 1 ( B i , k + l ) .

Proof. We show that the right-hand side contains the left. Equality then follows from Lemma 3.5, since Bi,k+lwill always be a union of sets of the form

u

Ak+l.T *

j - l < l < j nk

+

This is because t = r (mod u ) if and only if t 3 (r 1) (mod u ) , if r < u. Consider then an n E B,, . There is a t 3 1 congruent to an r (mod nlnz n k - J , with

0

<

such that n E Ak,t. T h e latter fact implies that ( t - 1)nk


<

tnk.

Now v k ( n )E Ak+l,,, where n is congruent to can represent n in the form Hence

T

(mod N / n k + , ) , SO we

118 Writting t

3. =

nk-l + r , with

ln,n,

we see that qn,

***

n,

where

REARRANGEABLE NETWORKS

+ (r

-

l)nk < T

< qn,

121%

T

+rnk,

aN

q=l---

I t follows that region

nk

***

%+l

is congruent (mod nlna ..- nk) to some integer p in the

(i - l)n,

"'

nk


<

in2

'**

nk

j

and thus p,(n) E Bi,k+l, completing the proof of Lemma 3.6. Now if R = m, the defining condition that t = r (mod n, ... n,,b-l) for some r with (i - l)n, ... nmp1< r < in, ... n,,-, , used in the definition of Bi, , can be put into a slightly different form. In this case we must have

and so t can only be congruent to r by being equal to r , that is

(i

-

l)n, .*.n,,-,


< in, ... nm-,.

Hence it can be seen that

{ B t m, i

=

1, ..., nl} = 1 7 1 .

Applying Lemma 3.6 ( m - 2) times we find that {Biz, i

=

1, ..., el} = 9;' ... pk'(I7') =

P)l(Y1)*

Now n = t (mod Nin,) if and only if pl(n) Also, by definition of B i z , Biz

=

(J

t=i(modn,)

Azt.

E

A,, , t

=

1,

..., N / n , .

16.

Let pl(n)

E

SUMMARY OF SECTIONS

17-21

119

Biz cp,(n) E AZt. Th en n has the form a=--

aN

filfl,

a1

+ t = (--aN + 6) n, + i, n1n2

so n = i (mod q).Since 1 Bi2 1 = I Bi,, 1 = n2 n, = N / n , , it follows that B,, is the cpl image of N/nl integers each of which is congruent to i (mod nl).Since each such integer must be in a different A,, , it follows that p;’ covers 17,from cpl(Y1). T h e remaining conditions in Theorem 3.10 can be demonstrated in essentially the same way; one has merely to identify the sets in question, and use Lemma 3.5 and an analog of Lemma 3.6. T h e details will be omitted. a.1

$16. Summary of Sections 17-21 T h e object of the remainder of Chapter 3 is to solve the synthesis problem of choosing, from a class of networks that are built of stages of square switches and satisfy some reasonable conditions on uniformity of switch size, a rearrangeable connecting network having a minimum number of crosspoints. Naturally, we do not pretend that minimizing the number of crosspoints (used to achieve a given end) is the only consideration relevant to the design of a connecting network. Other factors, like the number of memory elements, the amount and placing of terminal equipment, the ease with which a network is controlled (e.g., the possibility of reliable end-marking), etc., may be of overriding significance, depending on the technology used. Still, it is important to know the limits of the region of possible designs, and these are obtained by optimizing on one variable without attention to others. T h e problem of designing a good rearrangeable network was (probably first) considered in a paper (12) of C. E. Shannon investigating memory requirements in a telephone exchange. O n the networks that he considered he imposed the realistic “separate memory condition” to the effect that in operation a separate part of the memory can be assigned to each call in progress. This means that completion of a new call or termination of an old call will not disturb the state of memory elements associated with any call in progress. Shannon showed that under this assumption a two-sided rearrangeable network, with N inlets and N outlets, and N a power of 2, requires at least 2 N log, N

120

3.

REARRANGEABLE NETWORKS

memory elements (e.g., relays). He gave a design that actually realized this lower bound, using 4(2N - 1) log, N crosspoints (e.g., relay contacts). His design had the disadvantage of having very large numbers of contacts on certain relays. I t is to be noted that Shannon was concerned with minimizing the number of memory elements, without regard to the number of crosspoints. Shannon’s separate memory condition is actually met by modern connecting networks that are of current practical interest, viz., by the networks made of stages of crossbar switches, considered here. For indeed, an inlet relay on an n x n crossbar switch is used to close any and each of n crosspoints: the exact one that closes depends on what outlet relay is simultaneously activated. We consider the problem of minimizing the number of crosspoints in a network built of square switches, without attention to the number of relays. T h e following result (a consequence of Theorem 3.18) then complements Shannon’s: For N a power of 2 it is possible to design a rearrangeable network with N inlets and N outlets using 4N log,N - 6N relays and 4N(log,N - 2) crosspoints. T h e figure for relays is roughly twice Shannon’s, while that for crosspoints is much smaller than his, for N large. I n our design, no relay controls more than 4 contacts. I n Section 17 we discuss the notion of the combinatorial power or efficiency of a connecting network, and propose to define it as the fraction Y of permutations it can realize. According to this definition the four-stage No. 5 crossbar type of network with 10 x 10 switches has efficiency Y close to zero, although it turns out that for the same number ( ~ 1 0 0 0of ) terminals there are networks that achieve Y = 1 with a smaller number of crosspoints ( 4 ) . This greater efficiency is obtained by using many more stages than four. Preliminaries are treated in Section 18. Particular attention is drawn to the class C, of all networks having N inlets and N outlets, and built of stages of identical square switches symmetrically arranged around a center stage. T h e cost c(v) of such a network v is defined as the total number of crosspoints, divided by N. It is proposed to select rearrangeable networks v from C, that have minimal cost ~(v). This problem is attacked in Section 19 by defining (i) a map T from C,,, to a special set A such that ~ ( v )is a function of T(v)E A , and (ii) a partial ordering of A. I t is then shown (Section 20) that (roughly) a network v

17.

T HE COMBINATORIAL POWER OF A NETWORK

121

is optimal if and only if T(v) is at the bottom of the partial ordering of A. This result allows one to identify (Section 21) the optimal networks in C, . Their general characteristics are these : Except in some easily enumerated cases, the optimal network should have as many stages as possible, and switches that are as small as possible, the largest switches being in the middle stage; the cost ~ ( v )of an optimal network v is very nearly twice the sum of the prime divisors of N , while the number of its stages is 2x - 1 where x is the number of prime divisors of N . Our chief conclusion is that by using many stages of small switches it is possible to design networks that are rearrangeable and cost less (in crosspoints per terminal) than networks in current use, which are far from being rearrangeable. T h e price paid for this great increase in combinatory power is the current difficulty of controlling networks of many stages. This difficulty is technological, though, and will decrease as improved circuits are developed.

$17. The Combinatorial Power of a Network A principal reason why rearrangeable networks are of practical interest is (of course) that they can be operated as nonblocking networks. If the control unit of the connecting system using the rearrangeable network is made complex enough, it is in principle possible to rearrange calls in progress, repeatedly, in such a way that no call is ever blocked. At present this possibility is being exploited in only a few special purpose systems, because of the large amount of searching and data-processing it requires. However, there is another reason why rearrangeable networks should evoke current interest. Even if we do not care to exploit it, the property of rearrangeability in a connecting network is an indication of its combinatorial power or reach, and so can be used as a qualitative “figure of merit” for comparing networks. Other things being equal, a rearrangeable network is better than one which cannot realize all assignments of inlets to outlets. Rearrangeability expresses to some extent the efficiency with which crosspoints have been utilized in designing a connecting network for distribution, that is for reaching many outlets from inlets. If a numerical measure is called for, one can use the fraction of realizable maximal assignments. For a network v with the same

122

3. REARRANGEABLE NETWORKS

number N of inlets as outlets, and with inlets disjoint from outlets, this is just number of permutations realizable by Y r = N!

= combinatorial power of v.

<

It is apparent that 0 Y < 1, and that for a rearrangeable network 1. Also, 7 may be viewed as the chance that a permutation chosen at random will be realizable. We shall calculate a bound on the combinatorial power 7 of the kind of connecting network most commonly found in modern telephone central offices. This is the network illustrated in Fig. 10. We choose the switch size n = 10 as a representative value; the network then has N = 1000 inlets, as many outlets, and 4 x 104 crosspoints. Clearly, the network can realize at most all the permutations that take exactly n terminals from each frame on the inlet side into each frame on the outlet side. Now a frame has n2 inlets [outlets], and there are Y

=

n2! (n!)n

__

ways of partitioning n2 things into n groups of n each. Since there are 2n frames, there are

ways of choosing n groups of n each on each frame, and assigning inlet groups to outlet groups (one-to-one and onto) in such a way that for every inlet frame and every outlet frame exactly one group on the inlet frame is assigned to a group on the outlet frame. There are n2 groups on a side (inlet or outlet), and within each group (at most) n! permutations can be made, i.e., each inlet group can be mapped, terminal by terminal, in at most n! ways onto its assigned outlet group. Hence at most

permutations can be realized. There are N = n3 terminals on a side, and a total of n3! possible permutations in all. Thus

17.

T H E COMBINATORIAL POWER OF A NETWORK

123

FIG. 10. Structure of No. 5 crossbar network.

For n

=

10, with

20 log( loo!) = 3 159.4000 100 log(lO!) = 655.976 +log 2.rr = 0.39959 log(x!) log 2x (x we find roughly

-+

Y

+ + 4)log x - x log,,+, < 10-64.

Thus only a vanishingly small fraction of all possible permutations can actually be achieved by the No. 5 crossbar network (illustrated in Fig. 10) for n = 10, a reasonable switch size. I n the example calculated, the network has a “cost” of 40 crosspoints per terminal on a side. Much of the force of the example would be lost if it were in fact impossible to achieve high values of r (i.e., near 1) without incurring a great increase in the cost in crosspoints per terminal. This, however, is not the case. It follows from our Theorem 3.18 that a rearrangeable network ( r = 1) can be designed for N = 1024 terminals on a side using only 4(10g&’ - 2) = 32

124

3.

REARRANGEABLE NETWORKS

crosspoints per terminal. Th u s it is actually possible to achieve r = 1 for more than 1000 lines with fewer than 40 crosspoints per line. T h e network that does this turns out to have 17 stages instead of 4, an illustration of the way that allowing many stages can lead to vastly more combinatorially efficient network designs. T h e middle stage of this network consists of a column of 256 4 x 4 switches, and each of the other 16 stages, arranged symmetrically, consists of a column of 512 2 x 2 switches. For k = 1, ..., 8, the kth stage is connected to the (k 1)th as follows: T h e first outlet of the first switch of stage k goes to the first switch of stage (k I), the second outlet of the first switch of stage k goes to the second switch of stage (k l), the first outlet of the second switch of stage k goes to the third switch of stage (k l), etc., as in Fig. 11 with 1 k 7; when each switch of stage (k 1) has one link on it the process starts over again with the first

+

+

+ +

+

< <

STAGE

k

STAGE

FIG. 11. Link assignment.

k+i

18.

125

PRELIMINARIES

switch, and continues cyclically until all the links from stage k are 1 for k = 9, ..., 17 assigned. T h e connections between stages k and k are the inverses of those for k = 1, ..., 8, so that the network is symmetric about the middle stage.

+

$18. Prel iminaries T h e symbol C, , N 3 2, is used to denote the class of all connecting networks v with the following properties: (i) v is two-sided, with N terminals on each side. (ii) v is built of an odd number s of stages Y , , k = 1, ..., s, of square switches, i.e., there are permutations q l , ..., ys-l such that v = y,%yz,

*.a,

Cps-1ys.

(iii) v is symmetric in the sense that for K Y k= 9’s-k+l For v

E

=

1, ..., +(s - 1)

C, we use the notation s = s(v) = number of

stages of v n, = n k ( v ) = switch size in the Kth stage of

v.

(iv) v has Nln, identical switches is stage k, i.e., stage Y , is of the form U A x A AEII

for some partition 17 with 1 17 1 = N/n, .

IA 1

=

1 B 1 for all A , B E 17, with

T h e defining conditions of C, imply that nk = ns-lc+l

for K

=

1, ..., +(s - I)

and that

It is assumed throughout that nk(v) 2 2 for all v and all k

=

1, ..., s(v).

126

3.

REARRANGEABLE N E T W O R K S

T h e cost per terminal (on a side) c(v) of a network v E C, is defined to be the total number of crosspoints of v divided by the number N of terminals on a side. Since there are N/nk nk x nk switches in stage k, the total number of crosspoints is (using the symmetry condition)

and so

A network v is called optimal if c(v) = min{c(p): p

E

C,}.

I t is clear that the cost per terminal of a network v E C, depends only on the switch sizes, and not at all on the permutations that define the link patterns between stages. Also, it is apparent from Theorem 3.10 that given any network v1 E C, there is another network v2 E C, that is rearrangeable and differs from v1 only in the fixed permutations that are used to connect the stages; in particular, v1 and u2 have the same number of crosspoints. T hus the problem of selecting an optimal rearrangeable network from C, is equivalent to that of choosing an optimal network from C, , rearrangeable or not. A network in C, can be made rearrangeable by changing its link patterns at no increase in cost. We make Defiinition 3.77.

m

= m(v)=

n

=

s(v) ~

+1

2

=

numerical index of the middle stage

n(v) = nrn(”) = size of middle stage switches.

Definition 3.78. O(v) = {nl , ..., nm-l} = the set of switch sizes (with repetitions) in outer (i.e., nonmiddle) stages

19.

CONSTRUCTION OF T H E BASlC PARTIAL ORDERING

Definition 3.19.

Remark 3.3.

w(N) = {O(v): v

c(v) = n ( v )

E

127

CN}.

+ 22x. reO(v)

Theorem 3.11.

Let ( A ,n) be a point (element) of tu(N)

x

x

=

N

with UEA

Then there exists a nonempty set Y

c C, such that

T(v) = ( A ,n),

v E

Y.

T h e v’s in Y differ only in the permutations between the stages, and in the placing of the outer stages, and at least one of them is rearrangeable. This result follows from the definition of C, , and from Theorem 3.9.

$19. Construction of the Basic Partial Ordering T h e solution to the problem of synthesizing an optimal rearrangeable network from C, will be accomplished as follows: We shall define a mapping T of C, into w ( N ) x X , with X = (1, ..., N } , and a partial ordering of T(C,); the map T will have the property that C ( V ) is a function of T ( v ) ; then we shall prove that (roughly speaking) a network v is optimal if and only if T(v)is at the “bottom” of the partial ordering, i.e., that ~ ( vis) almost an isotone function of T(v). T o define a partial ordering of a finite set, it is enough to specify consistently which elements cover which others. Let 2, , Z , , be sets of positive integers N possibly containing repetitions.

<

<

Z , covers Z, ;f and on& if there are positive integers j and k such that k occurs in 2, ,j divides k, and Z, is obtained from Z , by replacing an occurrence of k with one occurrence each of j and Definition 3.20.

k,?.

128

3.

Definition 3.21.

T:

Definition 3.22.

Let p, v be elements of C ( N ) . T ( p ) covers T ( v )

A partial ordering of covering:

and only

if

REARRANGEABLE NETWORKS

v --z O(v),n ( v ) .

< of T(C,,,) is defined by the following definition

either

if

(i) n(v) < n ( p ) , n ( v ) divides n ( p ) , and O(v) results from O(p) by adding a n occurrence of n(p)/n(v),or (ii) n ( v ) = n ( p ) and O(p)covers O(v).

$20. Cost I s Nearly Isotone on T(C,,,) If T(v)

Theorem 3.12.

< T ( p ) ,and n ( p ) > 6 , then c(v)

Proof.

<: C(P).

It is enough to prove the result for

T(p)covers T(v).

p

and v such that

Case (i). n ( v ) < n ( p ) , n ( v ) divides n ( p ) , n(v) 2 2, and O(v) results from O(p)by adding an occurrence of n(p)/n(v).Then c(v) = n(v)

+2 2

XtO(V)

< c(p) if and only if

T hus ~ ( v )

that is if

x

20.

where x

= n(v)

T(C,)

COST IS NEARLY ISOTONE ON

and y

= n(p)/n(v).

129

Now n ( p ) > 6 implies that

or (ii) and (iii)

n(v)

3 3.

T h e condition 2 y / ( y - 1)

41 . G

< x is fulfilled in all three cases, and so

C(P).

Case (ii). n ( p ) = n ( v ) and O(p) covers O(v). There exist integers j, k such t h a t j divides k , j 2 2 in O(p),and O(v) results from O(p) by replacing one occurrence of k with one each of j and klj. Then C(V)

= n(v)

+2 2

x

xsuivt

2

= n ( p ) - 2 k + 2 j + - + 2k 2 3 = ~ ( p) 2k

Since j divides k and j

x

Xeub)

+ 2j +.2k:3

3 2, k

2j and k 3 2 k / j , so

and ~ ( v )< c(p). Theorem 3.13. If v E C, and O(v) does not consist entirely of prime numbers (possibly repeated), then there exists a network p in C, of s(v) 2 stages with c(p) < C ( V ) and v cannot be optimal in C, .

+

< <

Proof. There is a value of k in the range 1 k m(v) - 1 for which n,, is not a prime, say nk = ab. Define stages Y j ( p ) , j = 1, ..., ~ ( v ) 2 as follows:

+

Y j + l ( p )=

Yj(4

j =

+ 1,

a * * ,

4.);

130

3.

REARRANGEABLE NETWORKS

(1, ..., N } with

let U a ,17, be partitions of X

=

N n,= a

and

A

n,= b

and

R E I T , implies

Set yr;,l(tL) =

E implies ~ ~1 A

I

1B j

=

n

= b.

u u B2 A2

AEII,

5”&)

=

Benb

YJP)=

y”j4

j

=

1, ...,K - I

all l (j ~ =)1, ..., 4 Y ~ ( c=Ly) p * ( y ) - j +

+

~ ) 2.

By Theorem 3.10 permutations yl,..., y,q(,ll-l can be found so that the network IL = 9

1 %

f * . . f

P)s(fL)-1~Ps(li)

is in C, and is rearrangeable. I t is apparent that nm(@) = nn,(”) and that O(v) covers O(p). Hence the argument for case (ii) of Theorem 3.12 shows that p has strictly lower cost than v.

If N > 6 and is not prime, then a network v Corollary 3.1. consisting of one square switch is not optimal. $21. Principal Results of Optimization An element T(v) of T(C,) is ultimate if there are C, such that T ( v ) covers T ( p ) .

Definition 3.23.

no p

E

T ( v ) is ultimate if and only if n(v) is prime and O(v) Remark 3.4. consists entirely of prime numbers. Definition 3.24. An element T(v) of T(C,) is penultimate ;f it covers an ultimate element. Definition 3.25.

p , ,n

=

I, 2, ..., is the nth prime.

2 1.

131

PRINCIPAL RESULTS OF OPTIMIZATION

Definition 3.26. n ( n ) is the prime decomposition of n, that is, the set of numbers (with repetitions) such that

n

=p>p2

...p:i

if and

only if n(n) contains exactly cii occurrences of pi , i and nothing else.

=

1, ..., I,

p is the largest prime factor of N.

Definition 3.27.

Lemma 3.7. I f p = 3 and N conditions are equivalent:

> 6 is even, then the following

(i) v is optimal (ii) T ( v ) is penultimate and n(v) = 6 or 4 (iii) T(v) = ( ~ ( N i 66) ) , or (.rr(N/4), 4).

<

Proof. By Theorems 3.12 and 3.13 only v with n(v) 6 and O(v) consisting entirely of primes can be optimal. Writing N = 2"3u with x >, 1 and y >, 1, it is seen that such v must have a cost C(V) having one of the forms

+ 2 [ 2 ( ~ 1) + 3y] = 4x + 6-Y 2, 3 + 2[2x + 3(y l)] = 4~ + 6y 3, 4 + 2[2(x 2) + 3y] = 4x + 6y - 4 6 + 2[2(x - I ) + 3(y l)] 4x + 6y 4. 2

-

-

-

-

-

-

1

(only occurs if x

> l),

-

T h e least of these is either of the last two, which correspond to n(v) = 6 if x = 1, or to n(v) = 6 or 4 if x > 1. I t is apparent that (ii) is equivalent to (iii). Lemma 3.8.

equivalent :

If p

=

2, and N > 4, then the following conditions are

(i) v is optimal (ii) T(v)is penultimate and n(v) = 4 (iii) T ( v ) = (n(N/4),4).

132

3.

REARRANGEABLE N E T W O R K S

With N = 2" it can be seen as in Lemma 3.7 that only those Proof. v can be optimal whose cost c(v) has one of the forms 2 4

+ 2[2(. - 111, + 2[2(x

-

2)].

T h e second of these is the better, and corresponds to n(v) = 4. Theorem 3.14. Let p be a network such that a prime number r > n ( p ) occurs in O(p). Let M result f r o m O(p) by replacing one occurrence of r by n(p). Then f o r any network v with

i.e., v is strictly better than p. Among such largest.

V,

that is best for which r is

Proof. Existence of a rearrangeable v satisfying T(v) = ( M , r ) is guaranteed by Theorem 3.1 1. For the rest of the proof, we observe that r > n ( p ) and

<

I f n(p) 6 , n ( p ) = 2x3"52, some prime number r > 3 occurs in O(p), and if M results f r o m O(p) by replacing one occurrence of r by x occurrences of 2, y occurrences of 3, and x occurrences of 5 then for any network v E C, with

Theorem 3.15.

T(v) = ( M , r )

it is true that

4") G C ( P )

i.e., v is at least as good as p. Among such largest.

V,

that is best f o r which

Y

is

2 1.

PRINCIPAL RESULTS OF OPTIMIZATION

133

Proof. Existence of a rearrangeable v E C, satisfying T ( v ) = ( M ,r ) is given by Theorem 3.1 1. For the rest of the proof, we observe that r 3 5 and C(P) =

=

6 4+2 n(p)

+ 2r

+ n(p)

=Y

2

UEO(P)

-

-

4x

4x

-

-

64’ - 1Oz

6y

-

+2

u uEM

+

1 0 ~ c(v).

Since, x, y , and x can only assume the values 0 and I , with z = 1 if and only if x = y = 0, we have c ( p ) 3 c(v), the best v corresponding to the largest r . Definition 3.28.

Q

=

{ ( A ,Y ) :

Definition 3.29.

L

=

T-l(Q).

Remark 3.5.

ordering which

< of

r a prime and A

=

.rr(N/r)}.

Q consists of all the absolute minima in the partial T(CN),i.e., v E L implies that there are no p E C , for T(P) <

Theorem 3.16.

w4.

If p > 3, then all optimal networks belong to L.

Proof. Let p E CN - L be given. We show that there exits a v E L that is at least as good. Case 1.

v E L , n(p,)

There is a sequence p = p L 1p, z , ..., p n , v with p n # v,

>6

T(P1) 2 T ( P J

2

3 T(Pn)

and such that T(pn) covers T(v).Th en the numbers n ( p j ) , j = 1, ..., n are all > 6, and the result follows from Theorem 3.12. All sequences p = p1 , pz , ..., p n , v with p n # v , v E L , T ( p l ) 3 T(pJ 3 ... 3 T ( p n ) , and such that T ( p n ) covers T(v), are such that n(pn) < 6 . Consider such a sequence. Let i be the smallest index j for which n ( p j ) < 6, j = 1 , ..., n. T h e n Theorem 3.12 gives c ( p ) 3 c(pi). Since n(pJ < 6 and T(p,) covers T(v), it Case 2.

134

3.

REARRANGEABLE NETWORKS

follows that O(pi) contains an occurrence of p > 3. Hence by Theorem 3.15 there exists a network 7 E C, with n(7) = p and 47)

< 4 4 < 4P.).

Let E E L be such that n(5) = p and T ( 7 )covers T ( 0 .T h e n c(5) by case (ii) of Theorem 3.12. Hence

< c(7)

45) <4 4 {EL.

Theorem 3.17.

and c(v) Proof.

N

=

=

N.

If N

< 6 and v is optimal, then v is a square switch <

For prime N with 2 N < 6 the result is obvious. If E C, then exactly one of the following alternatives

6 and v

obtains:

T(v) = (% 6) T(v) = ({3}, 2)

C(V)

and and

T(v) = ((2},3)

=

6

c(v) = 8. c(v) = 7.

T h e first alternative is optimal, and there is exactly one v E C, such that T(v) = (6, 6), viz., the 6 x 6 square switch. Similarly, if N = 4 and v E C, , then T(v) = (6, 4) or ({2}, 2); the former has cost 4,the latter 6. Definition 3.30. 2, D(n) is the sum of the prime divisors of For n n counted according to their multiplicity; thus ;f n

=

2"13"2

.p p

then

c ( N ) = min(c(v):

Definition 3.31. Theorem 3.18.

N

p c ( N )=

if N < 6 or + 2D(N/p) if

2D(N/2)

if

v E

C,}.

N i s prime N > 6 and either p > 3 or N is odd N > 6 in all other cases (i.e., p = 2, or p = 3 and N is even).

135

REFERENCES

Proof. Putting together Lemmas 3.7 and 3.8 and Theorems 3.11-3.14, 3.16, and 3.17 we obtain the following values for the minimal cost in crosspoints per terminal on a side for networks in C x :

N

if

N

<6

or

N is prime

+ 6(log,N

3

+2 2

x =3

4

+2 2

x = 4(log,N

xen(Ni3)

e€n(N/4)

-

2)

- 1) =2

x z€n(N 1 2 )

if

p > 3, N > 6 ,

if

p

=

3, N > 6, N even

if

p

=

3, N > 6, N odd

if p

=

2, N

> 6:

simplification gives Theorem 3.18. REFERENCES

1. C. Clos, A Study of Non-Blocking Switching Networks, Bell System Tech. J .

32, 406-424 (1953). 2. V. E. BeneS, On Rearrangeable Three-Stage Connecting Networks, Bell System Tech. J . 41, 1481-1492 (1962). 3. A. M. Duguid, Structural Properties of Switching Networks, Brown Univ. Progr. Rept. BTL-7, 1959. 4. V. E. Ben&, Permutation Groups, Complexes, and Rearrangeable Connecting Networks, Bell System Tech. J. 43, 1619-1640 (1964). 5. V. E. BeneS, Optimal Rearrangeable Multistage Connecting Networks, Bell System Tech. J . 43, 1641-1656 (1964). 6. D. Slepian, Two Theorems on a Particular Crossbar Switching Network, unpublished manuscript, 1952. 7. M. C. Paull, Reswitching of Connection Networks, Bell System Tech. J . 41, 833-855 (1962). 8. P. Hall, On Representatives of Subsets, J . London Math. SOL.10, 26-30 (1935). 9. D. Gale, A Theorem on Flows in Networks, Pacific J . Math. 7, 1073-1082 (1957). 10. V. E. BeneS, Algebraic and Topological Properties of Connectipg Networks, Bell System Tech. J. 41, 1249-1274 (1962). 11. M. Hall, Jr., “The Theory of Groups.” New York, 1959. 12. C. E. Shannon, Memory Requirements in a Telephone Exchange, Bell System Tech. J . 343-349 (1950).

CHAPTER

4

Strictly Nonblocking Networks

A particularly strong combinatorial property of some connecting networks is that they are strictly nonblocking, i.e., that no call is ever blocked in any state of the network. Although these networks are not commonly used in contemporary telephone exchanges, they provide a convenient starting point for practical connecting network studies. The unique work of C. Clos on the design of nonblocking connecting networks made of stages of rectangular crossbar switches is described in this chapter.

$1. Introduction I n a strictly nonblocking connecting network, it is always possible to establish a connection from an idle inlet to an idle outlet, regardless of the state of the network. Because a simple square array with Ninlets, N outlets, and N 2 crosspoints meets this requirement, it can be taken as an upper design limit for the “two-sided” case in which there are as many inlets as outlets, and no inlet is an outlet. Similarly a triangular 1)/2 crosspoints gives a nonblocking network for the array of N ( N (6 one-sided” case in which every inlet is an outlet. Since the number of crosspoints in these simple nonblocking switches goes u p unpleasantly fast with N , it is desirable to consider strictly nonblocking networks in which the number of crosspoints goes up more 136

+

3.

NONBLOCKING CONNECTING NETWORK

137

slowly than N 2 . I n 1953, C. Clos published the first fundamental study ( I ) of nonblocking connecting networks; in it he elegantly solved the problem of designing a nonblocking network (with N inlets and N outlets, distinct from the inlets) having fewer than N 2 crosspoints. Indeed, he showed that by using enough stages one can design nonblocking networks for large N in which the number of crosspoints goes up more slowly than N1+f,for prescribed E > 0. T h e present chapter on strictly nonblocking networks is closely patterned on Clos’ work.

$2. Square Array A simple square array having N inputs and N outputs is shown in Fig. 1. T h e number of crosspoints equals N 2 ,and any combination of N N OUTPUTS

F

2 z z

NUMBER OF CROSSPOINTS N2

FIG. 1 .

Square array.

or fewer simultaneous connections can exist without blocking between the inputs and the outputs. T h e number s of switching stages is equal to 1. T h e number of crosspoints C(s) is C(1) = N2.

(1)

$3. Three-Stage Strictly Nonblocking Connecting Network A “two-sided” strictly nonblocking connecting network having N inlets, N outlets, and requiring fewer than N 2 crosspoints is exemplified in Fig. 2. This network has N = 36 inlets. There are three switching stages, namely, an input stage (a), an intermediate stage

138

4. STRICTLY

NONBLOCKlNG NETWORKS i

T-

-7 I

I I

I I I

I

I I

? I

9 O~HER INTERMEDIATE SWITCHES

N =36

OUTPUT SWITCHES

c

I

,,

4

----- - - _-__STAGE

(a)

NUMBER OF CROSSPOINTS

I.--

6-4

STAGE

= 6N3/2

(b)

!

,,

4

_--_- _ _ _ _ _ STAGE (C)

-3N (4488 CROSSPOINTS WHEN N =36)

FIG. 2. Three-stage connecting network.

(b), and an output stage (c). I n stage (a) there are six 6 x 11 switches, in stage (b) there are eleven 6 x 6 switches, and in stage (c) there are six 6 x 11 switches. I n total, there are 1188 crosspoints, which is fewer than the 1296 crosspoints required by eq. (1). Of interest are the derivations of the various quantities and sizes of switches. I n stage (a) the number n of inputs per switch was assumed to be equal to N1l2,thus giving six switches and six inputs per switch. I n a similar manner stage (c) was assigned six switches and six outputs per switch. T h e number of switches required in stage (b) must be sufficient to avoid blocking under the worst set of conditions. T h e worst case occurs when between a given switch in stage (a) and a given switch in stage (c); (i) five links from the switch in stage (a) to five corresponding switches in stage (b) are busy, (ii) five links from the switch in stage (c) are busy to five additional switches in stage (b), and (iii) a connection is desired between the given switches. T h u s eleven switches are required in stage (b). T h e remaining requirements, namely, eleven verticals per switch in stages (a) and (c) and 6 x 6 switches in stage (b) are then easily derived.

4. PRINCIPLE n

139

INVOLVED

T h e number of crosspoints required for three stages, where N 1 / 2 ,is summarized by the following formula:

=

C(3) = (2N1J2- 1)(3N) =

(2)

cw

6 N 3 J 2- 3N.

I n Table 4.1 it may be noted that the number of crosspoints is less than N 2 for all cases of N 3 36. TABLE 4.1

CROSSPOINTS FOR SEVERAL VALUESOF N Square array

N 4 9 16 25 36 49 64 81 100

...

1,000 10,000

N2

16 81 256 625 1,296 2,401 4,096 6,561 10,000

......

1,000,000 100,000,000

Three-stage network

6N3J2- 3N 36 135 336 675 1,188 1,911 2,880 4,131 5,700

......

186,737 5,970,000

$4. Principle Involved T h e principle involved for determining the number of switches required in the intermediate stage is illustrated in Fig. 3. T h e figure is for a specific case from which one can generalize for n inputs on a given input switch and m outputs on a given output switch. I n the figure it is desired to establish a connection from input B to output H . A sufficient number of intermediate switches is required to permit the ( n - 1) inputs other than B on the particular input switch and the ( m - 1) outputs other than H on the particular output switch to have connections to separate intermediate switches plus one more m-1 switch for the desired connection between B and H . T h u s n intermediate switches are required.

+

4.

I40

STRICTLY NONBLOCKING NETWORKS SWITCHES

n

m OUTPUTS ON A PARTICULAR

n

INPUTS ON A PARTICULAR INPUT SWITCH

-- E

A--

C

-\

Y.y"

SWITCHES REWIRED t (m-1) tc=nt m-1

\

(n-1)

WHEN

n=rn,

ABOVE =2n-1 ABOVE 3m-i

FIG.3. Principle involved.

$5. Five-Stage Network A five-stage switching network is illustrated in Fig. 4.T h e analysis of this network can be made in the following manner. Each input and output switch is assumed to have n = N1I3 inputs or outputs, respectively. Connection between a given input switch and a given output switch is made via levels, a level consisting of three intermediate switching stages. T h e number of levels required is (2N113- 1). Each level has N2I3 inputs and the same number o f outputs. T h e number of crosspoints for a three-stage nonblocking network for N213inputs and N 2 / 3outputs can be obtained from eq. (2) by substituting N 2 / 3for N in that equation. T h e total number of crosspoints required for the five-stage network is

+

C(5) = (2N113- 1)23N213 (2N113- 1)2N =

16N4'3- 14N

+ 3N2'3.

(3)

(34

T h e number of crosspoints required for several sizes o f the five-stage network is given in Table 4.2. T h e results are compared to the square and three-stage cases.

6.

141

SEVEN-STAGE NETWORK

1 OUTPUT SWITCHES

I '

2

r-

EEl El

El

i

*

f_ln

-7

SWITCH I NG LEV€ LS (EACH LEVEL CONSISTS OF THREE STAGES)

\\

1

n

\ FIG. 4.

141 2n- 2

\-

Five-stage connecting network.

TABLE 4.2

CROSSPOINTS FOR SEVERAL VALUES OF N N

Square array

Three-stage network

Five-stage network

64 729 1,000 10,000

4,096 531,441 1,000,000 100,000,000

2,880 115,911 186,737 5,970,000

3,248 95,013 146,300 3,434,488

$6. Seven-Stage Network A seven-stage switching network can be analyzed by considering paths requiring five intermediate switching stages as paths via switching aggregates. T h e number of such aggregates is (2N1/4- 1). Each aggregate has N3/4 inputs and a like number of outputs. From eq. (3) the crosspoints for each aggregate can be obtained by substi-

142

4.

STRICTLY NONBLOCKING NETWORKS

tuting N 3 / 4for N in that equation. T h e total number of crosspoints required for the seven-stage network is

57. General Multistage Switching Network Equations ( l ) , (2a), (3a), and (4a) are herewith tabulated as a series of polynomials together with the next polynomial: C(1) = N’, C(3) = 6N3” - 3N, C(5) = 16N413- 14N C(7) = 36N514- 46N C(9) = 76N615-

+ 3N213, + 20N314 3N1l2, 130N + 86N415- 26N3’5+ 3N215. -

(1) (24 f3a)

(44 (5)

These polynomials can be determined for any number of switching stages from the following formula where s is an odd integer:

qS) =2

k=( S i k 1 / 2 )

(2NZ/‘S+l’- l)(S+Y”Z-k NZl;/(S+l)

k=2

+ N4/(S+1)(2N”I‘Sfl’- l)(S-l)/Z*

(6)

An alternative expression equivalent to eq. (6) has been suggested by S. 0. Rice and J. Riordan”. T h e recurrence relation used in individually deriving the foregoing polynomials can be used to directly derive the following formula: C(2t

where

n2(2n 1) + 1) = ___-“5n n-1 -

s = 2t

N

=

-

3)(2n -

2nt]

(64

+ 1,

ntfl.

Table 4.3 gives comparative numbers of crosspoints for various numbers of switching stages and sizes of N . T h e data of Table 4.3 are

*

Private communication to C. Clos.

7.

GENERAL MULTISTAGE S W I T C H I N G NETWORK

143

plotted on Fig. 5 . T h e series of curves appears to be bounded by an envelope representing a minimum of crosspoints. T h e next section dealing with minima indicates that points exist below this envelope.

NUMBER OF INPUTS AND NUMBER OF OUTPUTS

FIG. 5.

Crosspoints versus switching stages.

4. STRICTLY

144

NONBLOCKING NETWORKS

TABLE 4.3 CROSSPOINTS FOR VARIOUSNUMBERS OF SWITCHING STAGESs, N

100 200 500 1,000 2,000 5,000 10,000

s = l

s=3

10,000 40,000 250,000 1,000,000 4,000,000 25,000,000 100,000,000

s-5

5,700 16,370 65,582 186,737 530,656 2,106,320 5,970,000

6,092 16,017 56,685 146,300 375,651 1,298,858 3,308,487

AND

VALUESOF N

s=7 7,386 18,868 64,165 159,9~4 395,340 1,295,294 3,159,700

5=9

9,121 23,219 78,058 192,571 470,292 1,511,331 3,625,165

$8. Most Favorable Size of Input and Output Switches in the Three-Stage Network T h e foregoing derivations were for implicit relationships between n and N , namely, n being the [(s 1)/2]th root of N . T o obtain a minimum number of crosspoints a more general relationship is required. For the three-stage switching network this is

+

N2 ( + 7)-

C(3) = (2n -- 1) 2 N

(7)

When n = " I 2 , eq. (7) reduces to eq. (2). For a given value of N , the minimum number of crosspoints occurs when dCjdn = 0, which gives 2n3 - nN

+N

= 0.

(8)

This equation has the following two pairs of integral values: n

=

2, N

=

16

and

n = 3, N = 27.

As N approaches large values, eq. (8) can be approximated by N

N

2n2.

(9)

Graphs of eqs. (8) and (9) are shown in Fig. 6. I n Table 4.4 the numbers of crosspoints are based on the nearest integral values of n for given values of N .

8.

BEST SWITCHES SIZE I N THE THREE-STAGE NETWORK

145

iooo 800 600

500 400

300

cn

2t 200 I-

3

0

a

0

cn

5

100

_z

80

g

60

n -J

I-

50

Z

40

II

30 20

f0

INPUTS OR OUTPUTS PER SWITCH

FIG. 6 . Relationship between N and n for minima in crosspoints. Three-stage

network.

Where comparisons can be made, Table 4.4indicates fewer crosspoints than does Table 4.1. This fact can be realized in another manner. By eleminating n in eqs. (7) and (9), the result for large values of N is C(3)

N

4(2)1'2"3'2- 4N.

Equation (10) indicates fewer crosspoints than does eq. (2).

(10)

146

4. STRICTLY

NONBLOCKING NETWORKS

TABLE 4.4 CROSSPOINTS FOR SEVERAL VALUESOF N Number of crosspoints

Nearest integral

N

16 27 40 44 55 60 65 78 84 98 105

value of n

N2

Eq. (7)

2 3 4 4 5 5 5 6 6 7 7

256 729 1,600 1,936 3,025 3,600 4,225 6,084 7,056 9,604 11,025

288 675 1,260 1,463 2,079 2,376 2,691 3,575 4,004 5,096 5,655

$9. Most Favorable Switch Sizes in the Five-Stage Network If n be the number of inputs per input switch and outputs per output switch, and m be the number of inputs per switch in the second stage and outputs per switch in the fourth stage, then the following equation gives the total number of crosspoints: 2N N2 7 XFjI [ + (2m - 1) (

C(5) = (2n - 1) 2N

+

*

(11)

T h e partial derivative of this equation with respect to m when set equal to zero yields n =

N(" - 1) 2m3

*

T h e partial derivative of this equation with respect to n when set equal to zero yields the following equation: N=

+

nm2(2n2 2m - 1) (2m - l ) ( n - 1)

Equations (12) and (13) can be solved for n and m in terms of given values of N . For example, for N = 240, we obtain n = 6.81 and m = 3.56.

11.

CASES WHERE

= r(mod n )

N

147

$10. Search for the Smallest N for a Given n for the Three-Stage Network For a given value of n, eq. (7) furnishes a means for locating that size of three-stage connecting network which has N 2 or fewer crosspoints. This can be done by setting eq. (7) equal to N 2 :

i

N 2 = (2n - 1) 2N

+ TI N2

and solving for N in terms of n. T h e solution is N>,

2n"2n - 1) (n - 1 ) 2 '

Minimum values of N for given values of n are listed in Table 4.5. This table also lists the next highest N exactly divisible by n. From this table it appears that when N = 24, we have the smallest switching array for which it may be possible to have fewer than N 2 crosspoints. However for N = 25, as shown in Table 4.1, eq. (2) g'ives more than N 2 crosspoints. T h e problem is one of finding a network for N = 25 with fewer than N 2 crosspoints. For this and all cases beyond, the next section indicates that it is profitable to consider situations where N is not exactly divisible by n. TABLE 4.5

MINIMUM VALUESOF N

FOR

GIVENVALUESOF n

n

N per Eq. (1 5)

2 3 4 5 6

24 22.5 24.9 28.1 31.7

N

= 0 (modn) 24 24 28 30 36

$11. Cases in the Three-Stage Switching Network Where N = r(mod n ) Table 4.1 indicated that for N = 25 and n = 5 a total of 675 crosspoints were required. A square array requires only 625. Figure 7

148

4. STRICTLY

NONBLOCKING NETWORKS

INPUT SWfTCHES

INTERMEDIATE SWITCHES

OUTPUT SWITCHES

#El$ #%

w PPPPL

1 1 1 1 1 ~

##k 9

9

9

8

FIG.7. Three-stage array. 25 = 1 (mod 3). An equivalent arrangement is to provide two 8 x 8 and three 9 x 9 intermediate switches. Two of the 9 x 9 switches need only 80 crosspoints.

11.

CASES WHERE

N = r(mod n)

149

shows a layout of switches where N = 25 and n = 3 . I n this case one input is left over when 25 inputs are divided into threes. T h e lone input requires three paths to the intermediate switches. This is in accordance with Fig. 3. T h e lone output also requires three paths to the intermediate switches. Also from Fig. 3 , the lone input to the lone output requires only one path. Hence there must be one switch capable of connecting the lone input to the lone output. T h e number of crosspoints required is 615, which is less than the 625 required by the square array. This scheme can be extended to any case where N = kn r , where the remainder Y is an integer greater than zero but less than n. T h e formula for the number of crosspoints where k input and k output switches of size n and one input and output switch of size r are used is

+

c = 2(2n

~

2I,+(

1)(N --

Y)

1)

(---N - Y + 1 y - n + Y. n

I. G. Wilson* has pointed out that for a lone input the crosspoints in the intermediate switches can be used to isolate its possible connections, hence no crosspoints are required in the input stage. This likewise applies for a lone output. With this modification the array in Fig. 7 requires six fewer crosspoints. For this case, when r = 1, the number of crosspoints is

N

J. Riordant has found a more efficient arrangement for cases where

+

= kn r . I n place of using k switches of size n and one switch of size r , he proposes that ( k + 1 - n + r ) switches of size n and ( n - Y) switches of size n - 1 be used. For this case the number of crosspoints is

C

*

=

2(2n

-

+1 + (2n

l)(k

- TZ -

+

3)(k

Y)TZ

+ 2(2n

-

2)(n - ~ ) ( n 1)

+ 1)2+ 2(k + l ) ( k + 1

Private communication to C. Clos.

t Private communication to C. Clos.

-

n

+ Y).

(17)

4.

150

STRICTLY NONBLOCKING NETWORKS

Equation (17) is identical to eq. (16) when r = n - 1. There are two cases, namely, when n = 2 and n = 3, for which eq. (16a) gives fewer crosspoints than does eq. (17).

$12. Search for the Minimum Number of Crosspoints between N = 23 and N = 160 T h e equations of the preceding sections furnish a means for searching for minimum crossnet arrays. Table 4.6 shows the results of such a search up to N = 160. Results are indicated in unit steps from N = 23 to N = 40 and for every tenth interval thereafter. At N = 161, a five-stage network requires the fewest crosspoints. Table 4.6 was computed by the use of finite differences. T h e equations were C[(K

C(Kn

+ l)n]

-

C(Kn) = (2n - 1)(2n

+ + 1) - C(Kn + r ) = 2(K + 3n C(kn + 1) C(kn) = 2Kn + 1. Y

-

-

+ 2k + l ) ,

(18)

l),

(19) (194

Equation (18) was derived from eq. (7) with N being replaced by

(k $- 1)" and by kn as required. Equation (19) was derived from eq.

(17) with Y being replaced by r 4- 1 as required. This equation applies for all values of n greater than 3 and for the particular case of n = 3 and 7 = 2. Equation (19a) was derived from eqs. (16a) and (7) and is for the particular case of r = 1, when n = 2 and n = 3.

$13. Search for the Minimum Number of Crosspoints for N = 240 For a case where N is large enough to require five switching stages, the search for the minimum number of crosspoints should be based on eqs. (12) and (13) and on the use of Table 4.6. T h e method is suggested by means of Table 4.7. T h e data in a previous section indicate that a minimum should occur for N = 240, when n = 6.81 and m = 3.56. I n Table 4.7 the minimum occurs when n = 6 and m = 4. I t fails to occur at n = 7 because 240 is not exactly divisible by 7.

13.

MINIMUM NUMBER OF CROSSPOINTS FOR

N

=

240

151

TABLE 4.6 CROSSPOINTS FOR VARIOUS VALCESOF N AND n

N

Square array

23 24 25 26 27 28 29

Three-stage array

n=2 540 576 625 663 716 756 813 855

3R 31

32 33 34 35 36 37 38 39 40

50

60 70 80 90 100 110 120 130 140

n = 5

861 899 935" 1,002 1,042 1,080 1,153 1,195 1,235 1,314

556 588 633 667 701 735 788 824 860" 896" 857 995" 1,033" 1,071" 1,140" 1,180" 1,220" 1,260"

864 911 951 891 1,031 1,071 1,128 1,170 1,212 1,254 1,286

n=7

n=8

4,499 5,315 6,100 7,044 7,923" 8,840" 9,968 10,979

6,156 6,975" 7,947 8,860 9,811" 10,800"

530 560" 609" 643" 675" 730" 766"

800"

n=4

n = 5

n=6

1,819 2,415 3,164 3,920

1,800" 2,376" 3,~24l 3,744" 4,536 5,400

1,879 2,420 3,056 3,764 4,455" 5,291" 6,199 7,040 8,076

150

160 a

n=4

n=3

Minimum values.

-

152

4.

STRICTLY NONBLOCKING NETWORKS

Except for this situation, the minimum would have occurred as predicted. T A B L E 4.7 CROSSI'OINTS

FOR

N

=

240

AND VARIOUS VALUES OF

Input and output stages

Intermediary stages .~

No. of

switches

n

2 3 4 5 6

120 80 60 48 40 \30

1 5 8

10

11 12

30

j2:

24 j20 ( 2 20

Size of switches

Crosspoints

2 x 3 3 x 5 4 x 7 5 x 9 6 x 11 7 x 13 6x12 8 x 15 9 x 17 8 x 16 10 x 19 11 x 21 10 x 20 12 x 23

1,440 2,400 3,360 4,320 5,280 5,460 720 7,200 7,344 768 9,120 9,240 800 11,040

No. of Inputs and levels outputs 3

5 7 9 11 2 11 15 2 15 19 2 19 23

Crosspoints per eq. (3) five-stage network Crosspoints per eq. (2) three-stage network Crosspoints per eq. (1) square network . a

n

120 80 60 48 40 30 35 30 24 21 24 20 22 20

x 120" x 80" x 60" x 48 x 40" x 35 x 35" x 30" x 27 x 27" x 24" x 22 x 22 x 20

.

m 8 5 5 4 4 3 4 3 3 3 3

~~

Crosspoints

20,925 18,720 16,632 15,120 13,860 1,826) 11,363 \ 12,000 1,230) 10,125/ 10,640 8801 9,196\ 9,200

Total crosspoints

22,365 21,120 19,962 19,440 19,140 19,369 19,200 19,467 19,760 20,116 20,240

. . . . . . . . . . . . . . 20,596 . . . . . . . . . . . . . 2 1,624 . . . . . . . . . . . . . . 57,600

See Table 4.6 for minimum number of crosspoints.

$14. Rectangular Array Referring to Fig. 1, if there were N inputs and M outputs, a simple rectangular array would result which would be capable of sustaining up to N or M , whichever is the lesser, simultaneous connections without blocking. T h e number of crosspoints is C(1)

=

A'M.

(20)

$15. N inputs and M Outputs in a Three-Stage Array For the case of a three-stage switching array with N inputs and M outputs, let there be n inputs per input switch and rn outputs per

16.

153

TRIANGULAR NETWORK

output switch. A particular input to be able to connect without blocking under the worst set of conditions to a particular output will 1) + ( m - 1) 1 available paths. Thus, by providing require ( n for that many intermediate switches, a nonblocking switching array is obtained. T h e number of crosspoints is

+

~

C(3) = (n

+m

-

1)

nm

1

Differentiating this equation first with respect to n and then to m yields two partial differential equations whose solution indicates that a minimum is reached when n = m. Replacing m by n in eq. (21), the equation for the number of crosspoints becomes C(3) = (2n -

Solving for the minimum number of crosspoints gives the following expression: NM NM n3 - ___ n -__ = 0. N+M N+M

+

When N = M this equation reduces to eq. (8). T h e three-way relationships of n, N , and M are shown in Fig. 8.

$16. Triangular Network For the “one-sided” case where all inputs are also outputs, an arrangement such as is shown in Fig. 9 can be used. T h e crosspoints in the intermediate switches permit connections between all switches on the left-hand side. For connections between two trunks on the same switch it is assumed that one of the links to an intermediate switch can be used to establish the connection but without affecting any of the crosspoints on the intermediate switch. If this is not so, then each input-output switch can be built up with an additional triangular portion, to permit connections within the switch, the intermediate switches being used only for connections between two distinct switches. For the case illustrated in Fig. 9 the number of crosspoints is C=(272-1)

where T is the number of two-way trunks.

4.

154

STRICTLY NONBLOCKING NETWORKS

5000 4000

3000

600 500 400 u)

t-

2 300 Z

;I 200

z

I0

20

30

40 50 60 80 (00 200 M =TOTAL OUTPUTS

300 400

600

I000

FIG.8. Relationship of n to N inputs and M outputs for a minimum in crosspoints in a three-stage network.

18.

COMPARISON WITH EXISTING NETWORKS

155

By differentiation, conditions for obtaining minimum numbers of crosspoints can be determined. T h e arrangement can also be extended to cases where extra switching stages are required.

$17. One-way Incoming, One-way Outgoing, and Two-way Trunks

A combination of the triangular network of Fig. 9 and of unequal inputs and outputs is shown in Fig. 10. I n this figure, one-way incoming, one-way outgoing, and two-way trunks can be freely interconnected without blocking. T h e number of crosspoints for this case is (25)

T h e comments concerning the triangular network also apply for this case.

$18. Comparison with Existing Networks Few existing connecting networks are nonblocking. An example is the Bell System’s four-wire intertoll trunk concentrating network. I n one of its standard sizes 4000 crosspoints are required for 100 incoming trunks and 40 outgoing intertoll trunks. From Fig. 8, for N = 100 and M = 40 it may be noted that the nearest integral value for n is 5. By substituting this value in eq. (22), a nonblocking three-stage connecting network of 2700 crosspoints is found, which could be used for the concentrating switch. I n this case the new approach to the connecting network problem may prove to be of value. Comparisons with existing networks having blocking are likely to be unfavorable because the grades of service are not the same. For instance, a No. 1 crossbar district-to-office layout of 1000 district junctors and 1000 trunks requires 80,000 crosspoints. This layout can handle 708 erlangs with a blocking loss of 0.0030. T h e minimum number of crosspoints with a nonblocking network is slightly less than 138,000. This, however, can handle 1000 erlangs without blocking. By introducing blocking into the design methods described in this chapter, a more favorable comparison with existing networks having blocking

156

4. STRICTLY

NONB L OC KI NG NETWORKS

INPUT AND OUTPUT SWITCHES

INTERMEDIATE SWITCHES

T=24

TWO- WAY TRU,NKS

I I II

I I I

I

I

I I

FIG. 9. Triangular network.

can be obtained. This can be done by omitting certain of the paths.

If done to an array requiring 1000 inputs and 1000 outputs a layout

can be obtained requiring 79,900 crosspoints with a blocking loss of

0.0022 for a load of 708 erlangs. For this example, at least, it appears that the new design methods may prove to be valuable, especially for use in the development of electronic switching systems where the control mechanism may not be dependent upon the particular connecting network used.

19.

157

CONCLUSION INTERMEDIATE SWITCHES

ONE-WAY OUTGOING TRUNKS

0 INCOM

-

7 I

I I

II

j

I

I

#{

I I

I I

N=!2

I

I

I I

I I I I

I

II

i-

PYPP?, I I I I

TWO-WAY

TRUNKS

I 11-

I I

I

T= 9 I I

I

IL p = 3 TI FIG. 10. One-way incoming, one-way outgoing, and two-w-ay trunks.

$19. Conclusion I n present day commercial telephone systems the use of nonblocking switching networks is rare. This is due in part to the large number of crosspoints required. With the design methods described herein, a wider use of nonblocking networks may occur in future developments. For the usual case of networks with blocking, new systems have generally been designed by an indirect process. Several types and

158

4. STRICTLY

NONBLOCKING NETWORKS

sizes of switching arrays are studied until the most economical one for a given level of blocking is found. With the new design methods, a straightforward approach is possible. Figure 5 indicates that a region of minimum values exists. By first designing a nonblocking system with a reasonable number of switching stages and then omitting certain of the paths, the designer can arrive at a network with a given level of blocking and be very close to a minimum in crosspoints. T h e possibility of the adoption of this direct design method is important. REFERENCE I . C. Clos, A Study of Non-Blocking Switching Networks, Bell System Tech. 3. 32, 406-424 (1953).

CHAPTER

5

A Sufficient Set of Statistics for a Simple Telephone Exchange Model

In this chapter we consider a simple telephone exchange model which has an infinite number of trunks and in which the traffic depends on two parameters, the calling rate and the mean holding time. We desire to estimate thebe parameters by observing the model continuously during a finite interval, and noting the calling time and hangup time of each call, insofar as these times fall within the interval. I t is shown that the resulting information may, for the purpose of this estimate, be reduced without loss to four statistics. These statistics are the number of calls found at the start of observation, the number of calls arriving during observation, the number of calls terminating during observation, and the average number of calls existing during the interval of observation. The joint distribution of these sufficient statistics is determined, in principle, by deriving a generating function for it. From this generating function the means, variances, covariances, and correlation coefficients are obtained. Various estimators for the parameters of the model are compared, and some of their distributions, means, and variances presented.

$1. Theoretical Problems and Methods of Traffic Measurement

Four important kinds of theoretical problems arise in the measurement of telephone traffic. These are: (1) the choice of a mathematical 159

160

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

model, containing parameters characteristic of the traffic, to serve as a description; (2) the devising of efficient methods of estimating the paramaters; (3) the determination of the anticipated accuracy of measurements; and (4) the assessment of actual accuracy, after measurements have been made. T h e present chapter deals with aspects of the second and third kinds of problems, for the simplest and least realistic mathematical model of telephone traffic. Specifically, for this model, we treat the problems of (i) complete extraction of the information from a given observation period, without regard to costs of observation, and (ii) determination of the anticipated accuracy of certain methods of estimation which arise naturally from the discussion of complete extraction. T h e method by which we attack problems (i) and (ii) has three stages. First we choose a small number of significant properties of, or factors in, t h e physical system we are studying. T h e n we abstract these properties into a mathematical model of the physical system. Finally, from the properties of the model, we derive results which may be interpreted as answers to the two problems treated. T h e advantage of this method is that we can use the precise, powerful apparatus of mathematics in studying the model; its limitation is that it yields results that are only as accurate as the model in describing reality. A method similar to the above forms the theoretical underpinning of telephone traffic engineering itself. T o design equipment effectively, the traffic engineer needs a description of the traffic that is handled by central offices. H e decides what properties of the entire system of telephone equipment and customers will be most useful to him in describing the traffic. He then designates certain parameters to serve as mathematically precise idealizations of these properties, and in terms of these parameters constructs a model of the traffic, upon which he bases much of his engineering. I n choosing a mathematical model for a physical system, one is confronted with two generally opposed desiderata: fidelity to the system described and mathematical simplicity. T h e model may involve important departures from physical reality; a mode'l that is sufficiently amenable to mathematical analysis often results only after one has introduced admittedly false assumptions, ignored certain effects and correlations, and generally oversimplified the system to be studied. However, the abstract model will be an exact and simple tool for analysis. We can construct a simple mathematical model for the operation

1.

TRAFFIC MEASUREMENT

161

of a telephone central office by leaving out of consideration many important facts about such systems, and by concentrating on factors most relevant to operation. Since we are interested in telephone traffic and in the availability of plant, it seems natural to require that a realistic model take account of at least the following five significant factors: (1) the demand for telephone service, (2) the rate at which requests for service can be processed and connections established, (3) the lengths of conversations, (4) the supply of central office equipment, and (5) the manner in which the first four factors are interrelated. Unfortunately, the mathematical complexity of such a realistic model precludes easy investigation. Therefore, the model used in this chapter is based only on factors (1) and (3). T h e demand for telephone traffic is usually made precise by describing a stochastic process that represents the way in which requests for telephone service occur in time. A realistic description will take account of the facts that the demand is not constant, but has daily extremes, and that in small systems the demand may be materially lessened when many conversations are in progress. * Since taking account of the first fact leads to a more complicated model in which our investigations are more difficult, we ignore it, with the proviso that the results we derive are only applicable to systems and observations for which the demand is nearly constant. T h e second kind of variation in demand becomes insignificant as the number of subscribers increases and the traffic remains constant. Hence, we further confine the applicability of our results to systems with large numbers of subscribers, and we assume that the demand does not depend on the number of conversations in existence. With these assumptions, a mathematically convenient description of the demand is specified by the condition that the time intervals between requests for service have lengths that are mutually independent positive random variables, with a negative exponential distribution. A telephone central office contains two kinds of equipment: control circuits which establish a desired connection, and talking paths over which a conversation takes place. T h e time that a request for service occupies a unit of equipment, be the unit a control circuit or a talking path, is called the holding time of the unit. A request for service affects the availability of both kinds of equipment but, except for special

-~ * This

is the so-called “finite-source” effect.

I62

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

cases, the holding times of talking paths are usually much longer than the holding times of control units such as markers, connectors, or registers. I n view of this disparity, we assume that the only holding times of consequence are the lengths of conversations; i.e., the holding times of talking paths. We assume also that these lengths are mutually independent positive random variables, with a negative exponential distribution. For the simplest mathematical model of telephone traffic, we may consider the arrangement of switches and transmission lines that constitutes a talking path in the physical office to be replaced by an abstract unit called a “trunk.” A trunk is then an abstraction of the equipment made unavailable by one conversation, and we may measure the supply of talking paths in the office by the number of trunks in a model. T h e word “trunk” is also used to mean a transmission line linking two central offices, but as long as we have explained our use of the word there need be no confusion. Often the number of transmission lines leading out of an office is a major limitation on its capacity to carry conversations, and in this case the two uses of the word “trunk” are very similar. Unfortunately, we do not take advantage of this similarity, since we make the mathematically convenient but wholly unrealistic assumption that the number of trunks in the model is infinite. T h e model we investigate thus depends on only two of the factors previously listed as essential to a realistic model, namely, (1) the demand for service and (3) the lengths of conversations. I n view of the simplicity and inaccuracy of this model, the question arises whether much is gained from a detailed analysis. Such scrutiny may indeed reveal little that is of great practical value to traffic engineers. I t is important methodologically, however, to have a detailed treatment of at least one approximate case. We undertake this detailed treatment largely for the insight that it may give into methods that could be useful in dealing with more complex and more accurate models. Once a designer has chosen a model and has specified the parameters he would like to have measured, it is up to the statistician to invent efficient means of measurement, by choosing, for each parameter, some function of possible observations to serve as an estimate of that parameter. One measure of efficiency that is of mostly theoretical interest is the observation time required to achieve a given degree of anticipated accuracy; the most realistic measure of efficiency is in terms of dollars and man-hours. I t may often be more efficient, in the

1.

TRAFFIC MEASUREMENT

163

sense of the latter measure, to spread observation over enough more time to compensate for the inability of an intrinsically cheaper method of measurement to extract all of the information present in a fixed time of observation. For example, periodic scanning of switches in a telephone exchange is usually less costly than continuous observation. As a result, telephone traffic measurement is usually carried out by averaging sequences of instantaneous periodic observations of the number of calls present, rather than by continuous time averaging, although it can be shown that continuous observation is more efficient for extracting information. T h u s statistical efficiency, which may be expensive in terms of measuring equipment, can be exchanged for observation time, which may be less costly. This exchange brings about a reduction in cost without impairing accuracy. Our concern in this chapter is with the less practical problems of complete extraction, and of the anticipated accuracy of estimation methods based on complete extraction. Let us consider how our mathematical model can shed light on these problems. A mathematical model may or may not be a faithful description of the behavior of real telephone systems. Nevertheless random numbers, with or without modern computing machines, enable one to make experiments and observations on physical situations that approximate, arbitrarily closely, any mathematical model. Th u s we can speak meaningfully of events in the model, and of making measurements and observations on the model. T h e mathematical model elucidates our problems in the following ways: (1) It enables us to state precisely what information is provided by observation; (2) it enables us to explain what we mean by complete extraction of information; and (3) it enables us to derive results about the anticipated accuracy of measurements in the model. These results will have approximately true analogs in physical situations t o which the model is applicable. T h e calls existing during an observation interval (0, T) fall into four categories: (i) those which exist at 0, and terminate before T ; (ii) those which fall entirely within (0, T);(iii) those which exist at 0 and last beyond T ; and (iv) those which begin within (0, T)and last beyond T . For calls of category (i), we assume that we observe the hangup time of each call; for category (ii), we observe the matching callingtime and hangup time of each conversation; for category (ii$ we observe simply the number of such calls; and for category (iv), we observe the calling times. Table 5.1 summarizes the kinds of calls and the information observed about each.

I64

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

TABLE 5.1 INFORMATIONOBSERVED Types of calls

End in (0, T )

End after T

Start before 0

Start in (0, T)

(i) Hangup times known, number of calls known

(ii) Matching calling times and hangup times known, number of calls known

(iii) Number of calls known

(iv) Calling times known, number of calls known

What we mean by the complete extraction of information is made precise by the statistical concept of su8ciency. By a statistic we mean any function of the observations, and by an estimator we mean a statistic that has been chosen to serve as an estimate of a particular parameter. Roughly and generally, a set S of statistics is sufficient for a set P of parameters when S contains all the information in the original data that was relevant to parameters in P. If S is sufficient for P, there is a set E of estimators for parameters in P, such that the estimators in E depend only on statistics from S, and such that an estimator from E does at least as well as any other estimator we might choose for the same parameter. Th u s we incur no loss in reducing the original data (of specified form) to the set S of statistics. It remains to state what it means for S to contain all the relevant information. We do this in terms of our model. T h e mathematical model we are adopting contains two distribution functions, that of the intervals between demands for service, and that of the lengths of conversations. We have supposed that these distributions are both of negative exponential type, each depending on a single parameter. Th u s we know the functional form of each distribution, and each such form has one unknown constant in it. Since the mathematical structure of the model is fully specified except for the values of the two unknown constants, we can assign a likelihood or a probability density to any sequence Z of events in the model during the interval (0, T ) . This likelihood will depend on the parameters, on X,and on the number of calls in existence at the start 0 of the interval. If the likelihood L ( E ) can be factored into the form L = F * H , where F depends on the parameters and on statistics from the set S only, and H is independent of the parameters, then the set

2.

DESCKIPTION OF T H E MATHEMATICAL MODEL

165

S of statistics may be said to summarize all the information (in a sequence C) relevant to the parameters. If L can be so factored, then S is sufficient for the estimation of the parameters. T h e mathematical model to be used in this paper is described and discussed in Sections 2 and 3, respectively. Section 4 contains a summary of notations and abbreviations that have been used to simplify formulas. I n Section 11 we show that the original data we have allowed ourselves can be replaced by four statistics, which are sufficient for estimation. I n Section 12 and Sections 5-8 we discuss various estimators (for parameters of the model) based on these four statistics. T o determine the anticipated accuracy of these methods of measurement, we consider the statistics themselves as random variables whose distributions are to be deduced from the structure of the model. ,4 primary task is the determination of the joint distribution of the sufficient statistics. I n view of the sufficiency, this joint distribution tells us, in principle, just what it is possible to learn from a sample of length 7’ in this simple model. By analyzing this distribution we can derive results about the anticipated accuracy of measurements in the model. T h e joint distribution of the sufficient statistics is obtainable in principle from a generating function computed in Section 13, using methods exemplified in Section 10. This generating function is the basic result of this chapter. T h e implications of this result are summarized in Section 9, which quotes the generating function itself, and presents some statistical properties of the sufficient statistics in the form of four tables: (i) a table of generating functions obtainable from the basic one, (ii) a table of mean values, (iii) a table of variances and covariances, and (iv) a table of squared correlation coefficients. ( T h e coefficients are all nonnegative.)

$2. Description of the Mathematical Model Throughout the rest of the paper we follow a simplified form of the notational conventions of J. Riordan’s paper ( I ) . A summary of notations is given in Section 4.T h e model we study has the following properties :

166

5.

A SIMPLE TELEPHONE EXCHANGE h1ODEI

(i) Demands for service arise individually and collectively at random at the rate of a calls per second. 'Thus the chance of one or more demands in a small time interval A t is adt

+ o(df),

where o ( d t ) denotes a quantity of order smaller than A t . T h e chance of more than one demand in A t is of order smaller than A t . It can be shomn [Feller ( 2 ) ,p. 364 et seq.] that this description of the demand is equivalent to saying that the intervals between successive demands for service are all independent, with the negative exponential distribution I

- e-nt

This again is equivalent to saying that the call arrivals form a Poisson process (2); i.e., that for any time interval of length t the probability that exactly n demands are registered is e-"'(at)'' n!

T h u s the number of demands in the interval has a Poisson distribution with mean a t . (ii) l ' h e holding times of distinct conversations are independent variates having the negative exponential distribution 1

- e-Yf

where y is the reciprocal of the mean holding time h . T h i s description of the holding time distribution is the same as saying that the probability that a conversation, which is in progress, ends during a small time interval At is y d t 4 .(At),

without regard to the length of time that the conversation has lasted [Feller (2, p. 375)]. (iii) T h e system contains an infinite number of trunks. Thus, at no time will there be insufficient central office equipment to handle a demand for service, and no provision need be made for dealing with demands that cannot be satisfied. (iv) T h e operating system is described by the random process x t = number of trunks in use at time 1.

3.

167

DISCUSSION OF THE MODEL

T h e original work on this particular model for telephone traffic is given by Palm ( 6 ) ,and Palm's results have been reported by Feller (3) and Jensen (4). T h e results have been extended heuristically to arbitrary absolutely continuous holding-time distributions by Riordan ( I ) , following some ideas of Newland ( 5 ) suggested by S. 0. Rice (private communication to J. Riordan). Let pil(t) be the probability that there are j trunks busy at t if there were i busy at 0. And let P2(t,x) be the generating function of these probabilities, defined by a;

Pdf,x)

.'P,,(t).

= j=o

T h e n Palm has shown (6, pp. 56 et seq.) that P,(t, x)

=

[l

+ (x

- 1)e-Ytlz

exp ((x

-

l)ah(l

- e-yt)).

This is formula (12) of Riordan ( I ) with his g replaced by ecYi. I t can be verified that the random process xi is Markov; the limit of Pi(t, x) as t + GO is exp{(x I)ah), ~

so that the equilibrium distribution of the number of trunks in use is a Poisson distribution with mean b = ah. T h e shifted random variable [ x l - b] then has mean zero, and covariance function be-?''. For additional work on this model the reader is referred to F. W. Rabe (7), and to H. Stormer (8).

$3. Discussion of t h e Model Let us envisage the operation of the model we have described by considering the random variable xt equal to the number of trunks busy at time t. As a random function of time, x 1 jumps up one unit step each time a demand for service occurs, and it jumps down one unit step each time a conversation ends. If xt reaches zero, it stays there until there is another demand for service. If x t = n, the probability that a conversation ends in the next small time interval A t is nydt

Io ( d t ) ,

because the n conversations are mutually independent. A graph of a sample of x 1is shown in Fig. 1.

168

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

t TIME,

t

-

FIG. 1. A graph of x t .

T h e model we described departs from reality in several important ways, which it is well to discuss. First, the assumption that the number of trunks is infinite is not realistic, and is justified only by the mathematical complication which results when we assume the number of trunks to be finite. (See, however, Chapter 6.) I t can also be argued that unlimited office capacity is approached by offices with ample facilities and low calling rates, and therefore, in some practical cases at least, the model is not flagrantly inaccurate. Second, the choice of a constant calling rate for the model ignores the fact that in most offices the calling rate is periodic. Thus, the applicability of our results to offices whose calling rates undergo drastic changes in time is restricted to intervals during which the normally variable calling rate is nearly constant. Finally, although the assumption of a negative exponential distribution of holding time affords the model great mathematical convenience, it is doubtful whether in a realistic model the most likely holding time would have length zero, as it does in the present one.

$4. Summary of Notations for Chapter 5 a Poisson

calling rate h Mean holding time y = h-l, hangup rate per talking subscriber

5.

T H E AVERAGE TRAFFIC

169

b = ah, average number of busy trunks x t Number of trunks in use at t (0, T )Interval of observation (I'> 0) n ==xo, number of trunks in use at the start of observation A Number of calls arriving in (0,T ) H Number of hangups in (0, T )

K=A+H Z

=

lTx,dt 0

Z / T , average of xt over (0, T ) {pn}The (discrete) probability distribution of n, the number of trunks found busy at the start of observation

M

=

An estimator for a parameter is denoted by adding a cap (^) and a subscript. T h e subscripts differentiate among various estimators for the same parameter. We use

Also, it is convenient to use the following abbreviations: Y for y T and C for (1 - e c r ) / Y , where Y is the dimensionless ratio of observation time to mean holding time. T h e symbol E is used throughout the chapter to mean mathematical expectation.

$5. The Average Traffic We have adopted a model that depends on two parameters, the calling rate a, and the mean holding time h or its reciprocal y. Before searching for a set of statistics that is sufficient for the estimation of these parameters, let us consider the product ah = b. This product is important because, as we saw in Section 2, the equilibrium distribution of the number of trunks in use depends only on b, and not on a and h individually. Indeed, the equilibrium probability that n trunks are busy is e-bbn P

T

l

=

T

>

and the average number of busy trunks in equilibrium is just b.

170

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

T h e average number of trunks busy during a time interval of length T is

M

j

1 = xt d t ; T o

=-

i.e., the integral of the random function xi over the interval, divided by T . This suggests that for large time intervals, M will come close to the value of b, and can be used as an estimator of b. Since M is a random variable, the question arises, what are the statistical properties of M ? This question has been considered in the literature, the principal references being to F. W. Rabe (7), J. Riordan ( I ) , and the author (9). Riordan’s paper is a determination of the first four semi-invariants of the distribution of M during a period of statistical equilibrium, but without restriction on the assumed (absolutely continuous) distribution of holding time. I t follows from Riordan’s results that M converges to b in the mean, which is to say that lim E{I M - b T+m

I*}

= 0.

I t also follows that M is an unbiased estimator of b ; i.e., that E { M } = b, and that M is a consistent estimator of b, which means that lim Pr(/ M - b

T+CC

for each

c

I > €1= 0

> 0.

$6. Maximum Conditional Likelihood Estimators As shown in Section 11, the likelihood L, of an observed sequence, conditional on x,, , is defined by lnL, = A l n a

+ H l n y - y Z -aT.

According to the method of maximum likelihood, we should select, as estimators of a and y , respectively, quantities 2, and 9, which maximize the likelihood L, . Now a maximum of L, is also one of In L, , and vice versa. Therefore 8, and 9, are determined as roots of the following two equations, called the likelihood equations:

a

-1n L, = 0; aa

7.

PRACTICAL ESTIMATORS

171

T h e solutions to the likelihood equations are H yAc = -

A a, = -, A

Z‘

T

These are the maximum conditional likelihood estimators of a and y . T h e estimator dc is the number of requests for service in T divided by T ; this is intuitively satisfactory, since dc estimates a calling rate. Since maximum likelihood estimators of functions of parameters are generally the same functions of maximum likelihood estimators of the parameters, we see that A Z / H T is a maximum likelihood estimator of b.

$7. Practical Estimators Suggested by Maximizing the Likelihood L, Defined in Section 11 We obtain as likelihood equations

These may be written as

and H

+ air

~ + n / y*

T h e first of these shows the estimated calling rate as a pooled combination of the conditional estimate A / T , considered in the last section, and an estimate n/h based on the initial state. This latter estimate has the form calls in progress

mean holding time ’ and so is intuitively reasonable, since b/h = a. T h e second equation exhibits our estimate of y as a pooled combination of the conditional estimate H / Z and the ratio ajn. This ratio is acceptable as an estimate of y , since a/b = y and b = E{n} is the average value of n.

172

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

If we substitute, in the right-hand sides of these equations, the conditional estimators AIT, HIZ, and ZIH for a, y , and h, respectively, we obtain simple, intuitive estimators that include the influence of the initial state n, and show how it decreases with increasing T . T h u s

+

+A ZjH T H AZ/l'H Z nZ/H

+ +

estimates

a,

estimates y .

$8. O t h e r Estimators Additional estimators may be arrived at by intuitive considerations or by modifying certain maximum likelihood estimators. Some estimators so obtained are important because they use more of the information available in an observation than do the conditional estimators dc and p,, without being so complicated functionally that we cannot easily study their statistical properties. I t seems reasonable, and can be shown rigorously (Section 13), that for an interval (0, T ) of statistical equilibrium, the distribution of A and that of H are the same. Th u s we can argue that, for long time intervals, A and H will not be very different. This suggests using

A S H q=--2T

-

K 2T

as an estimator of a. This estimator does not involve y , and it uses not only information given by A , but also information supplied by arrivals occurring possibly before the start of observation. Similarly, since b = a / y , and M is a consistent and unbiased estimator of b, we may use K 1 22 = x,

A y1=

to estimate y , and its reciprocal to estimate h. Finally, since for long (0, T ) we have A H , we may try N

A

i

Y3='=-

L

l

1

4

as an estimator of y , and its reciprocal as an estimator of h.

9.

J O I N T DISTRIBUTION OF SUFFICIENT STATISTICS

173

49. The Joint Distribution of the Sufficient Statistics

T h e basic result of this chapter is a formula for the generating function E(znyxTwAuHe"}

(1)

for the joint distribution of the random variables n, x T , A ,H , and 2. This formula is derived in Section 13, by methods illustrated in Section 10. For an initial n distribution {p,}, the generating function is

I t is proved in Section 11 that the set of statistics {n,A , H , Z } is sufficient for estimation on the basis of the information assumed, which was described in Section 1. Thus the generating function (2) specifies, at least in principle, what can be discovered about the process from an observation interval (0, T ) , for which xo has the distribution {p,}. All the results summarized in this section are consequences of (2). By substitution, and by either letting the appropriate power series variables approach 1, or letting i -+ 0, or both, we can obtain from (2) the generating function of any combination of linear functions of the basic random variables n, x T , A , H , and 2. Some of the generating functions thereby obtained are listed in Table 5.2, in which the entries all refer to an interval (0, T ) of equilibrium. Since, for equilibrium (0, T ) , the generating functions are all exponentials, it has been convenient to make Table 5.2 a table of logarithms of expectations, with random variables X on the left, and functions In E { X ) on the right. C as a function of Y is plotted in Fig. 2. Entry 1 of Table 5.2 is actually the cumulant generating function of 2 for equilibrium (0, T ) ;similarly, entry 2 is that of M , and depends only on the average traffic b and the ratio r . T h e form of the general cumulant of M is

174

5.

A S I M P L E TELEPHONE EXCHANGE MODEL

TABLE 5.2

X

In E { X )

2 . e-iM

5 . uKe-c*

b [(l

-

ru i+l) [ e - ( c t r ) - 11 - Y

9.

J O I N T DISTRIBUTION OF SUFFICIENT STATISTICS

I75

This result coincides with a special case (exponential holding time) of a conjecture of Riordan ( I ) . This conjecture was first established (for a general holding-time distribution) in unpublished work of S. P. Lloyd. T h e cumulant generating function permits investigation of asymptotic properties. We prove in Section 10 that the standardized variable 'u = (yT/2b)'/Z(M- b) =

(Y/2b)'/2(M - b)

is asymptotically normally distributed with mean 0 and variance 1. From entry 3 of Table 5.2 it can be seen that K is distributed as 2w v, where w and v follow independent Poisson distributions with the respective parameters a T ( 1 - C ) and 2aTC. T h e probability that K = n for an interval of equilibrium is

+

( ~ u T C ) + ~(aT ~ - aTC)j , (72 - 2j)! I!

r, = exp{aT(C - l)}

< <

where the sum is over j's for which 0 2j n. T h e estimator a", for a is equal to K / 2 T , and has mean and variance given by E(6,) = a, var(2,)

=

a

-( 2 - C ) . 2T

T h e distribution of 6, is given by

<

2Tx. the summation being over n From (2) one can obtain, by substitution of the stationary xo distribution for {p,}, and subsequent differentiation, the means, variances, covariances, and correlation coefficients of the sufficient statistics, for equilibrium intervals (0, T ) .I t has been convenient to display these in three triangular arrays, the first consisting of expectations of products, the second comprising the variances and covariances, and the third exhibiting, for simplicity, the squared correlation coefficients, since the correlation coefficients are never negative for these random variables. Y ) is E { X Y ) for I n Table 5.3, the entry with coordinates (X,

176

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

TABLE 5.3 EIX, Yl 1

1

n

1

b

b(l

n

A

+ b)

A

H

aT

aT

aT(C a T ( l + a T ) aT(1 - C baT

aT(1

H

Z

K

2a T

+ b) +

+ + uT(2- C + uT) 2aT(2- C + 2 a T )

aT(C 26) uT) aT(2- C uT)

+ uT)

K Z

bT bT(C bT(1- C

+ b)

+ uT) + uT) 2bT(1 C + u T ) bTh(2 - 2 c + a T ) bT(1 - C -

equilibrium (0, T ) . All three tables are expressed in terms of a , b, T , h, Y, and C, the last of which is plotted in Fig. 2. T h e variances and covariances of the sufficient statistics are listed in Table 5.4;the entries are of the form cov{X, Y ) = E / X Y } - E{X}B{Y}. TABLE 5.4 COVlX,

n It

A

H K Z

b

Y}

A

H

K

z

0

aTC aT(1 - C )

aTC aT(2 - C )

bT C bT(l - C )

UT

aT

aT(2

-

C)

2aT(2 - C )

bT(1 - C ) 2bT(1

-

C)

ZbTh(1 - C )

Table 5.5, finally, lists the squared correlation coefficients; i.e., the quantities COV2{X, Y}

pyx-, Y)= var(X) var(Y)

*

For any time interval (0, T ) , A has a Poisson distribution with parameter aT, so that TCi, has, also. Therefore the distribution of dc is given by

10.

THE DISTRIBUTIONS OF

AND

177

h'f

TABLE 5.5

P2(X Y) A

n

H

K

Z

1

2 - c 2

1-c 2

n

A H

1

K

z

1 - c 2 - c 1

where the summation is over n

< xT. Evidently,

E(2,)

= a,

and var(6,)

U

=-

T

,

so that dc is an unbiased and consistent estimator of a. We now compare the variances of estimators dc and d,. From Table 5.4 we have

so that aAl is a better estimator of a for any T

variance is less.

$10. The Distributions of Z and M Since we have defined Z

=

j T x t dt, 0

> 0, in the sense that its

178

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

we can regard Z as the result of growth whose rate is given by the random step function x,;when x, = n, Z is growing at rate n. An idea similar to this is used by Kosten, Manning, and Garwood (ZO), and by Kosten alone (ZZ). Now the Z(t)process by itself is not Markov, but it can be seen that the two-dimensional variable (x,,Z(t)} itself is Markov. Let Fn(z, t ) be the probability that x, = n and Z ( t ) z. Since the two-dimensional process is Markov, we can derive infinitesimal relations for Fn(z, t ) by considering the possible changes in the system during a small interval of time A t . If x, = n, then the probability is [ l - y n At - o(At)] that there is neither a request for service nor a hangup during A t following t, and that Z(t A t ) = Z ( t ) + n A t . Therefore the conditional probability , n and Z(t A t ) z , given that no changes occurred that x , + ~= in A t , is F,(z - n A t , t ) .

<

+

+

<

+

+

For x i = ( n + l), the probability is y (n 1) A t o(At) that one conversation will end during A t following t. T h e increment to Z ( t ) during A t will depend on the length x of the interval from t to the point within A t at which the conversation ended. T h e increment has n ( d t - x) = x n d t , as can be verified magnitude (n 1). from Fig. 3, in which the shaded area is the increment. Since x is n At distributed uniformly between 0 and A t , the increment x is distributed uniformly between n At and (n 1) A t . Therefore the

+

+

+

+

t

t +At FIG. 3.

Increment to 2 in A t .

+

10.

THE DISTRIBUTION OF

zA N D

conditional probability that xt+Af= n and Z(t one conversation ended in A t , is

I79

k!

+ A t ) < z , given that

By a similar argument it can be shown that the probability that o ( A t ) , and that the one request for service arrives in A t is a d t = n and Z(t At) z , given that conditional probability that one request arrived during A t , is

+ +

<

Define Fn(z, t ) to be identically 0 for negative n. Adding u p the probabilities of mutually exclusive events, we obtain the following infinitesimal relations for Fn(z,t ) :

Fn(z,t

+ At)

=

y(n

+ 1 ) J”

+ a J”

(n+l)dt nA t

nAt

+ F,(z

(n-l)dt

Fn+,(z

Fn-,(z

-

- u,

t ) du

u, t ) du

- n A t , t)[l - dt(yn

+ u ) ] + o(At),

for any n.

Expanding the penultimate term of the right side in powers of n A t , and the left side in powers of A t , we divide by A t , and take the limit as A t approaches 0. Now 1 lim A t 4 At

J”n d t

(n+l)At

Fn+,(z - u, t ) du = Fn+l(z,t ) .

Thus, omitting functional dependence on z and t for convenience, we reach the following partial differential equations for F J z , t ) :

a

at F,, = y(n

-

+ l)Fn+l + aFn-,

-

n

B az Fn

-

[yn

+ u]Fn ,

for any n.

(3) Since Z(0) = 0, we impose the following boundary conditions:

F,(O, t )

=0

Fn(z,0) = p ,

F,(z, 0)

=0

> 0 and for z 2 0, for z < 0,

for

n

t

> 0, (4)

where the sequence {pn} forms an arbitrary x,,distribution that is zero for negative n.

180

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

T o transform the equations, we introduce the Laplace-Stieltjes integrals

in which the Stieltjes integration is understood always to be on the variable z. We note that m

j

O-

and that P)n([,

e - c z ~ n ( ~t ), d z

t ) = FJO, t )

I

= - yn((, t>,

5

a F J ~ , t ) dx. + j" e-cz -8.z 0

Applying now the Laplace-Stieltjes transformation to (3), we obtain aVn at =~

+

( n

1)~n+l

+

a~fi-1- n5Vn

+

fi@n(O,

t ) - [P

+a l ~ n i

(5)

in which we have left out functional dependence on 5 and t where it is unnecessary. By the boundary conditions (4), n, 0 and t > 0; in ( 5 ) we may therefore omit this term in the region t > 0. Let y be defined by m

V ( Y , 5, 4 = Z Y % ( 5 , n=O

The series is absolutely convergent for 1 y 1

1 ~ ~ (t )5I < , 1,

4.

< 1, since

for all n.

T h e following partial differential equation for y is obtained from ( 5 ) :

If we integrate out the information about 2 by letting 5 approach 0 in this equation, we obtain the equation derived by Palm (6) for the generating function of x t . Therefore our equation has a solution of the same form as Palm's. For the boundary conditions (4),this solution is

10.

THE DISTRIBUTION OF

AND

h'f

181

Actually y contains more information than we want since it yields the joint distribution of xT and 2. We may integrate out the former variable by lettingy approach 1 in (7). Then,

is the Laplace transform of the distribution of 2 for an arbitrary x, distribution {p,}. This result is not restricted to an interval (0, T)of statistical equilibrium; however, if the sequence (p,} does form the stationary distribution discussed in Section 2, then

is the Laplace transform of the distribution of 2 for an interval (0, T ) of statistical equilibrium. T h e Laplace transform is a moment generating function expressible as

where m, is the nth ordinary moment of 2. Differentiation of (9) then gives a recurrence relation for the moments upon equating powers of (-c). Thus

and

+ 3 p ( n - l)mn-l - n(n - l)(n 2)mn-z ay2Tmn - (2a + ayT)nm,-, + 2ane-YT(m + T)"-l

y3mn+, - 3y2nmn =

+ n(n

where (m

-

l)nTe-YT(m

-

+ T)n-2 - n(n - l)(n

-

2)bTe-yT(m

+ T ) , is the usual symbolic abbreviation of 2( ) 3=0

+ T)n-3, (10)

182

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

From the recurrence (10) it is easily verified that m,

=

bT,

2bT + __ [l Y

m2 = (bT)2

-

C],

from which it follows that the variance of 2 is var{Z)

2bT

= -[l

Y

+

-

C].

Since is the Laplace-Stieltjes transform of the distribution of 2 over an interval of equilibrium, In is the cumulant generating function, and has the following simple form:

+

M is a linear function of Z, so we may obtain the cumulant generating function of M in accordance with CramCr (12, p. 187). This function is

and depends only on b and r . T h e mean and variance of M for an interval of equilibrium are respectively given by E { M } = b, 26 var{M} = [l r

-

C],

with

C

1

-

e-'

= ___

r

'

results which were first proved in Riordan (1).A normal distribution having the mean and variance of M has the cumulant generating function

which is to be compared to (11). Since var { M } goes to 0 as T approaches co,we may expect that a suitably normalized version of

11.

PROOF THAT

(n, A , H , 2 ) IS

SUFFICIENT

2 will be asymptotically normally distributed as T approaches

183 00.

T h e cumulant generating function of the normalized variable ( 2 b l ~ T ) - l / ~( Zb T ) is 5(2/aT)'/2

+2

exp(--t;(2b/r)-ll2 5(2b/r)-""+

-

r}

Y

-

1

1.

which approaches c2/2 as T -+00. I t follows that the normalized variable is asymptotically normal with mean 0 and variance 1, and ~ b( )Mis also asymptotically normal (0, 1). that ( ~ / 2 b ) ~ /-

$11. Proof that ( n , A, H , Z) I s Sufficient We observe the system during the interval (0, T ) , and gather the information specified in Section 1 , and summarized in Table 5.1. From this information we can extract four sets of numbers, described as follows:

S, T h e set of complete observed interarrival times, not counting the interval from the last arrival until T S , T h e set of complete observed holding times S, T h e set of hangup times for calls of category (i) S , T h e set of calling times for calls of category (iv) I n addition, our data enable us to determine the following numbers:

n T h e number x,,of calls found at the start of observation K T h e number of calls of category (iii); i.e., of calls that last throughout the interval (0, T ) x T h e length of the time interval between the last observed arrival and T I n view of the negative exponential distributions which have been assumed for the interarrival times and the holding times, and in view of the assumptions of independence, we can write the likelihood of an observed sequence of events as

184

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

It is easily seen that the summations and the two initial terms can be combined into a single term, so that we obtain

+

1nL = lnp,rL A In a

+ H In y

-y

Z

- aT.

This shows that L depends only on the statistics n, A , H , and 2; it follows that the information we have assumed can be replaced by the set of statistics {n,A, H , Z}, and that these are sufficient for estimation based on that information. T h e likelihood is sometimes defined without reference to the initial state, by leaving the factor p , out of the expression for L. Strictly speaking, this omission defines the conditional likelihood for the observed sequence, conditional on starting at n. We use the notation L L c -

Pn

A definition of likelihood as L, has been used by Moran (13). Clearly lnL,

=

A In a

+ H l n y - y 2 - aT.

$12. Unconditional Maximum Likelihood Estimators T h e definition of likelihood as L leads to complicated results which are of theoretical rather than practical interest. For this reason these results have been relegated to this penultimate section. T h e results of setting ajay I n L and ajaa In L equal to zero lead, respectively, to the likelihood equations

9=

H -n

& = -K -AM. T Y

-

M

+ {(H - n 22

-

M)2

+ 4MK}lI2,

13.

THE JOINT DISTRIBUTION OF X T ,

n, A, H ,

AND

z

185

These are the unconditional maximum likelihood estimators for y and a. Although a"c depended only on A and T , and 9, only on H and 2, the unconditional estimators depend on all of n, A, H , 2, and T . We may obtain a maximum unconditional likelihood estimator for b as well, either by considering L to be a function of b and y , or from general properties of maximum likelihood estimators. Since b = a/y, we expect that 6 = 219, as can be verified by an argument similar to that used above for a" and 9. T h e estimators 2, b, and 9 obtained in this section may turn out to be useful in practice, but their complicated dependence on the sufficient statistics n, A , H , and 2 makes a study of their statistical properties difficult. As a first step along such a study, we have derived the generating function of the joint distribution of the sufficient statistics in Section 13. T h e greater simplicity of the conditional estimators of Section 6 makes it possible to study their statistical properties. This fact gives them a practical ascendancy over the unconditional estimators, even though the latter may be more efficient statistically by dint of using all the information available in an observation.

$13. The Joint Distribution of x T ,n, A, H , and Z By methods already used in Section 10 one can obtain a generating function for the joint distribution of all the random variables n, x T , A , H , and 2. Let @

=

E(yxrwAuHe-cZ}.

Then @ satisfies the differential equation

whose solution has the form Q, =

+

R([
- yu]e-(c+yJt)

x exp

aw[(y

(

+ yy

-

p][1

(5 + Y Y

- e-(<+y)t]

aywut + __-at), 5+Y

where the function R is determined by the initial distribution (p,) through the relation

186

5.

A SIMPLE TELEPHONE EXCHANGE MODEL

From these results it follows that the generating function E(znyXTwAu

is given by

x exp

u ) [ l_ ( uw(5y + yy_(5 +y_YY -

- e-(-5+y)T]

_

~

+ a5uywuT +Y

--~ ~

“ T ).

If {pn}forms the stationary distribution, this reduces to

uw(5y

f-

+ yy +____-___ yu)[l uywuT + ___ - u T] . ( 5 + Y)*

- e--(i+y)T]

5+Y

If, in this last expression, we let y approach 1, z approach 1, and u approach 1, we obtain

as the generating function E{wAeriZ}for an interval of equilibrium. Alternately, if instead we let y approach 1, x approach 1, and w approach 1, we obtain (13) with u substituted for w ;this implies the not surprising result that, for an interval of equilibrium, the twodimensional variables { A ,Z } and ( H , Z } have the same distribution. From this and (13) it follows that for equilibrium (0, T ) , (i) A and H both have a Poisson distribution with mean aT, and (ii) the estimators hc and h2 have the same distribution. REFERENCES

I . J. Riordan, Telephone Traffic Time Averages, Bell System Tech.

r.

30, 11291144 (1951). 2. W. Feller, “An Introduction to Probability Theory and its Applications.” Wiley, New York, 1950. 3. W. Feller, On the Theory of Stochastic Processes with Particular Reference to Applications. Proc. 1st Berkeley Symp. Math. Stat. and Probability pp. 403-432. Univ. of California Press, Berkeley, California, 1949.

REFERENCES

187

4 . A. Jensen, An Elucidation of Erlang’s Statistical Works Through the Theory of Stochastic Processes. In “The Life and Works of A. K. Erlang” pp. 23-100. Trans. Danish Acad. Sci., Copenhagen, 1948. 5. W. F. Newland, A Method of Approach and Solution to Some Fundamental Traffic Problems, P.O.E.E. Journal 25, 119-131 (1932-1933). 6. C. Palm, Intensitatsschwankungen im Fernsprechverkehr, Ericsson Tech. 44 (1943). 7. F. W. Rabe, Variations of Telephone Traffic, Elec. Commun. 26, 243-248 (1949). 8. H. Stormer, Anwendung des Stichprobenverfahrens beim Beurteilen von Fernsprechverkehrsmessungen, Arch. Elektrischen Ubertragung 8, 439-436 (1954). 9. V. E. BeneS, A Sufficient Set of Statistics for a Simple Telephone Exchange Model, Bell System Tech. J. 36, 939-964 (1957). 10. L. Kosten, J. R. Manning, and F. Garwood, On the Accuracy of Measurements of Probabilities of Loss in Telephone Systems, Roy. Stat. SOC.11, 54-67 (1949). 11. L. Kosten, On the Accuracy of Measurements of Probabilities of Delay and of Expected Times of Delay in Telecommunication Systems, Appl. Sci. Research B2, 108-130 and 401-415 (1952). 12. H. Cramtr, “Mathematical Methods of Statistics.” Princeton Univ. Press, Princeton, New Jersey, 1946. 13. P. A. P. Moran, Estimation Methods for Evolutive Processes, J . Roy. Stat. SOC. 13, 141-146 (1951).

CHAPTER

6

The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement

Erlang’s classical model for telephone traffic in a loss system is considered: trunks, calls arriving in a Poisson process, and negative exponential holding times; calls that cannot be served at once are dismissed without retrials. Let x t be the number of trunks in use at t . An explicit formula for the covariance R(.) of x t in terms of the characteristic values of the transition matrix of the Markov process xt is obtained. Also, R(.) is expressed purely in terms of constants and the “recovery” function, i.e., the transition probability Prjxl = c 1 xo = c). I?(.) is accurately approximated by I?(0)erlt,with y1 the largest negative characteristic value, itself well approximated (underestimated) by --E{xt}/R(0).Exact and approximate formulas for sampling error in traffic measurement are deduced from these results. c

$1. Introduction A theoretical study of sampling fluctuations in telephone traffic measurements is useful both in designing procedures for measuring traffic loads and in interpreting field observations. Hayward ( I ) and Palm (2) have given an approximate formula for the sampling error 188

1.

INTRODUCTION

189

incurred when observations of the numbers of calls in existence are made at fixed intervals of time. Their formula has the disadvantage that it is derived for a probabilistic model (of the traffic) in which there is an infinite number of available trunks. (Cf. Chapter 5.) Thus there is no limit to the number of calls that can be in progress at one time, and no congestion. Two important parameters, the number c of trunks in the group, and the probability p , of loss, are left out of account. For this reason the practical application of this model is usually restricted to large groups of trunks that are lightly loaded. I n this chapter we derive and study the covariance function of the simplest stochastic model of a finite group of c trunks. T h e sampling error in traffic measurements can be calculated exactly from the covariance. We find formulas for the magnitudes of fluctuations of observed traffic for both periodic and continuous observation. T h e

a, OFFERED

TRAFFIC IN ERLANGS

FIG. 1. Probability p , of loss.

190

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

exact formulas lead to simple approximations similar to Hayward’s, which take account of the number of trunks. Our results are summarized and discussed in Section 2. We shall use A. K. Erlang’s classical probabilistic model for a group of trunks, described as follows: (i) Holding times of trunks are mutually independent, each with a negative exponential distribution. Time is measured in units of mean holding time. (ii) Epochs at which calls arrive form a Poisson process of intensity a > 0, independently of the holding times. T h e offered load is then a erlangs. (iii) There are c < GO trunks; calls that find all c of these trunks busy are “lost,” and are cleared from the system. These assumptions determine a Markov stochastic process x t , -a < t < GO, the number of trunks in use at time t. xiis a random step function fluctuating in unit steps between 0 and c. As is well known, x l has stationary probabilities { p , , n = 0, 1, ..., c} given by the (first) Erlang distribution Pn =

an n!

5 Zk?

k=O

=

equilibrium probability that n trunks are busy.

(1)

With this choice of absolute probabilities, x1 is a strictly stationary process, whose mean and variance are, respectively, 1121 = a(1

-PA

az = m1 - ap,(c - ml).

T h e probability p , of loss is shown in Fig. 1, the fractional occupancy c-lm, in Fig. 2, and the variance o2 in Fig. 3.

$2. Discussion, Summary, and Conclusions T h e covariance R(t, s) between samples xi, x, of the stochastic process under study is the average of the product of x1and x, , minus the product of the averages: R(t, s) = E{x,x,} - E{x,}E{x,}.

2.

DISCUSSION, SUMMARY, AND CONCLUSIONS

a, FIG.2.

OFFERED TRAFFIC

191

IN ERLANGS

Fractional occupancy ml / c.

Since xL is a stationary real process, we have R(t, s) = R(l t - s 1). T h e function R(.) is called the covariance function of the process xt. It can be written as R(t) = lim E{x,tux,} - E{x,+,}E{x,) u3m

where {prr8}are the stationary (or equilibrium) probabilities given by (11, and Prjx, = n \ xo = m} denotes the transition probability that n trunks are busy at time t if m were busy at time 0. T h e function R(.) expresses the average dependence or correlation between samples of x,?taken at times t apart.

192

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

a, OFFERED FIG. 3.

TRAFFIC IN ERLANCS

Equilibrium variance.

T h e principal practical use of the covariance function A!(*)in the theory of telephone traffic is in computing theoretical estimates of sampling error incurred in traffic load measurements. Two methods of measuring traffic, the switch count and the time average, are considered 37 ’ here. I n the switch count, n observations { x l , ..., x, , xi = x.

2.

193

DISCUSSION, SUMMARY, AND CONCLUSIONS

j = 1, ..., n} of the random process are made at intervals the average

T

apart;

is then used as an estimate of the carried load m, = a - u p , . This method is important economically because it is cheaper to scan trunk groups periodically than to observe them continuously. T h e number T is the scan interval, and the number S, = x1 x, is called the (total) number of paths in service, in n observations. Table 6.1 lists

+ +

TABLE 6.1

HOLDING TIMES, SCAN INTERVALS, AND VALUESOF Scan interval (seconds)

T

Ratio T of scan interval to holding time

Typical h o1ding time (seconds)

U.S.A.

Europe

U.S.A.

Europe

Local call

100-200

100

36

Long distance call

200-600

100

36

1to+ $to*

+to* l t o L

4 to 2

Type of call

Originating register holding time

No. 5 marker holding time

10-15

10or 100

36

1t o 3 or 10 to 7

0.25-1 .O

10

-

-

6

20

-

actual mean holding times, scan intervals used, and resulting values of T for various types of calls. T h e variance of n-IS, is expressible in terms of the covariance I?(.) as

I n the time average, the continuously recorded sample average M ( T ) = T-l

T 0

xt

dt

194

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

is used to estimate the carried load. T h e variance of this estimate is Var{M( T ) } = 2T-2

T 0

( T - t)R(t)dt.

(3b)

Thus the mean square error of both these methods of traffic measurement can be calculated theoretically if the covariance R( .) is known. I n formula (2) the covariance function is expressed in terms of the stationary probabilities {p,) given by the Erlang distribution, and the transition probabilities p,,(t)

= Pr{x, = n

1 x,,

= .I>.

I n the theory of telephone traffic, the particular transition probability

p c c ( t ) = Pr{x,

=c

1 xo = c }

has been singled out [by the author (3) following Kosten (41 as a suitable “recovery” or “relaxation” function that is characteristic of the dynamic behavior of the Markov process xl in point of the undesirable “all trunks busy” condition. We shall show that a much more cogent reason than this can be adduced to support the importance of the recovery function to traffic theory: T h e covariance function R(.) can be expressed entirely in terms of the recovery function and the offered load a. I n other transition probabilities appearing in words, a single one of the (c + formula (2) suffices for determining the covariance function, and this one is the recovery function p,,(.). This fact is a theoretical justification of the intuitive view that the recovery function is important, for now the variances of n-IS, and of M ( T ) are expressible using only the recovery function. We next give a summary of the contents of the remaining sections; this is followed by an account of specific results and conclusions. An exact formula for the covariance R(.) is stated and discussed in Section 3 , and derived in Section 7. T h e formula readily yields a rigorous upper bound which appears to give a close approximation to R(.) itself. In Section 4 the recovery function pee(-) is given, and it is shown how the covariance may be expressed in terms of the recovery function by a convolution integral. T h e variance of n-lS, is studied in Section 5 ; an exact formula, and an approximating upper

2.

195

DISCUSSION, SUMMARY, AND CONCLUSIONS

bound for I?(-),are both obtained. T h e variance of the time average M ( T ) is considered in Section 6; again, an exact formula and an approximating upper bound are found. T h e covariance function I?(.) is positive and is bounded from above and closely approximated by a single exponential function, 0

< R ( t ) < u2erit,

u2 = H(O),

r1 < 0.

Here = equilibrium variance of x t =

(load carried) - (load lost) (average number of idle trunks),

and the reciprocal time constant rl in the exponent is the dominant* characteristic value of the “rate” or “transition” matrix of the differential equations satisfied by the transition probabilities. Alternately, rl is related to the root of least magnitude of a PoissonCharlier polynomial. T h e root rl is shown as a function of offered traffic a for c = 1, ..., 8 in Fig. 4, and is tabulated in Table 6.2. A lower bound for r l , depending only on the mean and variance of x l , is derived in Section 8 by making use of the fact that the matrix TABLE 6.2 NEGATIVE OF DOMINANT CHARACTERISTIC

X‘ALUE Y I

~

a

N = 4

N = 5

N = 6

N = 7

N = 8

1 2 3 4 5 6 7 8 9 10

1.043967 1.249464 1.582363 2.000000 2.477548 3.000000 3.557618 4.143703 4.753426 5.383178

1.011448 1.112166 1.326321 1.629624 2.000000 2.422137 2.885474 3.382497 3.907677 4.456828

1.002421 1.045044 I. 172257 1.383389 1.663799 2.000000 2.38 1 627 2.800900 3.251918 3.730121

1.000421 1.015806 1.084025 1.222707 1.427870 1.689991 2.000000 2.350437 2.735363 3. I50052

1 .OW62 1.004800 1.037229 1.121762 1.265214 1.463798 1.710891 2.000000 2.325514 2.682770

* I.e., that

of least magnitude (among the nonzero characteristic values).

196

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

- rl

FIG.4. Negative of the root 1, ..., 8.

yI

of smallest magnitude as a function of load a for

c =

of the differential equations for the transition probabilities is symmetrizable. For low values of offered traffic per trunk, i.e., a/c < 1, this bound can be used to approximate rI . I n any case, the bound is a convenient starting place for the use of Newton's method. T h e bound is the ratio -ml/u2, which satisfies the inequality

2. DISCUSSION,

SUMMARY, AND CONCLUSIONS

197

with m,

= equilibrium =

load carried

mean of x t = a( 1 - p,),

a2 = equiiibrium variance of xt =

(load carried) - (load lost) (average number of idle trunks).

-ml/a2 is illustrated in Fig. 5. T h e approximation rl By the “infinite trunk” model we shall henceforth mean the stochastic model for telephone traffic (considered in Chapter 5 ) determined by all the same assumptions that we made in Section 1 of this chapter, except that c = co;i.e., an unlimited number of trunks is postulated. Riordan ( 5 ) and the author (6, 7) have considered this model; Hayward ( I ) based his sampling error formula on it.

FIG.5.

Illustration of the approximation rl

-rnl/ua.

198

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

It is widely believed that the “infinite trunk” model is applicable to large groups of lightly loaded trunks. Such a belief is gratuitous until comparisons with a model having a finite number of trunks are made. Studying the covariance function of the simple finite trunk group enables us to make some of the needed comparisons; e.g., the variances of n-lS, and M ( T ) in the two models are of particular interest. Knowledge of the covariance I?(.), however, is also relevant to the other three cases, to which engineers are loath to apply the “infinite trunk’’ model, viz.: (i) large groups of heavily loaded trunks, (ii) small groups of lightly loaded trunks, (iii) small groups of heavily loaded trunks. T h e variance of n-IS, is bounded from above and approximated by the formula Var{n-lS,}

< n-W

I

ctnh h -

(4)

where n is the number of observations, and T?, -2 = -I( scan interval) (dominant characteristic value). 2

T h e exact formula for the variance of n-lS, in the “infinite trunk” model is

T h e upper bound (4)for the finite group is compared with the exact formula ( 5 ) for the “infinite trunk” model in Fig. 6, which shows each formula as a function of the scan interval T for various n, for a = 20 erlangs offered to 20 trunks. T h e curves suggest that the upper bound for Var{n-lS,} for c < 00 is consistently less than the corresponding variance in the “infinite trunk” model. As might be expected, increasing the scan interval 7 improves accuracy for the same number of observations. This is because the covariance function is positive, and monotone in [ t I.

2.

199

DISCUSSION, SUMMARY, AND CONCLUSIONS

10 8

6 4

2

1.0

0.8 0.6 0.4

0.2

0.10

0.08

0.08 (DOMINANT CHARACTERISTIC VALUE )

0.04

0.01 QOt

0.02

0.04 0.06 0.1 0.2 0.4 0.8 1.0 2 SCAN INTERVAL, T, IN MEAN HOLDING-TIMES

4

8 10

FIG. 6 . Comparison of variance of S,/n for finite and infinite trunk models.

T h e variance of M( T ) is bounded from above and approximated by

where T is the length of the time interval of continuous observation, and 0 2 and rl are, as before, the variance of x t and the dominant characteristic value, respectively. T h e exact formula for the variance of M ( T ) in the “infinite trunk” model is 2a

ecT - 1 T2

+T

(7)

200

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

T IN MEAN HOLDINC-TIMES FIG. 7.

Comparison of variance of M ( T ) for finite and infinite trunk models.

2.

DISCUSSION, SUMMARY, AND CONCLUSIONS

20 1

Since rl < -1, and u2 is always less than a if c < CO, the “infinite trunk” model overestimates the variance of M ( T ) if applied to a finite group. This conclusion is illustrated in Fig. 7, which shows the formulas (6) and (7) for a load of 20 erlangs offered to 20 trunks. For an observation time of 10 mean holding times the “infinite trunk” formula (7) applied here would overestimate the variance by about 500 per cent. This is about as extreme a case as would occur in practice. Figure 7 also depicts a “mixed” formula obtained by replacing a by ~2 in the “infinite trunk” formula (6); for 10 mean holding times the “mixed” formula overestimates the variance by only about 100 per cent. Th u s most of the discrepancy is due to the difference between u2 and a. Our conclusions are set down in the following list:

(1) T h e average dynamic behavior of the process x l , as described by the covariance function I?(-), can be adequately determined from the dominant characteristic value rl and the variance u2. (2) T h e same parameters, rl and u2, suffice to give simple approximating upper bounds for the sampling error incurred in both periodic and continuous observation of x l . These bounds depend on the size c of the trunk group. (3) I n terms of rl and u2 it is possible to check the applicability, for theoretical estimates of sampling error, of the “infinite trunk” model which assumes c = 00. (4)T h e “infinite trunk” model, applied to finite trunk groups, consistently and often grossly overestimates the sampling error. T h e overestimation occurs largely because 02 is always less, and for heavy traffic is much less, than a, the (Poisson) variance of x t in the “infinite trunk” model. ( 5 ) I n terms of rl and uz it is possible to design sampling procedures for traffic measurement that depend explicitly on the number c of trunks in the group. By these methods, a given accuracy can be obtained with less observation, and thus at lower cost, than the “infinite trunk” model would require. (6) Hence for finite groups of trunks traffic, sampling procedures that are based on the “infinite trunk’’ model tend to be wasteful, particularly for heavy traffic. T h e parameters rl and u2 provide a systematic way of tailoring the measurement procedure to the number of trunks in the group.

202

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

$3. The Covariance Function T o state the formula R ( . ) we need the “sigma” functions* defined [see Riordan (9)] as am

a0(m) = m! ’

with m (but not k) a nonnegative integer. These functions are connected with the Poisson-Charlier polynomials

by the relation

uk(m) = ( -a1/2)m(m!)-1/2p,(

-k),

[See Szego (10, p. 33).] For fixed c and a, let r l , r p , ..., r, be (in order of increasing magnitude) the c zeros in the variable s of the polynomial O,+~(C). I n Section 7 the covariance is shown to be given by (the exact formula)

where p , is the probability of loss. It has been shown? that the zeros r j are all real, negative, and distinct; all are less than -1, and consecutive pairs are separated by at least unity. Figure 8 shows these roots for c = 1, 2, 3 as function of a. Now r j is always negative, and the terms of the produet satisfy 1 1 -___ rj

-

ri

> 0;

(9)

* The u notation is copied from unpublished work of H. Nyquist. The functions themselves were introduced into traffic theory by Palm (8). t The earliest reference appears to be Haantjes (11) in 1938. See also Ledermann and Reuter (12).

3.

THE COVARIANCE FUNCTION

203

FIG. 8. Roots of the first three o-functions.

hence the sum in (8) has all terms negative, so that R(t) > 0,

all t.

T h e correlation between successive samples is thus always positive. I t is obvious from (8) that

204

6.

COVARIANCE FUNCTION

OF A SIMPLE TRUNK GROUP

Since r1 is the root closest to zero, the value of (8) is only increased if the r j in the exponents of (8) are replaced by r1 . Using (9) and (lo), we conclude that 0

< u2erit

-

R(t) = [ ( t ) ,

where

(11)

and

< (0.3933)a2p&. T h e approximation R(t) u2erlt is illustrated in Figs. 9 and 10. I t appears to be fairly accurate, especially for light loads. T h e upper bound u2erlt for R(t) should be compared with the rigorous formula [see Riordan (5) and Beneg (7)] R(t) = a c t ,

which holds for the “infinite trunk” model. I n this model the equilibrium distribution of occupancy is Poisson, so that R(0) = 2

= Var{x,} = E{x,} = a,

and the “time constant” of the exponential is unity, since time is measured in units of mean holding time. T h e difference between the “infinite trunk” model and the “finite trunk” model in point of the covariance can be understood by considering the effect of congestion, which is present in the latter. Congestion affects the upper bound formula most directly through the value of the variance u2. It is obvious intuitively, and borne out in Fig. 3, that as a increases u2 must eventually decrease to zero. This behavior is not mimicked by the “infinite trunk’’ model, for which u2 = a. T h e finitude of c, i.e., congestion, affects the bound u2ePli} in two ways: (a) the “time constant” is not unity but the smaller number -(r1)-l, so that the rate at which dependence between samples of x t decreases (as a function of the interval between samples) is larger than in the “infinite trunk” model; this “time constant” decreases as

3.

205

THE COVARIANCE FUNCTION

0.80

0.75 0.70

0.65 0.60 0.55 0.50

0.45 h

(I:

0.40 0.35

0.30 0.25 0.20

0.1 5 0.10

0.05

0 0

0.1

0.2

0.3

-

FIG.9. T h e covariance R(t) for c mate formula R(t) uzerl*.

0.4

0.5

t

0.6

5 trunks,

a

0.7 0.0 0 . 9

1.0

10 erlangs, with the approxi-

the traffic a increases, because, as illustrated by Fig. 4, y1 is a monotone decreasing function of a ; (b) the value of R(0)(= uz) is not a but the generally much smaller number u i = a(1 =

- p c ) [l - u p ,

a[ 1- p,(l

+c

-

C

--- - l j ] ,

ia - up,

a

+

U P d .

206

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

3

t

-

FIG. 10. T h e covariance R(t) for c = 8 trunks, a = mate formula R(t) &?It.

4 erlangs, with the approxi-

T h e last form shows that u2 < a for all a and c. I n fact, it is obvious intuitively that 2 = m, - ap,(c - m,) < ml < a.

A simple approximation for the dominant root r1 can sometimes be used to make the approximation R(t) g u2ePI1) more useful. It is shown in Section 8 that - carried load _ _m1 02 load variance < r,; i.e., -ml/u2 is a rigorous lower bound to rl . Figure 5 suggests this bound gives a fairly good approximation to rl if a/c < 1. Hence a simple approximate formula for I?(*), valid for a / c < 1, is given by R(t)

&?-m,tio? N (load -

variance) exp

-

carried load 1 t . load variance !

(12)

3.

207

T HE COVARIANCE FUNCTION

<

We know that R ( t ) cr2ePlt1 and that -m,/02 < r l ; hence replacing rl by -ml/a2 tends to correct the error in the upper bound formula. T h e formula (12) is illustrated in Fig. 1 1.

FIG. 11.

Comparison of R(t) with 2-(mllu*Jt for c = 8 trunks,

u =

4 erlangs.

208

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

$4. The Covariance in Terms of the Recovery Function It has been shown (3) that the Laplace transform of p,,(.) is US(4

50SdC)

*

By expansion in partial fractions we find that

T h e sum assumes only negative values, and so pee(.) decreases monotonically to the loss probability p , . T h e recovery function is illustrated in Fig. 12. We now observe that for each j = 1, ..., c,

FIG. 12. Recovery function for c

=

5 trunks,

a

=

10 erlangs.

5.

VARIANCE OF THE NUMBER OF PATHS IN SERVICE

209

By comparison of formulas (8) and (13), and use of (14), one finds that R(t) = a2pc

t 0

( t - u)e-(t-u)[p,c(u) - pel du

+

u2ct

+ Ctect,

(1 5)

where

This formula expresses R(.) in terms of pee( .) by a simple convolution. T o evaluate C explicitly we note that

where a_, is the first coefficient in the power series expansion of the left-hand term in the bracket. One finds

=

- m1)

= (load

lost) (average number of idle trunks).

$5. The Variance of the Number of Paths in Service We assume that n observations (xi , j = 1, ..., n} of x i are made during an interval of equilibrium, so that cOV{Xi, Xj} where

T

=

R(I i -i

I T),

is the scan interval. Then with

s, = x1 + x2 + + x, ***

= number of paths

found in service,

210

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

we find that

j=-n

T o give an exact formula for Var{S,} we note that

z

e-lzmlu =

ctnhu,

m=-W

and

= $e-2nu

T he n also

csch2u.

m

j=-n

=n

ctnh u -

(1 - ecZnu) csch2u.* 2

Since the covariance can be seen that

(17)

I?(.) is a symmetric function given by (8), it

* U s e of this identity was suggested by unpublished work of which the author had access.

J. W. Tukey

to

5.

VARIANCE OF THE NUMBER OF PATHS I N SERVICE

211

This formula is exact, given the assumptions. I t is easily shown from formula (17) that the exact formula for the variance of n-lS, in the “infinite trunk” model is

illustrated in Fig. 6 (13). Returning to the case of finitely many trunks, we can obtain approximating upper bounds to formula (18) for Var{n-lS,) by using the results of Section 3 on the covariance function. I t can be seen from the arguments leading to (17) that replacing the roots yj by rl

h FIG. 13. Upper bound

to

(Var {S,})/no*.

212

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

in the hyperbolic functions in (18) increases the values of the expressions in square brackets; this replacement is equivalent to using the upper bound &?it

for R(t) in formula (16). Hence

where rr = -1 =

1

-I( scan interval) (dominant characteristic value).

Since a2erltis close to R(t),we may expect that the overestimate (19) gives a good approximation to the actual variance. This approximation is conveniently plotted as a function of h for various 1z in Fig. 13.

$6. The Variance of Time Averages It follows from formulas (3b) and (8) that

as T -+ co, where

is a negative constant, and

Note that c,, and c1 differ only in the power of ri that occurs in the denominators. T h e third term of (20) is positive, is given by

6.

THE VARIANCE OF TIME AVERAGES

213

equals -co at T = 0, and is of smaller order than e-= because < - 1. To evaluate c1 explicitly, we note that

rl

where a-2 and a_, are, respectively, the first and second coefficients in the power series expansion of the leftmost term in the bracket of (21); these are given by

T o find c1 we must compute

This equals 2(1 - P c ) - -

or

Now the generating function of the o-functions is

so that

a

- @(s,

as

2) =

@(s, z )

c

n=l

x”

-,

214

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

I t follows that c1

= 2azpc[a4 =

2a2

+ + (1 a-1

+ 2a2p,(l - pc)(1

-Pe@ - tc-1

+ tJ1 + t c ) + 2Wc. - tc-1

T h e constant c,, can be evaluated in a similar fashion. From the bounds (1 1) for R ( . ) we conclude that - Var{M( T ) )

and, since R ( t ) g

< (0.3933)a2p,

cT- 1 T2

+T

we may expect that the overestimate

a2er11,

1 - rl T T"12

2u2 e7ir -

is a good approximation to the variance of M( 7'). This approximation has the same form as the exact formula (7) for the "infinite trunk" model, because in both cases a single exponential is used for R ( * )in formula (8). T h e overestimate (22) is depicted graphically in Fig. 14.

f i = -P,T=- (DOMINANT CHARACTERISTIC VALUE)(OBSERVATION TIME)

-

N

(CARRIED LOAD) (OBSERVATION TIME) Q2

FIG. 14. Approximation to (Var (,'xt dt})/(ut)'.

7.

D E R I V A T I O N OF THE C O V A R I A N C E

215

I t was convenient to plot the ratio

__--Var(M(T)i UZ

as a function of the single parameter p =

r,T

=

-

(dominant characteristic value) (observation time).

A simpler form of (22), valid for aic < 1, results when we replace r1 by its lower bound carried load -m 1_ 0% load variance < y1. T h i s replacement decreases the value obtained, i.e., moves the approximation in the direction of Var{M( T ) } .

$7. Derivation of the Covariance T h e transition probabilities

p,,(t)

= Pr{x, =

n I xo = m}

satisfy the Kolmogorov equations

P,,(O)

d

= S,,I,l

,

Pm, = P7nl -

apnlo.

Multiplying the nth equation by n, and summing on the index n, we find d dt

-E{x,

1 ,yo = m} = - E { x , 1 xo = m} + a [ ] - p m e ( t ) ] ,

whence J O

216

6.

COVARIANCE FUNCTION OF A S I M P L E TRUNK GROUP

Ry formula (2), the covariance is then

where n=O

for i = 1, 2, and particular,

{pn} are the stationary probabilities given by (1). I n

m, = 4

1

u = (m,

(24)

- Pc), -

mlL)lI2= [m,

ap,(c

~

-

ml)]1/2.

(25)

‘The Laplace transform of Pr{x, = c \ xo = m} has been determined (3) to be ac-mm!us(m) ____c!sus+I(C)

Therefore that of R(

a)

*

is

By ( l ) , the last term of (26) is

It has been shown ( 9 ) that the “sigma” functions satisfy the recurrences

=

us+,(m) - us+Lm - 11,

ma,(m) = au,$(m- 1)

+ ~u,+l(m

-

(27) I),

(28)

7.

DERIVATION OF THE COVARIANCE

217

so that

T h e foregoing identities yield the following simplified formula for

R*(s):

From (27) we find that the partial fraction expansion, - 1) ____

.,+l(C

-

c c

-

JS1

u'+l(c)

_---_ (s

n

UT

- TI)

(c)c! -__-(TI - T I )

z 1 1

is valid, where { r j } are the zeros of O ~ , . ~ ( C ) . By a similar argument, since p,. = uo(c)/u~(c), -U F +---_ I ( C - 1)

$(I

+

s)u,,,(c)

-

------_1 -Pc

-

s

a(1

+

C

s)

Hence formula (29) can be inverted to give, for t R ( t ) = mZect

+ am,[l

-

ect] - acp,te-'

-

ml2

2 0,

218

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

where $=m

- m Z -- equilibrium variance,

and

K = z -c- - -l n > - 1- - L ,=I

r3 (1

+

Y *#,

-1-r r3 - rl

T o evaluate K explicitly we observe that

where a-2 and a_, are, respectively, the first and second coefficients in the power series expansion of the leftmost term in the bracket of (31). Thus K = a_, a-2 - 1 p , . Now

+

+

+$--LnL--r. + (s - Y)-I

,=I

(1

rJ2 z # l

r

-

YJ

1

-

- Yz

From the recurrence (28) for the cr-functions we find that

differentiating with respect to s and setting s

Clearly, a-2

uo(c - 1) c = ----OOfC) a’

and so

a2p,K

= --a2.

=

0, we obtain

r

8.

219

APPROXIMATION TO DOMINANT CHARACTERISTIC VALUE

T h u s the formula (30) for the covariance function R(*) simplifies to

$8. Approximation to the Dominant Characteristic Value T h e differential equations (23) can be written in the form

( t )where ) where P(t) is the matrix of transition probabilities { ~ , , ~ ~and

n = (qij)is the matrix of the “transition rates”*: -a

a 1 (-u-1) 0 2

0 a

... 0

0 0

(-a-2)

0 0

a

0

0 0

0 0

... ...

U

0

c-1

0

(-a-c+1)

0 a -

C

T h e characteristic values of Q are 0, r1 , r2 , ..., r, . We define

and we introduce an inner product for the spaceL2(p)of (c of complex numbers by the definition

+ 1)-tuples

n=O

T h e matrix Q represents a symmetric operator on L 2 ( p ) ,i.e., -

* We

( Q J L

=

are using the convention (QJt

Z&P,

’

=

C,q,$, , rather than the more usual

220

6.

COVARIANCE FUNCTION OF A SIMPLE TRUNK GROUP

It is easily seen that

VP = 0,

( P o Pl , ..., P c ) , i, j = 0, I , ..., c.

for

q*jp* = qjipj

P

,

=

(34)

3

(35)

T h e last identity implies that

so (as we already know) all characteristic values of Q are nonpositive, being of the form (Qx, x) for some x EL&). From the extremal properties of the characteristic values of symmetric operators [e.g., Zaanen (14, p. 383, Theorem 3)] we conclude that 7 , = max(Qx, x),

the maximum being over all x EL&) that are not identically zero, and satisfy (x, x) = 1, (x, p ) = 0, p being the vector of stationary probabilities, as in (34). We can now estimate y1 from below by choosing an appropriate vector x. We whoose

where m, and u are the mean and standard deviation of x 1 in equilibrium, given by formulas (24) and (25), respectively. Clearly, (x, X) = 1 and ( x , p ) = 0, and

2 p , z(----n - n z , r-1

(QX,

x)

= --a

U

n =O -

~

-

n+l-ml

---__-

4 1 -Pc) ____ UZ

-

_ m1 _ 0%

< 71

*

[See Kramer (25).] This approximation is illustrated in Fig. 5.

U

1

REFERENCES

22 1

REFERENCES 1. W. S. Hayward, Jr., T h e Reliability of Telephone Traffic Load Measurements by Switch Counts, Bell System Tech. J. 31, 351-317 (1952).

2. C. Palm, Accuracy of Measurements in Determining Traffic Volumes by the Scanning Method, Tekniska Medelanden f r i n Kungl. Telegrafstyrelsen Nos. 7-9 (1941). 3. V. E. Benei, Transition Probabilities for Telephone Traffic, Bell System Tech. J. 39, 1297-1320 (1960). 4 . L. Kosten, Over de invloed van herhaalde oproepen in de theorie der blokkeringskansen, Ingenieur (Utrecht) 47, 123 (1947). 5. J. Riordan, Telephone Traffic Time Averages, Bell System Tech. J. 30, 11291144 (1951). 6. V. E. Bene;, A Sufficient Set of Statistics for a Simple Telephone Exchange Model, Bell System Tech. J. 36, 939-964 (1957). 7. V. E. Benei, Fluctuations of Telephone Traffic, Bell System Tech. J. 36, 965973 (1957). 8. C. Palm, Calcul exact de la perte dans les groupes de circuits Achelonnb, Ericsson Tech. 4, 41 (1936). 9. J. Riordan, Appendix to R. I. Wilkinson, Theories for Toll Traffic Engineering in the USA, Bell System Tech. J. 35, 507 (1956). 10. G. Szego, Orthogonal Polynomials, Am. Math. SOC.Colloq. Publ. XXIII, 1938. 11. J. Haantjes, Wiskundige Opgaven 17 (1938). 12. W. Ledermann and G. E. H.Reuter, Spectral Theory for the Differential Equations of Simple Birth and Death Processes, Phil. Trans. Roy. SOC.(London) A236, 321 (1954). 13. K. M. Olsson, Calculation of Dispersion in Telephone Traffic Recording Values for Pure Chance Traffic, Tele (English ed.) 2, 71 (1959). 14. A. C. Zaanen, “Linear Analysis.” Wiley (Interscience), New York, 1953. 15. H. P. Kramer, Symmetrizable Markov Matrices, Ann. Math. Stat. 30, 149-153 (1959).

CHAPTER

7

A “Thermodynamic” Theory of Traffic

Two new theoretical models for representing random traffic in connecting networks are presented. T h e first is called the “thermodynamic” model and is studied in detail. T h e second model is formulated in an effort to take methods of routing into account and to meet certain drawbacks of the “thermodynamic” model in describing customer behavior; since it is more realistic than the first, it leads to results that are vastly more complicated. It is studied in Chapter 8. The “thermodynamic” model is worth considering for four reasons:

(1) It is faithful to the structure of real connecting systems. Indeed it is an improvement over many previous models in that it considers only physically accessible states of the connecting network, while the latter suffer the drawback that a large fraction of the network states on which calculation is based are physically meaningless, being unreachable under normal operation. (2) It gives rise to a relatively simple theory in which explicit calculations are possible. (3) T h e “thermodynamic” model provides a good simple description of traffic in the interior of a large communications network. (4) It has an analogy to statistical mechanics which permits us to be guided by the latter theory as we try to use the model to understand the properties of large-scale connecting systems. The two models to be described differ in only one respect. In the first (the “thermodynamic”) model, the system moves from any state x to a neighboring state y that has one more call in progress at a rate h ; the effective calling rate per idle inlet-outlet pair is thus proportional to the number of paths usable

222

1.

INTRODUCTION

223

in x from that inlet to that outlet. In the second model, the calling rate per idle inlet-outlet pair is set at A , and is then spread over the paths usable in x from that inlet to that outlet in accordance with some routing rule. This provides a mathematical description of routing, and avoids the unwelcome feature that a single customer’s calling rate depends on the state of the network. T h e “thermodynamic” model is based on the single postulate that the “equilibrium” probabilities of the states of the connecting network maximize the entropy functional for a fixed value of the traffic carried. These probabilities have the same geometric or exponential form as the canonical MaxwellBoltzmann distribution of statistical mechanics. T h e theory developed applies to any connecting network regardless of its structure or configuration. The number of calls in progress is analogous to the energy of a physical system. As in statistical mechanics, important averages can be expressed as logarithmic derivatives of a generating function analogous to the partition function of physics. I t is possible to give an interpretation of the maximum entropy postulate in terms of random behavior at the inlets and outlets of the connecting network; this interpretation leads to a stochastic process zt of the familiar Markov type, for which the canonical distribution is invariant. T h e transition rate matrix of z t is self-adjoint in a suitable inner product space, so that the approach of z t to equilibrium is easily studied, with resulting applications to traffic measurement.

$1. Introduction Like the physicist, the traffic engineer is faced with the study of an extremely complex system which is best described in statistical terms. T h e great success of the theoretical methods of statistical physics has given rise to a fervent hope, sometimes voiced among traffic theorists, that similar methods exist and can be found for the study of congestion. Indeed, the problems are much the same: one desires a small amount of “macroscopic” information about averages, based in a rational way on vast complexities of detail. A. K. Erlang was probably influenced by statistical mechanics when he introduced his method of “statistical equilibrium” into traffic theory. This method has had great success in dealing with problems of the birth-and-death type, like trunking and queueing, but as applied to more complex cases it has led mostly to algebraic and combinatorial difficulties. Nothing as elegant or powerful as statistical mechanics has resulted so far. We shall present two traffic models in this chapter. T h e first is the outcome of a deliberate attempt to ape the methods of physicists in statistical mechanics, and thus to realize, at least in part, the hope mentioned above. It is called the “thermodynamic” model, and it is treated in detail. T h e second model is introduced later in the chapter

224

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

in an attempt to avoid certain drawbacks that appear in the physical interpretation of the “thermodynamic” model. Since it has independent interest and leads to involved, more realistic results, it is studied in some detail in Chapter 8. T h e approach taken in the “thermodynamic” model bears a close analogy to the methods of statistical mechanics, being based on the single postulate that the “equilibrium” probabilities of the states of the connecting network maximize the entropy functional for a fixed value of the traffic carried. We develop a theory, briefly summarized in the next paragraph, by deducing the consequences and interpreting the meaning of the one basic assumption. T h e state probabilities that maximize the entropy for a given carried load form a distribution function over the set of states that has the same geometric or exponential form as the canonical (or Maxwell-Boltzmann) distribution of statistical mechanics. T h e theory applies to any connecting network, regardless of its structure or configuration. T h e number of calls in progress is analogous to the energy of a physical system. As in statistical mechanics, important averages can be expressed as logarithmic derivatives of a generating function analogous to the partition function of physics. It is possible to give an interpretation of the maximum entropy postulate in terms of random behavior at the inlets and outlets of the connecting network. T h i s interpretation leads to a stochastic process z1 of the familiar Markov type, and is such that any stochastic process based on it satisfies the maximum entropy postulate. T h e transition rate matrix A of z t is self-adjoint in a suitable inner-product space; its characteristic values are real and nonpositive, and can be studied by classical variational methods. I n terms of these characteristic values the approach of z t to equilibrium can be studied, with resulting applications to traffic measurements. I t turns out that the covariance of any function of z , is strictly positive. T h e chapter ends with a timedependent or nonstationary generalization of the maximum entropy postulate that has close analogies with the statistical “derivation” of thermodynamics.

$2. P re1im inar ies I n order to give an adequate summary and discussion of our theory in Section 3 , it is necessary to present first its concepts,

3.

SUMMARY OF CHAPTER

7

AND DISCUSSION

225

terminology, and notation. Virtually all the notions about to be discussed have appeared in earlier chapters and in papers by the author ( I , 2), so only a brief rCsumC is given here. Let S be the set of possible (or permissible, or both) states of a connecting network, and let x, y , ... be variables ranging over S. T h e where x y elements of S are partially ordered by inclusion means that x can be obtained from y by removing zero or more calls. Furthermore, the states x E S can be arranged in an intuitive manner in the state-diagram, the Hasse figure for the partial ordering This figure is constructed by partitioning the states in rows according to the number of calls in progress. T h e unique zero state (in which no calls are in progress) is placed at the bottom of the figure, above it comes the row of states with one call in progress, and so on. T h e figure is completed by drawing a graph with the states as nodes, and with lines between states (in adjacent rows) that differ in exactly one call. I n Chapter 2 and in an earlier work (2) we made the assumption that in a given state at most one call could be in progress between a given inlet and outlet; it is convenient to discard this assumption here. If the connecting network under study is in a state x, it can move only to states that are neighbors of x, i.e., are obtainable from x by adding a new call or terminating a call in progress. I t is useful to divide the neighbors of x into two sets A, and B, , where

<,

<

<.

set of states immediately above x, i.e., accessible from x by adding a new call, B, = set of states immediately below x, i.e., accessible from x by a hangup.

A,

=

For any set X , the notation I X I is used to denote the number of elements of X . T h e states x E S can be defined (2) as sets of chains on a graph, one chain for each call in progress. Hence it is natural to use i x 1 to mean the number of calls in progress in x. T h e kth level L, is the set of all states with k calls in progress, i.e., L,

= (x

E

s: I x I = k}.

$3. Summary of Chapter 7 and Discussion We start, in Section 4, with a brief informal discussion of what is meant, heuristically as well as precisely, by “equilibrium.”

7.

226

A “THERMODYNAMIC”

THEORY OF TRAFFIC

I n Section 5 we formulate and discuss the maximum entropypostulate, according to which a suitable “equilibrium” distribution {qz , x E S } of probability over S is obtained by choosing the probability vector 4 so as to maximize the entropy functional

for a given value of the average number of calls in progress, i.e., for XES

Various heuristic arguments are adduced to support the prima facie reasonableness of this principle. I n Section 6 it is shown that the maximizing probability vector q is given by

where

and X is a constant determined uniquely by the equation m = A(d/dA) log @(A).

Because of their close similarity to corresponding notions from are henceforth statistical mechanics, the vector q and the function called the canonical distribution and the partition function, respectively. I n Section 7 we have collected together various properties of the of S. partition function, most of them based on the partial ordering Among these are expressions for @(-)in terms of the Mobius function and in terms of several sets of “characteristic polynomials” for associated with < and S. T h e canonical distribution q is placed in a dynamic context in Section 8. This is done by defining a Markov process z t (taking values on S)for which q forms a stationary distribution. T h e transition rate matrix A (infinitesimal generator) of this process allows one to give interpretations of this dynamic context in terms of calling rates and mean holding times. An informal description of the process is @(a)

<

<,

3.

SUMMARY OF CHAPTER

7

AN D DISCUSSION

227

this: If it is in state x, it moves to a state y E A , at a rate A, and to a state y E B, at a rate set at unity by convention. A full discussion of the analogy between the “thermodynamic” theory of traffic and statistical mechanics is given in Section 9. For purposes of illustration, we mention that the number of calls in progress corresponds to the energy of a statistical mechanical system, and that the constant A is related to the calling rate and corresponds to the temperature (up to a monotone transformation). T h e reasonableness of z t as a description of an operating connecting network is discussed and criticized in detail in Section 10. Two possible interpretations of the inlets and outlets are considered: in one, the inlets and outlets are the ultimate terminals of the system, beyond which there is no more switching equipment; in the other, the inlets and outlets are switching centers such as PBX’s, frames, or individual crossbar switches, acting as sources of traffic for a network under study. I n the first interpretation, there can be at most one call in progress on an inlet or an outlet; in the second, there may be several. Regardless of which interpretation of the inlets and outlets is adopted, the transition rate matrix A for x t must be interpreted as saying that the calling rate between an inlet and an outlet in a given state x is proportional to the number of free paths that x provides between that inlet and outlet. This assumption is unacceptable for the interpretation of inlets and outlets as ultimate terminals; it is entirely reasonable if the inlets and outlets are local switching centers. Section 11 is devoted to describing, as an alternative to z t , a Markov stochastic process on S based on the assumption that the calling rate between an idle inlet terminal and an idle outlet terminal is a constant A. This calling rate is then spread over the possible ways of realizing the call in question in the current state of the network in accordance with some method of routing. A mathematical description of such a method of choosing routes for calls is given. This description leads directly to a transition rate matrix Q for a process x t (studied in Chapter 8) in which every idle inlet-outlet terminal pair has a calling rate A in every state. T h e possibility that z t may be a useful perturbation of x t is considered. I n Section 12 it is observed that the rate matrix A for z t is a selfadjoint operator in a suitable finite-dimensional inner product space. This implies that the characteristic values of A are real and nonpositive, and leads to bounds on the rate of approach of z t to equilib-

228

7.

A “THERMODYNAMIC” THEORY OF TRAFFIC

rium. These bounds can be applied to estimate the covariances of functions of z l , and the sampling error incurred in measuring carried loads by averaging. I n particular it is shown that the dominant (i.e., that of smallest nonzero magnitude) characteristic value rl of A satisfies -(m/aZ)

< Y1 < 0,

where m and are (respectively) the mean and standard deviation of the load associated with the equilibrium probability vector p for x l , so that (T

=C.l.1qxr XES

I n Section 13 we give a formula for the covariance of any process

fldefined by applying a function f(*)to x i ,i.e.,

This covariance is always positive. Applications of this formula to traffic averages are described briefly in Section 14. Finally, Section 15 considers a time-dependent generalization of the variational principle on which the “thermodynamic” theory of traffic is based. We conclude this section with an appraisal of the “thermodynamic” theory presented herein. This will take the form of a list of comments, first pro, then con, and then a defense.

(1) There exist theories ( 3 , 4 ) for connecting networks in which it is assumed that the links of the system are busy or idle with a given probability, all independently of one another. I t can be verified that an overwhelming fraction of the states of the system so considered are in fact not physically meaningful states that the system can reach under normal operation. T h e theory presented here is based only on permitted, physically meaningful states, and so is not open to this very serious objection. (2) T h e theory provides a uniform method of treating any connecting network in that the calculation of equilibrium probabilities always reduces to that of the partition function. In most other treatments the nature of the algebraic process of calculating probabilities depends

3.

SUMMARY OF CHAPTER

7

A N D DISCUSSION

229

heavily on a detailed account of the network configuration; in our theory it depends on the network only via the numbers 1 L , 1, I L, 1, ... . (3) T h e maximum entropy principle can be given a certain informal, a prior; justification. It provides a “conservative, worst possible case” approach to problems and processes of fantastic complexity. This is because it can be interpreted as enjoining that an “equilibrium” distribution of probability for given carried traffic correspond to a condition of maximum ignorance of the actual state of the connecting network. (4)T h e canonical distribution q that results from the maximum entropy postulate can be embedded in a dynamic model of traffic by defining a Markov process z t for which q is the invariant or stationary distribution. This dynamic model is described by a transition rate matrix which is a self-adjoint operator, a fact that makes it possible to study the time-dependent behavior of z t in a simple approximate way, with applications to traffic measurement, for instance. (5) A serious drawback of the “thermodynamic” theory is that its natural interpretation in terms of calling rates appears to be unreasonable in some practical cases. For this reason it may remain an amusing curiosity, rather than become an engineering tool. (6) T h e problem of calculating the partition function @(.) is, as in statistical mechanics, very difficult except in cases of unreaIistic simplicity. Thus, even if its assumptions are granted, our “thermodynamic” theory does not afford much progress in calculating quantities of interest. (7) T h e theory can take into account only one of the many different possible methods of routing calls in operating networks. T h u s it cannot help the designer choose among alternative methods. By way of defense against the objections just raised, these points can be made: (i) Comment 5 , that the interpretation of the “thermodynamic” theory in terms of calling rates is unreasonable, depends on a natural, but not necessarily valid or compelling, assignment of causes for new calls. is hard, it is at least a definite (ii) Although the calculation of combinatorial problem solvable in principle by counting; thus q t least part of the problem of obtaining state probabilities is disposed of. @(a)

230

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

(iii) I t is doubtful whether routing methods make as much as an order of magnitude of difference in carried loads in large systems; hence it is reasonable to ignore them in a relatively crude theory such as the present one. [See, however, the author’s paper (IZ).] T h e theory presented in this chapter should be judged by its success in practice as well as by its agreement with our preconceptions. I believe that, in spite of the major failings mentioned, the theory musters interest enough to warrant its presentation to traffic engineers, if only because its concepts and results may prove stimulating and useful in more realistic approaches.

$4. Equilibrium Quantities that are of interest in the design of a connecting network, such as the average load carried, the variance of the load, or even the probability of blocking, can often be calculated from a knowledge of some “equilibrium” or “stationary” state probabilities {qz , x E S>for the network of interest. These probabilities are usually assumed or proven to be of “equilibrium” type in the sense that they have some physically reasonable invariance property. Since the concepts of stationarity and equilibrium can assume many precise forms of varying strength, it is important to consider briefly some of the senses. T h e strongest notion, of course, is that of strict stationarity of a stochastic process, defined by the condition that all the finite-dimensional distributions be independent of time, i.e., be translation-invariant. A whole class of weaker notions can be obtained by requiring only that the distributions of dimension not greater than n be invariant. T h e notion of wide-sense stationarity, defined by the condition that the covariance depend only on the difference of its arguments, is still another concept of stationarity, formulated for a moment rather than a distribution. Again, Markov processes are described as homogeneous or stationary if their transition probability operators are time-invariant. “Equilibrium” is a word that usually connotes a stable, quasi-static random behavior which is perhaps a condition of attaction for a process, in the sense that a process not in equilibrium tends toward it. Ergodic Markov processes with denumerable state spaces are typical

4.

EQUILIBRIUM

23 1

examples. It is to be remarked, though, that use of the word “equilibrium” usually implies a nondegenerate limiting behavior for a process y 1 under study as t co. Th u s a time-homogeneous Markov process may not have a genuine “equilibrium” distribution because it in some sense “blows up,” e.g., the process may take values on the integers and the probability mass may move out toward fa, even though the transition probabilities are time-dependent. I n such a case, clearly, no first-order distribution can be assigned which is time-invariant. T h e analytical expression of “equilibrium” often takes the form of a statement to the effect that an operator has zero as a characteristic value. Perhaps the most familiar example of such a statement arises in the case of a Markov process in continuous time with a transition rate matrix A ; the equilibrium equation is Aq = 0, for a probability vector q. * This equation, together with its connections to semigroups, to Markov processes, and to the notion of statistical equilibrium used in traffic theory, is discussed immediately below. A traditional analytical method in telephone traffic theory is that of “statistical equilibrium,” due to Erlang (5). This method may be described heuristically as follows: A notion of equilibrium is defined by the property that the rate of flow of probability into (or onto) a state equals that out of (or from) the state; this equilibrium is expressed in a set of equations among the state probabilities, the so-called statistical equilibrium equations; the “equilibrium” state probabilities are then taken to be (or defined by) the solution of these equations. T h e method of statistical equilibrium can be interpreted in the mathematically rigorous context of semigroups of positive operators, here the matrices of transition probabilities {Q(t),t real} for a Markov process xitaking values in S , with ---f

a t ) = (qz!/(t)) qz:,(t) = Yr{state of system IS y at t if it was x at 0).

T h e generator A of the semigroup is the matrix of transition rates or the derivative

* We are using (AT),= aru9v . Y

n y r q v , rather than the more usual

the convention (Aq), = U

232

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

T h e matrix A expresses the relative probabilities of the various changes that can take place in a small amount of time, and indeed Q(t) = 1

+ At + o ( t )

as

t-0.

I n terms of the generator A the equation of statistical equilibrium can be written as Aq = 0, which expresses the fact that the vector q of state probabilities is an eigenvector of A corresponding to a zero eigenvalue of A. From the semigroup property Q(t

it follows that

+4

= Q(t)Q(s)

Q ( t ) = exp At

2 qUqyz(t) v

= qz ,

x

E

S , t real,

the last equation expressing the invariance of q under the transition probability matrices Q( .).

$5. The Maximum Entropy Principle I n the method of statistical equilibrium, the state probabilities are calculated a posteriori from a prior equation expressing an equilibrium or invariance principle. This equation is either postulated or is derived from assumptions that lead to a Marltov stochastic process as a model for the operating network. I n the present approach we use a variational principle rather than an equilibrium principle as a basis for calculating “equilibrium” state probabilities. I n drawing this distinction we refer only to the immediate form of the assumptions and derivations, and imply no absolute distinction, since an “equilibrium” principle can almost always be given a “variational” form. For example, if A is a transition rate matrix for an ergodic Markov process, and A is self-adjoint with .), then the “equilibrium” probability respect to an inner product vector q, i.e., the solution of Aq = 0, is equally well described as the vector which maximizes the Rayleigh quotient ( a ,

5.

THE MAXIMUM ENTROPY PRINCIPLE

233

I t will turn out that the probabilities {qz , x E S} derived from our variational principle also have an invariance property expressible, as in the example given, in terms of the self-adjoint generator A of a Markov semigroup by the equation Aq = 0. This equation can be interpreted as a “statistical equilibrium” equation, and the elements of A related to calling rates and hangup rates in the various states x E s. However, instead of starting with a suitable matrix A to represent the infinitesimal dynamic behavior, and solving Aq = 0 in order to obtain an equilibrium distribution {qz , x E S } over the states of the system, we shall directly choose a certain q, to be used as an “equilibrium” distribution for calculating quantities of interest, according to this criterion: T h e entropy functional

is to be as large as possible subject to the conditions qz 3 0,

c.I

xcs

.x

f

s,

2qx= 1,

X€S

1

x 41 = m,

where m is a given number, the average load carried. T h e first two conditions ensure that only bona jide probability distributions are considered, while the third enjoins that q give rise to m as the mean number of calls in progress in equilibrium. This criterion or method for choosing a probability distribution over S we call the maximum entropy principle; it is exactly analogous to that used in statistical mechanics, provided that the number of calls in progress is interpreted as the energy of the mechanical system. We have already stated that this principle leads to a unique q which is exactly the same as would be obtained by a particular choice of A , given later, and solving Aq = 0; this matrix A has a definite interpretation in terms of system behavior during small periods of time. A measure of justification for using the maximum entropy principle can be obtained from five arguments: (1) Insofar as a high value of the entropy functional is an indication of a low degree of information, so far can use of the principle be

234

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

interpreted as postulating that an equilibrium distribution {pr , x E S} corresponds to a condition of maximum ignorance subject to a given average number of calls in progress. T h e principle may thus be said to represent a “safe” or “worst case possible” approach to the problem. Exactly the same principle is used in statistical mechanics to obtain the canonical distribution. I n both cases it is a reasonable and systematic way of throwing up our hands. (2) T h e principle is appealing for the obvious reasons of unity, uniformity, simplicity, and elegance. (3) I t leads to a theoretical structure similar to that of statistical mechanics. As in the physical theory, statistical quantities of interest are calculated from a partition function, characteristic of the network under study, that depends on purely combinatorial properties of that network. (4) T h e principle of maximum entropy leads to a unified theory applicable to all connecting networks. That is, the resulting “equilibrium” distribution depends algebraically on the structure of the network in a way that in a sense is uniform for all networks. ( 5 ) T h e principle can be given a dynamic context in terms of Markov processes. This context permits the study of the approach to equilibrium in time, with important applications to sampling error.

56. The Canonical Distribution I n the next few sections we develop some of the principal consequences of the maximum entropy principle, and examine their similarity to Statistical thermodynamics. I n the present section we determine the distribution {p, , x E S } which maximizes H(q) for a given average load carried. T h e following lemma is no doubt wellknown, especially to physicists; since its application in traffic theory is relatively new, its detailed proof is included for completeness. Lemma 7.1.

Let f(.)be a nonnegative function dejined on S , and let ZES

The maximum of

6.

235

THE CANONICAL DISTRIBUTION

subject to the conditions 4x20,

c

XES

X E S

XES

qxf ( x ) = mf , ( m f a given positive number in the closed convex hull of the range of f ( * ),)

is

(2)

H ( q ) = log Y(w) - mf log W,

where w is the unique positive solution of w(d/dw) log

Y(w) = mf .

The maximum is achieved by the choice qx = [ W ” ” ’ / Y ( W ) ] , = exp{ -aJ(x)

n2 -

1j,

x

S,

(3) where a , , a2 are (the values of Lagrange’s multipliers) determined by any two of the relations -

E

a2 = log Y(ecal) - 1, mf

=

w =

C.fW exp( -%f (4- a2 - 1

)t

xss

exp{-a,).

Proof. With a, and a2 as Lagrange’s multipliers, we form the expression

differentiate with respect to each qs , and set the resulting derivatives equal to zero. This gives the equations ahjaq,

=

-(log qI

+ 1 + q1f (x) + a2)

= 0,

x

E

A!,

(4)

whose solution is (3). T h e multipliers ‘a, and u2 are to be determined from the conditions Z q Z=1

X€S

and

2f ( x ) q x X€S

=mf.

236

7.

A “THERMODYNAMIC”

T h e first gives 1 = e-az-1

c

xes

THEORY OF TRAFFIC

exp[--a,f(x)l,

u p = log Y ( e - a i ) - I,

while the second yields mf

= e-aa-1

c;f (4exp[- Qlf(41

XES

Zf(4exp[-%f(x>l

z

xeS -~

X€S

Setting w equation

=

exp(--a,f(x)l

exp{-aa,}, it is found that

w

w ( d / d w ) log Y(w) =

*

should be a solution of the

mf > 0.

(5)

From the fact that

> 0, it is easily shown [Khinchin (6, p . 77)] that there is exactly one solution of ( 5 ) , and that w is positive. A relative extremum of H(q) in p 3 0 subject to (1) and (2) must satisfy eqs. (4). Since these have only one solution there is only one such extremum. T o show that it is a maximum it is enough to show that the matrix of second derivatives of H(q) with respect to the components pz of p is negative definite. However, this is straightforward, since [ 0 if x f y

w

_qx

if x

= y.

I n Lemma 7.1 we let

f ( 4 = 1x1 =

and we obtain

number of calls in progress in state x

7.

PROPERTIES OF T H E PARTITION FUNCTION

Theorem 7.1.

Let m

237

> 0 ; let

and let h be the unique (positive) root of m = A(d/dA) log @(A).

The maximum of

WQ)= - 2 9%1% xes

Qr

?

subject to the conditions that q be a probability vector over S and that

is H,,

= log

@(A) - rn log h

and is achieved by the vector q with components

This is the distribution of probability over S that is determined uniquely by the maximum entropy principle; as noted before, it is the canonical distribution. T h e function @(*) is called the partition function of the connecting network whose states form the set S. Since m determines h uniquely and vice versa, we can use h as the parameter that determines the average traffic level instead of m. Indeed, m is a monotone increasing function of A 3 0. Also it can be seen that moments of the distribution of the number of calls in progress (other than the mean) can be calculated from @(.) by logarithmic differentiation.

$7. Properties of the Partition Function I n this section we exhibit various identities and relationships that are typical of the partition function @(-). This function is the

7.

238

A “THERMODYNAMIC”

THEORY OF TRAFFIC

generating function of the number of states in a given level; that is,

Thus the problem of calculating A, @(-),and q in our model reduces to the calculation of the sequence

I Lo I, 14 I)

and vice versa.

-.a

Remark 7.1.

T h e first part of this result was proven as Theorem 2.1 in Chapter 2, and it implies the second part (2). T h e Mobius function p ( * )of the partially ordered system ( S , <) is defined recursively by p(01 =

1,

p(x) = -

2 p(y)

if

x

>o,

~ E s .

Y i X

We have remarked previously (Chapter 2, Section 7 ) that if S is a class of network states, then p(.) takes on the simple form p(x) =

( - 1 p 1 x I!.

We define the generating function M ( . ) by

Since

it can be seen that (except for a change of sign in the generating variable) @( -) is the exponential generating function associated with & Thus I( wehave ).

7.

PROPERTIES OF THE PARTITION FUNCTION

Remark 7.2.

M(0

=

jm e-Q(

-&) du.

=

J

-&)

239

Proof.

e-U@(

du.

I n analogy with Birkhoff (7, p. 15, Eq. (12)), we define for each

x E S a characteristic polynomial by the recursion formula*

Px(5)

=

5'"' -

P,(O. Y<X

This is related to the Mobius function p ( * ) by the fact that if p,(*) denotes the Mobius function for the set (x:x 2 y } , then

y } is again of the However, the partial ordering of the cone {x: x same form as that of S ; i.e., there are exactly (I x - y I)! ascending chains between y and x, all of length I x - y I. Hence, from Birkhoff (7, p. 15, Eq. (11)),

and

* Actually, Birkhoff's polynomial p z [ f ] equals f p Z ( f ) .The definition we use is more convenient for our purposes.

240

7.

A "THERMODYNAMIC"

THEORY OF TRAFFIC

Let now Y <X

Y
T h e Mobius inversion formula gives

T o calculate qz( explicitly, we note that if 0 the cap n for set intersection, a)

i.e., there are exactly Hence

('

qr(4) =

< k < 1 x I, then, using

'1 states with k calls up below any state

cp

=

(1

x.

+ 4)'"'- p.

Y<X

Let us write

ti"'=

c.

ry(5)1

Y<X

where ry( are functions to be determined. Using the Mobius inversion formula once more, we find that one choice of the r,'s is 9)

so that

8.

A REVERSIBLE MARKOV PROCESS

24 I

and

-ztk 1x1

=

. number of elements of Lk less than or equal to x.

k=U

I t is apparent that

Since for 0

< k < 1 x 1 there are precisely

elements in L , that are below x, we have

@(I -k 8

=

CPX(&). Z€S

T h e preceding results yield the following identities for @( -); XE

s

ZtSi

%€S Y <%

y<x

XES

wcx

$8. A Reversible Markov Process for Which the Canonical Distribution Is Invariant We shall describe an ergodic reversible Markov process z l , taking values in the set ,S of states, and having the property that its stationary

242

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

distribution over S is preciseIy the canonical distribution derived from the maximum entropy postulate. This Markov stochastic process can be used to place the canonical distribution into a dynamic context by exhibiting it as invariant under a semigroup of positive operators, viz., transition matrices of the Markov process in question. T h e transition rate matrix A of this process, i.e., the generator of the semigroup, then provides several interpretations (cf., Section 10) of this dynamic context in terms of behavior at the terminals of the networks, i.e., in terms of calling rates and mean holding times. Let x E S be a possible state of the network. I n Section 2 we have introduced the sets of states A, and B, with A,

= set of

states immediately above x, i.e., accessible from x by adding a new call,

B,

= set of

states immediately below x, i.e., accessible from x by a hangup.

T h e process zt to be considered can be described heuristically by saying that if xi = x then z ( . )is moving to each y E A, at a rate X > 0, to each y E B, at a rate unity, and to any other state at a rate zero. Its transition rate matrix A = (azu)is given by

ax, =

--(x(--X(A,I 1

x

0

if y = x if Y E B , if Y E A , if y $ A , u B , and y f x .

With this matrix we can define a Markov process z t in the usual way. [Cf., Doob (S)] Since zt has only one ergodic class of states, it is an ergodic process. A discussion and critique of possible physical interpretations of the rate matrix A is given in Section 10. T h e probabilistic interpretation of the rate matrix A is that if zt = x there is a conditional probability Ah o(h) that x ~ = + y~, for y E A,; there is a conditional probability h o(h) that zt+h= y , for y E B,; there is a conditional probability 1 - Ah1 A, j - hj x I - o(h) that x ~ = + x;~ all other events have a conditional probability o(h), as h -+ 0. T h e constant X is the calling rate per idle path. An alternative informal description of the Markov process z t is as follows: the length of time spent in any state x is a random variable

+

+

8.

243

A REVERSIBLE MARKOV PROCESS

independent of all other lengths of time spent in a state, having a negative exponential distribution with a mean 1 1x1 + h l A x I

a

At the end of a stay in x, a new state is chosen (independently of everything except x) from A, u B, according to the probabilities

x

for elements of A ,

1x1 + h l A , l 1 1x1 + x l A x T

for elements of B,

.

T h e equation Aq = 0 is the matrix-vector form of the statistical equilibrium equations for the process z L .These equations can be written out and solved explicitly, as follows: Aq = 0 is equivalent to {!XI

+hIAxI)gx

=

z c. QY

YEA,

+A

qy7

YEB,

XES.

(6)

We find by substitution that q, taken proportional to hlxl gives a solution. Hence the unique normalized (to be a probability vector) solution is

xes

This is precisely the canonical distribution of probability over S which was obtained earlier from the maximum entropy principle. Thus, one sense in which the canonical distribution is an equilibrium distribution is that it is invariant under the transition probability matrices of z t . I t will be noticed that the vector q has components that satisfy not only the statistical equilibrium equation ( 6 ) for z l , but also the much stronger condition q x a x v = qyayx

?

x, Y E

s,

which is an analog of the principle of detailed balance. I n the language of probability, this condition is that of reversibility; that is, it is

7.

244

A “THERMODYNAMIC” THEORY OF TRAFFIC

equivalent to the condition that the process z t look the same whether seen forward or backward in the sense that for any two times t and s Pr{z,

=x

and

z,

= y } = Pr{z, = y

and

z , = x}.

T h e reversibility of z1has important statistical consequences, explored in Sections 12-14. However, an immediate consequence is the following form of the Boltzmann H-theorem for z t : Remark 7.3.

Let =

-

c

YES

where

qTY(t)= Pr(z,

q d t ) 1%

=y

Then (d/dt)H,(t)

QrnY(f)l

1 zo = x}.

> 0.

T h e proof of this is well-known, being just Pauli’s proof of the quantum-mechanical H-theorem from the principle of detailed balance. [See Tolman ( 9 , p. 464).]

49. Analogy with Statistical Mechanics As its name suggests, the canonical distribution of probability over S , implied by the maximum entropy principle, resembles the canonical ensemble of statistical mechanics and thermodynamics. This analogy extends to several other concepts arising either in traffic theory or in statistical mechanics, and will now be described. I t is assumed that the reader is familiar with the rudiments of statistical mechanics; a lucid account can be found in Khinchin (6). Let us consider a conservative mechanical system embedded in a heat bath, and assume that it is decribed by a canonical ensemble. It can exchange energy with its surroundings; its energy is a randomly varying quantity. T h e basic identification we make is of the number of calls in progress in a connecting network with the energy of this mechanical system. I n other words, new calls in the operating network are analogous to increments of energy in the mechanical system, while hangups represent decrements of energy. T h e average energy is identified with the average load carried by the network.

9.

ANALOGY WITH STATISTICAL MECHANICS

245

T h e surfaces of constant energy in the phase-space of the mechanical system are analogous to the levels L, , i.e., the sets consisting of the various states with k calls in progress for k = 0, 1, 2, ... . T h e number 1Lk I of ways of putting up k calls, on which our theory rests, is the analog of the area of a surface of constant energy. Just as the canonical density function is constant over the surfaces of constant energy and maximizes the entropy for a given average energy, so is the canonical probability vector q constant over each Lk and maximizes H(q). T h e partition function of statistical mechanics is defined [cf. Khinchin (6, p. 79)] by Jr

where f is the phase-space, x E I‘ is a typical state, H ( x ) is the total energy of state x (here given by the Hamiltonian function), and d V is the volume element of phase-space. I n a similar way, the partition function @(*) is the generating function of the numbers I L , I, K = 0, 1, 2, ... . T h e set S of states corresponds to the phase-space f, H(x)is analogous to I x 1, the volume measure on F is analogous to the counting measure on S , and e-x replaces 5. I n Khinchin’s development (6) of statistical mechanics the temperature is defined as inversely proportional to the unique root €’ of the equation (dldt)) log Z(0) = average energy. Specifically, the absolute temperature T is given by 0

=

(AT)-’,

where k is Boltzmann’s constant. I n our model for a connecting network the analog (modulo a logarithmic transformation) of 9 is the solution X of (dldh) log @(A) = average load carried. T hus it is tempting to identify (logX)-l as proportional to the << temperature” of the traffic system. T h e matrix A , introduced in Section 8 as the “transition rate” matrix for the process z t , provides a sense in which the canonical distribution q is of “equilibrium” type. T h e reversibility of z t is analogous to the detailed balance property of transition matrices in statistical mechanics. [Cf. Tolman ( 9 , pp. 165 and 521).] This

246

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

property also implies that a form of the Boltzmann H-theorem is valid for z t , as we saw in Section 8. T h e analogies between our thermodynamic model of traffic and statistical mechanics can be collected in the following tabulation: Statistical mechanics Energy Partition function

Traffic theory Calls in progress

\ Generating function @(.) of number of ( ways of putting up k calls, 0 < k < w

Entropy Temperature Area of surface of given energy Transition rate matrix Detailed balance Equilibrium Heat bath Phase space T Volume measure on T

jlog(cal1ing rate per idle path))-I 1 Lt 1 = number of ways of putting up k calls

A

Reversibility of z 1 Aq = 0 Idle customers’ needs Set S of possible states Counting measure on S

$10. Discussion and Critique I t is now reasonable to consider possible physical interpretations of the stochastic process z t and of the transition rate matrix A in terms of items describing behavior at the inlets and outlets of the connecting network, such as calling rates, holding times, and routing rules. Obviously, transitions of z t from a state x into B, represent hangups, while transitions from x into A, represent new calls; the entries of A indicate the “rates” at which these events occur in the different states. However, the reasonableness, and so the acceptability, of z t as a model for traffic depends on the interpretations of z1and A in physical terms. Hence we must inquire whether (and how) the rates entered in A can be viewed as realistically describing the terminations of calls in progress, the occurrence of new calls between inlets and outlets of the network, and their routing or disposition. I n general, to construct a Markov process as a model for traffic in a connecting network whose states form the set S, it is usually sufficient to give, for each state x E S , and each inlet u and outlet v ,

10.

DISCUSSION AND CRITIQUE

247

(i) the hangup rate for the various calls in progress in x, (ii) the calling rate between u and ZI in state x, (iii) the method for disposing of requests that encounter congestion, receive busy tone, etc., (iv) the method for choosing routes of new calls.

A particular choice of the items (i)-(iv) leads to a transition rate matrix, and so to a Markov process. We shall assess the reasonableness of z t as a model for traffic in terms of items (i)-(iv) above by exhibiting two choices of (i)-(iv) that both lead to the rate matrix A of z t . I n the dynamic model z t described in Section 8, the role of the inlets and outlets is open to (at least) two different interpretations, each of which induces a corresponding interpretation of the transition rate matrix A. One possible interpretation of the inlets and outlets is to take them seriously as actual terminals or customers’ lines. They are then the outermost portions of the network under study, the original sources for traffic that enters the system, beyond which there is no more connecting or switching equipment. From any inlet, or to any outlet, there can be at most one call in progress. I n this case the rate matrix A can be interpreted as saying that in a state x each call in progress is terminating at a unit rate, that the calling rate from an idle inlet u to an idle outlet ZI is X . number of available paths from u to v in state x =

X . number of states covering x which include a ( u , ZI) call,

and that of the possible routes for a new call one is chosen at random (with equal probability for all). T h e reader can verify that this choice of (i)-(iv) does in fact lead to the rate matrix A. Note that this description does not provide for the generation of blocked calls. T h e choice of a unit hangup rate per call in progress is tantamount to measuring time in units of mean holding time, with the convenience that carried and offered loads come out in the standard units of erlangs. This unit hangup rate can be obtained as a consequence of assuming that the holding times are negative-exponentially distributed with mean unity, mutually independent, and independent of the random process describing new calls. This assumption of “negative exponential holding times” is a standard one in congestion theory. [See e.g. Syski (10, p. 9).]

248

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

More interesting (and questionable!) is the fact that under this interpretation the calling rate in a state x between an idle inlet u and an idle outlet v depends on the number of ways in which a call from u to v could be put into the network in state x. This calling rate can therefore change in time as the state changes, even if u and v remain idle. I t can be argued that this is an unrealistic feature, and that therefore z 1 is not a wholly reasonable model for telephone traffic in a network whose inlets and outlets are interpreted as terminals or customers’ lines. For surely the idle calling parties do not know the state of the system, nor the number of paths available for a call between them, and so they cannot (let alone do not) adjust their calling rates accordingly. I n a sense, it would be more intuitive and reasonable to assign a calling rate h to each idle pair (u, v) of terminals (an inlet u and an outlet u ) irrespective of the state x of the system. This basic calling rate for each idle pair (u, u> is then distributed over the states that cover x and realize ( u , v) [assuming that ( u , v) is not blocked, so that there are such states] in accordance with some routing rule. A stochastic process xt on S based on this idea is described in Section 1 1 , and is studied in detail in Chapter 8 and elsewhere by the author (11). From an a priori viewpoint, x 1 is a more reasonable model for traffic than z i. T h e objection (described above) to letting the calling rate for an idle pair depend on the state is severe. Nevertheless it does not necessarily destroy the usefulness of the process z t for describing traffic. Three comments are relevant here:

(1) If all calls can be put up in at most one way, then x 1 and z t coincide. (2) If calls can be put up in only a few ways, it may often be possible or useful to regard z1 as a small perturbation of x i obtained by raising various calling rates. This idea is explored in Section 11. (3) Even if z t is not in any precise sense a small perturbation of the apriori reasonable model xt, it deserves to be considered as a model of traffic. I t must not be forgotten that the usefulness of a theory rests more on its success in predicting than on its meeting criteria of reasonableness that are adduced a priori. However, it is possible to give the inlets and outlets a second interpretation, different from the one that assigns them the role of “outermost terminals.” This interpretation makes z t a fairly reasonable

10.

DISCUSSION A N D CRITIQUE

249

model of traffic, in the a priori sense we are discussing. I t consists in letting each inlet or outlet represent a point from which several or many calls can be in progress to other points in the system. Physically, such an inlet or outlet might be a PBX or central office serving a locality. As such, it would itself contain a connecting network which is left out of account in the model. I t no longer necessarily makes sense to speak of busy and idle inlets, or outlets. T o give an intuitive rationale for this interpretation and for the assumption about calling rates that corresponds to it, let us pick an inlet-outlet pair ( u , u ) and think of u and u as (possibly geographically separated) points between which there may be several calls in progress. For example, the network under study might be a toll network, and u and u might be local central offices acting as sources of traffic for the toll system. Or, for a second example, u and z, might be distinct switches in a large network inside a central office. I n such situations, it is natural to expect that if in a state there are many paths available for a call from u to u , then there is a larger probability that a requested call from u to TI arise in the next small interval of time h than if there were very few paths between u and v available. I n other words, it is reasonable that the calling rate in x for (u,u ) calls be a monotone increasing function of the number of paths available in x for such calls. A particularly simple monotone function is the linear one, and we shall assume that the calling rate for an idle pair ( u , u ) in x is h * number of paths available in x for (u, u ) call,

and that of the available paths one is chosen at random. Again, no provision is made for the generation of blocked attempts, since these will not affect the state probabilities when blocked calls are refused. We observe that A, can be partitioned and written as

A,

=

u

A&,

4,

(U,V)

where with

A z ( u , v) = { y : y covers x and realizes (u, v))

I ‘q2(u,v) 1

=

number of paths available in x for a (u, v) call.

Since routes for new calls are chosen at random we find that the transition rate from x to y E A, is exactly A, so that this second interpretation also leads to the rate matrix A .

7.

250

A “THERMODYNAMIC”

THEORY OF TRAFFIC

$11. A Markov Model Based on Terminal-Pair Behavior We now revert to interpreting inlets and outlets as the ultimate terminals of the connecting network. I n Section 10 it was suggested that under this interpretation an apriori reasonable model (a stochastic process xi) can be obtained by postulating an effective calling rate h > 0 per idle inlet-outlet pair. This can be done by assuming that each idle inlet calls an arbitrary outlet at a rate A, with attempted calls to busy terminals rejected with no change of state. T h e total attempt rate in a state x (excluding calls to busy terminals) is A.

\number of idle inlet-) loutlet pairs in x j .

If I is the set of inlets, and SZ that of outlets, with I and SZ disjoint, this has the quadratic form A(l I

I

-Ix

MI 52 I

-

I x I>.

As before, we assume a unit hangup rate per call in progress, with blocked calls rejected. T h e description of x i can be completed, finally, by specifying a method of routing. This we do by introducing a “routing matrix” R = (rZy)with the following properties: Let x be a state, and let 17, be the partition of A, induced by the equivalence of “having the same calls up, possibly on different routes”; relation then y,y 2 0

-

rZy= 0 unless y

E

A,

for Y EII, We note that CyEsrxyis exactly the number s(x) of attempts which would be “successful” if they arose in state x, and that 17,consists of exactly the sets A,(u, v) for ((u, u)} idle and unblocked in x. T h e routing matrix R is to have this interpretation: Each time the call {(u, u ) ) is to be completed in state x, a state y is chosen independently from A,(u, u ) with probability rzy , and the call is routed so as to take the system to statey.

11.

MARKOV MODEL BASED ON TERMINAL-PAIR BEHAVIOR

251

T h e foregoing assumptions lead to a rate matrix Q for x i defined by 1 AT,,

-

1x1

- Xs(x)

0

if Y E B ~ if Y E A , if y = x if y + ( A 3 . wB,)

and y # x.

This matrix is exactly like A except that for y E A, the rate from x t o y is not h but (the in general smaller quantity) hr,, , and that the diagonal terms are correspondingly increased so as to keep row sums equal to zero. For each Y ETI,, r,, for y E Y represents a distribution of the calling rate of some idle unblocked pair (u, v) over A,(u, v) = Y . Indeed A results from Q if all the r,, are replaced by unity. T h e process x i can be defined in terms of its rate matrix Q. T h e assumptions leading to the rate matrix Q and to the process x L have much a priori appeal; x i itself is discussed in detail in Chapter 8 and in a paper (11)already mentioned. Here we shall merely consider whether x imay be regarded as a perturbation of x t . Since each process is determined by its respective rate matrix, and since we are interested mostly in equilibrium behavior, we direct attention to asking how different are the respective equilibrium distributions over S for x i and z t . Thus, if p and q are probability row-vectors satisfying Qp = 0 and Aq = 0, respectively, how different is p from q ? We restrict ourselves to looking at Q - A and, to give a precise estimate, we introduce the norm

for matrices. T h e norm of Q - A can be seen to be

I! Q - A !I

= 2h

2 2 (1 - rzy)

XES

YEA,

where s(x) = number of

pairs that are idle and not blocked in x.

252

7.

A “THERMODYNAMIC” THEORY OF TRAFFIC

Letting p = max number of ways a call can be realized,

we find I A,

1

< ps(x), and hence

Let

I 1,

w = max x xes

so that @’(I)

<WI

S 1, and

II Q - -4 /I < 2X(p - 1)w I T h e average contribution (per state) to

‘I

-A

IS1

”

s I.

1 I Q - A I I is then

< 2qp

-

1)w.

$12. The Approach t o Equilibrium I t is known from the theory of Markov processes that the matrix Q ( t ) = (qzl/(t))of the transition probabilities qmH(t= ) Pr{z(t

+ s) = 3’ I z(s) = x},

t >, 0

of the process z t satisfies the Kolmogorov equations (dldt)Q(t) = AQ(t) = Q(t)A,

Q(0) = 1,

and that the study of the time-dependent (as opposed to the asymptotic, or equilibrium) behavior of z t can be carried out in terms of the characteristic values of A. Knowledge of the transition probabilities is essential, for example, in calculating the sampling error incurred in such load averages as

12. where

THE APPROACH T O EQUILIBRIUM

253

is the interval between successive discrete observations of (0, T ) is an interval of continuous observation of I z t I. I n this section we study the manner in which z1 approaches equilibrium in terms of the two principal characteristic values of A, i.e., that of largest, and that of smallest nonzero, magnitude. Applications to estimating the covariances of functions of z t , and to studying sampling error for the traffic averages in (7), are described in Sections 13 and 14, respectively. Our study of the approach to equilibrium is based on the observation that the matrix A of transition rates for the process z t is symmetrizable, i.e., is a self-adjoint operator in a suitably chosen inner-product space of finite dimension 1 S I. T h e probabilities

\

T

z 1 1, and

4

-

A'"\ 1 @(A) -

are all strictly nonnegative, and we use their reciprocals pz as weights in defining an inner product, XES

and a norm,

\/ s 11

= (s, s)1'2.

We now remark that for all states x, y from S, qyavx = qxaxv

or alternatively

a,xpx = ax,p,.

Indeed, this remark is the basis for the solution q given in Section 8 for the statistical equilibrium equations ( 6 ) of the process z t ;it has the important consequence that A is self-adjoint with respect to the inner product defined by (8), viz. Lemma 7.2.

Proof.

( A r , s) = ( r , As),for any I S I-vector r , s.

A is a real matrix, so

I n a similar way we prove

254

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

Lemma 7.3.

Proof.

we have

Since the matrix whose elements are ayxpx is symmetric,

Now

because Aq

=

0, and X

Y

Y

X

=o

because

Xz aUr= 0. This proves the lemma.

Theorem 7.2. The characteristic values of A are real and nonpositive. Zero is a simple Characteristic value corresponding to the characteristic vector q, normalized to unity.

Proof. T h e result follows from the known properties of selfadjoint transformations [See Halmos (22, pp. 153-155).] T h e characteristic values of A will all be of the Rayleigh quotient form

for some vector v ; by Lemma 7.3 this form is nonpositive. T h e probability vector solution q of Aq = 0 is unique, so that zero is a simple characteristic value. Furthermore, if 0 > rmax = rl 3 ... >, Y , ~ = ~ - rmin ~ is an arrangement of the characteristic values in decreasing order, the variational description of the characteristic values (22, p. 111) implies that, with / I v 1 1 2 = ( u , v), rmax = 71 = max{(dv, v) rmin =

1 v Iq, II a /I

r,Sl--l = min((Av, v)

I /I a I/

=

=

1).

11

12.

T H E APPROACH TO EQUILIBRIUM

255

T h e alternative notations rmax and r,in identify the two “dominant” characteristic values, and are introduced for later convenience to enhance the symmetry of the theory. One can now estimate rl from below by substituting suitable trial vectors in the Rayleigh quotient. Choosing a vector ZI with components where

it is easily seen that (9, u )

=

0, that I I

u 1[ =

I , and that

I n equilibrium, the average rate of new calls equals the average rate of hangups, as can be verified from the equilibrium equations Aq = 0. That is,

CIYl!7,

Y€S

and we find

=JQ4.1/A,/? YES

_ _m d r1 < 0, lJ2

a generalization of a result known [BeneS (13, p. 147) and Chapter 6 herein) for the simple busy signal trunk group (classical Erlang model). I n general, letting f ( . ) be any function defined on the set S of states, but not identically a constant, we define

256

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

Choosing now a vector v with components v,

=

f’(4- mi ‘SfPX

we obtain (by repetition of previous reasoning)

as a lower bound for rl . W e now define a set of vector-valued function {cs(t), x the condition Cxy(t)

=

q.,,(t)

-

y

47, 1

E

E

S , t 3 0 ) by

L5,.

T h e function c,(*) describes the approach to equilibrium from the initial state x at time t = 0.

=

since (d’dt)c,

=

qc,,

Ac,)

Ac,; that is, for each y

E

S,

13. Hence,

11

COVARIANCES OF FUNCTIONS OF

zt

257

cx j I being nonzero, we find

< ( d / d t )log /I cz 11’ < 2rmax

2~min

t

and Theorem 7.3 follows by integration. T h e argument just given is essentially reproduced from Kramer (24).

$13. Covariances of Functions of z, For the purposes of this section it is convenient to introduce an inner product closely related to but different from (-, .) of the previous section, and defined by ( a ,

d)’,

T h e associated norm is denoted by 1 1 r 1 1 ’ = ( r , r)’ljZ. T h e point of the “prime” notation is explained by the fact that the transpose -4’ of A is self-adjoint with respect to (-, -)’. Remark.

Where A‘ is the transpose of A (A’Y,s)’

= (Y, A’s)’.

Proof.

X

Y

x

y

Let f(.)be a function defined on S , and define a stochastic process fi by the condition ft

Theorem 7.4.

=f(G

The covariance of ftis given by

where the vector ,f is defined by

258 Proof.

7.

A "THERMODYNAMIC"

THEORY OF TRAFFIC

T h e covariance off, is X

Y

with Q(t)' denoting the transpose, and not the derivative, of Q(t). T h e covariance off, is thus the exponential generating function of the series of numbers TZ = 0, 1, 2, ... . (f,A''y)', These can be calculated with the help of the following results: Lemma 7.4.

Let the matrix elements of A n be a:

Then

qxa'n' = q a'n). ZY Y YZ Proof.

q a'") = x xy

4,

2

c

... aun-lY

azu1au1u2

U1'. .9,un-1

-

2 1I . .

.

aulxau,ul

... aYun-l

= q a'"'. Y YX

Lemma 7.5.

Let Q be the diagonal matrix of elements qz , x ( w , A'"w)' = (AnQw,Qw).

Proof.

From the three preceding results we obtain

E

S. Then

13. Theorem 7.5.

COVARIANCES OF FUNCTIONS OF

259

zt

T h e covariance o f f is

where the vector f is as in Theorem 7.4, and Q is the diagonal matrix of elements qz , x E S .

>

It is readily seen that An,n 1, is again a self-adjoint operator .), and that its characteristic values are precisely with respect to 0, the nth powers of those of A. Also, for any vector v and n

>

( a ,

if if

j GO 20

('nu'

n isodd n is zero or even,

(9)

so that by the variational description of characteristic values we have

provided that ZI 1q (in the inequalities involving rmax). Returning now to the vector Q f of Theorem 7.5, we find

I1 Qf 112

=

c.

9r2f;2Px

X

=

2 (f(4 - mf)*qX xes

= Of2

and X

so that Q f

1q. Letting v

2 n

uf rmax

X

= Q f in

(lo), we obtain

< (AnQf, Qf)< U f r L n ,

n even.

Unfortunately, these inequalities do not give useful bounds for the covariance R f ( * ) .However, such bounds can be obtained from the formula of Theorem 7.5 in an elegant way by applying the spectral theorem to A.

2 60

7.

Theorem 7.6.

of A , and let

A “THERMODYNAMIC”

THEORY OF TRAFFIC

Let a l ,..., ak denote the distinct characteristic values Ei, i = 1, ..., k, denote the perpendicular projection on the

subspace of all solutions A r given by

=

air. Then the covariance R,(.) of

2 ( E t Q f ,Qf)e”tt

ftis

k

W t )=

< <

with 1 k 1S given by fz = f(x)

1, Q the diagonal matrix -

1

i=O

mf .

of elements qx , x

E

S , and f

Proof. By the spectral theorem for self-adjoint operators (12, p. 56) we can write

A

k

=

zctiEi i=O

and i=O

We can now calculate with formula (9) of Theorem 7.5:

i=O

This proves Theorem 7.6. Since we know that zero is among the characteristic values (indeed, it is a simple one), one of the a’s, say a1 , will be zero. We may reasonably expect R I ( . )to approach zero for large t ; hence the constant, i.e., a l , term of R,(.) should be zero. This can be seen as follows: the subspace associated with zero consists of vectors proportional to the equilibrium vector q, because zero is a simple characteristic value; but we have already verified that q 1Qf; hence all r. (Elr, Q f ) = 0, Using this we prove

14.

26 1

R,(t) 3 0 for all t , and in fact

Corollary 7.1.

0

APPLICATIONS TO SAMPLING ERROR

< uf2erminltl < R,(t) <

all t .

uf2eTmaxlt1,

Proof. Since the Ei of Theorem 7.6 are perpendicular projections, they are linear, self-adjoint, and positive in the sense given by Halmos (12, p. 140); the usual term for positive is nonnegative semidefinite. Hence (JQ, 4 2 0

for any vector r . Since (Elr,Qf)= 0 if El is associated with the zero characteristic value, the result follows from Theorem 7.6, using

2 Ei = I , k

$14. Applications t o Sampling Error Let us suppose that n samples of the processf,(=f(z,)) are observed during an interval of equilibrium of x 1 at intervals T apart, and that the normed sum

is used as an estimate of E { f , } . We find that

j=-n

where I?,(-) is the covariance o f f , . By using the identity

2 (n n

-

1j

l)e-21jlu = n ctnh u -

I=-n

1 - e-2nu

2

- csch”u

= v,(u),

together with Corollary 7.1 of Section 13, we find that uf2n,(-+wmfn)

< Var{Sn> < uf2vn(-+max).

262

7.

A “THERMODYNAMIC” THEORY OF TRAFFIC

I n a similar way, if fl is observed continuously over an interval (0, T ) of equilibrium of z1and the time average

is used as an estimate of E{f,}, then

and Corollary 7.1 gives

$15. A Generalization As an extension of the maximum problem posed and solved in Section 5 we shall seek functions qdt),

xE

s,

t,

< t < t,

7

t,

< t,

such that for each t in [tl , t,]

2qz(t) xes

2 I x I qdt)

q z ( 4 >, 0 ,

=

1,

=

m(t) > 0

XES

J:: H(q(t))dt = maximum. I n other words, we look for a time-dependent distribution of probability over S with prescribed mean values for the function 1 * 1 on S , such that the integral of the entropy functional over ( t l , t 2 ) is a maximum. T h e Euler equations for this problem assume the trivial form [with Ll( and L,( .) as Lagrange’s multipliers] : a)

(aHj3q.J

-

L,(t) I x I - L,(t)

xE

= 0,

s

or, writing out the H-derivative, log qz(t)

+ 1 +L,(t) I x 1 + L,(t)

=

0,

xE

s.

15.

A GENERALIZATION

263

T h e argument of Lemma 7.1 following Eq. (4)shows that qz(*) is given by

where A(.) is the unique solution of the equation

This solution has the form of the canonical distribution at each time point in [tl , t,], and Theorem 7.1 in effect is just the special case of this result that arises when m(t) = m. I t is apparent that the form of this solution does not depend on what interval [tl , t,] was considered, so we may assume that m( .), and hence also A( .) and q( -), are defined on the real axis. Let us define the matrix-valued function A(t) by A(t) = .(uzv(t)) where 1 if Y E B ,

I n other words A(t) is obtained from the transition rate matrix A of z t by replacing the constant X by the function A(.). T h e n for each t 4t)dt)=0

i.e.,

T hus an analog of the statistical equilibrium equation holds at each point in time, and in this sense, a system described by { ~ ( t ) , t, t t z } may be said to be locally in equilibrium throughout the interval (tl , t J .

< <

Let us now redefine the process z t to be the time-dependent Markov process corresponding to the (time-dependent) transition rate matrix A( .). We know that if A( .) were a constant function with the

264

7.

A “THERMODYNAMIC”

THEORY OF TRAFFIC

particular value h(u), then the process z t would have a stationary or equilibrium distribution over S given by

We may therefore expect that if A(.) is not constant, but changes only slowly with time, and if x,,has the absolute distribution (vector) p(O), then z t for t > 0 has a distribution approximately given by q(t). REFERENCES

1. V. E. BeneS, Heuristic Remarks and Mathematical Problems Regarding the Theory of Connecting Systems, Bell System Tech. J . 41, 1201-1247 (1962). 2. V. E. BeneS, Algebraic and Topological Properties of Connecting Networks, Bell System Tech. J . 41, 1249-1274 (1962). 3. C. Y. Lee, Analysis of Switching Networks, Bell System Tech. J . 34, 1287-1315 ( 1955 ) . 4. P. Le Gall, Mkthode de Calcul de 1’Encombrement dans les Systkmes T&phoniques Automatiques a Marquage, Ann. Tilieom. 12, 374-386 (1957). 5. A. Jensen, An Elucidation of Erlang’s Statistical Works Through the Theory of Stochastic Processes, in The Life and Works of A. K. Erlang, Trans. Danish Acad. Sci. No. 2, 23-100 (1948). 6. A. I. Khinchin, “Mathematical Foundations of Statistical Mechanics.” Dover, New York, 1949. 7. G. Birkhoff, Lattice Theory, Am. Math. SOL.Colloq. Publ. (rev. ed.) XXV (1948). 8. J. L. Doob, “Stochastic Processes.” Wiley, New York, 1953. 9. R. C. Tolman, “The Principles of Statistical Mechanics.” Oxford Univ. Press, London and New York, 1955. 10. R. Syski, “Introduction to Congestion Theory in Telephone Systems.” Oliver & Boyd, Edinburgh and London, 1960. 11. V. E. BeneS, Markov Processes Representing Traffic in Connecting Networks, Bell System Tech. J . 42, 2795-2837 (1963). 12. P. R. Halmos, “Finite-Dimensional Vector Spaces,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1958. 13. V. E. Benei, The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement, Bell System Tech. J . 40, 117-148 (1961). 14. H. P. Kramer, Symmetrizable Markov Matrices, Ann. Math. Stat. 30, 149 (1959).

CHAPTER

8

Markov Processes Representing Traffic in Connecting Networks

A class of Markov stochastic processes x t , suitable as models for random traffic in connecting networks with blocked calls cleared, is described and analyzed. These models take into account the structure of the connecting network, the set S of its permitted states, the random epochs at which new calls are attempted and calls in progress are ended, and the method used for routing calls. The probability of blocking, or the fraction of blocked attempts, is defined in a rigorous way as the stochastic limit of a ratio of counter readings, and a formula for it is given in terms of the stationary probability vector p of x t . This formula is

(0,B)

(P,4 ’

or

2

c

P Z P Z

-X-,€ S

lWx

ZSS

where sz is the number of blocked idle inlet-outlet pairs in state x, and oil is the number of idle inlet-outlet pairs in state x. On the basis of this formula, it is shown that in some cases a simple algebraic relationship exists between the blocking probability 6 , the traffic parameter X (the calling rate per idle inlet-outlet pair), the mean rn of the load carried, and the variance u2 of the load carried. For a one-sided connecting network of T inlets (==outlets), this relation is l - b = - 1 2m h ( T - 2m)Z - (T - 2rn) 402 ’

+

265

266

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

for a two-sided network with N inlets on one side and M outlets on the other it is nz l - b = - I X ( N - m)lM - m) o2 .

+

T h e problem of calculating the vector p of stationary state probabilities is fully resolved in principle by three explicit formulas for the components of p : a determinant formula, a sum of products along paths on S , and an expansion in a power series around any point X > 0. The formulas all indicate how these state probabilities depend on the structure of the connecting network, the traffic parameter A, and the method of routing.

$1. Introduction We recall that a connecting system is a physical communication system consisting of (i) a set of terminals, (ii) a control unit which processes the information needed to set u p calls, and (iii) a connecting network through which calls are switched between terminals. Connecting systems have been described heuristically and at length in a paper ( I ) , and in Chapter 1. Also, some of the algebraic and topological properties of connecting networks have been studied in another paper (2),and in Chapter 2. T h e models to be used here have been described (but not studied) in a third paper (3),and in Chapter 7. These papers and chapters are a source of background material fur reading the present chapter; familiarity with them is desirable, but is not presupposed. T h e principal problem treated here is the exact theoretical calculation of the grade of service (as measured by the probability of blocking) of a connecting network of given but arbitrary structure; the calculation is to be carried out in terms of a mathematical model for the operation of the network. T h e model used here is a Markov stochastic process x t defined by some simple probabilistic and operational assumptions. T h e problem is first reduced to calculation of the stationary probability vector p of x Lfrom the “statistical equilibrium” equations. From the form of this reduction it follows that in many cases of practical interest the probability of blocking is uniquely determined by the mean and variance of the carried load, a fact heretofore known only for very simple systems. In the past, the application of A. K. Erlang’s very natural method of statistical equilibrium has been visited by a curse of dimensionality, that is, by the extremely large number of equations comprised in the

2. P R E L I M I N A R Y

REMARKS A N D DEFINITIONS

267

equilibrium condition. This difficulty has not only put explicit solutions apparently out of the question; it has even made it effectively impossible to reach a reliable qualitative idea of the dependence of the blocking probability on the structure of the network, the method of routing, etc. Three explicit formulas for the solution p of the equilibrium equations will be given. One is based on purely algebraic considerations, and the others largely on combinatorial and probabilistic notions. Because of the generality of the model with respect to network structure, these formulas are of necessity rather complex. Except in simple cases, they cannot be regarded as giving a final (or even a working) solution to the problem of calculating equilibrium probabilities. Still, they expose the mathematical character of the problem, and provide a badly needed starting point for well-grounded approximations. For only after one has studied and understood this character can he seriously consider ignoring some of it in approximations.

$2. Preliminary Remarks and Definitions As we have seen, various combinatorial, algebraic, and topological features of the connecting network play important roles in the analysis of stochastic models for network operation. Some of these features are now be described, and terminology and notations for them introduced. Let S be the set of permitted (i.e., physically meaningful) states of the connecting network under study. I t has been pointed out in previous chapters and in earlier work ( I , 2) that these states are where partially ordered by inclusion

<,

means that state x can be obtained from state y by removing zero or more calls. Also, these states can be arranged (in fact, partitioned) in an intuitive manner in a state-diagram,the Hasse figure for the partial ordering 6. This figure is a graph constructed by partitioning the states in horizontal rows according to the number of calls in progress, the kth row consisting of all states with k calls in progress. T h e unique zero,or empty, state of the network, in which no calls are in progress, is placed at the bottom of the figure; above it comes the row consisting of states with exactly one call in progress, and so on. T h e figure is

268

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

completed by drawing a graph with the states as nodes, and with adjacency matrix determined by the condition that states differing by exactly one call are adjacent. This means that in drawing the graph we place lines between states (in successive rows) that differ in point of one call. A maximal state is one that forms a summit of the state-diagram, Le., has no states above it in the partial ordering. For most systems the state-diagram has the following heuristic description: There is a unique “point” at the bottom corresponding to the zero state; there are usually many “points” at the top corresponding to the maximal states; and the diagram is very “fat” in the middle, because of the multitude of states with a moderate number of calls in progress. We mention at this point that usually the number of states, i.e., the number of elements of S , is astronomically large. Indeed, this fact has been a principal obstacle to theoretical -progress on problems of congestion in large connecting systems. For an illustration, in the network of No. 5 crossbar type, illustrated in Fig. 1, made for

FIG. 1. Structure of No. 5 crossbar network.

3.

SUMMARY OF CHAPTER

269

8

1000 lines out of square 10 x 10 switches, the number of maximal states alone is ( 109400. T h e set of inlets of a connecting network is denoted by I,and the set of outZets by 9. It is possible that I n 9 = 8, that In 9 # 8, or even that I = 9, i.e., that all inlets are also outlets, depending on the community of interest” aspects of the structure of the network. It is assumed that every call or connection is made only between an inlet and an outlet. If x is a state, the notation 1 x 1 (read “the norm of x”) will denote the number of calls in progress in state x. If X is a set, then I X I will denote the cardinality of X,i.e., the number of elements of X . We define the levels (6

L,={xcS:

k = O , I ,..., m a x l x l

1x1 = k ) ,

Z€S

as the sets of states in which a specified number of calls is in progress. T h e (Lk)form a partition of S, UL,=S k

L , nL j

=

8,

k #j.

T h e “neighbors” of a state x are just those states that can be reached from x by adding or removing one call. These neighbors y of x can be divided into two sets according as y > x or y < x; so we are led to define A,

= set =

B,

of neighbors above x

set of states accessible from x by adding one call,

neighbors below x = set of states accessible from xy by remosing one call.

= set of

$3. Summary of Chapter 8 T h e basic probabilistic assumptions that define the randomness in the traffic models to be studied are given precise statement in Section 4.

270

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

They are, briefly, (i) the hangup rate per call in progress is unity, and (ii) the calling rate per idle inlet-outlet pair is a constant X > 0. Various operational aspects, such as the disposition of lost calls, and the method of routing, are specified and discussed in Section 5 . It is assumed that lost calls are refused without a change in state, and that routes for calls are chosen in a way that depends both on the call being set up or processed and on the current state of the system. I n Section 6 these probabilistic and operational assumptions are summarized in a transition rate matrix, Q. In Section 7, a Markov stochastic process x t (the mathematical model for the operating system) is defined, and the statistical equilibrium condition Qp = 0 for the stationary probability vector p of xl is formulated. I n Section 8, the probability b of blocking is defined as the (probability one) limit of a ratio of counter readings, and a formula for b is given in terms of the stationary vector p . From this formula it is shown, in Section 9, that a simple algebraic relationship often exists between the blocking probability b, the traffic parameter A, the mean load carried, and the variance of the load carried. T h e remainder of the chapter is devoted to the study and calculation of the vector p of stationary probabilities. Two explicit solutions, one algebraic and one combinatorial in character, are given in Section 10. I n Section 11 it is shown that the combinatorial solution is a special case of a general formula for the stationary measure of an ergodic Markov process. T h e dependence of p = p ( h ) on the network structure and the method of routing is analyzed in an elementary way in Section 12. I t is first shown that p(*)/p,(.)has components that are analytic in a neighborhood of the nonnegative real axis, and so are expressible in the form

For ,u = 0 and E = X sufficiently small, this gives an expansion of p in powers of A. It is then shown that, with 1 x 1 the number of calls in progress in state x, p , is of order A!"' as h + 0. This result renders possible a recursive calculation (Sections 12 and 14) of the coefficients c,(x, 0) from the partial ordering of S and a matrix used to specify the method of routing. Once p is developed as a power series in A, a similar expansion is readily given (Section 13) for the probability b of blocking.

<

4. PROBABILITY

27 I

I n Section 15, finally, we completely solve the problem of calculating the coefficients C , ~ ( X ,A) for arbitrary values of X > 0, giving each such coefficient both a combinatorial interpretation, and a n explicit formula, viz., a sum of products along paths through S which are trajectories for xLpermitted by the routing rule.

$4. Probability T o construct a Markov process for representing the random trajectory of the operating network through the set S of states, we shall make two simple probabilistic assumptions. T h e traffic models to be studied embody what has come to be known as a “finite-source effect,” that is, a dependence of the instantaneous total calling rate on the number of idle inlets, and on that of idle outlets. I n an attempt to describe this dependence in a simple rational way, let us imagine a customer located at one of the inlets [outlets] of the connecting network, and seek to assign him a calling rate, assuming that he is in an idle condition. We shall suppose that the traffic he offers is homogeneous in the sense that he calls every outlet [inlet] at the same rate, or with the same frequency. Indeed, we shall assume that all customers offer homogeneous traffic. Now on most occasions when he is making a call, a customer does not know whether the terminal he is calling is busy or idle. Thus, if he is on an inlet [outlet] it seems reasonable to suppose that there is a probability Ah

+ o(h),

A

>0

that he attempts a call to a particular outlet [inlet] (distinct from his own) in the next interval of time of length h, as h -+ 0, whether that outlet [inlet] is busy or not. T h e qualifying phrase “distinct from his own” is inserted to cover the case in which some inlets are also outlets, and in which it is reasonable to suppose that an idle terminal that is both an inlet and an outlet does not attempt to call itself. We therefore make these two probabilistic assumptions: (a) Holding times of calls are mutually independent random variables, each with the negative exponential distribution of unit mean. (b) If at time t the network is in a state x in which at least one member of the inlet-outlet pair ( u , u)E I x ,f2 is idle (that is, one of u or u is not involved in a call in progress), the time elapsing from t

272

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

until a call between u and TJ is attempted is a random variable having a negative exponential distribution with a mean l/X, X > 0. For different choices ( u , TI)and different occasions t , these times are all mutually independent and also independent of the call holding times. These assumptions can be rendered in the informal terminology of “rates” as follows: (i) T h e hangup rate per call in progress is unity. (ii) T h e calling rate between an idle inlet u (outlet TJ)and an arbitrary outlet TJ (inlet u ) with u # TJ is h > 0. Assumptions (a) and (b) provide all the “randomness” needed to construct our models. T h e choice of a unit hangup rate merely means that the mean holding time is being used as the unit of time, so that only the one parameter X need be specified.

$5. Operation T o complete the description of the traffic models to be analyzed we must indicate how the network is operated. Since in the present work we are taking into account only the network configuration, and omitting consideration of the control unit, it suffices to describe how calls to busy terminals are handled, how blocked calls are treated, and how routes or paths through the network are chosen. It will be assumed that attempted calls to busy terminals are rejected, and have no effect on the state of the system; similarly, blocked attempts to call an idle terminal are refused, with no change in the state of the system. All successful attempts to place a call are completed instantly, with some choice of route. T o describe how routes are assigned to calls, we introduce a routing matrix R = ( r Z l / )with , the following properties: For each x let 17, be the partition of A, induced by the equivalence relation of “having the same calls up,” or satisfying the same “assignment” (of inlets to outlets); then for each Y E 17,, rzl/ for y E Y is a probability distribution over Y ; in all other cases rxV = 0. (Cf., Chapter 7, Section I 1 .) T h e interpretation of the routing matrix R is this: Any Y E17, represents all the ways in which a particular call c not blocked in x (between an inlet idle in x and an outlet idle in x) could be completed

6.

TRANSITION RATES

273

when the network is in state x; for y E Y , rzy is the chance that if this call c is attempted, it will be routed through the network so as to take the system to state y. That is, we assume that if c is attempted in x, then a state y is drawn at random from Y with probability r z y , independently each time c is attempted in x; the state y so chosen indicates the route c is assigned. T h e distribution of probability {r,. , y E Y } thus indicates how the calling rate X due to the call c is to be spread over the possible ways of putting u p the call c. I t is apparent that

2 rry

=

number of calls that can actually be put up in state x

YEA,

= s(x)

(“successes” in x),

the second equality defining s(.) on S. This account of the method of routing completes the description of the traffic models to be studied.

$6. Transition Rates For the purpose of defining a Markov stochastic process it is convenient and customary to collect the probabilistic and operational assumptions introduced above in a matrix Q = (qxy) of transition rates, here given by Y E B, Y €A, y =x otherwise.

1 AT,

-

1 x 1 -As@) 0

T h e number q X u ,for x # y , has the usual interpretation that, if the system is in state x,there is a chance 9ruh

+ o(h)

that it will move t o y in the next interval of time of length h, as h Similarly 1 - pz,h 4h)

--f

0.

+

is the probability that the system will stay in x throughout the next interval of time of length h, as h + 0.

274

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

$7. Markov Processes I n terms of the transition rate matrix Q it is possible to define a stationary Markov stochastic process {xi, -a < t < +a}taking values on the set S of states. T h e matrix P(t)of transition probabilities PZY(t)

=Wx, =Y

I xo

= x:

of x t satisfies the equations of Kolmogorov d -P(t) dt

= QP(t) = P(t)Q,

P(0) = I ,

and is given formally by the formula P ( t ) = exptQ.

There exists a decomposition of the set S of states into Theorem 8.1. a transient set F and a single ergodic set S - F containing the zero state; members of F have the property

on S - F there is a unique stationary (or equilibrium) distribution - F } such that

(p, ,x E S

hhPZY(t)= py > 0,

c

Prl?r,(t) =

P,

1

y

E

S -F , x E S

Y E S- F ,

all t

xeS-F

2 95,PZ

= 0,

YES-F.

TES-F

T h e existence of the unique ergodic set S - F follows from Proof. the fact that the zero state is accessible from every other state by hangups. T h e existence and character of the limit of p,,(t) as t -+ GO is a consequence of the work of Feller ( 4 , p. 436, exercise 19), i.e., of the fact that the characteristic values Y of Q satisfy r = 0 or Re(r) < 0. [See also Bellman ( 5 , p. 294).]

7.

MARKOV PROCESSES

275

T o prove the uniqueness ofp, suppose that q is a different probability vector on S - F that also satisfies t& “equilibrium” condition

c

xES-F

qxv42

= 0,

Y

E

s

--*

Then by Kolmogorov’s equation

Integrating from 0 to t, and using P(0) = I , we find

Since S - F is the only ergodic set, the left-hand side approaches

p, as t -+ co. Hen cep

=

q.

I t is convenient to extend the dimension of p to I S 1 by adding zero components for states in F , so that pZY(t)-+p, >, 0 for all x, y E S. T h e consideration of the transient set F is not just a mathematical fillip, since a “good” routing rule R may explicitly make certain “bad” states unreachable from the zero state, and thus place them in F to good purpose. I n the notation of Halmos (6, p. 65), the stationary probability vector satisfies the equilibrium condition

Qp

= 0.

This is the classical equation of state, or equation of statistical equilibrium, familiar in traffic theory: For our process x t it takes on the takes on the rather simple form

T h e left-hand side represents the average rate of exits from x, while the right-hand side is the average rate of entrances into x, in equilibrium. We define

276

8.

Lemma 8.1.

MARKOV PROCESSES REPRESENTING TRAFFIC

For 1

< k < w = max,,,

I x 1,

2 PAX).

kPk =

XEL,--l

Proof.

obtain

From the statistical equilibrium equation for x

2 P,

= A4O)Po

?/,AO

=

0 we

J

which is Lemma 8.1 for k = 1. Assume that the lemma holds for a given k 3 1. Summing the statistical equilibrium equations over x E L,c we find kPk

+

z;

EL,

s(4Px =

cc

xeL, yEAZ

P,

z;c

+

x t L , vEB,

P J Y , .

T h e second sum on the right is the same as

and by definition,

z

=€A,

YYX =

4Y).

Hence (induction hypothesis) the second sum equals kp,, . I t is easy to see that in the first sum on the right each p , is counted exactly I y 1 times, i.e., ( K + 1) times, since for a given y E Lk+l there are exactly (k + 1) elements x € L A for which y E A, . T h u s the first sum is (k

+ 1)

c

P,

= (k

+

IlPk+l>

YELk+l

and Lemma 8.1 follows by induction. This result could also be obtained from the general observation that the statistical equilibrium equations are equivalent to the principle that for any set X of states the average rate of exits from X equals the average rate of entrances into X . [See Morris and Wolman (7).]

58. Probability of Blocking T h e fraction of calls that are refused because they are blocked, or the probability of blocking, is a quantity of particular interest to

8.

PROBABILITY OF BLOCKING

277

traffic engineers; they use it to assess the grade of service provided the customers by an operating connecting network. T h e rigorous theoretical calculation of blocking probabilities has long been an outstanding problem of traffic theory. This problem is outstanding in both senses of the word: it is conspicuous, and it is unsolved. I n fact, not even the definition (let alone the calculation) of the probability of blocking has received adequate treatment; for example, the otherwise monumental treatise of R. Syski (8) does not give a general account of blocking probability. Since it is desirable to have a close connection between t h oretical 5 quantities and their physical meanings in terms of measurements, we shall approach the study of blocking probabilities by asking how these probabilities might be measured “in the field.” T h e most natural method of measuring the fraction of blocked attempts seems to be this: T o the control unit of the connecting system under consideration we attach two counters; the first will count u p one unit every time an attempted call is blocked, and the second will register one unit every time a call is attempted; the ratio of the reading of the first counter to that of the second should, after a long time during which the system’s parameters remain constant, be an approximate measure of the fraction of blocked attempts. For mathematical convenience, one can then define the probability of blocking to be the limit (as time increases without end) of this ratio of the counter readings. This mathematical definition was first proposed by S. P. Lloyd, although, of course, the ratio has been the practical definition for 50 years, being the “peg count and overflow ratio.” A precise mathematical version of this measurement procedure can be given as follows: On the same sample space as that of the process x t that describes the operating network, we define two additional stochastic processes {b(t),t 3 0} and {a(t),t 3 0} by the (respective) conditions b(t) = number of blocked attempted calls in (0, t ] , a ( t ) = number of attempted calls in (0, t ] .

These stochastic processes are the mathematical analogs of the counter readings. I t is reasonable to use the limit

278

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

of the ratio of b(.) to a ( . ) as a mathematical definition of ability of blocking, provided that the limit exists in a suitable sense. W e show that this limit exists and is constant with probability one, and we give a formula for it.

The probability of blocking b, defined by

Theorem 8.2.

exists and is constant with probability one; its almost sure value is

X€S

where p is the vector of stationary probabilities, and

PT = number of idle inlet-outlet pairs that are blocked in state X, ciz =

Proof.

over S ,

number of idle inlet-outlet pairs in state x. I t can be seen that a ( t ) and b(t) can be written as sums

a(t) =

c

a,(t),

X€S

XES

where a z ( t ) = number of attempted calls made in (0, t ] with the system in state x,

b,(t)

=

number of blocked attempts made in (0, t ] with the system in state x.

Now a blocked attempt occuring at an epoch u such that xu = x does not change the state of the system. Such an epoch u is a regeneration point of the process x L . A successful attempt occurring at an epoch u at which xu = x does change the state of the system. T h e time interval from u back to the last previous epoch v at which a successful attempt occurred in state x, however, is independent of the

8.

279

PROBABILITY OF BLOCKING

behavior of x 1 for t > u ; it depends only on the fact that the system left x by adding a new call, not on what new call it was, nor on where into A, x 1 went as this new call was completed. This can be seen as follows: We have 11-

ZI

=T-u+u

- T

where T is the epoch at which x was last entered prior to u. Now T - v is independent of x 1 for t > T if x , + ~is known to be x, because x t is a Markov process. Let U be an event measurable on {xt, t > u } . Then

where

Thus the time intervals , ,B2, ... elapsing between successive blocked attempts in state x, and those a l ,a 2 ,... elapsing between successive attempts in state x, both form sequences of mutually independent, and except possibly for the first elements p1 and a l , identically distributed random variables. That is, the elements of each sequence are mutually independent, since one consists of partial sums over blocks of the other. Both these sequences can be studied, then, in terms of a sequence xl, x 2 ,... of mutually independent random variables, all (except possibly xl) identically distributed. We define for t 3 0 and k >, 0

so = 0

n(t) = k

if and only if

n ( t ) = czT(t) or

b,(t).

S,

< t < S,,,

,

280

8.

M A R K O V PROCESSES REPRESENTING TRAFFIC

We assume that n(.) is a separable stochastic process. It is now straightforward to show that t-ln(t) approaches a limit with probability one, and to find the limit. Let us put, for t > S , ,

T h e first factor converges to E-l{x,} with probability one, by the law of large numbers. T h e local suprema of t - Sn(t) ___-_ t

for t >:

S , occur at the points t=S,,

k = 2 , 3 ,...,

and have the values

Again, the first factor converges to E-l{x,} with probability one by the law of large numbers. Since E{x,) < co, and {xk , k >, 2) are identically distributed,

and it follows from the Borel-Cantelli lemma that, for any Pr{x, > ~k for infinitely many values of K >, 2) Hence xli = o ( k ) as k probability,

+ co, with

c

> 0,

= 0.

probability one, and with the same

__ sw,t, -1 t

as

t--tco.

It follows that with probability one, lim t-laT(t) = E-' {time interval between successive t+m attempted calls in x } , lim t-%,(t) = E-I (time interval between successive t +n blocked attempted calls in XI.

9.

28 1

A BASIC FORMULA

Furthermore [cf., Smith (10, p. 247, Eq. (1.2) and p. 249)],

However, by Feller's renewal theorem [cf., Smith (10, p. 246)], we know that lim t-lE{h,(t) thT

1 xo = y } = E-1 {time interval between siiccessive blocked attempted calls in x}.

Hence, with probability one, + XpT,Bz

t-%,(t)

as

t

--f

rx.

A similar argument shows that with probability one t-la,(t)

+ Xp,,

as

t

---f

\m

and completes the proof of Theorem 8.2.

59. A Basic Formula Engineers have recognized (at least) four quantities as significant for the study and design of connecting networks carrying random traffic. These are the calling rate, the average load carried, the variance of the load carried, and the probability of blocking. In our model these quantities are given respectively by X

=

m

=

calling rate per idle inlet-outlet pair

2 1 x I p,

=

average number of calls in progress

T€S

u2 =

C(~1 x

- m)2pp,

T€S

b

=

(P,P ) / ( P , a!.

I t is natural to ask whether there exist any systematic relationships between these quantities, or between these and (possibly) other simple parameters of the network under study. Such relationships

282

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

would be particularly useful and significant if they were largely independent of the structure or configuration of the connecting network, and were valid either for all networks or for large classes of them. We shall show that there often exists a simple algebraic relation among A’ m, u2, and b. Its exact form depends on which inlets are also outlets. First we prove Theorem 8.3.

The probability b of blocking can be written as b=l-

m

2Px% ZES

or, in words, as b

average load carried 1 1 -~ (calling rate per idle pair) x (average number of idle p a 3 *

I n equilibrium, the average rate of successful attempts must Proof. equal the average rate of hangups. Hence, intuitively,

Since p, = aZ - s(x), the result follows from Theorem 8.2. T h e actual validity of the identity (2) can be inferred from Lemma 8.1, by summation on k. Formula (1)) rewritten in the form 1-b=

average load carried average rate of attempts’

should be viewed as a direct generalization of Erlang’s classical loss formula for c trunks, blocked calls cleared, and calls arising in a Poisson process of intensity a > 0. I n that case the probability of loss is ac

-

9.

283

A BASIC FORMULA

and it can be seen that

average number of busy trunks -total calling rate

T o exhibit useful special cases of the general formula (1) of Theorem 8.2, we introduce a partial classification of connecting networks. A network is called one-sided if I = J2, i.e., if all inlets are also outlets; a network is two-sided if I n J2 = 8, i.e., if no inlet is an outlet.

For a one-sided network of T terminals

Corollary 8.1.

b Proof.

III

=

=

1

- -

1 X ( T - 2m)"

-

2m ___-( T - 2m) 4dL .

For the one-sided network in question, we have I

IB/= T , a n d s o

C;p,az = +{TZ- ( 2 -~1)2m - T 5XS

Corollary 8.2.

and N on the other

Proof.

+

= Q,

+ 4m' + 4021).

For a two-sided network with M terminals on one side

I t is clear that in this case

284

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

so that

Zp,a,

=

(M

-

m)(N - m)

X€S

+ u2.

Each of the foregoing corollaries exhibits an explicit algebraic relationship between A, m, U, and b, based only on the one- or twosidedness of the network. T he preceding corollaries can be used to show that, in a large system, the numerical value of the constant X will be small-indeed, of the order of the reciprocal of the number of inlets and outlets. This can be seen by the following heuristic argument, carried out for a one-sided network with T terminals: Suppose that each terminal carries q (0 < q < 1) erlangs and that the blocking probability b is so small that we can ignore it and set b

=

1 - -1 A (T

-

2m)2

2m ( T - 2m) + 4a2

~

-

0.

Since the network is one-sided, any load carried by one terminal is also carried by some other terminal, and so ql'

=

2m,

whence A=- 1

T (1

-

q)2

~

(1

4 -

Because

and

we have 0

< 4u2!T2 < 1, and so Am-

with

const T

~-

q ) / l ' $- 4a2/T2*

10.

EQUATIONS OF STATISTICAL EQUILIBRIUM

285

$10. Solution of the Equations of Statistical Equ iIibrium SO far, we have shown that the theoretical determination of the blocking probability b reduces to that of the stationary vector p or, in many cases, to that of the mean m and variance u2 of the carried load. I n either case, some knowledge of p is required. Most of the rest of this chapter, therefore, is devoted to the calculation of p and to the study of its properties. I n the past, the application of A. K. Erlang’s very natural method of “statistical equilibrium” to congestion in connecting networks has been visited by the curse of dimensionality, that is, by the extremely large number 1 S 1 of equations comprised in the stationarity condition Qp = 0. This difficulty has not only put explicit solutions apparently out of the question; it has even made it effectively impossible to reach a reliable qualitative idea of the dependence of the state probabilities (p, ,x E S } on the structure of the network and on the method of routing. T o be sure, it has always been possible in principle to solve Q p = 0 by successive elimination of unknowns; however, when the dimension o f p is of order IO4O or so, this remark is hardly helpful. Since successive elimination can be used to solve Qp = 0 for any “ergodic” transition rate matrix Q, it neither elucidates nor uses any of the special features of the matrices Q that arise in problems of congestion in networks. Thus, even were it is algebraically feasible, the method of successive elimination treats our matrices Q as indistinguishable from other matrices possessing a zero characteristic value. We shall give several explicit solutions of the equilibrium equations. One is based on purely algebraic considerations, and the others largely on combinatorial and probabilistic notions. Because of the generality of our model with respect to network structure, the formulas appearing in the solutions are necessarily rather complex. Except in simple cases, they cannot be regarded as giving a final (or even a working) solution to the problem of calculating equilibrium probabilities. Nevertheless, they expose the mathematical structure of the problem and provide a badly needed starting point for well-grounded approximations. For only after one has studied and understood this structure can he seriously think about throwing some of it away in approximations.

286

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

T o describe the solutions in full detail, we need various preliminary definitions and conventions. We suspect that the minimum value of the blocking probability b is achieved by a routing matrix R consisting entirely of zeros and ones, i.e., by a deterministic rule. So we assume henceforth that R has only zeros or ones for entries. A path on S of length 1 0 is any ordered sequence xo , x1 , ..., x I of (I 1) elements of S. A lower case pi, T , will be used as a symbol for a generic path on S , and we write

+

7T

= {xo ,XI,

...)x,},

1 = L(7-r)

to indicate that 71 is a path of length l(7r) consisting of xo , x1 , ..., x t in that order. Note that paths of length zero are countenanced. A path T is a loop if xo = XL , and also Xi

#

xj

<

whenever 0 < i < j l(71).A loop of length zero is a path of length zero. If 7r = (x,, , xl, ..., x,,] is a loop, each element xo,xl,..., etc. will be spoken of as being on 7 r . T h e elements x and y of S are called adjacent in the graph of ( S , <),i.e., in the state diagram, if one of the following equivalent conditions holds: (i) x covers y or y covers x, (ii) y E A, or x E A,, (iii) x and y differ by exactly one call in progress.

A path on S is called continuous if successive elements of the path are adjacent. I n order that x 1 have positive probability of following a path 71, it is not enough that 7r be continuous. For evidently the action of the routing matrix R (assumed to consist solely of zeros and ones) is to prohibit certain paths on S as (parts of) possible realizations of the process x i . Here “possible” of course means “having positive probability.” There exists then a class of those paths that are permitted by R, definable in several ways. One such way is as follows: A path

10. EQUATIONS

287

OF STATISTXCAL EQUILIBRIUM

= {x” , xl,..., xl}on S is permitted by R if for each i in the range 1
T

XI

or

E

r’*-lx,

=

1.

T h e set of paths permitted by R is denoted by P. With X a subset of S, perm(X) will denote the set of all permutations of X , i.e., one-to-one maps of X onto itself. We let Y1

TYZ

7

*a-

>YlSl

be an arbitrary simple ordering of S , and we define the ordinal number of a state x E S by the condition

W(X)

~ ( x= ) n,

if and only if

x

<

=yn ,

n = 1, 2, ..., 1 S

I.

<

For each m, n in the region 1 m, n 1 S 1, we define a function on the domain I i I S I by the condition &?%(a)

< <

1

< i < min(m, n)

We observe that Cmn(.) has an inverse for each m and n. Now let ?(*) be a permutation of the set of states with the mth and nth removed; then defines the permutation a,,(y) associated with g?. Also, sgn amn(cp) is 1 or - 1 according as the permutation a,,(y) is even or odd. T h e “hangup” matrix H = (hzu)is defined by the condition

+

hx‘

=

(1 10

if Y E B , otherwise.

Let x and z be states, and suppose that 7r = {xo , ..., x l } is a path in P beginning at x and ending at x, so that xo = z and x I = x. Suppose

288

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

also that the trajectory represented by .rr contains m new calls, Le., there are exactly m values of i in the range 1 i I such that

< <

X, E

Since n starts at z , in which I z 1 calls are in progress, and ends up at x,in which there are 1 x 1 calls in progress, it is evident that I(T)

=

2m

4-1.z 1

-

1x1.

T h e set of paths that start at a state z , never return to z , and end up at x f z , is denoted by Kzz.

<

T h u s n belongs to K,, if and only if x,,= z , xi f z for 0 < i l(.rr), and x l = x. Let n = (xo , x l ,..., be a path on S , and letf(.rr, .) be a function defined for x o ,xl,..., x l . I n terms off(^, -) and T , we define a product along the path .rr by the expression

XI>

iLji

f ( T , Xi).

It is convenient to abbreviate this product by the readily understandable expression

I n the special case thatf(.rr, -) has the form f ( T ,Xi)

= h(X,Gl , &),

i = 1, ..., 1,

we abbreviate the product by

T h e notation p . is supposed to represent the predecessor of an element in ?T. T h e first and simplest solution of the equilibrium equation Q p = 0 to be given is based on an observation made by I. W. Sandberg, namely, that the determinant of Q is zero, so that Q adj(Q) = 0, and

10.

EQUATIONS OF STATISTICAL EQUILIBRIUM

289

thus columns of the matrix of cofactors of Q should give solutions of Qp = 0. T h e author has not succeeded in elucidating the probabilistic significance of these simple algebraic facts. I t will be seen later that the other solutions to be given are, on the other hand, natural, plausible, or even obvious from a probabilistic viewpoint, but are algebraically involved.

< < 1 S 1.

m Let m be an integer in the range 1 A n unnormalized nonnegative solution p of Qp = 0 is given by

Theorem 8.4.

pun= ( - l ) l S l - l + m + n

r~

Plz)=z

(-

c;

wPerm(S-(u,,Y,)

I I -W )

)

sgn amn(Y)

rI

V(Z)#Z

+

(hrri(z) k Y Z + ( - ) ) .

Proof. Since det(Q) = 0, it follows [Birkhoff and MacLane ( 9 , p. 290)] that no matter what ordering of S is used,

Q adj(Q) = 0, where “adj” denotes the adjoint matrix, i.e., the transposed matrix of cofactors. Let C = (cz,) be the matrix of cofactors of Q corresponding to the ordering y , , y z , ...,y i S Iof S , and suppose that the entries of Q are also arranged according to this ordering. Then C

= adj(Q),

and we find that ZES

Thus any column of the matrix C of cofactors of Q gives a solution of the equilibrium equations. I t follows from a result of W. Ledermann [Bellman (5, p. 294, exercise lo)] that (-1)ISl-1

C3-y

3 0.

We see that all the cofactors czu have the same sign, and each column of the matrix C yields a nonnegative solution of Qp = 0. Hence all columns are proportional, because there is only one nonnegative solution, up to normalization. T h e theorem follows from the standard formula for a cofactor as a determinant.

290

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

If x E S is any state, then an unnormalixed solution of the statistical equilibrium equations Qp = 0 is given by

Theorem 8.5.

and, for x # x,

T h e formula given can be verified by direct substitution in Proof. the equations [Wx)

+Ix

llPx

=

c: P, + c

YEAZ

P J Y X

I

a! E

s.

YEB,

Convergence of the infinite sum follows from our Theorem 8.7 and from Feller ( 4 , p. 378, exercise 19).

$11. Stationary Probability Measures for Ergodic Markov Processes

I n order to shed light on Theorem 8.5 (and also to prove it by a probabilistic argument) we shall consider in this section the general problem of calculating the stationary probability measure of an ergodic continuous parameter Markov process on a finite number of states. Our object is to give an explicit formula for the measure in terms of the transition rate matrix. Again, it is needless to mention that a formula of such generality must be fairly complex. Applied to familiar Markov processes whose stationary measures are well known, the formula to be given yields some unexpected combinatorial identities, not pursued here. We shall now use the notations x t , S , Q, and P ( - ) to describe an arbitrary Markov stochastic process x t in continuous time, taking values in a finite set S of states with transition rate matrix Q = (qzl/) and transition probability matrices P ( t ) = pz,(t)), t real. I t is assumed that there is a single ergodic class of states. Such a general interpretation of notations already introduced (for specific processes describing traffic in connecting networks) is made to avoid defining new terminology; it is made in this section only, and should cause no confusion.

11.

29 1

ERGODIC MARKOV PROCESSES

If z is a state, a return to z is defined to be an epoch of time at which x 1 reaches x, i.e., u is a return to z if for some E > 0, x t # z for u - E < t < u and x - z for u < t < u E. A departurefrom 1 x is an epoch of time at which x t leaves z , i.e., u is a departure from z if for some E > 0, x t = z for u - E < t < u and xi # z for u < t <: u + E . A return time to z is a period of time elapsing between a departure from z and the next return to z. We set, for t >, 0,

+

H,(t)

= E{number

of returns to z in (0, t ] 1 x,,= z},

Eireturn time to z } , qz = -qzz = E-l{length of a stay in z}.

pz =

T h e notation H,(.) has been chosen because the defined quantity has an obvious resemblance to the classical renewal function. [See Smith (lo).] There is a simple relationship between the equilibrium probability of a state x, and the quantities p, and 4;, this is expressed in the next theorem which, though probably familiar, is included for completeness. Theorem 8.6.

For x E S , p ,

=

[1

+ q,pz]-l.

T h e transition probability p,,(t) approaches Proof. and is expressible as p z z ( t ) = e-Qzt

+

1

e-Qz(b-u) ~

p , as t -+ co,

, ( u ) .

Since stays in x and returns to x are all mutually independent, the stays being identically distributed, and the returns also, the renewal theorem [Smith (10, p. 247, formula (1.3))] implies that the right side approaches

Jr

e-qzt

dt

1

-

interval between successive El returns to x Thus p , can be calculated from p, where pz = J t u 0

=

dPr{return time to x

j:Prjreturn

1

+ qxcLx *

< u}

time to x > u} du.

292

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

For our purposes it is convenient to approach the calculation of p , in a slightly different way. Let z be any state, and let x be a state distinct from x, x f x. Define q,,(t) in t 3 0 to be the probability that if the stochastic process start at z at time zero, it be at x at time t without having returned to z. Th u s and (epoch of first return to z ) > t 1 xo = z}.

qZz(t)= Pr{x, = x

For convenience, we set q B Z ( t= ) 0 in t Lemma 8.2.

Proof.

t

For z # x,t

Let t i , i

=

2 0.

3 0,

1, 2, ..., be the epoch of the ith return to x in

> 0, and let A i ( t ) be the event {xi

=x

and

ti

< t < tiil}.

Then Pr{Ai(t) 1 xo = z> =

qzx(t- u ) d P r { t i

< u 1 xo = z}.

However [cf., Smith (10, p. 251, formula (1.7))],

and

T h e integration and the summation can be interchanged by the monotone convergence theorem, and the lemma follows. Lemma 8.3.

For z

+ x,

1 1. Proof.

T h e integral on the right exists, since

m

0

293

ERGODIC MARKOV PROCESSES

E{time spent in

qzx(u)du

x

between successive returns to z }

< pz

T h e lemma follows from Lemma 8.2 and the renewal theorem. T h e matrix A is defined by the condition A = (a,,) with

I t can be verified that A is a stochastic matrix, indeed, the one-step transition probability matrix of a Markov stochastic process {xn , n an integer) taking values on S ; x, is a discrete-time analog of xtobtained by ignoring the lengths of time spent in a state.

For z # x,

Lemma 8.4.

number of arrivals at x between successive returns to z T h e integral is the expected time spent in x between successive returns to z. Each stay in x has mean length l/q,, and the stays are independent of the rest of the trajectory followed. Proof.

For z # x,

Lemma 8.5.

E

inumber of arrivals at x between) successive returns to z i m

=

zI'r{xn

n=l

=x

and xj f x

for

1<j

< n I xo = z}.

Proof. We remark that the expectation on the left is the same for both x iand x, . T h e lemma is then a special case of the theorem that if ( A i i,= 1, 2, ...,} are any events, then the expected number of A, that occur is

294

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

Lemma 8.6.

Proof.

1

a.

0

This is an immediate consequence of q,,(.) = 0 and qzz(u)dn = E{time

spent in x between successive returns to z},

for z # x.

Let x # x. Then

Lemma 8.7.

Proof. T h e event in question can occur in as many ways as there n} are paths of length n in K,, . T h e probability that {xj , 0 j follow a path T from x to x is

< <

the quotient in front correcting for the end points. Combining Theorem 8.6 with Lemmas 8.4, 8.5, 8.6, and 8.7, we obtain the following explicit formula for the stationary probabilities:

-

Theorem 8.7.

If x # z , then 1

with

and

We remark that Theorem 8.5 follows from the above if we choose z = 0 = zero state, omit normalization, and observe that only

12.

EXPANSION OF THE STATIONARY VECTOR

p

295

products along permitted paths (TE P ) are nonzero. Theorem 8.7 is an analog for continuous parameter processes of a theorem of Derman (11) for Markov chains.

$12. Expansion of the Stationary Vector in Powers of h

p

We now turn to examining, in an elementary way, the analytical dependence of the state probabilities { p z , x E S}on the calling rate A, on the structure of the network, and on the routing matrix R. I t will be of the set S of states can be used shown that the partial ordering to calculate the elements of p by expanding the ratios

<

&, Po

x>o

in powers of the traffic parameter X in a neighborhood of X = 0, and then determining the coefficients of this expression from the structure of the network and the routing matrix by a recursive procedure. T h e solution so obtained is later (Section 15) extended to arbitrary real positive values of X by analytic continuation, and the coefficients are calculated. Our approach to studying the stationary probability vector p will be guided by these intuitive remarks: It is known that in various simple models (of connecting systems carrying random traffic with blocked calls refused) the probability that k calls be in existence is proportional to the kth power of a constant associated with the calling rate divided by k factorial. For example, in Erlang’s model for c trunks with blocked calls cleared, the chance that k calls are in progress is proportional to Uk

-

k! ’

O
where a is the calling rate. Note that the exponent of a is the number of calls in progress, i.e., the current difference between the cumulative number of new calls and that of hangups, assuming that the system started in the zero state. T h e factorial in the denominator is the number of orders in which the K calls in progress could all hang up, or alternatively, could all have arisen. T h e situation in our model is very similar. Each call still in progress

296

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

required an event occurring at the rate h to put it in existence; for each state x, there are exactly 1 x I! orders in which the 1 x 1 calls in progress in x could arise, or terminate. These circumstances suggest that for x > 0, p , might be of order XI"' as h 4 0, and that the coefficient of A'"'in p , might involve 1 x i ! in the denominator. These conjectures are true, and are the first step in the systematic calculation of p , by expansion in powers of A, to be carried out in this section. We first record some analytical properties of p as preliminary results. Some of these results could be obtained as consequences of the basic solutions given in Section 10. Most of the proofs to be given, however, are independent of Section 10, and proceed by simple arguments from the equilibrium equation. When we need to view p , as a function of the parameter A, we write 9, = p,(h), x E S, or in vector form, p = p(X). Lemma 8.8.

limp%(X)= S,, L O

.

Proof. Let x be a maximal state in the partial ordering set S of all states. The n s(x) = 0, and

I x PAX) ~

<

=

c

< of the

PV(4YlZ *

lJEBs

<

Since 0 p,(X) 1 for all X > 0 and a ll y E S, the lemma is true for maximal states. Assume, as a hypothesis of induction, that the lemma 1. The n for % E L !,, k > 0, is true for a l l y with l y , >, k

and SO p,(h) + 0 as X for each X > 0,

-+

0. T h e proof is completed by observing that, POP)

=

1-

x>o

Lemma 8.9. For each x E S , p , is the restriction to real positive argument of a rational function p , ( . ) of a complex variable p. The function p,( .) has no poles in a neighborhood of the half-line Re(p) >, 0, Im(p) = 0, and an expansion

12.

EXPANSION OF THE STATIONARY VECTOR

p

with real coeflcients c,&(x,h) is valid for Re(p) 3 0, Im(p) 1 E 1 small enough.

297 =

0, and

Proof. T h e equation Qp = 0 can be solved for a normalized (i.e., probability) vector p ( h ) by successive elimination or by use of Theorem 8.4. Either procedure gives rise to an algebraic expression for p,(h), x E S. Let p z ( p ) be that rational function of a complex variable p defined by substituting p for A in this algebraic expression. Since 0 p,(h) I , p,(p) has no poles in a neighborhood of the nonnegative real axis. T o justify the expansion we show that p,(.)/po(.) is also analytic in that neighborhood. But this is immediate because by Lemma 8.8,

<

<

and by Theorem 8.1, p,(h) > 0 for h 3 0 because the zero state belongs to the ergodic class S - F. Setting p = 0 and E = h in Lemma 8.9, we obtain an expansion of p J p , in powers of the traffic parameter A,

valid for h small enough. Theorem 8.8.

For k 3 0 and x E L , ,

Proof. We prove both results simultaneously by induction. By Lemma 8.8, the result is true for k = 0. Assume that it is true up through k - 1 3 0. From Lemma 8.1 and the induction hypothesis, we find

298

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

By Lemma 8.9, for h small enough p,/po can be written as a power series around h = 0 m

Thus

ZA~

m=o

c,(x, 0) = p,/po = O(P)

zeLk

as A + 0.

T h e first nonvanishing coefficient in the expansion on the left must be positive, else p , < 0 for h > 0 small enough, which is impossible since p,(X) >, 0 for h > 0. Hence

c;co(x, 0) 3 0.

X€L,

However, the first nonvanishing coefficient in the expansion of p , must also be positive, for the same reason as above, namely, that p,(h) 3 0 for h >, 0. Th u s co(x, 0 ) 3 0. Hence pk = O(hk)as h + 0 implies co (x, 0) = 0. We apply the same argument successively to show that for x 6 L k , the coefficients cl(x, o), ..., c k - l ( x , 0 ) are all zero, and the theorem is proved. Theorem 8.9.

For x

> 0,

where rx = (Rlxl)oz

Proof.

=

the 0, x entry of the I x Ith power of the routing matrix R

=

the number of permitted strictly ascending paths from 0 to x.

T h e equation of statistical equilibrium that defines p is

[I x I + A W I P X

=

c; PI! + c;

YEA*

PJI!z

x E s.

YEB~

For convenience, suppose that I x I = k. We divide the equation by P o , use Lemma 8.9 to expand the components of pip, in powers of

12.

p

EXPANSION OF THE STATIONARY VECTOR

299

A, and equate the coefficients of h k on each side of the equation. This gives kc&, 0) s(x)c,-,(x, 0) = c,(y, 0) CL-dY, 0)YYZ *

c

+

+ wn. 2

YEA,

By Theorem 8.8, ckPl(x,0 ) = 0 and ck(y,0) fore,

=

0 for y

E

A, . There-

or, in general, with y the vector with components cIV,(y, 0),

Iterating this relation 1 x I times, we find CISI(X,

0)

I

=7 (R1%),

I x I.

Now it is easily seen that the y , x entry of Rk is zero unless = I x I - I y I, and, in particular, if k = 1 x 1, this entry is zero 0. Th u s unlessy

k

= 1

c,&

0)

1

=7 (Rl”l)o,co(O,O),

I x I.

and it is obvious from the definition of the c,(y, 0) that c,(O, 0)

for

m

>Ix1

and

IC

=

1.

>O.

300

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

If f o r x > 0 h

< (lim sup I c,(x, m-tm

0)

then the component p , of p is given by

c m

P,

If

h

= Po

hn2Cm(X,

m=O

0).

< min (lim sup [ cnL(x,0) l'm)-l, x>O m+m

then the probability po of the zero state is determined by the normalization condition X Z E S p x= 1 as

Po

=

1 m

2 > 0 m=o

Proof. This result follows immediately from Lemma 8.9 and Theorem 8.9, using the standard formula for the radius of convergence of a power series.

$13. Expansion of the Probability of Blocking in Powers of A With a method of calculating equilibrium state probabilities for small h at hand (in principle, at least) we now show how the probability b of blocking can be calculated, to any desired degree of accuracy, by an expansion in powers of the traffic parameter A, assumed sufficiently small. I n most connecting networks of practical interest, none of the states near the bottom of the state-diagram has any blocked calls, so that it is necessary for a state x to have certain minimum-number of calls in progress before it can have any blocked idle pairs. To take advantage of this situation in our calculation, we let

n

=

least k such that some call is blocked in a state of L,

.

Theorem 8.77. The probability b of blocking can be expanded in a power series in h in a neighborhood of h = 0 ; only terms of order higher thun or equal to An appear.

13.

EXPANSION OF THE PROBABILITY OF BLOCKING

Proof. From Theorem 8.2 we have, since and ck(x, A) = 0 for k < I x 1,

2

Bz

0 for I x I

30 1

< a,

PXBX

Since the denominator is not zero in a neighborhood of h = 0, b = b(A) is analytic there and can be expanded in powers of A. U p to terms of order An+2 this expansion is

+

( A'(O)B"(O)

1

- A"(O)B(O) - A'(O)A(O)B'(O) - A"(O)B(O) 2[A(0)I2 [~(0)13

T h e coefficients in the first two terms can be obtained by the following calculations:

2

1 n! XEL,

=-

YXBX,

302

8. "0)

MARKOV PROCESSES REPRESENTING TRAFFIC

=

z

C,(X, 0)ax

iXlC1

T h e constants { C ~ + ~ ( XO), , I x I = k} can be determined by the following recurrence, obtained from Theorem 8.10:

Our results can be put in a slightly more explicit form by expanding log b rather than b, and using the fact that A(.)and B(.),as defined by (3), are generating functions. We have log n

= 71 log

x + log B(A) - log A(h).

Except for the systematic absence of factorials, the coefficients in the expansion of log B(h) are related to those in the expansion of B(h) as cumulants are to moments. Set

14.

INTERPRETATION AND CALCULATION OF CONSTANTS

303

so that W

B(h) = p b j ,

Then, by a standard formula [Riordan (22, p. 37)], W

log B(h) = z X ' K j ( b ) , j=O

where for u = a or b (sequences) K,L(U)

=

(- l)"-l(k - I)!

(U,)',

(U1)'l

(kl)!... (k,')!

+ +

with k = k , -I- R , k, , and the sum over all partitions of n, ... nk, = n. i.e., all solutions in nonnegative integers of R , 2k,

+

+ +

$14. Combinatorial Interpretation and Calculation of the Constants (c,(x, 0), x E S, m >, 0) We shall now evaluate the coefficients in the power series expansion of p around X = 0 explicitly as sums of products on paths in S . Additional combinatorial notions that enter this calculation are discussed first. A path is said to contain a loop if it returns (one or more times) to a place where it has been previously. Th u s T = {xo , ..., x l ) has a i <j I , such that loop if there are integers i and j , 0

<

<

x z. = x3. .

A circuit is a path that ends where it begins. T h u s the generic path = {xo, ..., x l } is a circuit if xo = x l . A circuit is a loop if it contains no subcircuits, i.e., T is a loop if xo = x l and i # j implies

T

x,

foranyO
#

xj

304

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

A circuit of length two is of necessity a loop. Furthermore, the circuits of length two on S can be partitioned into two classes according to the direction in which they are traversed. Each such loop can be thought of as obtained either by first adding a new call and then removing that same call, or by removing a call and then replacing that same call. T h u s if T = {x,,, x l , x2} is a loop of length two on S , then x,, = x2

and either

x2 = x0 E BZ1

or x2 = xo E

AZ1

and, of course, not both. I n the first instance we say that x is of the first kind. T h u s a loop of the first kind is a path T of the form {x,y , x} with y E A, , i.e., it is a trajectory in S obtained by starting at a state x, adding a new call to go to a state y , and then removing that very same call t o return to x. For a path T = {xo , ..., xl)and a state x we say that x is on n if x is one of x,,, x l , ..., x 1 . If x is on n,we say that a loop of the first kind on T ends at x if, for some i in the range 2 i I,

< <

that is, the subpath { x i - 2 , xi-l,xi} is a loop of the first kind, and xi is x. With T and x as above we define

g(7, x) =

'I

'' 1,

if x is on T and a loop of the first kind on T ends at x

otherwise,

and we denote by ~ ( nthe ) total number of loops of the first kind contained in T . With these combinatorial preliminaries behind us, we are ready to prove

14.

INTERPRETATION AND CALCULATION OF CONSTANTS

Theorem 8.12

and, fur x

> 0,

O), x

The coeficients {c,(x, Cm(0,O) =

E

305

S , m >, 0} aye given by

%I",

Before proving the theorem it is probably helpful to state it in words, thus: T o calculate c,(x, 0), consider all the paths T that are permitted by R, start at 0, never return to zero, and u p at x, and have length 2m - 1 x 1 (i.e., consist of m new calls and m - 1 x j hangups); along each such path T take the product of the reciprocals of the numbers of calls in progress in the states traversed by T , omitting states at which a loop of the first kind ends; weight the product positive or negative according as T has an even or an odd number of loops of the first kind; add up all the weighted products. and We already know that cnL(O,0) = Proof of Theorem 8.12. that c,,(x, 0) = 0 for x > 0 and m < I x 1. T h e latter result is consistent with Theorem 8.12 because no path from 0 can reach x in fewer than I x I steps. Consider then the case m - j x I >, 0. Any path z from 0 to x of length 2m - I x I = m consists entirely of new calk, and, for such a T ,

T h e number of such paths, summed over in (4), is easily seen to be r z , the number of permitted strictly ascending paths from 0 to x. T h u s for m = 1 x 1, (4) states that c,(x,

0)

=

YX ~

lxl!'

as was proven in Theorem 8.9. T h e remainder of the proof is by upward induction on m and finite downward induction on x, using the recurrence formula

8.

306

MARKOV PROCESSES REPRESENTING

TRAFFIC

given by Theorem 8.10. T h e starting step of the induction is the fact that formula (4)holds for x E S and m such that O
This is a consequence of the fact, already proven, that (4)holds for 1x1 = m . Let us assume, as a hypothesis of induction, that the theorem holds for all x E S and m such that (5)

O
We shall prove from this, by downward induction on x , that it also holds for x E S and m such that 0

< 1x1 < m = k + I .

T hi s last condition exactly describes the new cases covered in 1. extending ( 5 ) to R Where T is a path, we use the natural notation T X to denote the path obtained from T by adding x to T as a new ultimate element, assuming that x is adjacent to the last element of T . We now observe that if T X is a path that does not end in a loop of the first kind, then g ( x x , x) = 1, and, therefore,

+

Y(TX)

=

Y(T).

<

A state x is maximal in the partial ordering if no new calls can be put up in x, for whatever reason. If x is maximal, then A, is empty and s(x) = 0. A state x is maximal in ( a set) X c 8 if x >, y for every y E X , and x E X . Let 0 1 x I m = R 1 and suppose that x is maximal. Then

<

<

+

No path ending at a maximal state x can end in a loop of the first kind, and any such path must have a y E B, with rys = 1 as a penultimate element if it is to be a permitted path. Th en clearly {T

=

E

P n KO$: 1 ( ~ = ) 2m - I x

u

YEB,

{m:

T

E

I}

P n KO, and Z(T)

=

2m

-

Ix I -1

and

yvZ =

I}.

14.

INTERPRETATION AND CALCULATION OF CONSTANTS

307

Let nowy E B, , T E P n Koy, l(x) = 2m - 1 x 1 - 1, and ry, = 1. Then 7rx satisfies g ( m , x) = 1 and also formula (6). T h u s formula (4) 1 x 1 m = k + 1, by the hypothesis holds for maximal x and 0 of induction. Next, consider states x that are maximal in the set

<

<

These are just the elements ofLk+l , i.e., the states x with I x 1 = k + I . Since we are assuming m = k 1, the result (4) holds for these x by Theorem 8.10. Finally, assume as a hypothesis of downward induction (on I x I) that the result is true for y E S and m such that

+

and suppose that 1 x

I

=j.

Then

If T is a path on S the notation pe(T) denotes the penultimate element of T , i.e., pe(T) = zl-l for T = {z,,, zl,..., zz}. T h e notation ape(n) denotes the antepenultimate element of x , i.e., for T = {zo , z1, ..., 23, ape(.rr) = zz-2. A path of length 2m - 1 x j belonging to P n KO,reaches x either via A, or via B, . I n the latter case the path cannot end in a loop of the first kind. By the hypothesis of induction,

-

2

mPnK,,, l(n)=2m-/ll pe ( n m ,

(-l)V(T)

p M I I ’ SET

z

(7)

the second equality following from (6). Now consider a path x x of length 2m - I x 1, belonging to P n KO,, reaching x via A, , and not ending in a loop of the first kind, i.e., with

308

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

pe(n) # x. Using the hypothesis of downward induction, we find

the second equality again following from (6). Finally, we consider those paths of length 2m - I x I belonging to P n KO,which do end in a loop of the first kind. Such a path is of the form TYX

with n E P n K O , , l(n) = 2m - 1 x 1 - 2, and rZy = 1, for some y E A,. We observe that in this case 1 y I = 1 x I 1 and that

+

( - 1Ivin)

IxI

jrI dry, z )

____-

Len,

1zI

ztn

-

Yn) rIR(T, 4

(- 1 -~

IYI

En

lzl

We note that v(xyx) = v ( n ) + 1, and that v(ny) = v ( n ) . By summing formula (9) over paths of length 2m = 1 x 1 - 2 belonging to P n KO,and over y E A, such that rXy = 1, we obtain

15.

CALCULATION OF C,(X,

h)

309

By the hypothesis of induction

Also, it can be seen that, for y {r E P n Kov: Z(r)= 2m

u {ry: v E P n K O , , = {r E

-

Y,

E

A, ,

Ix 1 -1 =

P n KoY: 1 ( ~ = ) 2m

and

1,

-

and

]y

pe(r) # x}

Z(r)= 2m

I).

-I

x

1 - 2) (12)

Combining (8) and (lo), using (11) and (12), and applying the hypothesis of downward induction, we find

Together, (7) and (13) complete the inductive step.

$15. Calculation of c,(x, A) I n order to give an explicit formula for c,(x, A) for X > 0 we suppose that, for each path T on S , all the upward (in <)transitions on T are of two kinds, denoted by the symbols X and c. ( T h e calculation we present is more easily understood if the new calls labeled E are thought of as due to the increment E in calling rate, while those labeled X are due to the original calling rate X.) I n other words, we consider the set of all paths T on S as (partially) labeled by assigning either X or E to each upward transition. Formally, we define a labeling

310

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

A ( - ) of a path T = {x,, , xl, ..., xz> to be any function defined for x l ,x 2 , ..., xz with the property X(x,) =

;1

or X

if x i covers x i - l ,

otherwise.

T h e set of all possible labelings of a path rr is denoted by A(T),and membership therein by the notation A(*) E A(rr). T h e functions A(.) should not be confused with the constant A. A path rr together with a labeling A( of rr will be called a Zabeled path and denoted by (rr, A( -)). T h e index E ( T , A(.)) o f a labeled path is the number of times h(x) assumes the value E for x E T . Let A(.) E A(T) be a labeling of T = {xo , x l ,..., xl}. T h e function h(n, A(.), -) is defined by the condition a)

if x is on rr and a loop of the first kind on rr ends at x, with the first (i.e., upward going) leg of the loop labeled E by A(.), i.e.,

W x ) = 6, and by h(T, A(-), x) = 1

in all other cases. For a path rr and a labeling A(*) defined by S(X,

A(*))

Theorem 8.13.

and, f o r x

> 0,

=

E A(rr), the

function ( ( T , A(*))

the number of loops of the first kind on [I abeled E on the upward leg by A( .).

For A

> 0 and m 3 0

x

is

15.

CALCULATION OF

Note that by Theorem 8.5, with z reduces, as it should, to

=

c,,,(x, A)

31 1

0, formula (14) for m

=

0

Also, formula (14) agrees with formula (4), Theorem 8.12, as X is allowid to approach zero, since in this limit only (T, A(.)), which are 1 x I). full of E ' S , contribute, with m = *(Z(T)

+

of Theorem 8.13. T h e values of co(x, A) and C,,~(O,A) are consequences of the definition

Proof

T h e equilibrium condition Qp comprises exactly the equations

=

0 for traffic parameter h

x

+

E

+

E

s.

Dividing by po(X e) > 0, substituting the expansion (15), and collecting coefficients of like powers of E , we find that c,(x, A) for x E S satisfies the equation

[I x I

+ W X ) I C ~ ( X ,A) =

[ c m ( Y , A)

- rwcm-I(x,

YEA,

It can be verified, using the fact that for any labeling A(.) of a path nthe number of times A(x) has the value X for x E T is +

2

I

'

- E ( T , A(.)),

that formula (14) gives a formal solution of eq. (16). T o prove the theorem it suffices to show that the infinite sum over n- E P n KO, in formula (14) is absolutely convergent, and that the left-hand side of formula (15), with the c,(x, A) as given by the theorem, is absolutely convergent for E small enough.

312

8.

MARKOV PROCESSES REPRESENTING TRAFFIC

We first observe that for each T and A(.) summed over in formula the factors h ( ~A(.), , y ) in

are uniformly bounded, and that at most min(m, v ( T ) ) of them are greater than unity. Also, the number of upward transitions (new calls) along a path T E P n KO,of length Z ( T ) is just

T h u s there are exactly

ways of labeling the upward transitions on a path length Z(T) and index m. Hence for some constant a > 0

T

EP

n KO, with

By Lemma 8.7, with {xi,i an integer} of the lemma defined in terms of the matrix Q appropriate to our congestion problem,

= Pr{xi

# 0 for 1 < i

< k 1 xo = 0}

From Feller ( 4 , p. 378, example 19), we see that there exists 0 < q < 1 such that the probability on the right is at most qk for all k 3 I S 1. Hence

This proves that (14) converges absolutely, and that the left side of ( 1 5 ) converges absolutely for I E I small enough.

REFERENCES

313

REFERENCES 1. V. E. BeneS, Heuristic Remarks and Mathematical Problems Regarding the Theory of Connecting Systems, Bell System Tech. J. 41, 1201-1247 (1962). 2. V. E. BeneS, Algebraic and Topological Properties of Connecting Networks, Bell System Tech. J . 41, 1249-1274 (1962). 3. V. E. BeneS, A “Thermodynamic” Theory of Traffic in Connecting Networks, Bell System Tech. J . 42, 567-607 (1963). 4. W. Feller, “An Introduction to Probability Theory and its Applications,” Vol. 1, 2nd ed. Wiley, New York, 1957. 5. R. Bellman, “Introduction to Matrix Analysis.” McGraw-Hill, New York, 1960. 6. P. R. Halmos, “Finite Dimensional Vector Spaces,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1958. 7. R. Morris and E. Wolman, A Note on “Statistical Equilibrium,” Operations Research 9, 751-753 (1961). 8. R. Syski, “Introduction to Congestion Theory in Telephone Systems.” Oliver & Boyd, Edinburgh and London, 1960. 9. G. Birkhoff and S. MacLane, “A Survey of Modern Algebra.” Macmillan, New York, 1950. 10. W. L. Smith, Renewal Theory and its Ramifications, J. R o y . Stat. SOC. 20, 243-302 (1958). 11. C. Derman, A Solution to a Set of Fundamental Equations in Markov Chains, Prnc. Am. Math. Snc. 5, 332-334 (1954). 12. J. Riordan, “An Introduction to Combinatorial Analysis.” Wiley, New York, 1958.

Suggested Reading

1. J. L. Doob, “Stochastic Processes.” Wiley, New York, 1953. 2. M. Lokve, “Probability Theory,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1960. 3. E. Parzen, “Stochastic Processes.” Holden-Day, San Francisco, California, 1962. 4. A. T. Bharucha-Reid, “Elements of the Theory of Markov Processes and their Applications.” McGraw-Hill, New York, 1960. 5. E. Parzen, “Modern Probability Theory and Its Applications.” Wiley, New York, 1960. 6. P. LeGall, “Les Systkmes Avec Ou Sans Attente et Les Processus Stochastiques.” Dunod, Paris, 1962. 7. R. Syski, “Introduction to Congestion Theory in Telephone Systems.” Oliver & Boyd, Edinburgh and London, 1960. 8. J. Riordan, “An Introduction to Combinatorial Analysis.” Wiley, New York, 1958. 9. J. Riordan, “Stochastic Service Systems.” Wiley, New York, 1962. 10. H. J. Ryser, Combinatorial Mathematics, Carus Monograph No. 14, Math. Assoc. Am., John Wiley and Sons, New York, 1963. 11. G. Birkhoff, Lattice Theory, Am. Math. SOC.Colloq. Publ. (rev. ed.) XXV, 1948.

314

Author Index Numbers in parentheses are reference numbers, and are included to assist in locating a reference when the author’s name is not cited at the point of reference in the text. Numbers in italics indicate the page on which the full reference is listed. Bellman, R., 274, 289, 313 BeneS, V. E., 55(11), 56(2), 80, 85(2),

86(4, 5), 88(10), 89(10), 96(4), 97(5), 120(4), 135, 170(9), 187, 194(3), 197(6, 7), 204, 208(3), 221, 225(1, 2), 230(11), 238(2), 248(11), 251(11), 255, 264, 266(1, 2, 3), 267(1, 2), 313 Bharucha-Reid, A. T., 314 Birkhoff, G., 62(3), 63(3), 81, 239, 264, 313, 314 Blood, G. T., 3 Boltzmann, L., 244, 245, 246 Canceill, B., 5 , 52 Clos, C., 9, 24, 30, 52, 72, 81, 83, 85, 86,

Gale, D., 87, 135 Garwood,F., 5(17), 52, 178, 187 Haantjes, J., 202, 221 Hall, M., Jr., 100, 104(1 I), 135 Hall, P., 66, 81, 87, 107, 135 Halmos, P. R., 254, 260(12), 261, 264,

275, 313

Hayward, W. S., Jr., 188, 197, 221 Jacobaeus, C., 5, 52 Jensen, A., 4, 8(5), 51, 167, 187, 231(5),

104, 135, 137, 158

264

CramCr, H., 182, 187 Crommelin, C. D., 4, 51

Kelley, J. L., 70, 76(6), 81 Kharkevich, A. D., 72, 81 Khinchin, A. I., 4, S I , 236, 244, 245, 264 Kolmogorov, A. N., 4, 51 Kosten, L., 3, 5 , 51, 52, 178, 187 Kramer, H. P., 220, 221, 257, 264 Kuratowski, C., 70

DCjean, J. H., 86 Derman, C., 295, 313 Doob, J. L., 242, 264, 314 Duguid, A. M., 85, 86, 110, 135 Elldin, A., 5 , 20, 52 Engset, T., 4, 51 Erlang, A. K., 4(5), 8, 51, 231, 264 Feller, W., 5 , 52, 166, 167, 186, 274,

290, 312, 313

Fortet, R., 5, 52 Fry, T. C., 4, 51

Le Corre, J., 86 Ledermann, W., 202, 221,289, Lee, C. Y., 5 , 52, 228(3), 264 Le Gall, P., 5, 52, 228(4), 264, 314 Lloyd, S. P., 175, 277

315

316

AUTHOR INDEX

L o b e , M., 314 Lundkvist, K., 5, 52 MacLane, S., 313 Manning, J. R., 5(17), 52, 178, 187 Molina, E. C., 4, 51 Moore, E. F., 30 Moran, P. A. P., 184, 187 Morris, R., 276, 313 Newland, W. F., 167, 187 Nyquist, H., 202 O’Dell, G. F., 4, 51 Olsson, K. M., 211(13), 221 Palm, C . , 4, 5, 52, 167, 180, 187, 188, 202, 221 Parzen, E., 314 Pauli, W., 244 Paull, M. C., 87, 88, 89, 90, 95, 98(7), 110,135 Poisson, S. D., 4, 166, 190, 282 Pollaczek, F., 4, 51 Rabe, F. W., 167, 170, I 8 7 Reuter, G. E. H., 202

Reynolds, J. N., 5(18), 52 Rice, S. O., 142, 167 Riordan, J., 165, 167, 170, 175, 182, 187, 197, 202, 204, 216(9), 221, 303, 313, 314 Rorty, M. C., 4 Ryser, H. J., 314 Sandberg, I. W., 288 Scudder, F. J., 5(18), 52 Shannon, C. E., 119, 135 Slepian, D., 86, 89, 90, 110, 135 Smith, W.L., 281, 291, 292, 313 Stormer, H., 167, 187 Syski, R., 3, 6 , 51, 247, 264, 277, 313, Szego, G., 202, 221 314 Tolman, R. C., 244, 245, 264 Tukey, J. W., 210 Weisner, L., 66, 81 Wilkinson, R. I., 3, 4, 51 Wilson, I. G., 149 Wolman, E., 276, 313 Zaanen, A. C., 220, 221

Subject Index

Access, 12, 15 Algebraic, 85, 96, 267 Assignment, 68, 89 Birth-and-death process, 5 Blocking, 14, 15, 38, 71, 83, 265, 266, 276 Call, 69, 71 Canonical, 224, 226, 234, 241 Cardinality, 67 Chain, 61, 64 Circulation, 35 Closure, 70, 76 Common control, 5 Compatible, 61 Completion, 7, 9, 10, 11, 12, 14, 35 Complex, 98, 100 Complexity, 6, 7, 15, 18 Condition, 57 Configuration, 54, 56, 83, 282 Congestion, 2 Conjugate, 98 Connecting network, 5 , 16, 17, 18, 24, 38, 53, 56, 156 Connecting system, I , 2, 6, 7, 17, 19, 24 Connection, 61 Control unit, 7, 8, 9, 14, 15, 17, 121 Correlation coefficients, 175, 176 Covariance, 188, 189, 191, 215, 228 Cover, 65, 67, 79, 105, 127, 128, 249, 286 Critique, 3, 5

Crossbar, 5 , 8, 86, 96 Crosspoint, 17 Delay, 14 Dense, 53, 56, 74, 76, 78 Detailed balance, 243, 244, 245 Dial, 10, 11, 14, 15, 35 Dimension, 64, 65 Dominant characteristic value, 195, 219, 228, 255 Dual, 71 Efficiency, 163 Electromechanical, 11, 56 Electronic central office, 11, 84, 156 Elementary, 62 Energy, 244 Entropy, 223, 224, 229, 232, 233 Equilibrium, 225, 229, 230, 231, 267 Equivalence, 57, 63, 69, 70 Ergodic set, 242, 274 Estimator, 171, 172, 184 Extraction, 163 Fine structure, 13 Finite source, 4, 161, 271 Frame, 8, 15 Generator, 231, 242 Grading, 4 Graph, 57, 58, 60 Group, 86, 97, 100, 102, 109 317

SUBJECT INDEX

Hamiltonian, 245 Hasse figure, 63, 225, 267 Heuristic, I , 2 Hit, 105 H-theorem, 244, 246 Idle, 71, 89 Imprimitive, 100, 102 Inclusion, 64 Independent, 36, 161, 190, 271 Inlet, 8, 25, 59, 61, 71 Join, 59 Jordan-Dedekind chain condition, 66 Lagrange multipliers, 235 Laplace-Stieltjes transform, 180 Lattice, 62 Law of large numbers, 280 Level, 22, 63, 225 Likelihood, 8, 170, 183 Line, 8 Link, 59, 98 Loop, 286 Loss, 14, 16 Marker, 10, 11, 12, 35 Markov process, 4, 18, 24, 26, 167, 178, 190, 229, 231, 232, 265, 290 Maximal, 69, 71, 72, 87, 96, 99, 121, 268, 296, 306 Maxwell-Boltzmann distribution, 223 Measurement, 159, 188 Meet, 59 Microscopic, 19, 21 Mobius function, 66, 226, 238 Mobius inversion formula, 66, 240 Model, 16, 17, 18, 19 Negative exponential, 36, 161, 271 Network states, 58 Nonblocking, 9, 16, 24, 29, 30, 53, 55, 71, 72, 79, 83, 136 Norm, 251, 269 One-sided network, 265, 283 Optimal, 38, 40, 86, 97 Outlet, 8, 25, 59, 61

Packing, 29, 40, 83 Partial ordering, 22, 53, 62, 127 Partition function, 223, 224, 226, 234, 237, 241 Path, 286 Performance, 13, 15, 16, 24, 54 Permutation, 86, 87, 96, 97, 98, 109, 287 Poisson-Charlier polynomial, 195, 202 Policy, 42, 48 Predecessor, 288 Prime, 86, 131, 134 Pseudometric. 70 Random behavior, 1, 16 Randomness, 8, 15, 16, 269 Rayleigh quotient, 232, 254, 255 Reachable, 30, 33, 73 Realize, 68, 121, 249 Rearrangeable, 15, 40, 53, 55, 74, 75, 82, 121 Receiver, 12 Recovery function, 188, 194, 208 Recurrent process, 5 Register, 10, 15, 35 Renewal theorem, 291, 293 Retrials, 5 Routing, 6, 16, 26, 29, 38, 64, 222, 223, 229, 246, 250, 266, 270 Routing matrix, 250, 212, 286, 295 Rule, 29, 30, 72, 77 Sampling error, 193, 194, 197, 261 Scan interval, 193 Semigroup, 4, 231, 242 Sender, 12 “Sigma” or “Palm” functions, 202 Spectral theorem, 259 Stage, 98 State, 8, 17, 19, 20, 21, 36, 54 State-diagram, 22, 39,62, 63,70,225,267 Stationary measure, 4, 290 Statistic, 159, 164, 165 Statistical equilibrium, 4, 20, 223, 285 Statistical mechanics, 4, 8, 222, 244 Stochastic limit, 42, 277 Strategy, 42, 44 Structure, 1, 8, 18, 20, 21, 54, 56, 57, 282 Sufficient, 75, 159, 164, 183

SUBJECT INDEX

Symmetric, 70, 98, 102, 219 Symmetrizable, 196, 253 Symmetry, 8 Temperature, 245 Thermodynamic, 222, 234 Time average, 212, 262 Time-division, 56

i

Topological, 28, 53, 69, 70, 267 Toll, 8 Transient set, 274, 275 Trunk, 8, 18, 162 Two-sided network, 266, 283 Zero state, 22, 24, 37, 63, 66, 68

319

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