Performance Analysis of Multichannel and Multi-Traffic on Wireless Communication Networks

PERFORMANCE ANALYSIS OF MULTI-CHANNEL AND MULTI-TRAFFIC ON WIRELESS COMMUNICATION NETWORKS

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Performance Analysis of Multi-Channel and Multi-Traffic on Wireless Communication Networks Wuyi Yue Konan University, Japan

and

Yutaka Matsumoto I.T.S., Inc., Japan

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-47311-9 0-792-37677-3

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Visit Kluwer Online at: and Kluwer's eBookstore at:

http://www.kluweronline.com http://www.ebooks.kluweronline.com

The book will prove useful to

a post-graduate course in computer science or engineering. It is often a pre-requisite to some other more advanced courses like network design and

management based on queueing modeling with examples of their

applications to multimedia communication networks and computer networks. It can also be used for a course on stochastic models in mathematics and operations research departments.

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Contents

xiii xix xxi xxvii

List of Figures List of Tables Preface Acknowledgments 1. INTRODUCTION Overview of Multiple Access Communication Networks 1 Packet Communication Networks 2 2.1 Local Area Networks 2.2 Single Channel Networks 2.3 Multichannel Networks 2.4 Multi-Hop Networks Wireless Communication Networks 3 3.1 Cellular Mobile Telecommunication Networks 3.2 Mobile Packet Radio Networks 3.3 Wireless Local Area Networks Multiple Access Protocols 4 4.1 Fixed Assignment Protocols 4.1.1 TDMA 4.1.2 FDMA 4.1.3 CDMA 4.2 Random Access Protocols 4.2.1 Pure ALOHA 4.2.2 Slotted ALOHA vii

1 1 3 4 4 5 6 7 7 8 8 9 10 10 10 11 11 13 14

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

5

6

7

8

9

10

4.2.3 CSMA and CSMA/CD 4.2.4 Performance Comparisons for ALOHA and CSMA 4.2.5 CSMA/CA 4.2.6 Reservation Multiple Access Protocols Mobile Packet Radio Networks with Random Access Protocols 5.1 CSMA 5.2 CSMA/CD 5.3 Capture Multichannel Networks with Random Access Protocols 6.1 Slotted ALOHA 6.2 CSMA and CSMA/CD Multi-Hop Networks with Random Access Protocols Slotted ALOHA 7.1 7.2 CSMA and CSMA/CD 7.3 CDMA Channel Assignment Schemes in Cellular Mobile Networks 8.1 FCA Scheme 8.2 DCA Scheme HCA Scheme 8.3 8.4 Hand-Off Performance Analysis of Wireless Communication Networks Discrete-Time and Continuous-Time Markov Chains 9.1 9.2 Probability Distributions System Models and Performance Analyses in the Book 10.1 System Models and Performance Analyses 10.2 Performance Measures and Common Definitions

14 17 18 18 19 19 20 20 21 22 22 23 24 25 25 26 26 27 27 29 29 31 33 34 34 39

Part I Multichannel and Multi-Traffic Networks with Slotted ALOHA Protocol 2. OUTPUT AND DELAY PROCESS ANALYSIS OF SLOTTED ALOHA MULTICHANNEL NETWORKS 1 Introduction 2 System Model Performance Analysis 3

45 45 47 48

Contents

ix

Stationary Probability Distribution Average Performance Measures Packet Interdeparture Time Distribution of Type [c] Joint Probability Distribution of Packet Interdeparture Time and Number of Packet Departures Packet Delay Distribution 3.5 Numerical Results Conclusion

3.1 3.2 3.3 3.4

4 5

3. PERFORMANCE ANALYSIS OF MULTICHANNEL SLOTTED ALOHA NETWORKS WITH CAPTURE 1 Introduction 2 System Model 3 Performance Analysis 3.1 Stationary Probability Distribution 3.2 Average Performance Measures 4 Numerical Results Conclusion 5

48 51 52

55 60 65 67 71 71 72 74 74 76 77 78

4. OUTPUT AND DELAY PROCESS ANALYSIS OF MULTICHANNEL 83 NETWORKS WITH VOICE/DATA 1 Introduction 83 2 System Model 85 87 3 Performance Analysis 87 3.1 Analysis of Nonreservation System Analysis of Reservation System 3.2 90 92 3.3 Packet Departure Distribution 94 3.4 Average Performance Measures Packet Delay Distribution 98 3.5 4 103 Numerical Results 106 Conclusion 5

5. PERFORMANCE ANALYSIS OF PRIORITIZED MULTIHOP PACKET RADIO NETWORKS 1 Introduction 2 System Model

113 113 116

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

3

4 5

Performance Analysis Joint Probability Generating Function 3.1 Analysis for Three-Hop Network 3.2 3.2.1 Fully Connected Three-Hop Network 3.2.2 Tandem Three-Hop Network Analysis for N-Hop Network 3.3 3.3.1 Queue Length Distribution 3.3.2 Average Performance Measures 3.3.3 Transmission State Distribution Two-Hop Parallel Network 3.4 Numerical Results Conclusion

117 117 119 119 123 126 128 132 132 139 141 141

Part II Multichannel and Multi-Traffic Networks with CSMA Protocols 6. OUTPUT AND DELAY PROCESS ANALYSIS OF SLOTTED CSMA/CD MULTICHANNEL LANS 1 Introduction System Model 2 3 Performance Analysis Stationary Probability Distribution 3.1 Average Performance Measures 3.2 Packet Interdeparture Time Distribution of Type [c] 3.3 Joint Probability Distribution of Packet Interdeparture 3.4

4 5

Time and Number of Packet Departures Packet Delay Distribution 3.5 Numerical Results Conclusion

7. PERFORMANCE ANALYSIS OF CSMA/CA WIRELESS LANS 1 Introduction 2 System Model Performance Analysis 3 3.1 Analysis for Second State Transition Analysis of IFT for First State Transition 3.2 Analysis of DFT for First State Transition 3.3

149 149 152 153 153 155 155 157 159 162 163 169 169 172 173 175 175 177

Contents

4 5

xi

3.4 Analysis for Third State Transition 3.5 Stationary Probability Distribution 3.6 Average Performance Measures Numerical Results Conclusion

8. OUTPUT PROCESS ANALYSIS OF WIRELESS CSMA/CA LANS WITH VOICE/DATA 1 Introduction System Model 2 3 Performance Analysis Analysis for Second State Transition 3.1 3.2 Analysis of IFT for First State Transition 3.3 Analysis of DFT for First State Transition 3.4 Analysis for Third State Transition 3.5 Stationary Probability Distribution 3.6 Average Performance Measures 3.7 Packet Interdeparture Time Distribution 4 Numerical Results 5 Conclusion

178 179 179 180 182 187 187 189 191 193 195 196 198 199 200 201 203 210

Part III Multichannel and Multi-Traffic in Wireless Communication Networks with CDMA Protocol and Hybrid Channel Assignment Scheme 219 9. OUTPUT AND DELAY PROCESS ANALYSIS OF CDMA NETWORKS WITH VOICE/DATA 1 Introduction System Model 2 3 Performance Analysis Stationary Probability Distribution 3.1 Packet Departure Distribution 3.2 Average Performance Measures 3.3 3.4 Packet Delay Distribution Numerical Results 4 Conclusion 5

221 221 224 227 227 230 231 233 236 239

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10. PERFORMANCE ANALYSIS OF CELLULAR MOBILE

NETWORKS WITH HCA SCHEME Introduction 1 2 System Model Performance Analysis 3 Interoverflow Time Distribution 3.1 Analytical Method I 3.2 Analytical Method II 3.3 3.3.1 Blocking Probability 3.3.2 Binomial Moment Numerical Results 4 Conclusion 5

245 245 248 250 250 252 255 255 257 261 262

11. OUTPUT AND DELAY PROCESS ANALYSIS OF CELLULAR MOBILE NETWORKS WITH HAND-OFF 1 Introduction 2 System Model 3 Performance Analysis 3.1 Stationary Probability Distribution Average Performance Measures 3.2 Waiting Time Distribution 3.3 Numerical Results 4 Conclusion 5

267 267 269 271 271 277 279 282 285

12. CONCLUDING REMARKS 1 Summary of the Book Topics for Future Research 2

289 289 289

Appendices Derivation of Equation (5.16) Derivation of Equation (8.4) Numerical Calculation Method to Solve Row Vector x in Chapter 11

298 299 301 305

References Index

309 321

List of Figures

2.1 2.2 2.3 2.4 2.5 2.6 2.7

2.8 2.9 2.10 3.1 3.2 3.3 3.4 4.1 4.2

Multichannel system configuration Imbedded Markov points and channel state transition Packet interdeparture time Packet interdeparture time T Initial delay and backlog delay Average channel utilization versus packet arrival rate Average channel utilization versus packet retransmission rate Average packet delay versus packet arrival rate Coefficient of variation of packet interdeparture time versus packet arrival rate Coefficient of variation of packet delay versus packet arrival rate Average channel utilization versus packet arrival rate with Average packet delay versus packet arrival rate with Average channel utilization versus packet arrival rate with M Average packet delay versus packet arrival rate with M System state transition diagram Imbedded Markov points and channel state transition xiii

47 50 53 56 61 68 68 69 69 70 80 80

81 81 86 89

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

4.3 4.4 4.5 4.6 4.7

4.8 4.9 4.10 5.1 5.2

5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 6.1 6.2

Average channel utilization of data traffic versus data packet arrival rate Average data packet delay versus data packet arrival rate Coefficient of variation of data packet delay versus data packet arrival rate Average channel utilization of voice traffic versus data packet arrival rate Loss probability of voice traffic versus data packet arrival rate Effect of selection probability on reserved channels Effect of reserved channel on correlation coefficient of packet departures Effect of selection probability on correlation coefficient of packet departures System configuration System state transition rate diagram for fully connected three-hop network Tandem three-hop network Tandem N-hop network Imbedded Markov points and channel state transition at hop i Decomposition at hop N Decomposition at hop i Two-hop parallel network Average packet delay versus packet arrival rate Average packet delay versus packet transmission error rate Average packet delay characteristics (N = 3) Average packet delay characteristics (N = 4) Average packet delay characteristics (N = 5) Average packet delay characteristics for three systems Average channel utilization versus total offered rate Average packet delay versus total offered rate

108 108 109 109 110 110 111 111 117 122 124 127

128 133 137 139 143 143 144 144 145 145 165 165

List of Figures

6.3 6.4 6.5 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11

Coefficient of variation of packet interdeparture time versus total offered rate Coefficient of variation of packet interdeparture time T versus total offered rate Coefficient of variation of packet delay versus total offered rate Imbedded Markov points and channel state transition Second state transition diagram for both IFT and DFT First state transition diagram for IFT First state transition diagram for DFT Average channel utilization versus packet arrival rate with Average packet delay versus packet arrival rate with Average channel utilization versus packet arrival rate with D Average packet delay versus packet arrival rate with D Average packet delay versus average channel utilization Imbedded Markov points and channel state transition Second state transition diagram for both IFT and DFT First state transition diagram for IFT First state transition diagram for DFT Average packet delay versus total offered rate with Total packet delay versus total channel utilization with Average voice packet delay versus total offered rate with D Average data packet delay versus total offered rate with D Total packet delay versus total channel utilization with D Average packet delay versus total offered rate for IFT and DFT with Total packet delay versus total channel utilization for IFT and DFT with

xv

166 166 167 174 175 176 177 184 184 185 185 186 192 193 195 197 212 212 213 213 214 214

215

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

8.12

8.13 8.14 8.15

8.16 8.17 9.1 9.2 9.3 9.4 9.5 9.6 9.7 10.1 10.2 10.3 10.4 10.5 11.1 11.2 11.3 11.4 11.5

Average voice packet delay versus total offered rate for IFT and DFT with D Average data packet delay versus total offered rate for IFT and DFT with D Total packet delay versus total channel utilization for IFT and DFT with D Throughput of voice or data traffic versus total offered rate Coefficients of variation of packet interdeparture time versus total offered rate with Coefficient of variation of data packet interdeparture time versus total offered rate with D System state transition diagram Throughput of voice traffic versus data packet arrival rate Throughput of data traffic versus data packet arrival rate Blocking probability of voice traffic versus data packet arrival rate Average data packet delay versus data packet arrival rate Correlation coefficient of packet departures versus data packet arrival rate Coefficient of variation of data packet delay versus data packet arrival rate Cellular mobile telephone network State transition rate diagram upon arrival Flowchart of simulation model Blocking probability versus arrival rate with method I Blocking probability versus arrival rate with method II Flow diagram of system model with hand-off System state transition diagram Average channel utilization versus total offered load Blocking probability versus total offered load Average queue length of hand-off calls versus total offered load

215 216 216 217

217 218 226 241 241 242 242 243

243 249 253 264 265 265 270 273 286 286 287

List of Figures

11.6 11.7 11.8

Average waiting time of hand-off calls versus total offered load with M Average waiting time of hand-off calls versus total offered load with and Standard deviation of hand-off waiting time versus total offered load

xvii

287 288

288

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List of Tables

0.1 1.1 1.2

Classification of performance model in each chapter Related wireless communication networks Specific model types used in the book

xix

xxiii 2 34

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Preface

With the rapidly increasing penetration of laptop computers and mobile phones, which are primarily used by mobile users to access Internet services like e-mail and World Wide Web (WWW) access, support of Internet services in a mobile environment is an emerging requirement. Wireless networks have been used for communication among fully distributed users in a multimedia environment that has the needs to provide real-time bursty traffic (such as voice or video) and data traffic with excellent reliability and service quality. To satisfy the huge wireless multimedia service demand and improve the system performance, efficient channel access methods and analytical methods must be provided. In this way very accurate models, that faithfully reproduce the stochastic behavior of multimedia wireless communication and computer networks, can be constructed. Most of these system models are discrete-time queueing systems. Queueing networks and Markov chains are commonly used for the performance and reliability evaluation of computer, communication, and manufacturing systems. Although there are quite a few books on the individual topics of queueing networks and Markov chains, we have found none that covers the topics of discrete-time and continuous-time multichannel multitraffic queueing networks. On the other hand, the design and development of multichannel multihop network systems and interconnected network systems or integrated networks of multimedia traffic require not only such average performance measures as the throughput or packet delay but also higher moments of traffic departures and transmission delay. xxi

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The purpose of this book, therefore, is to offer detailed exact and approximate analytical solution methods and techniques using queueing theory to model the complex multichannel and multi-hop network systems with procedures of multiple access schemes and reliably evaluate the performance of the systems. In particular, this book presents methods of approximating the system performance of discrete-time and continuous-time multimedia networks, the probability distribution of the interarrival time of internetwork transmissions at the adjacent network and the higher moments of the transmission departure distribution and delay distribution in wireless multimedia communication environment. The generally accepted view is that discrete-time multimedia communication systems can be more complex to analyze than equivalent continuoustime ones, because of the finite size of a time-unit, multiple state changes can occur from one time-unit to the next. This complicates the resulting analysis of the model. In this book, numerical results that illustrate the applications of the theory and various properties are also discussed. We organize the book to three parts. In Part I, we discuss wireless communication networks with the multiple random access slotted ALOHA protocol with several system performance analyses for multichannel, multi-hop and multi-traffic network systems. Part I includes 4 chapters, Chapters 2-5. In Chapter 2, we consider packet radio communication systems that employ a set of M parallel channels under the slotted ALOHA protocol and exactly derive the moment generating functions of the packet interdeparture time, number of packet departures and packet delay for both IFT and DFT protocols. In Chapter 3, we present an exact analysis to evaluate the effect of capture on the multichannel slotted ALOHA protocol and give the improved system performance such as the channel utilization and average packet delay. In Chapter 4, we propose two different procedures of multichannel multiple random access schemes with slotted ALOHA operation for integrated voice and data traffic and present exact analyses to numerically evaluate the average performance measures and high moments of these systems. In scheme I, there is no limitation on access between voice transmission and data transmission, i.e., all channels can be accessed by all transmissions. In scheme II, a channel reservation policy is applied where a number of channels are used exclusively by voice packets, while the remaining channels are used by both voice and data packets, and voice packets select the reserved channels with a given probability.

PREFACE

xxiii

In Chapter 5, we analyze a multi-hop packet radio communication network which consists of a finite number of hops with infinite buffer capacities. In this chapter, two major results are presented. First, through an exact analysis, the average queue length and packet delay of the system are explicitly derived, and performance of the system with and without transmission error is compared. Then, an approximate analytic method, based on a decomposition approach in which the total system is divided into subnetworks of the generalized M/G/1 type, is proposed to simplify the analysis of the queue length and packet delay in packet radio communication networks with a large number of hops. In Part II, we provide the analyses and various properties for local area networks (LANs) and wireless LANs (WLANs) with the multiple random access CSMA/CD and CSMA/CA protocols for multichannel and multitraffic. There are 3 chapters, Chapters 6-8, in Part II. In Chapter 6, we present the system performance analysis of slotted multichannel non-persistent CSMA/CD local area networks with a finite number of users. Channel utilization, delay performance and higher moments of the packet interdeparture time, number of packet departures and packet delay are then calculated in the terms of the number of network users, the number of network channels and the channel access rate. In Chapter 7, we present an exact analysis to numerically evaluate the performance of high-speed and realizing fully distributed WLAN systems with a multiple random access method named non-persistent CSMA/CA protocol. The collision avoidance portion of CSMA/CA in this system model is performed with a random pulse transmission procedure, in which a user with a packet ready to transmit initially sends some pulse signals with random intervals within a collision avoidance period before transmitting the packet to verify a clear channel. WLANs as in Chapter 7 have been used for communication among fully distributed users in a multimedia environment that has the needs to provide real-time bursty traffic (such as voice or video) and data traffic. In Chapter 8, we present a detailed system model and an effective analysis for the performance of WLANs which support multimedia communication with the non-persistent CSMA/CA protocol. In this chapter, we also present an exact analysis to derive the moment generating function of the packet interdeparture time for the output process. In Part III, we present performance analyses and evaluations for personal communication networks and cellular mobile networks with various chan-

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nel access process methods as the CDMA protocol, fixed and hybrid channel assignment schemes for supporting multi-traffic transmission and handoff. Part III has 3 chapters, Chapters 9-11. In Chapter 9, we present the output and delay process analysis of integrated voice and data slotted CDMA network systems with a multiple random access protocol for personal wireless communications. In the system, the allocation of codes to voice calls is given priority over that to data packets, while an admission control, which restricts the maximum number of codes available to voice sources, is considered for voice traffic so as not to monopolize the resource. In addition, the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use the voice codes for transmission if the voice sources are silent. In Chapter 10, we provide two approximate techniques to evaluate the performance of large scale cellular mobile wireless network systems using a hybrid channel assignment scheme. The two approximate analyses give the steady-state probability distributions of the system which are used to obtain expressions for the blocking probabilities. Analytical results are compared with simulation results and good agreements are observed for both fixed and hybrid channel assignment schemes. In Chapter 11, we present an exact analysis and an efficient matrixanalytic procedure to numerically evaluate the performance of the large scale cellular mobile wireless network systems with hand-off. This chapter considers such a priority scheme that some channels and buffers are reserved for hand-off calls to reduce the forced termination of calls in progress. Performance characteristics included blocking probability, channel utilization, average queue length, average waiting time and high moments. Finally, in Chapter 12 we offer a summary of our conclusions in this book and suggest topics for future research. We classify the performance models analyzed in each chapter in Table 0.1 with regard to whether the analysis is exact or approximate, what types of communication network and protocol are considered, and whether the performance measure obtained is with respect to average or distribution. In Table 0.1, we use some abbreviated signs as follows: “Ch.” for “Chapter”, “M.C.” for “Multichannel”, “M.T.” for “Multi-Traffic”, “M.H.” for “Multi-Hop”, and “Perf.M.” for “Performance Measure”. Furthermore, we point out that Chapters 2-9 analyze discrete-time networks operating on the basis of time slotting, and transmitting information

PREFACE

xxv

in fixed length units such as packets (but in Chapter 5, arrivals of packets can possibly occur at any given time instant on the time axis). Chapters 10 and 11 analyze continuous-time networks at which arrivals and departures can possibly occur at any given time instant on the time axis. The analysis offered in each chapter is independent of that offered in other chapters, although, depending on the class of queueing system involved, there is some common ground between the techniques employed.

Each chapter contains its own system model and offers important equations and specific numerical results. The reader will find it helpful to refer to Chapter 1 initially, but after that the remaining chapters are stand-alone units which can be read in any order. The book should prove useful to a post-graduate course in computer science or engineering. It is often a pre-requisite to some other more advanced courses like network design and management based on queueing modeling with examples of their applications to multimedia communication and computer networks. It can also be used for a course on stochastic models in mathematics and operations research departments.

Y UTAKA

W UYI Y UE MATSUMOTO Kobe, Japan

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Acknowledgments

The authors would like to thank Itochubei Foundation of Konan University, Kobe, Japan, for supporting this publication. The authors are also grateful for the support for the research which led to this book received from GRANT-IN-AID FOR SCIENTIFIC RESEARCH.

xxvii

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Chapter 1 INTRODUCTION

In this introductory chapter, we describe the background technology and analytical models for the studies of this book.

1.

Overview of Multiple Access Communication Networks

Multiple access communication networks with common single channel or multichannel usually are used by many heterogeneous users to communicate with each other or transmit information to the center stations. Multiple access communication networks specify how signals from different sources can be combined efficiently for transmission over a given bandwidth band and then separated at the destination. Multiple access communication networks include packet radio networks, cellular mobile networks, wireless communication networks, local area networks (LANs), and wireless local area networks (WLANs). Multiple access communication networks can be classified into two main groups: (1) Fully connected networks: all users are in the same communications environment and can thus hear all channel activity such as packet radio networks, LANs and WLANs. (2) Multi-hop networks: not all users are able to hear a particular transmission and must use an intermediate station as a store-and forward repeater such as multi-hop packet radio networks and cellular mobile networks. 1

2

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

One of the first multiple access packet-switched radio networks was the ALOHAnet at the University of Hawaii and developed in 1971, under the sponsorship of ARPANET (Advanced Research Projects Agency Network) [Klei74]. This concept has been generalized to include mobile ground wireless networks and local area computer networks. Packet-switched radio networks have been in use for about 30 years since the introduction of the ARPANET. The current generation of wireless communication systems are all digital. The growth of mobile telephony has been extraordinary, reaching 200 million subscribers worldwide by early 1998. Wireless local area networks (WLANs) operated in the 2.4 GHz band are now available with bit rates on the order of 11 Mbps. The need for WLANs may grow with the demand for mobile computers. Wireless communication networks must provide connectivity to the Internet, integrated services, especially voice and data, as well as multimedia Web access. The wireless communication systems studied in this book for computer communications, wireless LANs and mobile telecommunications are summarized in Table 1.1.

The differences among these types of networks are still big. However, the table reveals that each type of network is now able to provide services that were formerly the exclusive province of other networks.

Introduction

2.

3

Packet Communication Networks

In this section we describe related packet communication networks such as local area networks and packet radio networks. These systems are mainly differentiated by their application (voice or data), support for user mobility, channel access method (fixed or random), channel number (single channel or multichannel), coverage area (local area or wide area, shingle hop or multi-hop) and network type (wire or wireless). Functionally, a packet switching network includes two types of devices: user and station. A station differs from a user, in that it is not a source for information flow, and its objective is to extend the effective communication range of users to achieve wide coverage. It provides area coverage for fixed and mobile users, global control functions, gateway functions for interfacing with other networks, and initialization functions. In centralized hierarchical routing algorithms, all packet transmissions between users in a network are routed via a station. Moreover a station initializes and periodically updates parameters for routing. A packet radio network is a store and forward packet switching system employing radio channel(s). It consists of geographically dispersed users

communicating with each other or with a central station over single or multiple radio channel which may be stationary or mobile (computers, mobile users, etc.). Packet switching networks, however, still employ point-topoint communication channels and large multiplexing switches for routing and flow control in a fashion similar to conventional circuit switched networks. Packet radio is simply a data communication architecture which combines the features of packet switching with those of radio channels for data communication networks. However, packet radio networks operating today also consist of voice communication networks and integrated voice and data communication networks in which voice, video and data are digitalized and packetized. Various multiple access protocols have been proposed such as fixed assignment, controlling assignment, and multiple random access methods. Important performance measures of interest in the fully connected networks and the multi-hop packet radio networks with multiple access protocols are the throughput, average packet delay, channel utilization, blocking probability and high moments for packet departures, packet interdeparture time and packet delay.

4

2.1

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Local Area Networks

A local area network (LAN) provides interconnection of a variety of data communicating devices within a local area such as a single department in a company, a university, a hospital and so on. LANs are used to provide communications between computers and devices, workstations, distributed processing, rapid access to data banks, telephones, facsimile, peripheral devices, local storage, sensors, and sharing of expensive devices and resources. Future LANs may carry a variety of traffic, such as voice, video, and data. One reason for utilizing a local area network is to share expensive resources, such as central data files. Another reason is to exchange data and information among systems. LANs are generally privately owned by single organizations and are usually designed so that all users are connected by a single (or multiple) highspeed shared channel(s). Data or messages are packetized and then transmitted through the common channel(s) such that a given packet can be received by all users on the network. Thus, routing is not required, and it is possible to install high-speed, low-noise channels. A typical example of the LAN technology is the Ethernet. Since the early 1980s, LANs have been able to provide significantly higher data rates than the switched telephone network. Thus, interconnecting these high data rate LANs has been a problem. Two metropolitan area network (MAN) solutions for distributed interconnection are the fiber distributed data interface (FDDI) and the distributed queue dual bus (DQDB). The primary issues for local area network design are congestion control, channel access protocol, and network architecture. In the performance analysis of LANs, network topology, transmission media, network access techniques, and network interfaces are given. In particular, the CSMA, CSMA/CD protocols have been analyzed and numerically evaluated in a large number of papers such as [Klei75c], [Toba80a], [Toba82a], [Coyl83], [Medi83], [Taka85a], [Taka85b], [Apos86], [Taka86], [Takah86], [Tasa86], [Kama87], [Mats90a], [Mats90b], [Mats90c] and [Onun91] for important performance measures among which are throughput, packet delay, response time and higher moments of the output distribution and delay distribution.

2.2

Single Channel Networks

Single channel packet communication networks have for a long time been studied for their operational properties and potential for computer

Introduction

5

communications [Kahn77]. Many performance studies have been made of these traditional networks (using point-to-point wire connections), starting with the pioneering work of Kleinrock [Klei64]. It was in this work that the independence assumption among users was introduced. This assumption can fairly be said to be the basis of all later work in performance modeling of computer communication networks. The reason why this assumption is so important is that it allows a communication network to be modeled as an open network of queues which leads to simple closed form solutions for many measures of interest, such as average queue length, network average delay and so on. Several authors have faced the analysis problem in the above context by employing suitable models and mathematical tools. The models used can be distinguished on the basis of specific hypotheses about the user population size and the packet length distribution. The population size is assumed either finite or infinite, while the packet length is considered either fixed or variable.

2.3

Multichannel Networks

Multichannel packet communication networks are expansions of the single channel packet communication networks. In a multichannel system, there are several independent channels, say, M channels. A user can transmit to or receive from any of M channels in some transmission protocols. It is based upon frequency domain design techniques which reallocate the system bandwidth. One method is to divide the entire available bandwidth into homogeneous partitions operating with the same multiple access protocol on each channel. It essentially allows for full sharing of the channels over the entire user population. Each user having an arriving packet can transmit the packet at a randomly chosen channel in using random access protocols cases. In such multichannel systems, the sum of the multiple channel bandwidth of the multichannel networks is equal to the bandwidth of a single channel network, i.e. all M channels are considered to have the same bandwidth in Hz, where V is the total available bandwidth of the system. The time slot size over a channel with bandwidth is given by where is the time slot size in a channel with bandwidth V. Therefore, as the number of channels is changed, the bandwidth of each must also change to satisfy the fixed total bandwidth constraint.

6

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

It can be seen that this may result in less packet retransmitting and an improvement in the throughput performance. In general, multichannel broadband radio networks have advantages over single channel broadband radio networks in the following points [Yung78], [Mars83], [Okad84] and [Mars87]:

High effectiveness of channel utilization by keeping the capacity per channel to be rather low rate Low waiting delay owing to distribution of traffic load on multiple channels Large total capacity by increasing the number of channels and also easy expansion ability Easy organization by frequency division multiplexing technology

High reliability and fault tolerance

2.4

Multi-Hop Networks

A network in which every user can communicate directly with all the others is called the one-hop network; otherwise, repeaters relay packets, thus creating a network with multiple hops that is called the multi-hop network. Due to the limited transmission power of each user, not all users will be able to hear a particular transmission, and multi-hop transmission then become necessary. For packet radio networks to operate in large geographical areas, stations are commonly used to ensure connectivity of users, thus forming multi-hop packet radio networks. A packet or a message from source to destination will typically be relayed over multiple hops before arriving at its destination. A number of access protocols have been proposed for multi-hop packet radio networks, in view of providing improved performance of the system. For one-hop networks, several protocols have been proposed and thoroughly studied. The main issue in these protocols is how to coordinate packet scheduling temporally, to avoid collisions and improve throughput. If these protocols are applied to multi-hop networks, however, timing and spatial coordination jointly determine their performance. This makes the analysis significantly more complex, due to the fact that multi-hop networks are spatially distributed, operationally decentralized and involve multiple access communication channels.

Introduction

3.

7

Wireless Communication Networks

Modern wireless communication networks include cellular telecommunication systems, wide area wireless data systems such as mobile packet radio networks, wireless LANs (WLANs). These systems are also mainly differentiated by their application, support for user mobility, channel access method, number of channel and coverage area.

3.1

Cellular Mobile Telecommunication Networks

In regard to mobile telephone communication service, the increasing demand for mobile telephone communication service and the finite spectrum allocated to this service lead to the proposal of the cellular structure that is to divide spatially the entire service region into a number of small cells each of which contains a base station. The concept of the cellular mobile telecommunication system has been studied intensively in [MacD79]. A cellular mobile telecommunication network is a wireless network that provides communication services for users. The cellular mobile systems, also referred to as Personal Communication Systems (PCS), are high capacity land mobile systems in which the available frequency spectrum is partitioned into discrete channels which are assigned in groups to geographic cells covering a cellular Geographic Service Area (GSA). Because the spectrum allocated to the mobile telecommunication system is an important resource, it is necessary to use this frequency spectrum most efficiently. It leads to frequency reuse that is the use of the same carrier frequency channel in different cells within the tolerable level of signal to interference radio constraints. Distinct frequency channels are assigned to each base station for communication, and the users in a cell can communicate with the base station by these radio channels. Reuse of frequency allows one to serve a very large number of users with a fixed number of allocated frequency channels. A major concern in a cellular mobile radio system is co-channel interference. Any technique to reduce interference in the cellular systems leads directly to an increase in system capacity and performance. Some methods for interference reduction in use today or proposed for future systems include cell sectorization, directional and smart antennas, multiuser detection, and dynamic channel and resource allocation. Cellular mobile radio communication system will also play an important role in predicting the flavor of future mobile wireless data networks.

8

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

3.2

Mobile Packet Radio Networks

A mobile packet radio network can generally be defined by the following

features: its host computers and users, communication processors, topological layout, communication equipment and transmission media, switching technique, mobile unit and protocol design. Mostly, investigations have been limited to the study of stationary packet radio networks. There exists a number of differences between the stationary and mobile data communication systems. The mobile packet radio system generalizes the stationary system by allowing every element of the hardware to be independently in motion. In the mobile data communication system, users are assumed to be in motion, and covered cells vary with time. In this system, the system may become inoperative due to motion of the users. Consequently, a frequent monitoring operation is required. When the performance of the systems is analyzed, it is necessary to consider the effect of channel noise and transmission error. These features are chosen to accomplish the functions of the network subject to specified performance requirements. The performance measures most commonly quoted include packet or message delay, throughput, error rate including channel noise and transmission error, reliability, and cost. When mobile operations are involved, the measurements indicate temporary degradation in the performance, affecting both throughput and delay. The first analysis of mobile packet radio performance assumed that packet collisions were the major cause for loss of a packet and subsequent retransmission. More recently, efforts to design mobile data communication systems to operate over degraded channels have been undertaken. The throughput and packet delay, the two primary performance criteria in data communications, have been extensively studied for basic system concepts such as pure ALOHA, slotted ALOHA, CSMA and CSMA/CD. However, we need to consider the effect of channel errors due to both noise and transmission error.

3.3

Wireless Local Area Networks

Nowadays, most of large offices, campus classrooms, conference registration are equipped with wiring for conventional LANs, and the inclusion of high-speed wireless LAN (WLAN) in the planning of a large new office building is done as a standard procedure, along with planning for telephone and other different wireless medias [Pahl95].

Introduction

9

Mobile data communication services provide a low-speed solution for wide area coverage, but for high-speed and local communications, a portable user with wireless access can bring the processing and database capabilities of a large computer directly to specific locations for short periods of time, thus opening a horizon for new applications, such as the wireless multimedia campus and hospital for instructional purposes and medical diagnosis. The two primary standards that have emerged are IEEE 802.11 in the United States and HIPERLAN (High Performance European Radio LAN) in Europe. IEEE Standard 802.11 specifies the physical and MAC layers for operation of WLANs and addresses the direct sequence spread spectrum (DSSS) and frequency-hopping spread spectrum (FHSS) access methods for the radio medium. It also allows for a third option for the infrared medium, which is still under development. It specifies support for asynchronous as well as synchronous data transfers. The asynchronous data transfer applies to applications that are not time sensitive (e.g., e-mail and file transfers). The synchronous mode supports time-bounded applications like video and packetized voice.

Using wireless network interfaces, mobile devices can be connected to the public telecommunication network in the same way as wired telecommunications, or to the Internet in the same way as desktop computers are connected, using the Ethernet, token ring, or point-to-point links. Current trends in communication service applications indicate that there will be an increasing demand in future WLANs for multimedia communication such as voice, full-motion video and data. And multimedia applications will provide users with the means for truly universal portable communications and computing.

4.

Multiple Access Protocols

In computer communication networks and wireless telecommunication networks, efficient channel access protocols must be employed to utilize the limited spectrum among all the users efficiently. In this section, we discuss various related communication protocols. There have been different access methods that are appropriate for use with different network topologies in these networks including fixed assignment protocols and random access protocols.

10

4.1

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Fixed Assignment Protocols

Multiple access techniques assign dedicated channels to multiple users through bandwidth division. The main fixed assignment protocols with multiple access methods currently in use in wireless communication networks are Time Division Multiplexing Access (TDMA), Frequency Division Multiplexing Access (FDMA) and Code Division Multiple Access (CDMA) (for spread spectrum networks). These protocols allocate a portion of the channel (in the time, frequency or code space), to each user (or link) in a predetermined fixed manner. Thus the common resource is divided into a set of disjoint dedicated components. The performance of these schemes can be modeled using standard queueing theoretic approaches since each component acts independently of the other. While fixed assignment protocols have excellent performance (in terms of throughput) in steady heavy traffic, the delay performance in light burst traffic (typical of data) is very poor. Some of the disadvantages of these methods for users with high peak-to-average data rates have been discussed by Carleial and Hellman [Carl75]. 4.1.1

TDMA

In TDMA system, time is divided into nonoverlapping time slots that are allocated to different users. All the users have access to the total band, and simultaneous communications are possible when the users transmit during different time slots. Assuming all users to be identical, TDMA assigns fixed predetermined channel time slots to each user; it results in assigning a fraction 1/N of the total channel capacity where N is the number of users in the system and is also requires buffering capabilities. 4.1.2

FDMA

In FDMA systems, the total band is divided into orthogonal channels that are nonoverlapping in frequency, and each of these channels is allocated to one and only one user for the time of its communication. Under an assumption that all users are identical, FDMA consists of assigning to each user a fraction V/N, where V is total radio channel bandwidth and N is the number of users in the system. Such techniques are widely used in wire packet switching networks, and they have already been applied to some mobile systems.

Introduction

11

A number of disadvantages of FDMA exist when compared with TDMA: wasted bandwidth for adequate frequency separation, lack of flexibility in achieving dynamic allocation of bandwidth, lack of broadcast operation. The only major disadvantage in TDMA is the need to provide time synchronization and sufficient separation between slots to avoid time overlap. 4.1.3

CDMA

The CDMA systems utilize the spread spectrum technique, whereby a spreading code (called a pseudo-random noise code or PN code) is used to allow multiple users to share a block of frequency spectrum. CDMA allows more than one user to simultaneously use all the available frequency bands. In CDMA time and bandwidth are used simultaneously by different users, modulated by orthogonal or semi-orthogonal spreading codes. In order to obtain distinguishable radio signals, the users must have different possibilities of encoding digital information into modulated waveforms. Transmissions are coded with receiver specific codes, using spread spectrum wave-forms such as pseudo-noise modulation or frequency hopped patterns. A radio is then able to receive the first transmission coded for it,

and any subsequent transmission can be rejected as noise, during the reception. This ability is of course dependent on the coding and signal to noise ratio, which dictate the number of simultaneous transmissions a receiver can tolerate while successfully receiving one transmission. One advantage of CDMA over the more classical multiple accessing techniques of either FDMA or TDMA is the relative ease of adding additional users, even a system initially designed to satisfy the peak distortion constraint might at some point be forced to operate in an environment where the constraint is violated. All digital cellular standards are undergoing enhancements to support high rate packet data communication. The systems will support higher data rates by evolving to the wideband CDMA standard in International Mobile Telecommunications 2000 (IMT-2000).

4.2

Random Access Protocols

Efficient techniques to share the available bandwidth among many heterogeneous users are needed due to the scarcity of wireless spectrum. Applications requiring continuous transmission such as voice and full-motion video generally allocate dedicated channels for the duration of the call.

12

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Sharing bandwidth through dedicated channel allocation is called multiple access. In contrast to voice or video transmission, transmission of data tends to occur in bursts. Bandwidth sharing for users with bursty transmission generally employs some form of random channel allocation that does not guarantee channel access. Random multiple access protocols have been proposed for data communication networks, wide wireless mobile communication networks and wireless LAN networks because they present low delay than fixed assignment techniques when the traffic is light or burst. Random access schemes are characterized by not requiring a centralized control mechanism to regulate the transmission of messages. The random access protocols are based on the fact that any number of users may transmit packets or messages over the common single channel or multichannel at any particular point in time. In such an environment it makes sense not to bother to maintain a global queue, but rather let each user make its own decision when it thinks to use the channel.

In these networks, multiple distributed users share a single channel or multiple channels to communicate with each other. Occasionally, more than one user will attempt to use the channel at the same time, resulting in destruction of one or both packets or messages (collision). This is usually resolved by some kind of acknowledgment scheme and retransmission.

Various multiple access protocols have been proposed. The main multiple random access protocols include pure ALOHA, Slotted ALOHA and Carrier Sense Multiple Access (CSMA), Carrier Sense Multiple Access with Collision Detect (CSMA/CD), Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA), etc. Random access methods provide a more flexible and efficient way of managing channel access for communication of short messages. However, it is a common characteristic of these protocols that they have good delay performance in light traffic and poor throughput performance in heavy traffic. What constitutes light and heavy traffic will depend on the specific protocol. [Klei76] contains a discussion of the relative merits of protocols in each category. In the researches on packet radio networks and wireless communication networks using a multiple access broadcast channel, the system performance and applications of multiple random access schemes have been widely studied.

Introduction

4.2.1

13

Pure ALOHA

The ALOHA system employs two different transmission strategies: the pure (or unslotted) ALOHA protocol and the slotted ALOHA protocol. The pure ALOHA protocol is one of the best known multiple random access protocols. Using a Markov model, we can find the throughput-delay relationship for the pure ALOHA protocol. The pure ALOHA protocol was developed in the early 1970 to allow communications between remote users and the central computer at the University of Hawaii [Abra70]. In the pure ALOHA technique, all packets have the same length requiring T times for transmission. All users begin their packet (re)transmission upon receiving a packet immediately to a central base station with a common single channel. When two or more packets arrive at the station simultaneously, we say that there is a collision. If there is a collision, users involved in it will retransmit their packets. To avoid repeated collisions with the same users, each retransmission will be attempted after a random time. A packet will be successful if no other users start a transmission at least within this packet vulnerable period T. If we assume that packets are of a fixed length, we can define the throughput as the number of packets transmitted successfully per packet transmission time. Note that the throughput thus defined takes on a value between 0 and 1. Let be the probability a packet is successfully transmitted, and let G be the offered traffic on the channel (including new transmissions and retransmissions, and measured in packets per unit time) by an infinite population of users. Then as a well known result the throughput is given by

We assume that the users generate packets including both new transmissions and retransmissions as a Poisson process. The number k of packets generated in T unit time has a probability mass function

and the probability no other packet is generated is thus Since is the probability that no additional packets are generated during the vulnerable interval of length 2T time, so

14

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

The maximum throughput is 1/2e, or 0.184 and corresponds to an offered load of 0.5 packet per unit time. These results were first obtained by Abramson [Abra70]. 4.2.2

Slotted ALOHA

Roberts [Robe75] extended Abramson’s result to develop a protocol with minimal coordination between the users called slotted ALOHA. The slotted ALOHA multiple random access protocol is among the simplest and oldest multiple random access protocols for communication systems, and is a viable access technique in certain applications. Under the slotted ALOHA protocol, the time axis is divided into fixed length time slots of one packet transmission time, and all users with a packet are synchronized for transmission. A packet will be successful if no other user starts a transmission at the beginning of the same slot, and the collided packets are retransmitted after random retransmission delays. The throughput is defined to be the average number of successful packet transmissions per time slot by using the same notations defined in the part of the pure ALOHA protocol, that is given by The maximum throughput is doubled to about 37 percent and occurs when the offered load is one. 4.2.3

CSMA and CSMA/CD

The CSMA protocol has now been widely accepted in multiple access broadcast local area networks. In the CSMA protocol, users of a network first sense the channel before transmitting their packets. If the channel is sensed busy, the sensing user refrains from transmitting (to avoid a collision) and reschedules his transmission according to one of several strategies. If the channel is sensed idle, the user transmits his packet immediately. Collision of packets only occurs if two or more users start transmitting packets within a propagation time of each other. Each user data source regulates his own use of the communication medium by use of an access protocol which requires only information available to the users by sensing the channel. In CSMA/CD, collisions can be detected by letting users monitor the channel to see if they agree with the packets being transmitted. When a collision is detected, the detecting user will immediately abort his transmission and send a noisy pulse informing all the users which in turn will

Introduction

prevent others from transmitting. If no collision is detected in

15

then this

user is assured of a successful transmission. is the propagation delay between the two farthest users. We call the end-to-end propagation delay. Thus, collision detections is feasible only when the propagation delay is short compared to the transmission time of a packet on the channel. Collision detection improves the system performance by preventing a user from wasting time in transmitting the remainder of a collided packet. The CSMA/CD protocol is one of the most widely used access protocols over different local area network topologies. The first study of the CSMA protocol was carried out by Kleinrock and Tobagi [Klei75a]. That paper implicitly assumed that the rescheduling de-

lay is infinite, so that all packets that find the channel busy or that are destroyed in a collision are abandoned. In a subsequent paper by Tobagi and Kleinrock [Toba77], the rescheduling times were modeled as independently geometrically distributed random variables. They used a finite number of users and assumed the arrivals to be quasi-random (i.e., finite-source Poisson). The CSMA/CD protocol was first introduced by Metcalfe and Boggs

[Metc76] as an extension to the ALOHA network presented by Abramson [Abra70] and the CSMA protocol without collision detection proposed by Kleinrock and Tobagi [Klei75a]. Tobagi and Hunt [Toba80a] added collision detection to the model of Tobagi and Kleinrock [Toba77]. The model in [Toba80a] allowed variable length packets. There are variations to CSMA and CSMA/CD basic strategy as follows:

(1) Non-persistent CSMA: If the channel is idle, send; if the channel is busy, wait a random time and try again (2) p-persistent CSMA: This is used for slotted channels. If the channel is idle, send with probability p and defer until the next slot with probability 1 – p. This is repeated until the packet is successfully sent, or until another user is sensed to have begun transmitting, in which case wait a random time and try again (3) 1-persistent CSMA: This is a special case of the p-persistent CSMA in which case p = 1. A ready user senses the channel, if the channel is idle, it transmits the

16

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

packet. If the channel is busy, it waits until the channel becomes idle and then the user transmits the packet with probability one. A. Non-persistent CSMA Because of the simplicity and the good performance of the non-persistent protocol, its analysis has received a wide attention. The basic equation for the throughput was expressed in terms of (the end-to-end propagation delay) and G (the offered traffic rate) as follows [Klei75a]:

and the throughput equation for the slotted non-persistent CSMA is also given by [Klei75a], it is

Tobagi and Hunt [Toba80a] provided an analysis for the slotted nonpersistent CSMA/CD protocol with finite population. The same protocol under the unslotted operation with an Erlangian distribution of the message transmission and collision detection times was also analyzed by Coyle and Liu [Coyl83].

B. p-persistent CSMA The throughput of the p-persistent CSMA/CD protocol has been studied in [Taka85a] by Takagi and Kleinrock for both infinite and finite population using the assumption that the backlogged users join the idle users (the strong Poisson assumption). They also assumed that the number of ready users at the start of a transmission is independent of the number of ready users at the start of the preceding transmission. The same assumption of having all backlogged and idle users generate packets according to the same process was used in [Apos86] to analyze the performance of buffered CSMA/CD. Kamal [Kama87] analyzed the slotted p-persistent CSMA/CD protocol with finite population assuming that each user has a single buffer and it operates in a synchronous manner. For a given offered traffic G and a given value of the parameter p, Kleinrock and Tobagi have given the throughput (see [Klei75a]).

Introduction

17

C. 1-persistent CSMA The throughput equations for 1-persistent CSMA and slotted 1-persistent CSMA were also given by [Klei75a]. The throughput equation for the 1persistent CSMA protocol case was given by

and the throughput equation for the slotted 1-persistent CSMA case was given by

Shacham [Shac82] provided an analysis of the 1-persistent CSMA protocol in the case of finite population with heterogeneous traffic and capture. They assumed single buffer users, with constant length packet.

4.2.4 Performance Comparisons for ALOHA and CSMA The main advantages of ALOHA and CSMA systems are that they are inherently robust due to the distributed access protocol and passive channel coupling. The throughput-offered load curves for different multiple access protocols to compare were showed in [Hamm86]. The throughput-delay relationship of the ALOHA and CSMA protocols was given by [Klei75a] and [Hamm86]. In these papers, a certain propagation delay is given for the CSMA protocols. It was shown that when the optimum p-persistent CSMA has the best performance; on the other band the performance of the non-persistent CSMA is quite comparable. We observed that as the offered load increases, the networks become unstable and the delay may become infinite. This is because the retransmitted packets keep colliding with each other and no packet is successfully transmitted. There are various proposals to stabilize the ALOHA protocol, by controlling the retransmission process [Klei75a]. They also caused some data packets to suffer delays. Meditch and Lea [Medi83] presented a comprehensive study of the stability and optimization of the infinite population, slotted, nonpersistent CSMA and CSMA/CD channels.

18

4.2.5

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

CSMA/CA

Carrier sensing and collision detection require a transmitting user to detect packet transmissions from other users to his intended receiver. However, the user is unable to listen to the channel for collisions while transmitting, so carrier sensing and collision detection are not very effective in wireless channels. In the result, it is impossible to apply CSMA/CD protocol to WLAN systems for fully distributed users. The current proposal is based on a carrier sense multiple access with collision avoidance (CSMA/CA) protocol by IEEE 802.11, a standard for WLANs, see, for instance, [Davi94], [IEEE96], [LaMa96] and [Crow97]. WLANs with CSMA/CA protocol work by a “listen before talk” scheme. The collision avoidance portion of CSMA/CA is performed by such a procedure that lets some randomly chosen users talk and the others listen, so that the users who hear somebody talking will refrain from talking. Despite

using the “listen before talk” scheme, packet collision can still occur as it is a random process to select talkers and listeners. Collision avoidance significantly increases the throughput of the system and is currently part of several wireless LAN standards. However, the efficacy of collision avoidance can be degraded by the effects of path loss, shadowing, and multipath fading on the busy tone. 4.2.6

Reservation Multiple Access Protocols

Reservation multiple access protocols assign channels to users on demand through a dedicated reservation channel. Reservation multiple access protocols include several channel access methods. Packet Reservation Multiple Access (PRMA) is a relatively new random access technique that combines the advantages of reservation protocols and ALOHA. In PRMA time is slotted and the time slots are organized into frames with T time slots per frame. Active users with packets to transmit contend for free time slots in each frame. Once a packet is successfully transmitted in a time slot, the time slot is reserved for that user in each subsequent frame, as long as the user has packets to transmit. When the user stops transmitting packets in the reserved slot the reservation is forfeited, and the user must again contend for free time slots in subsequent packet transmissions. PRMA is well suited to multimedia traffic with a mix of voice (or continuous stream) traffic and data traffic. Once the continuous stream traffic has been successfully transmitted, it maintains

Introduction

19

a dedicated channel for the duration of its transmission, while data traffic only uses the channel as long as it is needed. PRMA requires little central control and no reservation overhead, so it is superior to reservation based protocols when there is a mix of voice traffic and data traffic. Another reservation multiple access method is a channel reservation policy where a number of channels (called reserved channels) can be used exclusively with a given probability (called selection probability) by higher priority traffic such as voice packet, while the remaining channels are used by both higher priority traffic and lower priority traffic.

5.

Mobile Packet Radio Networks with Random Access Protocols 5.1 CSMA The classic derivation of the CSMA channel utilization and delay performance [Klei75a] assumed that only user packet collisions over the single packet radio channel result in the failure of an acknowledgment to be returned. Also, the channel for acknowledgment was assumed to be separate from the message channel and both were assumed perfect. However, for

the mobile packet radio channel, it is necessary to modify the fundamental throughput-delay analysis, as the assumptions of no transmission error, and noiseless channel are not applicable. Sinha and Gupta have given the throughput and packet delay for the non-persistent CSMA scheme in mobile packet radio considering channel noise [Sinh84a]. The throughput of the system was given by

The average packet delay was given by

where G : offered channel traffic end-to-end propagation delay average channel noise radio

20

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

It is assumed that the two-way propagation delay equals Q packet lengths, and that the packet retransmission times are selected from a uniform distribution of delays ranging from 1 to K packet lengths.

5.2

CSMA/CD

Sinha and Gupta [Sinh85] extensively studied and analyzed the performance of CSMA/CD for mobile packet radio channels considered a spread spectrum scheme. Spread Spectrum Multiple Access (SSMA) allows a packet to be captured at the receiver, while CSMA allows a user to capture the channel. Therefore, by using CSMA in conjunction with SSMA Sinha and Gupta [Sinh84a] achieve the benefit of keeping away all users within hearing distance of the transmitter and thus help to keep the capture effect. By accommodating the collision detection capability in the CSMA protocol and applying it to Frequency Hopping (FH)/Frequency Shift Keying

(FSK) mobile packet radio, the overall improvement in the system performance is accomplished by exploiting the channel capturing capability of CSMA and packet capturing capability of the Spread Spectrum technique. The modified throughput expression for CSMA/CD is stated below

where T : transmission time (in slots) of a packet

collision recovery time (the time until all user stop transmission given that a collision has occurred)

5.3

Capture

In mobile packet radio channel environments, due to topological and environmental conditions, the relative distances between transmitters and receivers, or the transmission power levels from transmitters, can be vastly

different. Thus, data packets from different transmitters arrive at a receiver with substantially different power levels. On the other hand, in packet radio communication systems with mobile

users, the users may occasionally be hidden because of physical obstacles or fading problems. The near and far phenomenon and the shadow fading

Introduction

21

and Rayleigh fading for mobile users give rise to a capture effect at the receivers. The packet arriving with the highest power has a good chance of being detected accurately, even when other packets are simultaneously presented. The capture effect in slotted ALOHA systems and in framed ALOHA systems has been considered before in [Good85], [Cido87], [Du87] and [Wies87] in terms of average packet delay and throughput. These studies have been concerned with how capture improves the overall system performance. They derived explicit expressions for the average and variance of the number of packet departures and considered two optimization issues, i.e., the maximization of the throughput of one marked group and the maximization of the throughput of the system. Wieselthier, Ephremides, and Michaels [Wies87] considered the operation of framed ALOHA on channels that are characterized by a general capture model, under which the probability that one packet is received successfully depends on the number of packets involved in the collision. Cidon, Kodesh and Sidi [Cido87], presented tree-based collision resolution algorithms that are capable of handling captures and erasures. They suggested and analyzed two schemes in [Cido87], the Wait scheme and the Persist scheme for a communication system that consists of an infinite population of users accessing a common receiver. In their analysis, they concluded that the Wait scheme is better. Nelson and Kleinrock [Nels84] analyzed two models of capture in a random planar system where slotted ALOHA was used to broadcast packets on the channel. They showed, under ideal circumstances, the expected fraction of stations in the system that are engaged in successful traffic in any slot does not exceed 21%.

6.

Multichannel Networks with Random Access Protocols

There are many researches in developing the advanced multiple access communication networks. But, most of them aimed to improve the performance on the single channel networks. Recently, some studies have appeared in the literature on the analysis of performance of a number of access protocols implemented in multichannel packet radio communication systems [Yung78], [Mars83], [Okad84] and [Mars87]. In these studies, the network performances were analyzed considering ALOHA, CSMA and CSMA/CD protocols. Another practical appeal is that this multichannel

22

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

system can be best implemented in a cellular mobile telecommunication environment. In this section, we centrally discuss some classical analyses of multichannel packet data networks.

6.1

Slotted ALOHA

Under the assumption of an infinite user population, and a Poisson offered traffic, fully connected multichannel local area networks using the ALOHA protocol can yield significant throughput and delay improvements over the traditional single channel architecture for a fixed total data rate. Yung has analyzed the performance of a multichannel slotted ALOHA system [Yung72]. Marsan and Bruscagin used finite population to investigate the behavior of the slotted ALOHA protocol on multichannel systems with reduced connections between users and channels [Mars87], in which each user can access only a subset of the M available channels. The connection pattern was represented with a connectivity matrix. In [Mars87], it was shown that under the fixed total data rate assumption, this architecture can yield throughput improvements from those of a fully connected multichannel slotted ALOHA network. In multichannel pure or slotted ALOHA networks, a user with a packet can transmit the packet at a randomly selected channel. If two or more packets are simultaneously transmitted over the same channel, the collision of packets occurs.

6.2

CSMA and CSMA/CD

The multichannel CSMA/CD system is suitable for both baseband and broadband bus type networks. In [Todd85], Todd presented an exact analysis of the throughput for slotted multichannel CSMA/CD systems. However instead of the fixed total bandwidth constraint he employed a fixed normalized propagation delay constraint. He concluded that in terms of channel utilization, increasing the number of channels results in an increase in the throughput only over a restricted range of applied loads and always leads to a decrease in the maximum throughput achievable. In [Okad84], Okada et al. also showed the quantitative superiority of multichannel CSMA/CD system over single channel CSMA/CD system on throughput and packet delay characteristics with a different approximate analysis. Multichannel CSMA/CD has been shown not only to preserve the

Introduction

23

preferable features of the single channel CSMA/CD but also to have the following advantages over the single channel CSMA/CD [Okad84].

Wide bandwidth and extensiveness of capacity Effectiveness (especially on throughput) Load distribution Reliability Adaptability and serviceability

Some priority schemes have been proposed and analyzed for multichannel CSMA/CD protocols [Ko86] and [Okad87]. In [Ko86], Ko and Lye presented and analyzed a simple priority CSMA/CD multichannel local area networks. They estimated the priority mechanism of the scheme considerably reduces the total performance where low priority messages cannot be transmitted if the number of idle channels is less than a certain threshold value. In [Okad87], Okada and Ikebata have proposed two effective priority schemes of multichannel CSMA/CD named Busy Channel Priority and Ordered Channel Priority, which impose some functional restrictions on the channel selection, for multimedia local area networks.

7.

Multi-Hop Networks with Random Access Protocols

When multiple random access protocols are applied to multi-hop networks, however, timing and spatial coordination jointly determine their performance. This makes the analysis significantly more complex, due to the fact that multi-hop networks are spatially distributed, operationally decentralized and involve multiple access communication channels. Nevertheless, over the past several years, a number of performance analyses of multi-hop packet radio networks have been proposed by various groups of researchers, and a brief review follows. Because of colliding transmissions from multiple packet radio users, we have not found any exact solution in a product form for the average packet delay of a general class of multi-hop packet radio networks. One of the reasons that a discrete-time queueing network (modeled on the slotted ALOHA system) does not lend itself to a product form solution is that more than one event can occur in a single slot [Bhar80]. Analyses of simplified versions of unslotted ALOHA and unslotted CDMA were proposed in [Toba83].

24

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

In [Chen85], an approximation was developed for throughput analysis in CDMA networks in the absence of noise.

7.1

Slotted ALOHA

Kleinrock and Silvester studied randomly dispersed multi-hop packet radio networks under Slotted ALOHA and concluded in [Klei78] that “Six is a Magic Number”. Gitman [Gitm75] has given the equations for the total throughput on hop 1 and hop 2 as a function of the average transmission rates in a two hops slotted ALOHA packet radio communication system, in which hop 1 has m repeaters and hop 2 has a single station.

where is throughput of hop i (i = 1, 2) and is the average rate of packet transmissions per slot in hop i (i = 1 , 2 ) , I is the interference level, Takagi and Kleinrock [Taka85c] analyzed the throughput-delay characteristics for slotted ALOHA multi-hop packet radio networks where the hearing configuration of packet radio units (users and repeaters) and sourceto-sink paths of packets were given and fixed. The problems are formulated as discrete-time Markov chains and then solved numerically. The total throughput of the network was given by [Taka85c]

where

where s is the state of the whole network in a slot represented by the states of user’s buffers, N is the total number of users involved. is the equilibrium probability that the network is in state s.

Introduction

When the buffer of user i is empty,

packet which belongs to path

25

and when the buffer contains a

c is that

and it is given as follows:

7.2

CSMA and CSMA/CD

System performance of two-hop networks operating under the CSMA protocol was evaluated in [Toba80a]. Models of continuous-time Markov chain were presented for multi-hop CSMA networks in [Boor80] and extended in various ways in [Magl80], [Boor82] and [Kers84]. These Markov models led to automated procedure capable of analyzing networks of realistic size and arbitrary topology, and provided approaches for tackling the analysis of large multi-hop networks under the CSMA protocol interesting

access schemes. In [Toba83], an analysis of throughput in multi-hop packet radio networks with zero propagation delay was presented to compare the performance of various channel access schemes such as pure ALOHA, CSMA,

BTMA etc., and a discussion of numerical methods for the application of the analysis to general configurations was given. The performance of the CSMA access scheme with a Busy-Tone and collision detection protocol (CSMA/BT-CD) in a multi-hop packet radio environment was studied in [Roy84] where the approach was based on a continuous-time Markov chain. The packet length, retransmission time, Busy-Tone detection time, collision detection time and collision termination time in the system were assumed to be exponentially distributed.

7.3

CDMA

An approximate procedure for the analysis of throughput in multi-hop packet radio networks using the CDMA protocol has been presented in [Chen85]. This approximate procedure provided a tractable tool for handling large networks and has been verified to be accurate to within five to ten percent for small and regular networks via exact analysis and simulation. Sen [Sen86] provided an alternative approximation for analyzing large networks operating under a CDMA scheme. Analyses of simplified versions of unslotted ALOHA and unslotted CDMA were proposed in [Toba83]. CDMA techniques using spread spectrum waveforms such

26

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

as direct sequence pseudo-noise modulation, frequency hopped modulation were studied in [Kahn78].

8.

Channel Assignment Schemes in Cellular Mobile Networks

Important performance measures of the system are grade of service and the number of users which can be served by the system. To improve the performance of the system using cellular concept and frequency reuse, some particular methodology for channel assignments to base stations is needed. Different channel assignment schemes have been studied, such as fixed channel assignment scheme, dynamic channel assignment scheme and hybrid channel assignment scheme. Important performance measures in the cellular mobile telephone communication systems are channel utilization and blocking probability which we define as the probability that an arrival call to a cell finds all channels to be busy. Many different channel assignment schemes have been studied in order to increase channel availability. Previously studied major techniques of channel assignment are in the following:

Fixed Channel Assignment (FCA) scheme Dynamic Channel Assignment (DCA) scheme Hybrid Channel Assignment (HCA) scheme

8.1

FCA Scheme

In a cellular mobile communication system with fixed channel assignment (FCA) scheme, a group of channels is fixedly assigned to the base station of each cell [Schi70] and [Cox72a]. Only channels from the assigned set of channels can be used to serve a call in a certain cell. If all the channels in one set of channels are busy, service will not be provided to other users in the cell even though there may be vacant channels in the neighboring cells. If a certain set of channels (for upstream or downstream traffic) is assigned to the base station of a particular cell, then that set will not be reassigned to any base station which lies in l belts of cells around the cell considered to prevent co-channel interference which is called the co-channel reuse distance. The choice of l depends on the propagation delay and also on how much interference rejection is desired.

Introduction

27

The analysis of the fixed assignment multichannel system is simple, because every cell operates just like a independent multiple channel system. If the system is a loss system, the traffic characteristics of each cell are simply given by the Erlang B formula. The basic theories of Erlang on the multi-server loss and delay systems are the first examples of a successful traffic modeling and performance analysis. Although these results date back to more than 80 years, they are still the most frequently used tools and formulas today.

8.2

DCA Scheme

In the dynamic channel assignment (DCA) scheme, there is no fixed relationship between the channels and the cells [Ande73]. All channels are kept in a central pool, and any channel can be used at any cell if no constraints are violated. A channel in use in one cell in the system can be used in another cell if the separation distance between the two cells is larger than the co-channel reuse distance. If no idle channel exists which does not violate constraints, the call is blocked. Systems that employ dynamic channel assignment schemes exhibit considerable service deviation, especially at large blocking rates. The service deviation denoted by SD has been defined by Anderson [Ande73] as follows:

and

where L is the average blocking probability in the entire system, is the ratio of calls blocked in cell i to offered calls in a given time interval, and K is the total number of cells in the system. Therefore a well-designed mobile communications system should have a service deviation value of close to zero.

8.3

HCA Scheme

A hybrid channel assignment (HCA) scheme, where some channels are permanently assigned to each cell while some channels are dynamic has

28

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

been suggested. DCA is superior when traffic demand is low while FCA is superior when traffic demand is high. HCA is suitable under more general conditions as the ratio of fixed and dynamic channels is chosen appropriately [Kahw78]. In hybrid channel assignment systems, if all fixed channels are assigned to a particular cell are busy when a new call arrives in this particular cell, then borrowing may take place from the dynamic channel set, shown in brackets in that cell, provided no interference will result as a consequence of this borrowing. It is of interest to note that the set in brackets may contain many channels, and therefore the decision on which channel of the set will be borrowed is important. Some papers have been published on this subject, advancing different criteria for channel borrowing and investigating the system performance [Ande73] and [Enge73].

A general conclusion reached by most authors on this subject was that adopting a simple test for borrowing yields performance results quite comparable to systems which do a lot of exhaustive searching for channels that are the ultimate best for borrowing, thus giving rise to a lot of processing per call. The simplest borrowing algorithm is that borrowing the first available channel that satisfies the co-channel reuse distance. The first available channel assignment has been shown to be cost effective, simple to implement, and to have acceptable performance. The designs of mobile radio communication systems are based on an estimated traffic demand. The traffic demand may not be uniformly distributed among the cells and may be distributed according to various factors such as mobile population distribution and calling patterns. This in fact may change slowly according to the gross shifts of the mobile population with times of the day and days of the week. One example of this would be the shift to the suburbs on a weekday afternoon. Therefore the spatial distribution of traffic demand should be considered for channel assignment. DCA scheme or HCA scheme is one of the ways to cope with this problem [Cox72a]. But the real time application of these schemes may not be practical for large systems with many cells and a large number of channels. Also these schemes will have worse performance when the traffic demand is high due to relatively large average reuse distance.

Introduction

8.4

29

Hand-Off

The cellular systems for the next generation of wireless multimedia networks will rely on cells that are smaller than those used today. In particular, the cellular mobile telecommunication systems evolving to smaller cells with base stations are close to street level or inside buildings transmitting at much lower power. These systems have evolved to support lightweight handheld mobile users operating inside and outside buildings at both pedestrian and vehicle speeds. These smaller cells are called microcells or picocells, depending on their size. For a given coverage area, a system with many microcells has a higher number of users per unit area than a system with just a few macrocells. Small cells also have better propagation conditions since the lower base stations have reduced shadowing and multipath. In addition, less power is required at the mobile users in microcellular systems, since the users are close to the base stations. However, the evolution to smaller cells has complicated network design. However, the evolution to smaller cells has complicated network design. When a user moves from one cell to another while a call is on progress, the call requires a new channel (in the new cell) to continue. If no channel is available in the new cell, the call will be dropped or forced terminated. A

cellular system should provide the capability to hand-off calls in progress, as the mobile user or user moves between microcells or picocells. The forced termination probability and blocking probability are important criteria in the performance evaluation of cellular mobile telecommunication networks. Forced termination of an ongoing call is considered less desirable than blocking of a new call attempt. Several handoff schemes have been proposed (see [Hong86] and [McMi95] for a survey) to reduce forced termination. Performance modeling of the handoff schemes has been intensively studied. Most modeling attempts either approximate analysis or simulation.

9.

Performance Analysis of Wireless Communication Networks

Wireless communications is a rapidly emerging field of activity in multimedia communication networks. The unrelenting growth of wireless communications continues to cause new research and development problems in performance analysis.

30

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

The need for traffic analysis and engineering arises whenever a wireless communication network with finite resources supporting multimedia services is subjected to random service demands. For example, in a wireless communication network the random service demands are the call origina-

tions and terminations, voice or data transmissions by the users. The random service demands and multimedia service demands complicate wireless communication systems to process traffic loads. Furthermore, modern wireless communication systems with multichannel and multi-traffic process complicate workloads, since the available wireless channels are finite resource, and these determine call blocking and data packet waiting. The three entities that are the main important points to traffic analysis and design are external traffic load, engineered resources, and observed performance. Given the amount of resources and the traffic load, we should evaluate the system performance such as channel utilization, throughput, packet delay, call blocking, and high moments about traffic characters. Wireless communication system designers need methods for the quantification of system design factors such as performance and reliability. We usually want to know which network protocol gives the best system performance as the best throughput-delay characteristic under specified conditions, how coefficients of variation of the packet delay and the packet interdeparture time change with the offered traffic load, what size buffers must be employed by a network to keep the probability of buffer overflow below a particular value, and what is the maximum number of voice calls that can be accepted by a network in order to keep the voice packet transfer delay within reasonable bounded. We can answer these and other related questions for the wireless communication networks by developing performance models, and then analyze these models to obtain such performance measures. Probabilistic and statistical methods are commonly employed for the purpose of performance and reliability evaluation. The most direct method for performance evaluation is based on actual measurement of the system under study. However during the design phase the system is not available for such experiments and yet performance of a given design needs to be predicted to verify that it meets design requirements and to carry out necessary trade-offs. Hence abstract models are necessary for performance prediction of designs. To evaluate and improve the system performance, efficient performance analysis methods must be provided.

Introduction

31

The purpose of this book, therefore, is to offer detailed exact and approximate analytical solution methods and techniques using queueing theory to model the complex multimedia and multichannel systems with procedures of multiple random access schemes, and reliably evaluate numerically the performance of the systems. In particular, in this book we mainly present analysis methods to evaluate the system performance of discrete-time wireless communication network models and a few continuous-time network models, and derive the probability distribution of the interarrival time of internetwork packets at the adjacent network and the higher moments of the transmission departure distribution and delay distribution in wireless multimedia communication environment.

9.1

Discrete-Time and Continuous-Time Markov Chains

Successful applications of queueing theory have been widely researched for modeling, performance analysis and numerical evaluation for discretetime and continuous-time traffic systems, such as computer systems, packet radio communication networks, telecommunication systems. The analytical models can be broadly classified into state space models and non-state space models. A Markov process with a discrete state space is referred to as a Markov chain. Most commonly used discrete state space models are discrete-time Markov chains and continuous-time Markov chains. First introduced by Markov in 1907, Markov chains have been in use in performance analysis since around 1950. In the past decade, considerable advances have been made in the numerical solution techniques, methods of automated state space generation, and the availability of software packages. These advances have resulted in extensive use of Markov chains in performance and reliability analysis. Discrete-time Markov chains permit the particle to occupy discrete positions and permit transitions between these positions to take place only at discrete times. On the other hand, continuous-time Markov chains permit the particle to change positions or states at any point in time. A Markov chain consists of a set of states and a set of labeled transitions between the states. A state of the Markov chain can model various conditions of interest in the communication systems being studied. These could be the number of packets of various types waiting to use each resource, the number of resources of each type that have failed, the number of concurrent tasks of a given job being executed, and so on. After a sojourn in a state, the Markov

32

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

chain will make a transition to another state. Such transitions are labeled either with probabilities of transition (in case of discrete-time Markov chains) or rates of transition (in case of continuous-time Markov chains). The steady-state dynamics of Markov chains can be studied using a system of linear equations with one equation for each state. Transient (or time dependent) behavior of a continuous-time Markov chain gives rise to a system of first-order, linear, ordinary differential equations. Solution of these equations results in state probabilities of the Markov chain from which desired performance measures can be easily obtained. The number of states in a Markov chain of a complex system can become very large and, hence, automated generation and efficient numerical solution methods for underlying equations are desirable.

Some concise notations (based on queueing networks) have evolved, and software packages that automatically generate the underlying state space of the Markov chain are now available. These packages also carry out efficient solution of steady-state and transient behavior of Markov chains. In spite of these advances, there is a continuing need to be able to deal with larger Markov chains and much research is being devoted to this topic. If the Markov chain has a nice structure, it is often possible to avoid the generation and solution of the underlying (large) state space. For a class of queueing networks, known as product-form queueing networks, it is possible to derive steady-state performance measures without resorting to the underlying state space. Such models are therefore called non state space models. Most performance modeling techniques for information communication network systems are based on discrete-time Markov chains and continuoustime Markov chains. Discrete-time Markov chains can often apply to digitized computer communication networks, where the basic time unit can be used to reflect such things as time slotting in the physical communication system. And continuous-time Markov chains can often apply to telecommunication base networks, where users can communicate at any point in time. In this book, both discrete-time and continuous-time Markov chains will be the main theoretical modeling tool. We can therefore describe and present discrete-time approach for the modeling of digitized computer communication networks and similar systems such as packet radio communication networks, and continuous-time approach for the modeling of traffic flow networks such as cellular telecommunication systems.

Introduction

33

We note that in this book Chapters 2-9 analyze discrete-time networks operating on the basis of time slotting, and transmitting information in fixed length units such as packets (but in Chapter 5, arrivals of packets can possibly occur at any given time instant on the time axis). Chapters 10 and 11 analyze continuous-time networks at which arrivals and departures can possibly occur at any given time instant on the time axis.

9.2

Probability Distributions

The analysis of the output process is important in relation to the interconnection of local area networks or multi-hop networks. Namely, since the output process from a network may constitute part of the input process to an adjacent network, the output process should be analyzed to predict the performance of interconnected network systems or multi-hop networks. In particular, the first and second moments of probability generating functions of the packet interdeparture time distribution may be used to approximate the probability distribution of the interarrival time of internetwork packets at the adjacent network by an appropriate independent phasetype distribution. Until recently, studies on the probability distributions of

the packet interdeparture time and packet delay have only been concerned with the single channel systems in a number of access protocols [Toba82a], [Taka85b], [Taka86], [Mats90a], [Mats90b] and [Mats90c]. In this book, we will first briefly summarize some analyses and rationales of the multichannel and multi-hop packet radio communication systems, personal communication networks and cellular mobile telephone wireless systems in the following sections, then will give the abstract of our new exact analyses and approximate analyses to evaluate the performance of some multi-hop and multichannel packet radio systems and cellular mobile telephone wireless communication systems using multiple random access protocols and hybrid channel assignment scheme, respectively, by developing classical queueing theory. There are many important performance measures that can be used for evaluating the performance of communication or computer networks. They are throughput, channel utilization, average voice and data packet delay, probability of voice packet loss, coefficient of variation of random variables such as packet interdeparture time, packet departures and packet delay. We consider these as the key performance measures for all the networks presented in this book.

34

10.

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

System Models and Performance Analyses in the Book

This book is organized in three parts. Part I includes 4 chapters, Chapters 2-5, Part II includes 3 chapters, Chapters 6-8, and Part III includes 3 chapters, Chapters 9-11.

10.1

System Models and Performance Analyses

We classified the performance models analyzed in each chapter in Table 0.1 regarding whether the analysis is exact or approximate, what types of communication network and protocol are considered, and whether the performance measure obtained is with respect to average or distribution. The models used in this book can be distinguished on the basis of specific hypotheses about the user population size (either finite or infinite), buffer size (either finite or infinite), the time axis (either slotted or not) and so on. The specific model types used in this book are summarized in Table 1.2.

In Part I (Chapters 2-5), we present wireless communication networks by using the multiple random access slotted ALOHA protocol with several performance analyses for multichannel networks, multi-traffic networks and multi-hop networks, respectively. In Chapter 2, we consider packet radio communication systems that employ a set of M parallel channels under the slotted ALOHA protocol. Multichannel radio network is based upon frequency domain design techniques which reallocate the system bandwidth. We exactly derive the moment gen-

Introduction

35

erating functions of the packet interdeparture time, number of packet departures and packet delay for both IFT and DFT protocols. We can calculate not only the averages of these performance measures but also their higher moments by numerical differentiation to discuss traffic characters on multichannel packet radio networks. These results can then be used to evaluate the performance of interconnected networks, multi-hop multichannel networks or integrated networks of voice and data. In Chapter 3, we present an exact analysis to evaluate the effect of capture on the multichannel slotted ALOHA protocol. We derive the probabilities of the successful transmission, then by using these probabilities, we calculate the throughputs, average packet delays for both IFT and DFT protocols and numerically compare the performances of the systems with and without capture. Numerical results show that when we consider a quantitative capture restriction u, in a multichannel system having fixed total bandwidth, depending on parameter u and channel number M, the improvement of system performance such as the channel utilization and average packet delay can be obtained. We can also calculate coefficients of variation of the packet delay and the packet interdeparture time of this capture system by using the analysis of probabilities of successful transmission given in this chapter and the analysis in Chapter 2. In Chapter 4, we first propose two different procedures of multichannel multiple access schemes with slotted ALOHA operation for both voice and data traffic and presents an exact analysis to numerically evaluate the performance of the systems. In scheme I, there is no limitation on access between voice transmission and data transmission, i.e., all channels can be accessed by all transmissions. In scheme II, a channel reservation policy is applied where a number of channels (called reserved channels) are used exclusively by voice packets, while the remaining channels are used by both voice and data packets, and voice packets select the reserved channels with a given probability (called selection probability). We call the system using scheme I the nonreservation system and call the system using scheme II the reservation system. Then probability distributions for the numbers of voice and data departures and for the data packet delay are derived. Numerical results compare some cases with different numbers of channels, different numbers of reserved channels and different selection probabilities to discuss what effects they may have on channel utilization, loss probability, average packet delay, coefficient of variation of the data packet delay, and correlation coefficient of the voice and data packet departures.

36

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

In Chapter 5, we consider a multi-hop packet radio communication network which consists of N hops with infinite buffer capacities. In this sys-

tem, each hop has an access to a synchronous channel for transmitting its packets with one another or outside the system, using priority-based slotted ALOHA access scheme. External arrivals at each hop are assumed to form a general arrival process with independent and identical interarrival time distribution. In this chapter, we present two major results. First, through an exact analysis of the joint probability generating function of the steadystate distribution of queue lengths, the average queue length and average packet delay of the system are explicitly derived, and performance of the system with and without transmission error is compared. Then, an approximate analytic method, based on a decomposition approach in which the whole network is divided into subnetworks of the generalized M/G/1 type, is proposed to simplify the analysis of queue length and packet delay in networks with a large number of hops in tandem. The method is exact for the lowest priority hop. In numerical results, approximate results for the other hops are tested by simulation. It is demonstrated that this approximation method is particularly useful, simple, and accurate in treating large networks. Finally, we offer a parallel packet radio communication network with N users and two hops to evaluate the system performance by applying the approximation technique. In part II (Chapters 6-8), we present the analyses and various properties for local area networks (LANs) and wireless LANs (WLANs) with the multiple random access protocols, CSMA/CD (Carrier Sense Multiple Access with Collision Detect) and CSMA/CA (Carrier Sense Multiple Access with Collision Avoidance), for multichannel and multi-traffic network systems. In Chapter 6, we consider slotted non-persistent CSMA/CD LANs with a finite number of users in a multichannel communication environment. Markov chain analysis is used to exactly derive the moment generating functions of the packet interdeparture time, number of packet departures and packet delay. Channel utilization, delay performance and higher moments of the packet interdeparture time, number of packet departures and packet delay are then calculated in the terms of the number of network users, the number of network channels and the channel access rate. Calculated results show that multichannel CSMA/CD not only preserves preferable features of the single channel CSMA/CD but also has advantages such as extensibility of capacity, effectiveness, reliability, adaptability and serviceability over the single channel case.

Introduction

37

In Chapter 7, we present an exact analysis to numerically evaluate the performance of high-speed and realizing fully distributed WLAN systems with non-persistent CSMA/CA IFT and DFT protocols. The collision avoidance portion of CSMA/CA in this system model is performed with a random pulse transmission procedure, in which a user with a packet ready to transmit initially sends some pulse signals with random intervals within a collision avoidance period before transmitting the packet to verify a clear channel. The system model consists of a finite number of users to efficiently share a common channel. The time axis is slotted, and a time frame has a large number of slots and includes two parts: the collision avoidance period and the packet transmission period. A discrete-time Markov chain is used to model the system operation. The number of slots in a frame can be arbitrary, dependent on the chosen lengths of the collision avoidance period and packet transmission period. The influence of possible length of the collision avoidance period and packet transmission period, and pulse transmission probabilities on the network performance are discussed, based on the results of the channel utilization and average packet delay for different packet generation rates. In Chapter 8, we analyze wireless LANs presented as in Chapter 7 for realizing fully distributed users in a multimedia environment that has the ability to provide both real-time bursty traffic (such as voice or video) and data traffic. We first present a detailed system model and an effective analysis for the performance of wireless LANs which support multimedia communication with non-persistent CSMA/CA IFT and DFT protocols. Then we present an exact analysis for the protocol to derive the moment generating function of the packet interdeparture time for the output process. The results obtained in this chapter also include those in Chapter 7 for the single traffic models. In general it is considered that the real-time traffic such as voice needs a higher priority to transmit than data traffic. In this system, the larger pulse transmission probability a user uses, the higher priority this user will get. In the numerical results, (1) we discuss the optimal network design parameters such as the pulse signal transmission probability and ratio of the collision avoidance period and transmission period to maximize the channel utilization and minimize the average packet delay of the whole system. (2) We make some comparisons with other previously known schemes for data traffic only using the fixed pulse transmission probability and fixed collision avoidance period length. (3) We compare the average performance measures of CSMA/CA systems with the IFT protocol and the DFT proto-

38

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

col. (4) We make comparisons between the single traffic system models and the multi-traffic system models in term of the coefficient of variation of the packet interdeparture time by differentiating the moment generating function. In part III (Chapters 9-11), we present performance analyses and evaluations for personal communication networks and cellular mobile telecommunication networks with various channel access process methods such as CDMA protocol, fixed and hybrid channel assignment schemes to support multi-traffic transmission and hand-off. In Chapter 9, we present the output and delay process analysis of integrated voice and data slotted CDMA network systems with the multiple random access protocol for wireless communications. The system model consists of a finite number of users, and each user can be a source of both voice traffic and data traffic. The allocation of codes to voice calls is given priority over that to data packets, while an admission control, which restricts the maximum number of codes available to voice sources, is considered for voice traffic so as not to monopolize the resource. Such codes allocated exclusively to voice calls are called voice codes. In addition, the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use the voice codes for transmission if the voice sources are silent. A discrete-time Markov chain is used to model the system operation, and an exact analysis is presented to derive the moment generating functions of the probability distributions for packet departures of both voice and data traffic and for the data packet delay. For some cases with different numbers of voice codes, numerical results display the correlation coefficient of the voice and data packet departures and the coefficient of variation of the data packet delay as well as average performance measures, such as the throughput, the average delay of data packets, and the average blocking probability of voice calls. In Chapter 10, we present two approximate techniques to evaluate performance of the large scale mobile radio systems using a hybrid channel assignment scheme and a cellular telecommunication structure. The two approximate analyses give the steady-state probability distributions of the system which are used to obtain expressions for the blocking probabilities. In the first method, the blocking probability is obtained by finding the interarrival time probability distribution function of one composite interrupted Poisson process (IPP) stream consisting of several IPP streams overflowing from the cell of interest and its co-channel interference cells. The second

Introduction

39

method is proposed to solve the blocking probability of the system by re-

garding each call as a GI/M/m(m) model. Two analytical results are compared with simulation results and good agreements are observed for both fixed and hybrid channel assignment schemes. The methods presented in this chapter will be not only useful for the performance prediction and the optimum design of the cellular mobile radio communication system with a hybrid channel assignment scheme, but also applicable to the study of the system with a dynamic channel assignment scheme when the traffic offered to a group of dynamic channels forms a Poisson arrival process. In Chapter 11, we present an exact analysis and an efficient matrixanalytic procedure to numerically evaluate the performance of cellular mobile telecommunication networks with hand-off. In high-capacity microcell cellular radio communication networks, a cell boundary crossed by moving users can generate many hand-off attempts. This chapter considers such a priority scheme that some channels and buffers are reserved for hand-off calls to reduce the forced termination of calls in progress. Performance characteristics we obtained include blocking probability, channel utilization, average queue length and average waiting time for hand-off calls. Using the matrix-analytic solution for the stationary state probability distribution, we also derive the probability distribution of the waiting time of a hand-off call. Numerical results show how priority can be provided to hand-off calls according to the number of reserved channels and buffer size. They also clarify the effect of the hand-off priority scheme on the standard deviation of waiting time of a hand-off call. Finally, in Chapter 12 we offer a summary of our conclusions in this book and our considerations on topics for future research. The result discussed in Chapter 2 is mainly taken from [Yue89], Chapter 3 from [Yue91b], Chapter 4 from [Yue00a], Chapter 5 from [Yue87a], Chapter 6 from [Yue93], Chapter 7 from [Yue02], Chapter 8 from [Yue01b], Chapter 9 from [Yue00c], Chapter 10 from [Yue91a] and Chapter 11 from [Yue96].

10.2 Performance Measures and Common Definitions We define some of the important performance measures and list below common symbols to denote them:

40

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

(1) Average channel utilization: The average portion of a channel that is used for successful transmission of packets, that will be denoted by U. (2) Throughput: The average number of packets successfully transmitted per unit time (slot or frame), that will be denoted by

(3) Average packet delay: The average time from the epoch of the packet arrival to the epoch of the completion of packet transmission, that will be denoted by E[D]. (4) Average number of collision packets: The average number of packets which are involved in collision on an arbitrary channel per slot, that will be denoted by (5) Loss probability in packet communication networks or blocking probability in cellular mobile networks and personal communication networks: The probability that a packet in packet communication networks or a call in cellular mobile networks and personal communication networks cannot seize a channel upon arrival, that will be denoted by L.

(6) Moments of a random variable: It is usually sufficient to describe a random variable by a set of numbers, known as moments, that summarizes the essential attributes of the random variables. These moments are defined in terms of the probability distribution functions (PDFs) and the probability density functions (pdfs) for the packet interdeparture time, packet delay and so on. (i) Moment generating function (m.g.f.) of a corresponding discrete random variable X is denoted by

(ii) Laplace-Stieltjes transform (LST) of the PDF of a corresponding continuous random variable X is denoted by (iii) The nth moment of a random variable X: It is denoted by We define the nth moment of a continuous random variable X with pdf given by f(x), as follows:

Introduction

41

and the nth moment of a discrete random variable taking values with probabilities respectively, as follows:

We can also obtain the nth moment of a continuous random variable X by a rather simple way as follows:

By this, we can thus find the nth moment of the continuous random variable X. Moreover, we can also obtain the nth moment of a discrete random variable X by differentiating with respect to z and evaluating the result at z = 1 as follows:

The right hand side of the above equation is called the factorial mo-

ment of X, of order n. The nth factorial moment of X is denoted by

By expanding this, we can thus find the nth moment of the

discrete random variable X. (iv) The average of a random variable X such as the average waiting time, average number of backlogged users and average queue length is denoted by E[X]. It can be obtained by the first derivative of at z = 1 or by the negative first derivative of at s = 0. (v) The variance of X: It is defined by the following:

(vi) The coefficient

of variation for a random variable X:

It is obtained by

In particular, the coefficient of variation of the packet interdeparture time will be denoted by and coefficient of variation of the packet delay will be denoted by

42

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Next we give some common definitions as follows: (1) IFT (Immediate-First-Transmission) protocol: In the IFT protocol, a newly generated packet makes a transmission attempt in a slot (or a frame) when it is generated. (2) DFT (Delayed-First-Transmission) protocol: In the DFT protocol, a newly generated packet enters the backlogged state, and its transmission attempt might be delayed by a geometrically distributed time following its generation.

(3) In all analyses in the remaining chapters, we define

where

as

when a > b.

(4) In the following chapters, the transition probabilities, m.g.f.s and LST are defined to be zero out of the region denoted in the parentheses.

I MULTICHANNEL AND MULTI-TRAFFIC NETWORKS WITH SLOTTED ALOHA PROTOCOL

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Chapter 2 OUTPUT AND DELAY PROCESS ANALYSIS OF SLOTTED ALOHA MULTICHANNEL NETWORKS

1.

Introduction

Fully connected multichannel packet radio networks whose operational characteristics depend on the structure of the network are expansions of the fully connected single channel packet radio networks. A number of studies have appeared in the literature on the analysis of performance of some access protocols implemented in multichannel radio communication systems [Yung78], [Mars83], [Okad84], [Mars87], etc. An approximate analysis was given by Yung for the throughput in a multichannel slotted ALOHA system in which each user always randomly chooses a fixed number of channels from the available channels to transmit packets simultaneously [Yung78]. Multichannel carrier sense multiple access with and without collision detection (CSMA and CSMA/CD) have also been analyzed to evaluate throughput and packet delay characteristics [Mars83] and [Okad84]. In [Mars83], Marsan et al. have shown that since the ratio of propagation delay to packet transmission time is smaller on each subchannel than it is on the wide single channel, the system can achieve significant throughput and delay improvement over the single channel architecture for a fixed total data rate. In [Okad84], Okada et al. have also shown the quantitative superiority of multichannel CSMA/CD system over single channel CSMA/CD system on throughput-packet delay characteristics with a different approximate analysis. Marsan and Bruscagin investigated the behavior of the slotted ALOHA protocol on multichannel systems with reduced 45

46

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

connections in which users are connected only to an appropriate subset of channels [Mars87]. It was shown that using the technique the throughput, stability and reliability of this type of network increase. However, in most of these studies, multichannel systems have been approximately analyzed to obtain average performance measures. The design and development of multichannel multi-hop systems and interconnected network systems require not only such average performance measures but also higher moments of packet interdeparture time and the number of channels having successful transmission in parallel. We have interests in characterizing the packet interdeparture time and the number of packets successfully transmitted in parallel of multichannel networks, so as to enable evaluation of the performances of the interconnected networks, since the output process from the system constitutes a part of the input process to another interconnected neighboring network or another set of channels. We note that closed-form expressions exist for all moments of the moment generating functions in this chapter. Using the first and second moments of the packet interdeparture time we can determine the parameters in the diffusion approximation to the input process of another interconnected systems. Furthermore, to evaluate the performance of communication networks such as multi-hop system and digitized voice system, the analysis has to be extended so as to provide delay distributions. Until recently, the studies on the probability distributions of the packet interdeparture time and packet delay have been only concerned with the single channel systems in a number of access protocols. In particular Tobagi [Toba82a] has investigated the packet departure process of the slotted ALOHA and CSMA systems using a finite user and idle/backlogged model. More recently Takagi and Kleinrock [Taka85d], Takagi and Murata [Taka86] have derived the Laplace-Stieltjes transform of the output processes for pure ALOHA channel, non-persistent CSMA and CSMA/CD, and p-persistent CSMA and CSMA/CD systems with a Poisson arrival process for finite user population models. Also, Matsumoto et al. [Mats90a] and [Mats90b] have considered an asynchronous (unslotted) CSMA/CDIFT and CSMA/CD-DFT systems and exactly derive the Laplace-Stieltjes transforms of the probability distribution functions of the packet and message interdeparture times, and the message response time. The performance analyses of multichannel systems are quite different from those of single channel systems [Toba82a]. This chapter exactly analyzes the transmission behavior of the multichannel slotted ALOHA sys-

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

47

tems for both IFT and DFT protocols. The analysis is based upon Markov chain techniques and gives the moment generating functions of the packet interdeparture time, number of packet departures and packet delay. We can explicitly calculate their averages and higher moments by differentiating these moment generating functions to numerically compare with those of the single channel case. These results are useful for analyzing multichannel multi-hop networks and integrated networks of voice and data employing these multiple random access protocols. The chapter is organized as follows. In Section 2, we define the model and its assumptions. In Section 3, we present the moment generating functions of the packet interdeparture process, number of packet departures and packet delay, and describe the average performance measures and higher moments. In Section 4, we present some applications and numerical results together with the average performance measures such as channel utilization, packet delay and higher moments. Finally, we offer a conclusion in Section 5.

2.

System Model

The multichannel system configuration is shown in Fig. 2.1. The system is a fully connected network: all users are in the same communications environment and thus can transmit to or receive from any of M channels that have equal bandwidth in Hz.

48

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

The system consists of a finite population of N users sharing M parallel channels with equal probability 1/M. Users can transmit to or receive from any of M channels according to the slotted ALOHA protocol. The time axis is slotted into segments of equal length seconds corresponding to the transmission time of a packet. All users are synchronized and all packet transmissions over the chosen channel are started only at the beginning of a time slot. If two or more packets are simultaneously transmitted over the same channel at the same slot, a collision of the packets occurs. It is assumed that the user will know about his success or collision immediately after the transmission. The users whose transmissions are unsuccessful retransmit their packets in future time slots. Each user has his own buffer which can store at most one packet at any time. Once a packet is accommodated at a buffer, it remains there until it is successfully transmitted. Each user is in either of the two states: the idle state if he does not have a packet in his own buffer to transmit; the

backlogged state if he has a packet awaiting or undergoing transmission. We call a consecutive pair of idle and backlogged periods a renewal cycle. In the IFT protocol, an idle user generates a new packet only at the beginning of a slot with probability and upon a new packet arrival, the user transmits the packet with probability one. In the DFT protocol, a packet is generated only at the end of a slot with probability and when a new packet arrives, the user joins the backlogged state at once, then transmits the packet after slots with mean of where is the retransmission probability.

A backlogged user cannot generate a new packet and retransmits the old packet according to a geometrical distribution with mean

slots.

3. Performance Analysis 3.1 Stationary Probability Distribution We model the system as a finite state and discrete-time Markov chain where the imbedded Markov points are chosen at the end of slots. We define

the state of the system by the number of backlogged users at the imbedded Markov points. Let x(t) represent the number of users backlogged at the end of slot t. The system state is said to be in state i if x(t) = i, and it is denoted by

The process {x(t), t = 0,1,2,...} clearly forms a finite

Markov chain. We assume that the steady-state of the system exists.

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

49

To obtain the one-step state transition probabilities, we first define conditional probabilities for the IFT protocol and the DFT protocol, respectively, as follows: For the IFT protocol:

for the DFT protocol:

We then define a conditional probability

as follows:

The imbedded Markov points and transitions from system state to state for both IFT protocol and DFT protocol are shown in Fig. 2.2. In the IFT protocol case, a users which generate new packets at the beginning of slot t + 1 transmit their packets with probability one, immediately. Among n users including a users who have new packets, the users who do not successfully transmit their packets enter the backlogged state at the end of slot t + 1. On the other hand, in the DFT protocol case, n users out of i backlogged users retransmit their packets at the beginning of slot t + 1, then a users out of N – i + c idle users generate new packets at the end of slot t + 1 and join the set of backlogged users at once. Therefore, the points where the new packets are generated are different for the IFT protocol and for the DFT protocol, so, the method for deciding the number of backlogged users at the end of each slot is also different. The conditional probabilities for both IFT protocol and DFT protocol can be obtained as follows:

50

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

For the IFT protocol:

for the DFT protocol:

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

51

is given by [Fell57], [Szpa83] and [Poun92] as follows:

The one-step state transition probability

is defined by

By summing up the conditional probabilities on n and c, from (2.1) and (2.2), we obtain the one-step state transition probabilities as follows: For the IFT protocol:

for the DFT protocol:

Let P be an ( N + 1) × ( N + 1) matrix of the one-step state transition probability where N + 1 is the total number of the system states. Next let denote an (N +1) -dimensional row vector of the stationary probability distribution. can be calculated by solving the set of linear equations as

3.2

Average Performance Measures

The throughput is obtained from the stationary state probability as follows:

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

For the IFT protocol:

For the DFT protocol:

The average channel utilization U is given for the IFT protocol and the DFT protocol, respectively, by

where is given by (2.7) for the IFT protocol, or by (2.8) for the DFT protocol. The average packet delay E[D] can be obtained by considering the renewal cycle defined in Subsection 3.1. As all N users are stochastically homogeneous, it is easy to see that the following balance equation holds true:

Solving (2.10), we can obtain the average packet delay E[D] for the IFT protocol or the DFT protocol, as follows:

3.3

Packet Interdeparture Time Distribution of Type [c]

In this subsection, we consider the packet interdeparture time distribution of type [c]. We define the packet interdeparture time of type [c] as the time interval between two consecutive successful transmissions that have the same number of successful channels as Let represent the packet interdeparture time of type [c]. Clearly, is an independently, identically distributed random variable. The packet interdeparture time of type [c] is shown in Fig. 2.3.

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

Let the m.g.f. for

represent the moment generating function of To obtain we first define a conditional transition probability that is

It can be obtained numerically as follows: For the IFT protocol:

for the DFT protocol:

53

54

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

On the other hand, let represent the time interval from the end of slot t to the end of the slot where c packets are successfully transmitted, and let represent the conditional m.g.f. of random variable on the condition that x(t) = i. Then is obtained by using the transition probability defined in Subsection 3.1 as follows:

The above equation can be considered as follows. Under the condition that x(t) = i,

(1) the first term in the large parentheses expresses that the number of successful channels in parallel is c (0 < c) at the end of slot t + 1 and therefore the packet interdeparture time is one slot;

(2) the second term in the large parentheses expresses that when the number of successful channels in parallel is not c at the end of slot t + 1, then from the end of slot t + 1 to the end of the slot in which successful transmission of type [c] occurs, the distribution of the time interval has a moment generating function given by By summing up the all probabilities on j, we can obtain Moreover, we let denote an (N+1)dimensional column vector of the conditional m.g.f. be calculated by solving the following equation:

can

therefore,

where e is a column vector with N + 1 elements, all of which equal 1 and E is an identity matrix. P(c) is an (N + 1) × (N + 1) matrix whose elements are But,

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

55

for the DFT protocol. P is the matrix of the one-step state transition probabilities defined in Subsection 3.1. Let denote the conditional stationary state probability distribution immediately after successful packet transmission of type [c] where the state of the system is i. We can obtain the unconditional m.g.f. of as follows:

where

is an (N + 1)-dimensional row vector

of the conditional stationary state probabilitydistribution evaluated numerically by solving

can be

where is the state transition probability, and the (–1) notation implies the matrix inverse. The average packet interdeparture time of type [c] is given by differentiating (2.17) and substituting 1 in z as Similarly, the second moment and substituting 1 in z, then adding

3.4

is also given by differentiating (2.17) as

Joint Probability Distribution of Packet Interdeparture Time and Number of Packet Departures

In this subsection, we analyze the joint probability distribution of the packet interdeparture time and the number of packet departures. We define

56

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

the packet interdeparture time as the time interval between two successive successful transmissions in which at least one packet is successfully transmitted, and define the number of packet departures as the number of packets successfully transmitted in a slot through all the channels when the packet departure satisfying the above definition occurs. Let T represent the random variable of such packet interdeparture time, and let represent the number of packet departures in parallel when the departure satisfying the above definition occurs. The packet interdeparture time T and number of channels having at least one successful transmission in parallel are shown in Fig. 2.4. It is different from the definition in the previous subsection, in which the packet interdeparture time of type [c] is defined as the time interval between two consecutive successful transmissions that have the same number of successful channels as c. We first denote, by , the conditional m.g.f. for the system state transition in one slot when the system state changes to j at the end of slot

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

57

t +1 and the number of successful channels at the end of slot t + 1 in which at least one packet is successfully transmitted given x(t) = i. For the IFT protocol:

For the DFT protocol:

To obtain the m.g.f. of the joint probability distribution of packet interdeparture time and number of packets successfully transmitted, we also define a conditional transition probability as follows:

at least one packet is successfully transmitted in slot

Next let represent a random variable of the time interval from the end of slot t to the instant in which at least one packet is successfully transmitted, and let represent a random variable of the number of channels having successful transmission in parallel when at least one packet is successfully transmitted. Now we consider the conditional joint m.g.f. of the joint probability distribution of the random variables and Cs given Under the condition that has the following meaning:

(1) When the system state changes to j at the end of slot t + 1, if at least one packet is successfully transmitted at the end of slot t + 1, then the packet interdeparture time is just one slot, and the joint m.g.f. of the transition in this slot and the number of successful channels in parallel is given by (2) If there is no successful transmission at the end of slot t + 1, the joint m.g.f. of the joint probability distribution of the following three parts:

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

(i) the transition in this slot; (ii) the transition in the time interval from the end of slot t + 1 to the end of the slot in which at least one packet is successfully transmitted, and (iii) the number of successful channels in parallel at the end of the slot being when (ii) occurs, is given by Similarly, summing over all probabilities on j,

is obtained by

In the same manner as that given in Subsection 3.2, let be an (N + 1)-dimensional column vector, then

therefore,

where

) and are (N +1) × (N+1) matrices, whose elements are and respectively. But and for the DFT protocol. P is the matrix of the one-step state transition probability and E are defined in the previous subsection. Let denote the unconditional joint m.g.f. of the packet interdeparture time T and the number C of packet departures in parallel. To obtain we also need to give the conditional stationary state probability distribution immediately after successful packet transmissions with the condition that at least one packet is successfully transmitted. Let be an (N+1 )-dimensional row vector representing the conditional stationary state probability distribution. It is given by solving the following equations:

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

Based on

and

we can finally give

59

as follows:

The average E[T] of the packet interdeparture time in which at least one packet is successfully transmitted is obtained by differentiating (2.27) for z and letting as follows:

By differentiating (2.27) twice in terms of z, taking E[T], the second moment is obtained as

, then adding

For random variable C of the number of channels having successful transmission in parallel, E[C] and are also obtained by differentiating (2.27) in terms of taking as (2.28) and (2.29).

_

_

where and and are the first and the second factorial moments of the joint probability generating function given by (2.21) for the IFT protocol or by (2.22) for the DFT protocol. Using (2.12) or (2.13), and can be given by

and

where

is the matrix form of

60

3.5

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Packet Delay Distribution

In the same manner as for the analysis of the probability distribution of the packet interdeparture time, we now give an expression for the moment generating function D *(z) of the packet delay. Let us call a packet (user) which we focus on a tagged packet (user). Since all packets (users) are homogeneous in terms of channel access rate and transmission time, we can choose any packet (user) as the tagged packet (user). The packet delay for the IFT protocol is shown in Fig. 2.5. Upon packet arrival, the packet delay involves two parts as follows: (1) The first part is the delay that a newly generated packet finds itself in the system upon arrival among backlogged packets at the end of the slot where it arrives. For the IFT protocol, we consider that a newly arriving packet is successfully transmitted in its first transmission, in which its delay is just one slot.

(2) The second part is the delay from the end of the slot to the completion of this packet transmission. We refer to the former as the initial delay and the latter as the backlog delay. We first derive the backlog delay. We denote, by the conditional m.g.f. of the probability distribution of the backlog delay that the tagged user finding himself among i backlogged users at the end of slot t given x(t) = i. In order to obtain we consider the conditional transition probabilities as follows:

Under the assumption that users of the system are of a uniform nature, the probability that i backlogged users are contained by l successfully transmitted is l/i for the IFT protocol case. The conditional transition probabilities can be obtained by the equations for the IFT protocol as follows:

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

61

IFT protocol case

Case 1: success in arrival slot

Case 2: no departure in arrival slot

where is the probability that the backlogged users successfully transmit l packets under the condition that n packet transmissions are started, among which b packets are old and c packets are successfully trans-

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

mitted. It can be obtained by

For the DFT protocol:

Using the above conditional transition probabilities, the conditional mo-

ment generating function log delay is thus given by

of packet delay for the back-

That is if the tagged user, finding himself in a backlog of size i at the end of slot t, successfully transmits his packet in slot t +1, then his delay is exactly one slot, but if the tagged user does not successfully transmit his packet in slot t + 1 and finds himself in a backlog of size j, then his delay distribution has a m.g.f. given by Writing in vector form, that is given as follows:

then

where is an N × N matrix, whose elements are defined in (2.34) for the IFT protocol or (2.36) for the DFT protocol. Next we consider the initial delay part. We let represent the probability of the tagged packet being successfully transmitted in its first transmission upon its arrival. For the DFT protocol, upon a new packet arrival,

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

63

the user first joins the backlogged state, then transmits the packet after slots with mean of Let represent the probability that the tagged packet finds itself among j backlogged packets at the end of slot t+1 upon its arrival given x(t) = i. These probabilities can be obtained by the following expressions: For the IFT protocol:

and

For

where is the probability that k new packets are successfully transmitted with the condition that n packet transmissions are started, among which a packets are new, and c packets are successfully transmitted. It is given by

K in (2.40) and (2.41) is a normalizing constant such that

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

For the DFT protocol:

Finally, we obtain the unconditional m.g.f.s of the packet delay for the IFT protocol and the DFT protocol, respectively, as follows: For the IFT protocol:

for the DFT protocol:

where is a row vector determined by (2.41) for the IFT protocol and by (2.44) for the DFT protocol. is an N-dimensional column vector excluding element Those elements are as given in (2.37). The average E[D] and the second moment of the packet delay are obtained from (2.45) for the IFT protocol by

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

65

and from (2.46) for the DFT protocol by

where we note that E[D] obtained in (2.47) and (2.49) should equal E[D] in (2.11) for the IFT protocol or the DFT protocol, respectively.

4.

Numerical Results

Using the equations presented Section 3, the coefficients of variation of a random variable X for the packet interdeparture time, packet delay and the number of packets successfully transmitted are also obtained by where Var[X] is the variance of the random variable X, Now, we numerically compare the average performance measures and the coefficients of variation for the multichannel system with those of the single channel system under the slotted ALOHA protocol. For all the numerical results, we consider the case N = 30. We consider all channels to have the same bandwidth υ, υ = V/M, where V is the total available bandwidth of the system. A packet transmission time over a channel having bandwidth υ therefore becomes where is the transmission time of a packet in a channel having bandwidth V. In Fig. 2.6, Fig. 2.7 and Fig. 2.8, we compare the average performance measures of the multichannel system with those of the single channel case under the same available bandwidth. In Fig. 2.6, we plot the average channel utilization U versus packet arrival rate for M = 1, 3, 5, respectively, with retransmission rate The single channel case offers a higher throughput than the multichannel cases for small packet arrival rate. However, we can see that in the single channel case throughput severely decreases to a lower level when the packet arrival rate increases. On the other hand, in the multichannel system case, the maximum utilizations decreases only slightly for large values of Fig. 2.7 shows the average channel utilization U versus packet retransmission rate for the M = 3, 5 cases, for packet arrival rate

66

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

and , respectively. The average channel utilization U of the M = 3 case is higher than U of the M = 5 case when retransmission rate is lower, while when retransmission rate becomes higher, U of the M = 5 case is higher than that of the M = 3 case. Fig. 2.8 shows the average packet delay E[D] characteristic for M = 1, 3, 5. The unit time of the average packet delay E[D] is denoted by a packet transmission time in the single channel system. In general, the average delay changes when the retransmission rate changes, so, in Fig. 2.8, we first change the retransmission rate for each packet arrival rate, then show the smallest packet delay values among all packet delay values, values are given from 1/N to 1.0, and the average delays E[D] are multiplied by M for M > 1 systems. The average packet delay E[D] is given by (2.11). The average packet delay E[D] is smaller when is small in the single channel case, but becomes very large when increases. The average packet delay E[D] can be kept smaller in the multichannel case when increases from packet arrival rate 0.01 to 0.1. The reason is that when is small, in the multichannel case, the transmission time of a packet is longer than in the single channel case, so delays are larger than in the single channel case; however, packet collisions are fewer and therefore retransmissions are fewer than in the single channel case when increases, and performance improvement of the average delay is obtained.

It is interesting to observe that for small arrival rate the IFT protocol offered a higher throughput and lower packet delay than the DFT protocol, but for large the DFT protocol performs better. This is due to the fact that the rate of user channel access for all channels in the IFT protocol case is in practice greater than in the DFT protocol. When increases over some value, in the IFT protocol, then there are much more packet collisions than in the DFT protocol. The curves of the coefficient of variation of the packet interdeparture time T in which at least one packet is successfully transmitted, and the coefficient of variation of the packet delay are showed in Fig. 2.9 and Fig. 2.10, respectively. Fig. 2.9 shows that with multichannel the value of coefficient of variation decreases to low levels when the arrival rate of new packets is very large. However, for the single channel case, the coefficient of variation is very large even when is large. Fig. 2.10 shows that for all cases the coefficients of variation of the packet delay are very stable when arrival rate increases.

Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

5.

67

Conclusion

In this chapter, we exactly derived the moment generating functions of the packet interdeparture time, packet delay and number of packets successfully transmitted with the slotted ALOHA protocol for multichannel system. The performance measures have been calculated under both IFT and DFT protocols. Numerical results show that channel utilization, packet delay and other important performance measures can be improved by em-

ploying multichannel in slotted ALOHA. Interesting performance characteristics have been observed through numerical results. These results are useful for analyzing multi-hop multichannel networks and reservation channels employing these multiple random access protocols.

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Output and Delay Process Analysis of Slotted ALOHA Multichannel Networks

69

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Chapter 3 PERFORMANCE ANALYSIS OF MULTICHANNEL SLOTTED ALOHA NETWORKS WITH CAPTURE

1.

Introduction In a packet radio communication system with mobile users, the relative

distances between the users and the receiver can be varied greatly and there is shadow fading and Rayleigh fading. Thus, packets from different transmitters arrive at the receiver with substantially different power levels. These

phenomena give rise to a capture effect at the receiver. The packet arriving with the highest power level has a good chance of being detected accurately, even when other packets arrive at the same time. This capture effect improves the system throughput and other performance measures.

The capture effect in slotted ALOHA networks and in framed ALOHA networks has been considered before in references [Nels84], [Good85],

[Cido87], [Du87] and [Wies87], respectively. However, in most of these studies, multichannel systems with capture have not been investigated. Multichannel packet radio networks with capture are expansions of the single channel packet radio networks with capture. In the multichannel system, there are several independent channels; a user can transmit to, or receive from, any of M channels according to some transmission protocols. In this chapter, we precisely evaluate the performance of the slotted ALOHA access scheme with capture in a multichannel wireless communication environment for the DFT and the IFT protocols. This system can be characterized by a general capture model. The analysis is based upon Markov chain techniques and the use of combinatorial techniques. To eval71

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

uate the basic performance of a multichannel slotted ALOHA system such as the throughput and average packet delay, we first derive the probabilities of successful transmission in the system. By using these probabilities, we numerically compare the performance of each system with and without capture. We can also calculate the coefficients of variation of the packet delay and the packet interdeparture time of this capture system by using the probabilities of successful transmission given in this chapter and the analysis in Chapter 2. The chapter is organized as follows. In Section 2, we define the model and its assumptions. In Section 3, we present the performance analysis to obtain performance measures such as channel utilization and packet delay. In Section 4, we present some applications and numerical results to compare

the performance of systems with and without capture. Finally, we offer a conclusion in Section 5.

2.

System Model

The system consists of a finite population of N users sharing a set of M parallel channels that have equal bandwidth in Hz (see Fig. 2.1). The time axis is slotted into segments of equal length seconds corresponding to the transmission time of a packet. All users are synchronized and all packet transmissions over the chosen channel are started only at the beginning of a time slot. Every user has his own buffer which can store at most one packet at any time. Once a packet is accommodated at a buffer, it remains there until it is successfully transmitted. Each user can transmit a packet on a randomly chosen channel. It is assumed that the user will know about his success or collision immediately after the transmission. If two or more packets are simultaneously transmitted on the same channel, collision of the packets occurs. The users whose transmission is unsuccessful retransmit their packets in future time slots. Therefore, each user is in either of the two states: the idle state if he does not have a packet in his own buffer to transmit, or the backlogged state if he has a packet awaiting or undergoing transmission. We call a consecutive pair of idle and backlogged periods a renewal cycle. In the IFT protocol, an idle user generates a new packet only at the beginning of a slot with probability and when the new packet is generated, the user transmits the packet with probability one; while in the DFT protocol, a packet is generated only at the end of a slot with probability andwhen

Performance Analysis of Multichannel Slotted ALOHA Networks with Capture 73

the new packet is generated, the user joins the backlogged state at once. A backlogged user cannot generate a new packet and retransmits the old packet according to a geometrical distribution with mean slots, where is retransmission probability. We assume that each packet is transmitted with fixed power, and all N users in the system transmit with the same power on the fixed total frequency band. In practice, one particular user, say a, will be captured by a receiver if the ratio of the received power of his packet to the total received power of all other packets simultaneously heard by the receiver is greater than a given capture ratio. If a receiver captures the strongest of a number of colliding packets, the packets of the weaker power are essentially considered to be noise. Let represent the capture ratio, and let denote the level of power at the receiver caused by transmitting user In the presence of noise of level the receiver is assumed to be able to capture this transmission if where and is the total received power of all other packets simultaneously heard by the receiver. Well-designed FM receivers have a capture ratio approximately equal to 0.7 [Robe75]. In this chapter, a simplifying assumption we make is that capture is a deterministic phenomenon such that if less than or equal to packets are transmitted on a channel, the receiver can detect one of them and the packet is successful in the transmission with probability one. We call parameter u a quantitative capture restriction. The definition for u is similar to that defined in [Du87]. The relation between the capture ratio transmission power and capture restriction parameter u is given as follows. Suppose that there are u users transmitting simultaneously and causing the levels at the receiver, respectively, because the distances between users and the receiver are different. For any the user a will succeed in his transmission if his level alone is at least times the sum of the other levels, that is, if

where We let denote the average power level of packets that are not captured at the receiver. Therefore, we have and

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

We assume that the power levels of all users of the system heard by the receiver are random variables with the same probability distribution. We also assume that we know how the mobile users are spread over the geographic area covered by the system and we have a map of that area which shows the average power levels produced at the receiver by a user transmitting from different regions of the area. (Using a topographical database of the Federal Republic of Germany, Dr. Lorenz of Fernmelde-technisches Zentralamt der Deutschen Bundespost has created maps of this kind as a planning tool for the mobile telephone network [Nami84].) Then, and can be previously obtained. When we assume that FM receivers in the system have a fixed capture ratio then for fixed and the effect of capture of the system depends on the capture restriction parameter u. u is obtained by

By the definition for u, corresponds to the case of no capture, and corresponds to the case of capture without any restriction.

3. Performance Analysis 3.1 Stationary Probability Distribution We model the system as a finite state and discrete-time Markov chain where the imbedded Markov points are chosen as the end of slots. We define the state of the system by the number of backlogged users at the imbedded Markov point. Let x(t) represent the number of users backlogged at the end of slot t. We assume that the system is in the steady-state. The one-step state transition probability is defined by

We give the one-step state transition probabilities as follows: For the IFT protocol:

Performance Analysis of Multichannel Slotted ALOHA Networks with Capture 75

For the DFT protocol:

where

c is the number of successful channels. is a conditional probability that there are packets successfully transmitted, given that packets are simultaneously transmitted over channels. It is given by [Zhan93]. By conditioning on the number of packets being simultaneously transmitted over the first channel (arbitrarily chosen) and using total probability, we have as follows:

where

The term

is the probability that i packets out

of n are transmitted over the first channel. The initial conditions are given by

76

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Let be an (N + 1)-dimensional row vector of the stationary probability distribution. can be calculated by solving the set of linear equations

where P is an matrix of the one-step state transition probability for the IFT protocol and the DFT protocol, respectively.

3.2

Average Performance Measures

The throughput can be simply obtained from the stationary state probability as follows: For the IFT protocol:

For the DFT protocol:

The average channel utilization U is given by

where is given by (3.10) for the IFT protocol, or by (3.11) for the DFT protocol. The average packet delay E[D] can be obtained by considering the renewal cycle. As all N users are stochastically homogeneous, it is easy to see that the following balance equation holds true:

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77

therefore, we obtain that

for the IFT protocol or the DFT protocol with given by (3.10) or (3.11).

4.

Numerical Results

In Figs. 3.1 and 3.2, we numerically compare the average performance measures of the multichannel system with and without capture under the slotted ALOHA protocol. Here we present the case N = 30, M = 3, with retransmission rate . We consider all channels to have the same bandwidth v, v = V/M, where V is the total available bandwidth of the system. A packet transmission time over a channel having bandwidth v therefore becomes where is the transmission time of a packet in a channel having bandwidth V. In Fig. 3.1, we plot the average channel utilization U and in Fig. 3.2, we plot the packet delay E[D] versus packet arrival rate for u = 1, 3, N, respectively. We can see that the capture cases offer higher throughputs and lower packet delays than the no-capture cases for all packet arrival rates. It is interesting to observe that for small arrival rate the IFT protocol offered a higher throughput and lower packet delay than the DFT protocol, but for large the DFT protocol performs better. This is due to the fact that the rate of the user channel access for all channels in the IFT protocol case is in practice greater than in the DFT protocol. When increases over some value, in the IFT protocol, then there are many more packet collisions than in the DFT protocol. In general, the average channel utilization and the average delay change when the retransmission rate changes. In Fig. 3.3 and Fig. 3.4, we plot the average channel utilizations and the packet delays for u = 3, N = 30 and in the M = 3 and M = 5 cases. The average channel utilization U of the M = 3 case is higher and the average packet delay is smaller than the average channel utilization and the average packet delay of the M = 5 case when arrival rate is lower, while when arrival rate becomes higher, the average channel utilization of the M = 5 case is higher and the average packet delay is smaller than that of the M = 3 case for both protocols in the u = 3 case. However, we see that in the case, the average channel utilization of the M = 5 case is lower and the average packet delay is larger than

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that of the M = 3 case whether packet arrival rate increases or not. The reason can be considered as follows. In the multichannel system, for fixed total bandwidth of the system and the same packet arrival rate and retransmission rate, the rate of user channel access for a channel in the smaller channel number case is in practice greater than in the larger channel number case (note that in the larger channel number system, the transmission time of a packet is longer than the smaller number case), and the number of packet collisions over a channel is greater than in the larger channel number case. When the parameter u = N (that corresponds to the case of capture without any restriction, i.e., when the collision of packets occurs, a packet is always captured accurately), the system with more packet collisions has more chances of being captured than the larger channel number case. Therefore, for the same packet arrival rate and retransmission rate when u = N, the smaller channel number system has better system performance in terms of average channel utilization and packet delay, compared to the large channel number system.

5.

Conclusion

In this chapter, we have analyzed the performance of the multichannel packet radio system with capture where the slotted ALOHA protocol was used to transmit packets on the channel. The performance measures have been calculated under both IFT and DFT protocols. Numerical results showed that channel utilization, packet delay and other important performance measures can be improved by employing capture. In a multichannel system having fixed total bandwidth, depending on parameter u and channel number M, the improved system performance such as the channel utilization and average packet delay has been obtained. We have seen the quantitative capture restriction u is a very important parameter and increasing u can improve the performance of the system. There are several practical ways to increase the capture restriction parameter u. Considering a simple way, we use the relation and let For fixed and we need let the capture ratio of FM receivers of the system and for fixed we need let We can also calculate the coefficients of variation of the packet delay and the packet interdeparture time of this capture system by using the anal-

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ysis of probabilities of successful transmission given in this chapter and the analysis in Chapter 2.

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Chapter 4 OUTPUT AND DELAY PROCESS ANALYSIS OF MULTICHANNEL SLOTTED ALOHA NETWORKS WITH INTEGRATED VOICE AND DATA TRANSMISSION 1.

Introduction

Traditionally, wireless communication services have primarily carried data traffic such as portable computing, paging, personal email, etc. But in recent years, there has been growing interest in supporting real-time traffic such as interactive voice and video applications in a wireless communication environment. Wireless communication has become an important field of activity in telecommunications to support a wider range of telecommunication applications, including voice, packet data, image and full-motion video. By implementing a truly multimedia and Personal Communication System (PCS), the user will be released from the bondage of the telephone line and will enjoy greater freedom of telecommunications. To this end, the system must be able to provide diverse quality of service to each type of traffic. In any communication system it is desirable to efficiently utilize the system bandwidth. In the recent literature, random channel access protocols in wireless communications have been actively studied to utilize the limited spectrum among all the users efficiently. In particular, the slotted ALOHA protocol has been considered in a number of papers [Toba82a], [Szpa83], [Nami84] and [Poun92]. Slotted ALOHA allows a number of relatively uncoordinated users to share a common channel in a flexible way. It is therefore interesting to 83

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investigate it in more depth and to exploit some existing characteristics in wireless communication environments to obtain some gain in performance. There are some studies on multichannel radio communication systems for a number of multiple random access protocols to achieve higher channel utilization [Szpa83], [Nami84] and [Poun92]. In the multichannel systems, the available total bandwidth (called single-channel) is divided into a number of (say, M) equal independent sub-bandwidths (called multichannel) in order to be used. Therefore, in such multichannel systems, users can transmit to or receive from any of these channels according to a certain transmission protocol, but the transmission time of a packet over a channel is M times as long as in the single-channel system. The studies on the multichannel systems showed that the multichannel scheme can improve the throughput, packet delay and other important performance measures by reducing the number of users who make simultaneous access on the same channel. In addition, it has many favorable characteristics such as easy expansion, easy implementation by frequency division multiplexing technology, high reliability and fault tolerance. We note that in most of these studies, multichannel systems have been analyzed to obtain performance measures for data traffic only. To the author’s knowledge, there is no work on performance analysis of multichannel multi-traffic wireless networks supporting real-time traffic and data traffic simultaneously. On the other hand, the design and development of systems such as multichannel multi-hop systems and interconnected systems require not only average performance measures but also higher moments of packet interdeparture times and packet delay, etc., because the output stream from one system often forms the input stream to another. In Chapter 2, we have derived probability distributions of packet interdeparture time and packet delay in a multichannel slotted ALOHA system with single traffic by extending the analytical technique in [Toba82a]. This chapter first proposes two different procedures of multichannel multiple access schemes with the slotted ALOHA operation for both voice and data traffic and presents the exact analysis to numerically evaluate the performance of the systems. In scheme I, there is no limitation on access between voice transmission and data transmission, i.e., all channels can be accessed by all transmissions. In scheme II, a channel reservation policy is applied where a number of channels are used exclusively for voice packets while the remaining channels are used for both voice packets and data

Output and Delay Process Analysis of Multichannel Networks with Voice/Data 85

packets. We call the system using scheme I the nonreservation system and call the system using scheme II the reservation system. Then probability distributions for the numbers of voice and data departures and for the data packet delay are derived. Numerical results are compared in some cases with different numbers of channels, different numbers of reserved channels and different selection probabilities to discuss what effects they may have on channel utilization, loss probability, average packet delay, coefficient of variation of the data packet delay, and correlation coefficient of the voice and data packet departures. The results obtained in this chapter contain those for the single traffic system as for a special case if the arrival rate of voice packets is zero. The analysis presented in this chapter can also be applied to evaluate performance measures of other networks such as priority networks, cellular mobile radio networks and multi-hop packet radio networks. The chapter is organized as follows. In Section 2, we describe the multichannel multi-traffic slotted ALOHA system model with and without reserved channels. In Section 3, we first derive the stationary probability distribution for both systems. Then we present the analysis of the probability distributions of the voice and data packet departures, and the data packet delay, and describe the average performance measures and higher moments for both systems. In Section 4, we discuss the numerical results and in Section 5, we conclude this chapter.

2.

System Model

The system consists of a finite population of N users accessing a set of parallel M channels that have equal bandwidth in Hz (see Fig. 2.1). The time axis is slotted into segments of equal length seconds corresponding to the transmission time of a packet. All users are synchronized and all packet transmissions over the chosen channel are started only at the beginning of a time slot. If two or more packets are simultaneously transmitted on the same channel at the same slot, a collision of the packets occurs. It is assumed that the user will know about his success or collision immediately after the transmission. In general, traffic is classified into two types: real-time bursty traffic (such as voice or video), which is strict with regard to instantaneous delivery but relatively tolerant of bit errors, and data traffic, which is strict with regard to bit errors but relatively tolerant of instantaneous delivery. In this

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chapter we assume that each user can generate both voice and data packets. Namely, the user can be in one of three modes: idle mode, speech mode, or

data transmission mode. At the beginning of a slot, each user in the idle mode enters the speech mode by generating a voice packet with probability or enters the data transmission mode by generating a data packet with probability or remains in the idle mode with probability Users in the speech mode transmit a voice packet in the slot when they enter the speech mode with probability one (IFT) and in the succeeding slots with probability until they quit the speech mode to return to the idle mode. The length of a user’s speech mode obeys a geometric distribution with mean slots. We assume that if voice packets are involved in collision, they are just discarded and no retransmission is necessary. Users in the data transmission mode transmit a data packet in the slot when they enter the data transmission mode with probability one. If this

first transmission attempt is successful, they return to the idle mode at the end of the slot. Otherwise, they schedule their retransmissions after a time interval that obeys a geometric distribution with mean slots. The state transition diagram for this system is shown in Fig. 4.1.

The nonreservation system and the reservation system differ in channel access method. In the nonreservation system, all M channels can be ac-

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87

cessed by both voice and data packets which choose one of the channels for transmission with equal probability 1/M. In the reservation system, voice

packets can access all M channels as in the nonreservation system, while access of data packets is restricted to channels and they choose one of the channels for transmission with equal probability Users with voice packets are assumed to use either the reservation channels with probability or the common channels with probability In the following, is called the selection probability. Once they decide to access the reservation (common) channels, they choose one of the channels with equal probability respectively. It is noted that when the system is a nonreservation system where all channels can be accessed by all transmissions with equal probability 1/M. If the whole channel is split into voice channels and data channels. We define the system state as a pair of the number of users in the speech mode and the number of users in the data transmission mode. We observe the system state at the end of slots and denote it by Then we can model the present system as a finite state and discrete-time Markov chain. Let denote a system state that there are users in the speech mode and

users in the data transmission mode at the end of a slot. We note that because the total number H of the system states is given by

3. Performance Analysis 3.1 Analysis of Nonreservation System In this subsection, we first analyze the nonreservation system. In the nonreservation system, all M channels can be accessed by both voice and data packets which choose one of the channels for transmission with equal probability 1/M. Let and represent the numbers of users in the speech mode and in the data transmission mode, respectively, who transmit their packets at the beginning of a slot. Among packets, let and represent the numbers of voice and data packets, respectively, which are transmitted successfully in the slot. Because one of the M channels is chosen randomly with equal probability and the transmission is successful if no other users transmit packets over the same channel at the same time slot, the probabil-

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ity that voice and data packets are successfully transmitted, given that voice and data packets access one of M channels randomly, is given by

The derivation of (4.2) is straightforward by extending the proof of (2.3). Let us consider the transition of the system state in equilibrium. Given that there are and users in the speech mode and in the data transmission mode, respectively, at the beginning of the tth slot (notationally, the following five events must occur to be and users in the speech mode and in the data transmission mode, respectively, at the beginning of the (t + l)st slot (notationally, where (1) At the beginning of the slot, users in the idle mode enter the speech (data transmission) mode by generating new voice (data) packets, respectively, where and are integer random variables with range (2) At the beginning of the slot, among users in the speech mode, users transmit voice packets where is an integer random variable with range (3) At the beginning of the slot, among users in the data transmission mode, users retransmit data packets where is an integer random variable with range (4) Given

voice (data) packets are transmitted over M channels, voice (data) packets succeed in transmission where is an integer random variable with range

(5) At the end of the slot, among users in the speech mode, users quit the speech mode and return to the idle mode.

Output and Delay Process Analysis of Multichannel Networks with Voice/Data 89

In Fig. 4.2, we show the imbedded Markov points and channel state transition in an arbitrary slot, where and “other transmission users” in Fig. 4.2 includes the users who are in the speech mode and transmit their voice packets and the users who are in the data transmission mode and retransmit their data packets, respectively.

We define the probabilities for events (1) to (5) by and respectively, as follows:

and is given by (4.2). The one-step state transition probability for the nonreservation system is denoted by it can be obtained as follows:

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The one-step state transition probability matrix for the nonreservation system is given by where and P is an matrix, where H is the total number of the system states defined in (4.1). Let be an H-dimensional row vector of the stationary probability distribution as It is determined by solving P as

3.2

Analysis of Reservation System

In the reservation system, the slot structure is the same as in the nonreservation system, but a channel reservation policy is applied where a number of channels can be accessed exclusively by voice packets, while the remaining channels can be accessed by both voice and data packets. Let denote the number of reserved channels that can be accessed by voice traffic only. Let denote the conditional probability that voice packets are successfully transmitted given that n voice packets are transmitted over channels. can be obtained by (2.3) as follows:

Output and Delay Process Analysis of Multichannel Networks with Voice/Data 91

As in the case of the reservation system, let us consider the transition of the system state in equilibrium. Given that there are and users in the speech mode and in the data transmission mode, respectively, at the beginning of the tth slot, the following seven events must occur to be and users in the speech mode and in the data transmission mode, respectively, at the beginning of the (t + l)st slot, where (1) At the beginning of the slot, users in the idle mode enter the speech (data transmission) mode by generating new voice (data) packets, respectively, where and are integer random variables with range (2) At the beginning of the slot, among users in the speech mode, users transmit voice packets, where is an integer random variable with range (3) At the beginning of the slot, among users in the data transmission mode, users retransmit data packets, where is an integer random variable with range (4) Among voice packets which are transmitted over M channels, n voice packets belong to reserved channels, where n is an integer random variable with range (5) Given n voice packets are transmitted over channels, voice packets succeed in transmission, where is an integer random variable with range

(6) Given data and voice packets are transmitted over free channels, data and voice packets succeed in transmission, where is an integer random variable with range (7) At the end of the slot, among users in the speech mode, users quit the speech mode and return to the idle mode.

We note that the probabilities for events (l)-(7) are given by respectively.

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Therefore we can write the one-step state transition probability the reservation system as follows:

for

The one-step state transition probability matrix for the reservation system is given by where and P is an matrix, where H is the total number of the system states defined in (4.1). Let be an H-dimensional row vector of the stationary probability distribution as It is determined by solving P as

3.3

Packet Departure Distribution

In this subsection, we derive the joint moment generating function of the probability distribution for packet departures of both voice and data traffic. We define the number of packet departures as the number of either voice or data packets successfully transmitted in a slot through all the channels. Let denote the joint m.g.f. of both voice and data packet departures. Then it is derived as follows:

Output and Delay Process Analysis of Multichannel Networks with Voice/Data

For the nonreservation system:

where

is given by (4.4).

For the reservation system:

where

is given by (4.7).

93

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By differentiating we can obtain higher factorial moments for both nonreservation and reservation systems as follows:

All moments of the joint m.g.f. of both voice and data packet departures have closed-form expressions. Using these higher factorial moments, in particular, we can also evaluate the correlation coefficient of the numbers of voice and data packet departures which is defined by

3.4

Average Performance Measures

The throughputs and for voice traffic and data traffic, respectively, are obtained by differentiating in (4.8) and (4.9) at as follows: For the nonreservation system:

Output and Delay Process Analysis of Multichannel Networks with Voice/Data

where is given by (4.4). For the reservation system:

95

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where is given by (4.7). We define the channel utilization for voice traffic and data traffic, respectively, as the average number of voice packets or data packets successfully transmitted on an arbitrary channel per slot. The average channel utiliza-

tions

and

for voice traffic and data traffic, respectively, are given as

follows: For the nonreservation system:

For the reservation system:

where is Dirac’s delta that it equals 1 if where l = 0 or 1, and 0 otherwise. The average number of voice packets which are involved in collision per slot is given in the same manner as in the derivation of as follows: For the nonreservation system:

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97

where is given by (4.4). For the reservation system:

where

is given by (4.7).

By using and the loss probability of voice packets for both nonreservation and reservation systems is given as follows:

where and are given by (4.12) and (4.20) for the nonreservation system, or by (4.14) and (4.21) for the reservation system. As all N users are stochastically homogeneous, it is easy to see that the following balance equation holds true:

where is given by (4.13) for the nonreservation system, or by (4.15) for the reservation system. Solving (4.23) for E[D], we can obtain the average data packet delay as follows:

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for the nonreservation system or the reservation system with (4.13) or (4.15).

3.5

given by

Packet Delay Distribution

In this subsection, we analyze the moment generating function of the probability distribution of data packet delay for the reservation system. It is noted that when the system is a nonreservation system, where all channels can be accessed by all transmissions with equal probability 1/M. Therefore, the analysis presented in this subsection can be applied to analyze the probability distribution of data packet delay for the nonreservation system with We focus on a user who has a data packet called a tagged data user. Since all users who have data packets are homogeneous in terms of channel access rate and transmission time, we can choose any such user as the tagged data user. We note that the data packet delay consists of two parts as follows: (1) The first part is the delay from the arrival instant of a new data packet to the end of the first slot where the data packet might be successfully transmitted. (2) The second part is the delay from the end of the first slot to the departure instant of the data packet if the first transmission attempt is unsuccessful. We refer to the former as the initial delay and the latter as the backlog delay. We first derive the backlog delay. We let represent the conditional moment generating function of the backlog delay that the tagged data user finds himself among iv users in the speech mode and users in the data transmission mode at the end of the tth slot. In order to obtain we first define conditional transition probabilities as follows:

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99

can comprise an matrix. These are transition probabilities that, given the tagged data user finding himself among users in the speech mode and id users in the data transmission mode at the end of the tth slot, he transmits his data packet successfully in the (t + l)st slot. Since the behavior of users with data packets is stochastically homogeneous, the tagged data user retransmits his data packet with probability

and the transmission is successful with probability

The other probabilities in (4.25) are the same as in (4.6). Let denote conditional transition probabilities that the tagged data user does not transmit his packet successfully, given x(t) = I. We have

where is the one-step state transition probability defined in (4.6). It is noted that can comprise an matrix. By

using follows:

and

we obtain the moment generating function

as

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The above equation is derived from the following observation: the first term expresses that if the tagged data user in the data transmission mode transmits his packet successfully in the (t + l)st slot, then his delay is just one slot; the second term expresses that if the tagged data user does not transmit successfully in the (t + l)st slot and finds himself among users in the speech mode and users in the data transmission mode at the end of the (t + 1)st slot, then the distribution of his backlog delay has a m.g.f. given by By summing up all the probabilities on J, we can obtain We can write (4.27) in matrix form as

where

is an (H – N – l)-dimensional column vector with elements and are matrices of and and e is a column vector with H – N – 1 elements, all of which equal 1 and E is an identity matrix. Let represent the probability of a data packet being successfully transmitted in the first slot upon its arrival, and let represent the probabilities that the tagged data user fails in transmission at the first attempt to find himself among users in the speech mode and users in the data transmission mode at the end of the (t + l)st slot. They can be obtained by the following:

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101

and

where is the stationary state probability distribution given in (4.7). We note that

in (4.29) and (4.30) is the probability that k new data transmissions and backlogged data transmissions succeed in transmission

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under the condition that new data packets and backlogged data packets are transmitted, among which packets are successful. It can be derived from the following observation: (1) The number of distinct ways of choosing successful new data packets k out of

new data packet transmissions is given by

(2) The number of distinct ways of choosing successful backlogged data packets out of backlogged data packet transmissions is given by (3) The number of distinct ways of choosing successful transmissions out of

transmissions is given by

K in (4.29) and (4.30) is a normalizing constant such that

where Finally, we can obtain the m.g.f.

of data packet delay as follows:

Closed-form expressions for all moments of can be obtained in the following manner. Let represent the nth factorial moment of the data packet delay distribution. Taking account of we have

Output and Delay Process Analysis of Multichannel Networks with Voice/Data 103

where is an (H – N –1) -dimensional row vector. We note that the average data packet delay, obtained in (4.34) should equal E[D] in (4.24) for the reservation system. Using the second moment we can obtain the coefficient of variation of the data packet delay.

4.

Numerical Results

Using the equations in the previous subsections, we can numerically compare the performance of the systems with different numbers of channels, different numbers of reserved channels and different selection probabilities of reserved channels to discuss what effects of multichannel, channel reservation policy and selection probability may have on the channel utilization, loss probability, average packet delay, coefficient of variation of the data packet delay and correlation coefficient of the voice and data packet departures. For all cases, the number of users is N = 30 and all channels have the same bandwidth v = V/M where V is the total bandwidth available to the system. A packet transmission time over a channel with bandwidth v, therefore, becomes where is the transmission time of a packet over a channel with full bandwidth V. The other parameters for the voice traffic are determined by referring to [Good89], where the mean talkspurt duration is 1.00 s, the mean silent gap duration is 1.35 s, and slot duration is 0.8 ms. In our model, the mean silent gap duration corresponds to 1,688 slots, and the mean length of being in the speech mode corresponds to 1,250 slots, so that we use and to provide performance curves. In general, numerical results change with the retransmission rate of data packets, and the generation rate of voice packets in the speech mode, As an example, we use the retransmission rate of data packets, and the generation rate of voice packets in the speech mode, for all cases. In Figs. 4.3 and 4.4, we plot the channel utilization of data traffic and the average data packet delay E[D], respectively, for the non-reservation system as a function of the arrival rate of data packets ranging from 0.0 to 1.0 to examine the effects of the number of channels on the performance of data traffic. The unit time of E[D] (in slots) in Fig. 4.4 is taken from the transmission time of a packet in the single-channel system,

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so that E[D] is multiplied by M in the case of M > 1 for fair comparison. In these two figures, we observe that: for smaller the single-channel system offers higher channel utilization, but as increases, it has a very low channel utilization and a very long delay of data packets. On the other hand, the channel utilization of multichannel systems (M > 1) can rise to higher levels and the delay of data packets can maintain shorter levels even if increases to a very large value. These are due to the fact that when is small, the access rate of users per channel in the case of M = 1 is actually greater than that in the case of M > 1, so that the utilization per channel becomes higher. However, with becoming larger, collisions occur more frequently and the number of retransmission attempts increases in the single-channel system while the channel utilization of data traffic increases gradually and the data packet delay is still kept significantly shorter in the case of M > 1. From these plots we can conclude that the multichannel systems perform better than the single-channel system for the integrated voice and data transmission and they can achieve performance improvements in the channel utilization and the average delay of data packets. We next investigate the coefficient of variation of the data packet delay versus for M = 1, 3, 5 in Fig. 4.5. As can be observed, all cases have larger values of when data arrival rate is small and they decrease gradually as becomes larger. Compared to the single-channel case for the same the multichannel systems (M > 1) have larger values of and all values of in the case of M = 5 are the largest for all This can be explained as follows. A new data packet may be successfully transmitted in the first slot upon its arrival, which tends to increase the variation of the delay. As the number of channels increases, more data packets may be successfully transmitted upon their arrivals, so that the variation of the delay also increases. Under higher data traffic, there is less probability of succeeding in transmission on arrival, so that values of are going to be smaller. In Figs. 4.6 and 4.7, we show the average channel utilization and the loss probability for voice traffic respectively, by setting the number of reserved channels for M = 5 to compare the effect of the number of reserved channels on the performance as a function of where the selection probability is given by The system with is a nonreservation system. We can observe that when is small, there is

Output and Delay Process Analysis of Multichannel Networks with Voice/Data 105

no big difference in the channel utilization or loss probability for different values of As reaches more than 0.05, the channel utilization and the loss probability for larger become smaller. Such a phenomenon is not surprising. The reason is because for larger collisions between voice and data packets would be more frequent over channels while some voice packets are transmitted more easily over the reserved channels without any interference of data packets, so that the utilization per (accessible) channel goes down while the loss probability can be reduced. The fact that the case of has the smallest channel utilization means that the reserved channels have not been used as effectively. If is increased more, there would be more chances to access the reserved channels for voice packets and the performance would be improved, which will be seen in Fig. 4.8. In Fig. 4.8, we examine the effects of the selection probability of reserved channels on the loss probability for and It is interesting to observe that depending on the number of reserved channels, the optimal that achieves the minimum loss probability is different. Obviously, the greater the number of the reserved channels, the bigger the optimal It is also interesting to see that the more reserved channels, the smaller the minimum loss probability. This is because the more reserved channels there are, the better the system can be adjusted for the convenience of voice packets. To observe the effects of reserved channels and the selection probability on the correlation coefficient of the voice and data packet departures, Figs. 4.9 and 4.10 depict versus . for the case of and for the case of respectively. The system with a small absolute value of shows a weak correlation between the number of voice departures and the number of data departures in a slot. In Fig. 4.9, it is observed that if is smaller than 0.01, the correlation coefficients for all cases are small and almost the same. However as further increases, the correlation coefficient once reaches a negative peak and then becomes weaker and weaker in each case. The reason why the correlation coefficient is negative is that voice packets and data packets compete for the limited bandwidth with each other. Among the three cases, the case of has the strongest negative correlation while the case of has the weakest negative correlation. This is because the more reserved channels for voice packets there are, the less interference in transmission between voice and data packets there is. The effect of using

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the reserved channels in the multichannel multi-traffic system on the interference between voice and data transmission is more significant for higher

data traffic load. In Fig. 4.10, we can also see a similar phenomenon as in Fig. 4.9. Namely, in each case, as increases, the negative correlation coefficient becomes stronger first, then reaches a negative peak and finally becomes weaker. The reason for this can be considered the same as in Fig. 4.9. Among the three cases, the system with shows the weakest correlation for all The reason is that users in the speech mode access only the reserved channels in the case of so that the output process of voice packets is almost independent of the output process of data packets. Here the correlation coefficient cannot be zero, however, because the number of users is finite and each user can generate both voice and data packets.

5.

Conclusion

This chapter proposed two different procedures of the multichannel slotted ALOHA protocol for the integrated voice and data transmission in wireless information networks and presented an exact analysis to numerically evaluate the performance of the systems. In scheme I, there was no limitation on access between voice transmission and data transmission, i.e., all channels can be accessed by all transmissions. In scheme II , we applied a channel reservation policy with the selection probability of reserved channels, where a number of channels are used exclusively by voice packets while the remaining channels are used by both voice and data packets. The probability distributions for voice and data departures and for the data packet delay were derived. The analysis contains the case of nonreservation systems as a special case if the number of reserved channels is set equal to zero. The results obtained in this chapter also include those of the system presented in Chapter 2 as a special case where multi-traffic and reservation policy had not been considered.

We examined some cases with different numbers of channels, different numbers of reserved channels and different selection probabilities to discuss their effects on the average performance, coefficient of variation of the data packet delay, and correlation coefficient of the voice and data packet departures in the numerical experiments. From the numerical results, we can conclude the following:

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(1) The proposed multichannel system for integrated voice and data transmission can improve the system’s performance considerably in terms of the channel utilization and data packet delay. (2) The reservation policy we proposed in this chapter is very useful to improve the system performance for voice traffic. The effect of reserved channel is more significant under higher data traffic load.

(3) The selection probability of reserved channels can make efficient use of reserved channels to decrease the loss probability of voice traffic. (4) The reserved channels and their selection probability have significant effects on the correlation coefficients of the voice and data packet departures. The analysis presented in this chapter can also be applied to evaluate performance measures of other networks such as priority networks, cellular mobile radio networks and multi-hop packet radio networks.

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Chapter 5 PERFORMANCE ANALYSIS OF PRIORITIZED MULTI-HOP PACKET RADIO NETWORKS

1.

Introduction

Packet radio broadcast techniques have been proposed as a method providing communication media for geographically distributed users. Many studies have appeared in the literature on the analysis of performance of a number of access protocols implemented in computer communication systems [Sidi83a], [Taka83a], [Taka83b] and [Roy84]. Repeaters which are commonly used to ensure connectivity of users of packet radio networks which are to operate in large geographical areas, are forming multi-hop packet radio networks. For the latter, a large number of access protocols have been proposed in order to improve the performance of the systems such as ALOHA multiple random access method and CSMA with and without busy-tone, etc. The system with ALOHA employs two different transmission strategies, namely, the pure (or unslotted) ALOHA protocol and the slotted ALOHA protocol. In the slotted ALOHA protocol, the start of the transmission and that of the time slot are synchronized. The slotted ALOHA random access scheme is one of the simplest and oldest multiple random access protocols for communication systems, and is a viable access technique in certain applications (e.g., a system supporting a large number of light users). When this multiple random access scheme is used, it may happen that two or more packets are simultaneously received by the receiver due to independent transmissions of several users. In such a situation, it is assumed that 113

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none of the packets was correctly received, and the corresponding devices have to retransmit their packets. In the CSMA protocol, the users of the network before transmitting their packets first sense the channel. If the channel is sensed to be busy, the sensing user refrains from transmitting (to avoid a collision) and reschedules his transmission according to one of several strategies. If the channel is sensed to be idle, the user transmits his packet immediately. Collision of packets occurs only when two or more users start transmitting their packets within a propagation time of each other. The basic measures for the efficiency of the system are the throughput, average queue length and average packet delay. The performance of the CSMA access scheme with busy-tone and collision detection protocol in a multi-hop packet radio environment is studied in [Roy84]. The approach is based there on a continuous-time Markov chain, while the packet length, retransmission time, transmission detection time, busy-tone detection time, collision detection time and collision termination time are assumed to be exponentially distributed. In [Magl80], the throughput of a multi-hop packet radio network, operating according to the CSMA protocol with perfect or imperfect capture assuming zero propagation delay and Poisson arrivals, is obtained using a mathematical model corresponding to a continuous-time Markov chain for exponentially distributed packet length with a product form solution. In [Toba83], an analysis of the throughput in multi-hop packet radio networks with zero propagation delay is presented in order to compare the performance of various channel access schemes, such as pure ALOHA, CSMA, busy-tone multiple access (BTMA), etc., and numerical methods for the application of the analysis to general configurations are discussed. In [Taka83b], a three-hop packet radio communication system with mobile users using the slotted ALOHA protocol and the CSMA protocol is analyzed. Four cases differing in repeater allocation and bandwidth division (namely, regular hexagon scheme, square scheme under the slotted ALOHA access protocol and the CSMA access protocol) are considered and analyzed to obtain the throughput and the average packet delay of the system. A multi-hop centralized packet radio network using the slotted ALOHA access scheme with a fixed configuration of users and fixed source to link paths for packets have been analyzed by [Taka85c], where a Markov chain approach is taken to find the throughput and packet delay characteristics.

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Throughput-delay performance of a discrete-time queueing model with priority using the slotted ALOHA channel access protocol has been studied in [Sidi83b] by applying the piecewise Markov theory. In the above studies, however, in order to simplify the analysis of the throughput and the packet delay, channel noise is neglected and zero transmission error rate is assumed. In reality, hops communicate with one another by packets which may be destroyed by transmission errors. The hop incurring a transmission error must retransmit its packet at a later slot, according to a given retransmission policy. In this chapter, we evaluate the performance of the slotted ALOHA access scheme with priority in multi-hop packet radio communication environments. The system consists of N hops with infinite buffer capacities, the intervals of external arrival at each hop from the corresponding source and the transmission times are generally distributed. Here, two major results are presented. First, considering transmission error, we present an exact analysis of the joint probability generating function of the queue length distribution of the system in steady-state, in order to obtain the average queue length and average packet delay. Then, we give a system example in which the analysis can be applied in order to calculate numerically the average performance. It is a tandem packet radio network having three hops the priorities of which are determined on the basis of the network’s topology. Since for systems with a large number of hops this method involves computational difficulty, we also present an approximate analysis to numerically evaluate the system performance such as the average packet delay and average queue length in tandem systems with an arbitrary but finite number of hops. In this analysis we suppose that external arrivals to each hop occur in a Poisson stream of rate and transmission time is constant. The analysis is based on the method of decomposition, where the system is divided into two subnetworks at hop i. The first one includes i – 1 hops which have higher priorities than the hop i, and the second has N – i – 1 hops, including hop i. We can model these two subnetworks as the generalized M/G/1 queueing system, respectively, and the transmission process of the second subnetwork corresponds to the transmission process of hop i. In this approximation method, we first analyze the busy period of the first subnetwork, then find the probability distribution function of transmission period and idle period of the second subnetwork by analyzing the queue length probability distribution of the second subnetwork at imbedded Markov points, when

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the second subnetwork is given a transmission chance. Thus, we derive the average queue length and the average packet delay at hop i. The analysis is exact for the lowest priority hop, because the arrivals at the hop are only external, which are assumed to form Poisson process. The chapter is organized as follows. In Section 2, we give a mathematical model for a multi-hop packet radio network of N hops with transmission error. We assume a general arrival process and constant transmission time. In Section 3, after having analyzed the Markov chain describing the system, we first derive the exact expression for the average packet delay and average queue length of the system. Then we present a tandem packet radio communication system having three hops. Furthermore, in the same section we present a new approximation technique to derive the average packet delay and average queue length of the system. We also consider a parallel packet radio communication system with N hops and two hops as an example of multi-hop communication networks to evaluate the performance of the system by applying the approximation technique presented in this section. Numerical results of the analysis and simulation results for the system to test the approximate analysis are shown in Section 4 and a conclusion is drawn in Section 5.

2.

System Model

In this section, we present a slotted ALOHA model with priority in multihop packet radio communication environments. The model considered in this chapter consists of N hops with infinite buffer capacities. Hops use the priority based slotted ALOHA scheme to access a common channel and transmit their packets to one another or to the outside of the system. The time axis is slotted into equal length segments with a duration corresponding to the transmission time of a packet. All hops are assumed to be synchronized and to start their transmissions only at the beginning of a slot. Its configuration is shown in Fig. 5.1 where the numbers in circles are the numbers of hops. New packets arrive at each hop from outside of the system (external arrival) with rate according to a general process, and the arrival processes are assumed to be independent of one another. This system has N different priority classes denoted by 1,2,..., N, in which the increasing numbers indicate decreasing priorities, and each hop has a different priority. We assume that a packet is transmitted by hop i only when

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all hops 1,2,..., i – 1 are empty. Since the access appropriate to the channel is allocated according to the priorities preassigned to the hops, a hop is not allowed to interfere with the transmission of a hop having a higher priority. Moreover, it is impossible for a hop to transmit a packet to other hops and simultaneously receive a packet from other hops. If a packet is transmitted successfully in a slot then it is removed from the queue of the hop. In the

case of a transmission error the packet must be retransmitted by the hop.

3. Performance Analysis 3.1 Joint Probability Generating Function Let

represent the number of packets which arrive at the hop from the outside of the system during the time interval We assume that is an independently identically distributed (i.i.d.) random vector sequence with integer valued elements. For the system, the probability distribution of the arrival process from the corresponding sources and the generating function for the joint probability

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distribution of the number of arrivals in any slot are defined by the following expression:

where . For example, for Poisson arrivals, and for Bernoulli arrivals, where r is an external arrival rate. We note that external arrivals never interfere with the transmission, but a transmission might interfere with other transmissions. To represent the system state, we use the following notations: : the branching probability of a packet from the hop i to the hop j, where is the probability of transmission from i to outside of the system : the number of packets at the hop i at the beginning of the tth slot : a binary random variable taking the value 1 or 0 defined as follows:

: error rate of the transmission from the hop i to the hop j

(in packet per slot), where in general, With the above notations, the number of packets in the hop i at the beginning of the (t + l)st slot can be written in the following form:

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The steady-state joint probability generating function of the queue length is defined by

where We note that is a continuous-time Markov chain. It can be shown that this chain is ergodic as long as the probability that there are no packets in the system is positive and that there exists a unique steady-state probability distribution. Thus, the Markov chain is ergodic if and only if

On the other hand, the successful transmission routing polynomials from the hop is defined as follows:

and routing polynomials from the hop transmission due to error as follows:

3.2

for unsuccessful

Analysis for Three-Hop Network

3.2.1 Fully Connected Three-Hop Network In this subsubsection, we consider a fully connected three-hop network with transmission priority. Let us assume henceforth that in (5.2) and (5.4)

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then we obtain

where (5.8) can also be represented as

The marginal generating functions become as follows: At hop 1:

at hop 2:

at hop 3:

In order to determine G(z) we must also determine the boundary functions G G and the constant G(0,0,0). G(0,0,0) is the probability of zero packet in the system.

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We define emission rate of each hop as the average sum of successful and unsuccessful transmissions in a slot, and denote it by is given by

To determine G(0,0,0), we take in (5.8), then, using we obtain G(0,0,0) from (5.13) as follows:

For the existence of steady-state we need G(0,0,0) > 0. Let denote the average arrival rate (packet per slot from the corresponding source into the hop then satisfies that

Fig. 5.2 is the state transition rate diagram for the system with three hops. We note that transmitted packets from the hop i to the hop j which are destroyed by transmission error have to still remain in the buffer of the hop and wait for retransmission. Next, let us determine G and G . Since the joint probability generating function G(z) is analytic in the polydisk , there, by applying Rouche’s theorem [Cops48], we can show that for the equation in given by

has a unique solution in the unit circle The existence and uniqueness of such a solution is proved in Appendix A. Let this solution be denoted by

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Then, since G(z) is analytic in the polydisk, we obtain from (5.8)

where For G

the equation in

given by

has a unique solution in the unit circle. Let this solution be denoted by the following:

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We obtain from (5.18)

where Using (5.14), (5.18) and (5.21) in (5.8), we see that G(z) is uniquely determined. In addition, let the average queue length in hop i at steady-state be denoted by which is defined as the average number of packets in hop i and is evaluated by the following equation:

We define the average packet delay as the average time from the epoch of the packet arrival to the epoch of the completion of packet transmission at hop i. By applying Little’s result to each hop we can derive the average packet delay for hop i as follows:

The total average packet delay in the system, E[D], is given by

3.2.2 Tandem Three-Hop Network The purpose of this subsubsection is to show how the analytical method presented in Subsection 3.1 is applied. A tandem packet radio network with three hops is illustrated in Fig. 5.3. The basic assumptions and parameters and taken from Section 2 and Subsection 3.1. In the present network, we additionally assume the slotted ALOHA transmission protocol with transmission errors and independent Bernoulli external arrival processes. All the hops are assumed to use the same transmission channel. The routing policy in the network is formally stated as follows:

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Therefore, from the system topology we find the routing polynomials as follows:

In the above system, we assume that packets do not arrive at hop 1 from its source and that the external arrival processes to hops 2 and 3 are independent Bernoulli processes with average rates and respectively. The probability generating function for the joint distribution of the number of arrivals in a slot is given by

where for i = 2,3. Therefore, the emission rates including unsuccessful transmissions in a slot to other hops are given by

The condition for steady-state in the system is

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The average queue length and the average packet delays in the hops are obtained as follows:

where

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and

3.3

Analysis for N-Hop Network

In this subsection, we propose an approximate method for evaluation of the average queue length and the average packet delay in the model involving an arbitrary but finite number of hops. In Fig. 5.4, we show a packet radio network involving N hops in tandem. The system considered here is an extension of the model with three hops, illustrated in Fig. 5.3, having N single server queueing hops and satisfying the assumptions and conditions stated in the preceding Section 2. The external arrivals to hop i are assumed to form independent Poisson processes with parameter and the transmission time of a packet is fixed and equal to one slot. At the beginning of a slot, each hop is assumed to know whether a packet may or may not be transmitted. Except for the

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lowest priority hop, the internal arrival at each hop is the output from the hop with a lower priority than its own. The total arrivals at each hop are assumed to be approximately Markovian. This assumption is not unreasonable for a large network. For the hop of the lowest priority, the assumption is always true because its only arrivals are external. For this reason, the total arrival rate at each hop become a sum of the external arrival rates from the sources that are associated with all the hops of lower priorities than the hop in question. Thus, under the assumption of a stationary state, the total arrival for hop i is obtained from

The total arrival rate for hop N is

where is the average external arrival rate. For such a network, we note that because of the dependence in the transmission of packets among hops, the steady-state queue lengths at the hops are not independent of one another. The interdependence of the hops complicates the analysis to the point of intractability. Therefore, to simplify the computation of the average queue length and packet delay for a system involving a large number of hops, this network is approximately treated as a combination of N independent single queueing facilities of the generalized M/G/1 type with an appropriate modification. The analysis is based on the method of decomposition, where the network is divided at the hop into two subnetworks modeling by the generalized M/G/1 queueing system, iteratively. Each generalized M/G/1 type subsystem is related to its network surroundings by consider-

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ing the relevant arrival and transmission processes. To derive the average

queue length and the average packet delay of each hop, we first consider the Markov renewal process imbedded in the queue length process of a single queue model at hop . Then by analyzing the probability distribution functions of transmission period and idle period of the second subnetwork, we get the average queue length and average packet delay of the hop i. 3.3.1 Queue Length Distribution We now analyze the queue length behavior of hop at those instants where hop i is given a transmission chance and may commence its transmission. The imbedded Markov points and channel state transition at hop i are shown in Fig. 5.5. Let us introduce the notations showed in Fig. 5.5, where

: idle period : period between the completion epoch of transmission and the next imbedded Markov point : transmission time of a packet (1 slot) : transmission period : transmission chance point (imbedded Markov point) (j = 0,1,2,...)

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We also define the following notations to analyze as follows:

: the probability of k packets arriving at the hop i in an idle period : the probability of k packets arriving at the hop i in a transmission period : the probability of k packets arriving at the hop i in one slot : the stationary state probability that hop i has k packets at an imbedded point, k = 0,1,2,.... F : probability distribution function of idle period F : probability distribution function of the transmission period Obviously, these instants can be thought of as imbedded Markov points which correspond to slot boundaries for each hop and are different due to the priorities. The behavior of the chain of these imbedded points can be completely described as an imbedded Markov process. If the buffer of hop i, at those imbedded Markov points, is nonempty it transmits a packet to hop i – 1 at once with one slot. If it is empty the transmission never occurs and its chance is passed at hop . To analyze the queue length of the hop i at these imbedded Markov points, let us first define the idle and transmission periods of hop i. The idle period is the time interval between two successive imbedded Markov points, and at the beginning of the interval no packets are waiting in the queue for the transmission. The transmission period is the time interval started with the transmission, and contains the transmission time of one slot and the remaining time interval until the next imbedded Markov point. We note that forms a continuous-time Markov chain under the present assumptions, accordingly satisfies

Let H(z) be the stationary probability generating function of the number of packets at the imbedded Markov points in the queue, and let it be given by

We note that is assumed to be independent of so the numbers of the packets arriving at the hop i are independent during and If we

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define as the probability generating function of the number of packets arriving at hop i, during the idle period then we have

Similarly, we define as the probability generating function of the number of packets arriving at the hop i and during a transmission period we get

Since the hops having a lower priority than the hop i cannot transmit during a transmission period of the hop i, the total arrival rate at the idle period of the hop i can be obtained approximately by and the total arrival rate at the transmission period of hop i is just equal to the rate of external arrivals where is given by (5.39) and (5.40). The probability is given by

then

can be given by

where E denotes the expectation. Hence (5.41) yields the probability generating function H(z) of the imbedded Markov chain is

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Therefore,

The solution given in (5.50) contains the undetermined probability which is obtained by the normalization condition Allowing in (5.50) and noting that and we have

where and are the first moments of and respectively. Consider now the process of transmission at the hop i and let G(z) define the steady-state probability generating function of the number of packets in the hop i, as seen by a packet transmitted at the end of its transmission. It follows that

Since the transmission of a packet occurs only when at an imbedded Markov point (see Fig. 5.5), the probability is given by

or equivalently

We define by the probability generating function of the number of packets arriving at hop i during the transmission time when the packet is transmitted, and it is obtained as follows:

for

(1 slot), we have

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Then G(z) becomes

From (5.50) and (5.57) we can obtain

For the lowest priority hop, the assumption is always true because its arrivals are external only, so the analysis is exact.

3.3.2

Average Performance Measures

The average queue length

for the hop i is given by the first moment

of G(z), and it is given by

where and are the second moments of and respectively. If we apply Little’s theorem to the hop i, then we find that the average packet delay is

However, to calculate the average queue length and the packet delay of hop i, we still have to determine the first moments and the second moments of the idle period and the transmission period respectively. These analyses are considered in the next subsubsection.

3.3.3 Transmission State Distribution To analyze the average idle period and the average transmission period of the hop we define the following notations:

: z-transform of the probability distribution function for : z-transform of the probability distribution function for : the busy period of the first subnetwork which starts with a packet

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arrival which finishes an idle period and finishes when the queue of the first subnetwork becomes empty again A. Exact Analysis for Hop N

Since the arrivals at hop N are only external, we can exactly obtain the exact probability distributions of the idle period and the transmission period at hop N. We first divide the network into two subnetworks modeled on the generalized M/G/1 type queueing system at hop N, as shown in Fig. 5.6. The first subnetwork includes N – 1 hops which have higher priorities than the hop N, and the second subnetwork has only a single queueing hop N.

Here, the assumptions and parameters are the same as in the previous subsections. The external arrival processes to the two subnetworks are Poisson processes with mean arrival rates and respectively, which are assumed to be independently and identically distributed in each

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slot. However, the first subnetwork has both internal and external arrivals, because the output of the second subnetwork enters its buffer. The transmission time of the second subnetwork is N slots per packet, this time being required for transmission of the packets to the outside of the system. First, we present an exact analysis method for the average packet delay and queue length of hop N. Packets from N – 1 corresponding sources and from a single corresponding source arrive randomly at the buffer of the first subnetwork and at the buffer of the second subnetwork, respectively. If the second buffer has any packets at an imbedded Markov point, it transmits one packet through the first subnetwork to the outside of the system at once with transmission time of N slots. If the second buffer is empty at an imbedded Markov point, it loses this transmission chance. The arrival of a packet from the second subnetwork and the external arrivals at the first subnetwork terminate the idle period of the first subnetwork and a new transmission period begins. Then, the second subnetwork cannot transmit until the first subnetwork becomes empty (namely at the next imbedded Markov point). We see that the idle period and transmission period of the second subnetwork can be obtained from the busy period analysis of the first subnetwork. We note that external packets arriving at the first subnetwork with different Poisson arrival rates need different transmission times in order to leave the first subnetwork, e.g., the packets arriving with rate will need i slot transmission time. Following this consideration we note that, though the transmission order among N – 1 different hops differs from that of original system, the time interval between imbedded Markov points at the hop N is the same. Therefore, the analysis for the hop N is exact. Let be the number of slots needed to transmit packets arriving in one slot from Poisson stream with rate Therefore, for the probability generating function of denoted by we have the expression

Let L(z) be the probability generating function of the total number of slots needed to transmit the packets arriving in one slot from Poisson stream with rate we can obtain

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Thus, it is clear that the total number of time slots needed for transmission of packets entering the first subnetwork during N time slots, corresponding to the time slots needed for a single packet arriving at the second subnetwork to leave the system, is To calculate the means of the idle period and transmission period, it is necessary to know the expected values of the busy period at the first subnetwork. The z-transform of the busy period defined by at the first subnetwork satisfies

By these arguments, we can give the z-transform of the idle period and transmission period of the second subnetwork, and calculate the first and second moments. Since the idle period of the second subnetwork depends on the number of external arrivals of different rates from outside of the system to the first subnetwork during the time interval only j = 0,1,2,..., the z-transform of the idle period of second subnetwork can be given directly by

During the transmission time of the second subnetwork with N slots, other packets may arrive at the first subnetwork from outside the system. The z-transform of the probability distribution function of the transmission period which is itself the sum of N independent random variables, is equal to the product of the z-transform for the probability distribution function of packets arriving at the first subnetwork in each of N slots. Consequently, we have

From (5.64) and (5.65), we have

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Using (5.59), we can obtain the average queue length

as follows:

and using Little’s formula the exact average packet delay of the hop N is given by

B. Approximate Analysis for Hop i We first consider the analysis for the hop 1. As the hop 1 has the highest priority and can always transmit a packet, if any, with constant transmission time of 1 slot, it is easy to see that the means and second moments of idle period and transmission period are all equal to 1, and the average queue length is obtained by using the results obtained from the analysis as follows:

In order to apply Little’s theorem to get the average packet delay of the hop 1, we assume that all internal arrivals take place at the end of a slot. The

Performance Analysis of Prioritized Multi-Hop Packet Radio Networks

average packet delay follows:

137

of the hop 1 obtained from Little’s theorem as

In the hop the idle period and the transmission period become random variables with general probability distributions

and the parameters of the latter depend on the stochastic behaviors in hops 1,2,..., i – 1 for . Now we consider the idle period and the transmission period for the hop using the same method as for the hop N case. We divide the system into two subnetworks at the hop i (see Fig. 5.7).

The first subnetwork consisting of i – 1 different hops and the second subnetwork consisting of N – i +1 different hops are analyzed respectively as single queueing systems of the generalized M/G/1 type. The packets

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for the first subnetwork arrive according to an independent Poisson process with the different arrival rates and the times needed for transmission are different. For the second subnetwork the arrival process is also independent Poisson process with the arrival rate being Since all the packets departing from the second subnetwork need the same time for transmission through the first subnetwork to the outside of the system from hop i, the time periods and between two successive Markov points depend only on the transmission state of the first subnetwork. The z-transforms of the idle period and the transmission period under the assumption of generalized M/G/1 model, are then calculated as

Using (5.74), (5.75) with (5.59) and (5.60), we can obtain where

and

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E[D] is defined as the total average packet delay in the system and it is given by

3.4

Two-Hop Parallel Network

In this subsection, we consider a parallel packet radio communication network with N users and two hops shown in Fig. 5.8 to evaluate the system performance by applying the approximation technique presented in Subsection 3.3.

The system is different from that of the previous subsection. In this system we assume that a packet is transmitted out of the system needs two slots. This system is a parallel system and has N different priority classes numbered as 1,2, ..., N, such that a small number indicates higher priority and each user has a different priority. We divide this network at user

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into two subnetworks that are modeled by the generalized M/G/1 type queueing system. The first subnetwork includes i – 1 users which have higher priorities than users in the second subnetwork. The second subnetwork has N – i + 1 queueing users. The external arrival processes to the first subnetwork are Poisson processes with mean arrival rate . The total arrival rate at each user is the sum of the external arrivals to all users with a lower priority than its own. With these definitions, the average idle period and the average transmission period are obtained as follows:

and hence, the second moments of

The average queue length and the second moment of

and

are given by

for user i is obtained using the first moment and by

Performance Analysis of Prioritized Multi-Hop Packet Radio Networks

Finally, we obtain the average packet delay average packet delay of the system, they are

4.

141

of user i and the total

Numerical Results

In Fig. 5.9 and Fig. 5.10, we illustrate the numerical results for the networks with and without transmission error. In these figures E[D] indicates the total average packet delay (in slot) for the system with (packets/slot). The total average packet delay E[D] versus arrival rate r with (packets/slot) is plotted in Fig. 5.9. Fig. 5.10 shows the total average packet delay E[D] versus transmission error rate characteristics of the network shown in Fig. 5.3. It is found that with the increasing transmission error rate, the system throughput decreases. The curves show that when the system is heavily loaded a slight improvement in error rate markedly reduces the packet delay. The average packet delay for each hop with N = 3,4,5 versus (packets/slot) are plotted in Figs. 5.115.13, respectively. The average packet delay for three system cases versus r are plotted in Fig. 5.14. These approximate results being also tested in Fig. 5.11-5.14, total by comparing them with simulation results for N = 3,4,5 and three model, respectively, seem to be very accurate.

5.

Conclusion

In this chapter, we presented an exact method and an approximate method for analyzing the average packet delay and the average queue length in a class of priority based multi-hop packet radio networks using the slotted ALOHA scheme to access a common channel. Using the exact method we derived the average queue length and the average packet delay of the system having a general arrival process and a transmission process with fixed trans-

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mission time through the analysis of joint probability generating function of the steady-state distribution of queue lengths with transmission error. This method can be also used for comparing the performance of systems with and without transmission error. Due to the inter-dependence in the transmission of packets among hops, the analysis of the average packet delay and the average queue length for a large number of hops becomes very complicated. Therefore, we also presented an approximate method for the performance analysis of the tandem multi-hop packet radio network with an arbitrary but finite number of hops. In our approximate method, the model consisting of N hops was divided into two subnetworks. Each subnetwork with Poisson external arrival process is analyzed as a queueing system of the generalized M/G/1 type. We also presented an analysis for a parallel packet radio communication system with N hops and two hops as an example of multi-hop communication networks to evaluate the system performance by applying the approximation technique. The major assumption made in the approximation is that the total arrivals at each hop are Poissonian. This assumption, not unreasonable for large networks, readily yields a simple expression for the average queue length and the average packet delay at each hop. By the approximate method we have obtained the probability generating function of the queue length for each hop at imbedded Markov points. The average queue length and the average packet delay are given from the moment of the probability generating function of the queue length for each hop. For the hop of the lowest priority, the assumption is always accurate because its only arrival is the external one. The obtained approximate expressions, after having been tested through simulation results, seem to yield very accurate numerical results. The approximate method can be used for simple and accurate computation of the average queue length and average packet delay of tandem multi-hop radio systems involving an arbitrary but finite number of hops under prioritybased access scheme using slotted ALOHA.

Performance Analysis of Prioritized Multi-Hop Packet Radio Networks

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Performance Analysis of Prioritized Multi-Hop Packet Radio Networks

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II MULTICHANNEL AND MULTI-TRAFFIC NETWORKS WITH CSMA PROTOCOLS

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Chapter 6

OUTPUT AND DELAY PROCESS ANALYSIS OF SLOTTED CSMA/CD MULTICHANNEL LOCAL AREA NETWORKS

1.

Introduction A packet radio network is a store-and-forward packet switching system

employing radio channel(s). It consists of geographically dispersed hops communicating with each other or with a central base station over a single or multiple radio channels, which may be stationary or mobile (computers, mobile users, etc.). The CSMA and CSMA/CD protocols have now been widely accepted in multiple access broadcast LANs. In CSMA/CD, collisions can be detected by letting users monitor the channel to see if they agree with the packets being transmitted. When a collision is detected, the detecting user will immediately abort his transmission and send a noisy pulse informing all the users, which in turn will prevent others from transmitting. If no collision is detected in then this user is assured of a successful transmission, where is the end-to-end propagation delay between the two farthest users. The CSMA/CD protocol is also one of the most widely used access protocols over different LAN topologies. Many performance studies of CSMA and CSMA/CD systems can be found in [Klei75a], [Metc76], [Toba77], [Toba80a], [Coyl83] and [Taka85a]. In [Klei75a], Kleinrock and Tobagi assumed that the rescheduling delay is infinite, so that all packets that find the channel busy or that are destroyed in a collision are abandoned. In a subsequent paper by Tobagi and Kleinrock [Toba77], the rescheduling times were modeled as independently geomet149

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rically distributed random variables. They used a finite number of users and assumed the arrivals to be quasi-random (i.e., finite-source Poisson). The CSMA/CD protocol was introduced by Metcalfe and Boggs [Metc76] as an extension to the ALOHA network presented by Abramson [Abra70] and the CSMA protocol without collision detection proposed by [Klei75a]. Tobagi and Hunt [Toba80a] added collision detection to the model of Kleinrock and Tobagi [Toba77]. The model in [Toba80a] allowed variable length packets. There are three variations of the CSMA and CSMA/CD basic strategy:

(1) non-persistent CSMA: If the channel is idle, send; if the channel is busy, wait a random time and try again. Because of the simplicity and the good performance of the non-persistent protocol, its analysis has received wide attention. (2) p-persistent CSMA: This is used for slotted channels. If the channel is idle, send with probability p and defer until the next slot with probability 1 – p. This is repeated until the packet is successfully sent, or until another user is sensed to have begun transmitting, in which case wait a random time and try again. (3) 1-persistent CSMA: This is a special p-persistent case, in which . A ready user senses the channel, and if the channel is idle, he transmits the packet. If the channel is busy, he waits until the channel becomes idle and then the user transmits the packet with probability one.

Recently, some studies have appeared in the literature on performance analysis of a number of access protocols implemented in multichannel radio communication systems [Yung78], [Mars83], [Okad84], [Todd85], [Ko86] and [Okad87]. In these studies, the system performance of networks were analyzed for ALOHA, CSMA and CSMA/CD. The control procedure of a multichannel CSMA or CSMA/CD system is almost the same as that of a single channel CSMA or CSMA/CD system, except that the multichannel CSMA or CSMA/CD system has an additional mechanism, channel selection. In [Mars83], Marsan and Roffinella considered two ways to select one from a number of identical channels to transmit packets:

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(1) CSMA/CD-RC: A user with a ready packet randomly selects one of M available channels. That particular channel is then accessed using the conventional CSMA/CD protocol, independent of activities on the other M – 1 channels before sensing it. (2) CSMA/CD-IC: A user with a ready packet first senses all M available channels. Then out of all the channels sensed idle, the user chooses one randomly and accesses it using the conventional CSMA/CD protocol.

They showed that the multichannel scheme improves the channel throughput rate by reducing the number of users that can make simultaneous demands on the same channel, in addition to its preferable characteristics such as easy expansion ability, easy implementation by frequency division multiplexing technology, high reliability and fault tolerance. In [Todd85], Todd presented an exact analysis of the throughput for slotted multichannel CSMA/CD systems. Some priority schemes have also been proposed and analyzed for multichannel CSMA/CD protocols [Ko86] and [Okad87]. However, in most of these studies, multichannel systems have been analyzed to obtain average performance measures. The design and development of multichannel multi-hop systems, and interconnected network systems or integrated networks of voice and data, require not only such average performance measures as the throughput or packet delay, but also higher moments of the packet interdeparture time and number of departures in parallel. We have an interest in the traffic characteristics of the packet interdeparture time and the number of packets successfully transmitted in parallel in multichannel networks with CSMA/CD to evaluate of the performance of the interconnected networks, since the output process from the system constitutes a part of the input process to another interconnected, neighboring network or another set of channels. Using the first and second moments of the packet interdeparture time, we can determine the parameters in the diffusion approximation to the input process of interconnected systems. Furthermore, to evaluate the performance of communication networks such as a multi-hop system and a digitized voice system, the analysis has to be extended to provide delay distributions. Until recently, studies on the probability distributions of the packet interdeparture time and packet delay have only been concerned with the single

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channel systems in a number of access protocols [Toba82a], [Taka85b], [Taka86], [Mats90a], [Mats90b] and [Mats90c]. However, the performance analyses of multichannel systems are quite different from those of single channel systems. In this chapter, we study the performance analysis of a slotted nonpersistent CSMA/CD multichannel LAN system with a finite population. The analysis is based upon Markov chain techniques, and gives the moment generating functions of the output process and the packet delay. We can calculate their average and higher moments by differentiating these m.g.f.s to compare with those of the single channel case. We note that closed-form expressions exist for all moments of the m.g.f.s in this chapter. These results are useful for analyzing multi-hop networks and integrated networks of voice and data employing these multiple random access protocols. The chapter is organized as follows. In Section 2, we describe the system model with slotted non-persistent CSMA/CD multichannel. In Section 3, we present the transition probability of the system state, the analysis for probability distributions of the packet interdeparture time, packet departures and packet delay. In Section 4, we discuss the numerical results with the average performance measures and higher moments. Finally, we offer a conclusion in Section 5.

2.

System Model

There is a finite population of N mutually independent users which communicate over M broadcast channels. All channels are assumed to have the same bandwidth in Hz (see Fig. 2.1). The users’ access is synchronized to the start of time slots of seconds each. The time slots are defined as synchronous across all M channels and amongst all users. Each user sends his packets one-by-one, thus he is not allowed to transmit the next packet before he has finished the previous one. Without loss of generality, we assume M < N. In mobile packet radio systems, it is well known that the time varying channel suffers fading, shadowing ignition noise, co-channel interference and from the Doppler effect due to mobile users. It is possible to introduce a CSMA/CD with spread spectrum scheme or an FH/FSK CSMA/CD (Frequency Hopping/Frequency Shift Keying) [Sinh84a] in the presence of partial-band noise interference. However, it would significantly increase the complexity of performance analysis, so

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numerical experiments would be intractable. For this reason, this chapter employs a fixed normalized propagation delay, and assumes that packet retransmissions are only due to collisions. The slot size is assumed to be fixed and equal to the end-to-end propagation delay denoted by . The transmission time of a packet is denoted by is assumed to have a geometrical distribution with rate and is considered to contain both packet service time and propagation delay for each transmitted packet. All users are synchronized, and start their transmissions at the beginning of a slot. It is assumed that each user attempts to access a channel with probability at the beginning of a slot, and then one of the free channels is selected with equal probability (the IC strategy). Owing to the decentralized nature of this protocol, simultaneous transmission on a single channel by two or more users can occur. When this happens, it is assumed that the transmission of all such users is unsuccessful, and a collision is said to have taken place. The time between the occurrence of a collision on a channel and recovery from the collision is called the collision detection time. The time required for collision detection is assumed to be one slot. We model the system as a finite state and discrete-time Markov chain where the imbedded Markov points are chosen at the beginning of slots. We define the state of the system by the number of successfully transmitting users at the imbedded Markov point. Let x(t) denote the number of successfully transmitting users at the beginning of the tth slot. Under the above assumptions, we see that the process {x(t),t = 0,1,...} has Markovian property. In the following analysis, we take x(t) as the system state.

3. 3.1

Performance Analysis Stationary Probability Distribution

To obtain the one-step state transition probability following probabilities and as:

and

we first define the

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They are given by

For transition from i to j in a slot to occur, the following events must be satisfied at the same time, where

(1) Among N – i non-transmitting users, n users access channels. (2) Among n accessing users, j – i + c users succeed in transmission. (3) Among j + c users, c users complete their transmissions. Expressing these events mathematically, we obtain the conditional transition probability as

where is the conditional probability that c packets are successfully transmitted given that n packets are transmitted over m accessible channels. is the same as (2.3) and given by

The one-step state transition probability

is given by

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155

Let be an (N + 1)-dimensional row vector of the stationary probability distribution. It is determined by solving P as

where P is an .(N + 1) x (N + 1) matrix of the one-step state transition probability Pij .

3.2

Average Performance Measures

denote the m.g.f. of the number of packet departures at an arbitrary slot boundary. We define the number of packet departures in this subsection as the number of packets successfully transmitted in a slot through all the channels. Then we obtain

The throughput (defined as the average number of packets successfully transmitted per slot over the all channels in the system) is obtained by differentiating at z = 1 as follows:

The average channel utilization is given by

3.3

Packet Interdeparture Time Distribution of Type [c]

In this subsection, we derive the packet interdeparture time distribution of type [c] defined in Subsection 3.3 of Chapter 2 for the system considering in this chapter. Namely, the packet interdeparture time of type [c] is defined

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as the time interval between two consecutive slot boundaries at which the number of successful departures is equal to c (see Fig. 2.3). We denote by the packet interdeparture time of type [c]. Clearly, is an independently, identically distributed random variable. Now consider slot t such that x(t) = i. On the condition that x(t) = i, the time interval from the beginning of slot t to the first slot boundary, where c packets are successfully transmitted, is denoted by , and its moment generating function is denoted by is derived from the following observation: (1) if c packets depart successfully at the end of slot t, then (2) if the number of successful departures in slot t is not c and x(t + 1) = j, then the distribution of the time interval until the next departure of type [c] has a moment generating function given by By summing all probabilities on the above events, we can obtain as follows:

Let dimensional column vector of the conditional m.g.f. (6.10) in matrix notation, is given by

denote an (N + 1). Expressing

where e is a column vector with N + 1 elements, all of which equal 1, and E is an identity matrix. P(c) is an matrix of the conditional transition probability given by (6.3). P is the matrix of the one-step state transition probability given by (6.5). Let denote the conditional stationary state probability distribution of state i at the slot boundary with type [c] departure. We can obtain that

Output and Delay Process Analysis of Slotted CSMA/CD Multichannel LANs

where

157

is an (N + 1)-dimensional row vector

of the conditional stationary state probability distribution , Let denote the unconditional m.g.f. of the packet interdeparture time of type [c]. Finally, we obtain as follows:

The average packet interdeparture time and the second moment of the packet interdeparture time of type [c] can be obtained by (2.19) and (2.20), respectively, with the probabilities and m.g.f.s given in this subsection.

3.4

Joint Probability Distribution of Packet Interdeparture Time and Number of Packet Departures

In this subsection, we analyze the joint probability distribution of the packet interdeparture time and the number of packets successfully transmitted. The definition of the packet interdeparture time in this subsection is the same as in Subsection 3.4 of Chapter 2. Namely, the packet interdeparture time is defined as the time interval between two consecutive slot boundaries, at which at least one packet is successfully transmitted (see Fig. 2.4). We also define the number of packet departures in this subsection as the number of packets successfully transmitted in a slot through all the channels. Note that the packet interdeparture time is different from the definition in the previous subsection. Let T represent the random variable of such packet interdeparture time, and let C represent the number of packet departures in parallel when the departure satisfying the above definition occurs. We define the conditional moment generating function of the number of packets successfully transmitted and x(t +1) = j, given x(t) = i

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as follows:

where is the conditional transition probability given by (6.3). To obtain the joint m.g.f. of the joint probability distribution of the packet interdeparture time and the number of packet departures, we also define a conditional transition probability as at least one packet departs successfully

in slot

Next let denote a random variable of the time interval from the end of slot t to the instant in which at least one packet is successfully transmitted and let denote a random variable of the number of channels having successful transmission in parallel in the instant. Now we consider the conditional joint m.g.f.

of

the joint probability distribution of the random variables and given has the following meaning. When the system state changes to j at the beginning of slot t + 1, if at least one packet departs

successfully at the end of slot t, then the packet interdeparture time and the joint m.g.f. of the transition in this slot and the number of successful channels in parallel is given by

On the other hand, if there is no

packet successfully transmitted at the end of slot t, the joint m.g.f. of the transition in this slot, and the time interval from the end of slot t to the

end of the slot in which at least one packet is successfully transmitted, and also the number of departures in parallel at the end of that, is given by Summing over all probabilities on j, ing expression:

is obtained by the follow-

Output and Delay Process Analysis of Slotted CSMA/CD Multichannel LANs 159

In the same way as in Subsection 3.3, let be an (N + 1)-dimensional column vector, then we have

where and are matrices whose elements are given by (6.14) and given by (6.15), respectively. Note that . e and E are defined as in Subsection 3.3 and P is the matrix of the one-step state transition probability given by (6.5). Let denote the unconditional joint m.g.f. of the packet interdeparture time T and the number C of successful departures in parallel. To obtain , we also need to give the conditional stationary state probability distribution immediately after at least one packet departure. Let be an (N + 1)-dimensional row vector representing the conditional stationary state probability distribution. It is obtained by solving a set of the following equations:

From

and

we can finally obtain

as

The average E[T] and the second moment of the packet interdeparture time in which at least one packet is successfully transmitted, the average E[C] and the second moment of the number of successful departures in parallel can be obtained by using (2.28)-(2.31) in Chapter 2 with the probabilities and m.g.f.s given in this subsection.

3.5

Packet Delay Distribution

In the same manner as for the analysis of the probability distribution of the packet interdeparture time, in this subsection, we derive the m.g.f.

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of the probability distribution of the packet delay for each user. We focus on a user called a tagged user. Since all users are homogeneous in terms of channel access rate and transmission time, we can choose any user as the tagged user. We note that the data packet delay can be divided into two parts as follows: (1) The first part is the delay from an arrival point of a new packet until the end of the first slot where the packet might be successfully transmitted. (2) The second part is delay from the end of the first slot until the departure instant of the data packet if the first transmission attempt is unsuccessful. We refer to the former as the initial delay and the latter as the backlog delay. First we derive the conditional m.g.f. of the backlog delay. Suppose that the tagged user finds himself among i backlogged users at the end of slot t given x(t) = i. Let and denote the random variable and its conditional m.g.f. for the time interval from the beginning of slot t +1 with x(t) = i to the next departure from the tagged user, respectively. is obtained according to the following observation: (1) If the tagged user succeeds in transmission in slot t + 1, then

(2) If the tagged user does not access or fails to transmit in slot t + 1, and the system state changes from i to j, then

Considering the probabilities for each case, we get

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161

Let DL denote an (N+1) -dimensional column vector whose ith element is given by

and let V denote an given by

Writing

matrix whose (i,j)th element is

in vector form, we have

where is an N-dimensional column vector. Let be defined as the probability for the initial delay that the tagged user sees the system in state j when he has just finished his transmission. By using the stationary probability distribution, defined in Subsection 3.1, , the conditional transition probability, and the throughput given in Subsection 3.1, we obtain as

Finally, we obtain

The average E[D] and the higher moments of the packet delay can be obtained by differentiating (6.27) for z and letting z = 1.

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4.

Numerical Results

In the previous section, equations for calculating the probability distribution functions of the packet interdeparture times and number of departures in parallel for slotted non-persistent CSMA/CD-IC were given. Using these equations, we obtain their average performance measures. By using Var[X], the variance of the random variable X, and the coefficient of variation of the random variable X, the coefficients of variation for the packet interdeparture times and number of successful departures are also obtained. We compare systems where the sum of the multiple channel bandwidth is equal to the bandwidth of a single channel network, i.e. all M channels are considered to have the same bandwidth v in Hz, v = V/M, where V is the total available bandwidth of the system. In dividing the entire available bandwidth of the system into homogeneous partitions, the time slot size over a channel with bandwidth v is given by where is the time slot size in a channel with bandwidth V. Therefore, as the number of channels is changed, the bandwidth of each must also change to satisfy the fixed total bandwidth constraint. When comparing system performances with different values of M, we have considered the fact that slot sizes are different. In numerical examples, we consider a system with and with the propagation delay for the single channel system and for the multichannel systems. Fig. 6.1 shows the average channel utilization U versus total offered rate in the system for the cases, respectively, to see how the various channel utilizations depend upon M. The channel utilization for M = 1 is higher than the channel utilization for M = 3 and M = 5 when is low, while as increases, the channel utilization for M = 5 is higher than that for M = 1 and M = 3.

Fig. 6.2 shows the average packet delay E[D]. We can obtain average packet delay E[D] by differentiating the m.g.f. . The average packet delay is small in the M = 1 case when

is light, but it becomes

very large as increases. The average packet delays are kept smaller in the multichannel cases because the transmission times of a packet for the multichannel cases are longer than those for the single channel case, so that their packet delays are larger than that for the single channel case as long as the traffic is light. However as

increases, multichannel systems suffer

Output and Delay Process Analysis of Slotted CSMA/CD Multichannel LANs 163

fewer collisions than do single channel systems. Our results agree with the conclusion in [Mars83], i.e. the multichannel scheme can improve channel utilization by reducing the number of users that can make simultaneous access to the same channel. Curves of the coefficients of variation of the packet interdeparture time of type [c] (where c = 3), the coefficients of variation of the packet interdeparture time T in which at least one packet is successfully transmitted, and the coefficients of variation of the packet delay are shown in Figs. 6.3, 6.4 and 6.5, respectively. Fig. 6.3 shows that with M the coefficient of variation of the packet interdeparture time of type [c] (where c = 3) decreases as increases from zero. In Fig. 6.4, the coefficient of variation of the packet interdeparture time T is plotted as a function of We can see that the coefficient of variation of the packet interdeparture time T begins to decrease from 1.0, reaches a minimum value, and is sustained at a value that depends on M. These behaviors can be explained as follows. When is small, the interdeparture time consists of a very long idle period and a packet transmission

time, where it should be noted that the coefficient of variation for the former period is 1.0. Fig. 6.5 shows that for the single channel case, coefficient of variation of the packet delay fluctuates the most. This is because as increases, packet collisions happen more easily in the single channel case than in multichannel cases.

5.

Conclusion

In this chapter, we have given a detailed study of the finite population with slotted non-persistent CSMA/CD multichannels. We exactly derived the m.g.f.s of the packet interdeparture times, number of packet departures and packet delay. In numerical examples, we examined the effect of channel number on the averages and coefficients of variation of the packet interdeparture time of type [c], the packet interdeparture time T, the number of successful departures and the packet delay. From numerical examples, we see that multichannel broadband radio networks have advantages over single channel broadband radio networks, such as higher channel utilization by distributing traffic load on multiple channels in addition to higher reliability and stability. These results are useful

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for analyzing multi-hop multichannel networks, interconnected networks or integrated voice and data networks employing CSMA/CD.

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Chapter 7 PERFORMANCE ANALYSIS OF CSMA/CA WIRELESS LOCAL AREA NETWORKS

1.

Introduction

Wireless local area networks (WLANs) are a rapidly emerging field of activity in computer communication networks. WLANs are being developed to provide a high bandwidth to users in a limited geographical area. It can be used within many local area places like wireless offices, campus classrooms, conference registration and so on. Using wireless network interfaces, mobile devices can be connected to the Internet in the same way as desktop computers are connected, using the Ethernet, token ring, or pointto-point links. The current generation of wireless communication systems are all digital. The growth of mobile telephony has been extraordinary, reaching 200 million subscribers worldwide by early 1998. Wireless local area networks (WLANs) operated in the 2.4 GHz band are now available with bit rates on the order of 11 Mbps.

Several contention-based protocols that could be adapted for use in wireless local area network environment currently exist. The protocols currently being looked at by IEEE 802.11, a standard for WLANs, include a protocol based on CSMA. The CSMA and CSMA/CD protocols have been used with great success in the Ethernet. Many performance studies of CSMA and CSMA/CD systems are found in [Klei75a], [Toba80a], [Toba82a], [Coyl83], [Mats90a], [Mats90b], [Mats90c] and [Onun91]. 169

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Unfortunately, the CSMA/CD protocol is not used in a wireless domain because the user is unable to listen to the channel for collisions while transmitting. The current proposal is based on a carrier sense multiple access with collision avoidance (CSMA/CA) protocol, see, for instance, [Davi94], [IEEE96], [LaMa96] and [Crow97]. WLANs with the CSMA/CA protocol work by a “listen before talk” scheme. The collision avoidance portion of CSMA/CA is performed by such a procedure that let some randomly chosen users talk and the others listen, so that the users who hear somebody talking will refrain from talking. Despite using the “listen before talk” scheme, packet collision can still occur as it is a random process to select talkers and listeners. Recently, some studies have appeared in the literature on the CSMA/CA protocol implemented in WLAN communication systems, see the references in [Sobr96] and [Deng99]. In [Sobr96], a simple priority scheme for IEEE 802.11 has been proposed, where a high priority user has a shorter waiting time when accessing the channel. In [Deng99], another priority scheme for IEEE 802.11 PCF (Point Coordination Function) to support asynchronous data transmission access method has been proposed. The priority scheme proposed in [Deng99] is very similar to that in [Sobr96], except that when a collision occurs, a high priority user can take advantage in accessing the channel too. However, in most of these studies, the performance of CSMA/CA with some additional mechanisms has been evaluated by simulation or has been approximately analyzed by applying the analyses of the CSMA/CD protocol to obtain performance measures for fixed carrier sensing intervals. In this chapter, we present a new realistic detailed system model and an effective analysis for the performance of wireless LANs by using CSMA/CA IFT and CSMA/CA DFT protocols based on the IEEE 802.11 WLAN standard presented in [Davi94], [IEEE96], [LaMa96] and [Crow97]. The access method that we consider in this chapter is different from those in [Sobr96] and [Deng99]. The present system based on the multiple random access is used to support data transmission without adopting additional mechanisms which might cause additional delay for a lightly loaded media. The method is simpler, more efficient and easier to implement in comparison to the methods presented in [Sobr96] and [Deng99] for data transmission. Until now, there has been no paper to present effective analysis methods for the performance of WLANs, so that we have not been able to inquire

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about the effect varying the length of the collision avoidance period and pulse transmission probabilities on the system performance. It is important to present effective mathematical analysis methods to numerically evaluate the system performance in such complex networks. The proposed method in this chapter allows the number of slots in a frame to be arbitrary depending on the chosen lengths of the collision avoidance period and packet transmission period. So, the influence of the possible length of the collision avoidance period and packet transmission period, and pulse transmission probabilities on the network performance can be discussed, based on the results of the channel utilization and the average packet delay for different packet generation rates. The results calculated for our system examples show that the choice of collision avoidance period and pulse transmission probability is the most significant issue for better performance. In this system, a user with a packet ready for transmission randomly senses the channel by pulse signals within a fixed interval (called collision avoidance period) to verify a clear channel before transmitting. When users do not transmit pulse signals in the collision avoidance period, they can listen continuously to pulse signals transmitted by other users from the channel. The system model consists of a finite number of users to efficiently share a common channel. The time axis is slotted, and a time frame has a large number of slots and includes two parts: the collision avoidance period and the packet transmission period. A discrete-time Markov chain is used to model the system operation. The number of slots in a frame can be arbitrarily dependent on the chosen lengths of the collision avoidance period and packet transmission period. In the numerical results, the performance of the CSMA/CA IFT protocol and the effects of the design parameters, namely, the collision avoidance period, packet transmission period, and pulse transmission probability are evaluated, based on results of the channel utilization and the average packet delay for different packet generation rates. The chapter is organized as follows. In Section 2, we give a system model. In Section 3, after analyzing the Markov chain describing the system, we derive the exact expression to obtain the channel utilization and average packet delay of the system for both IFT and DFT protocols. In Section 4, exact numerical results for the IFT protocol based upon the analysis are given along with some graphs and discussions. Finally, in Section 5, we present our conclusion.

172

2.

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System Model

In this section, we present the detailed system model for the system analysis. In the analysis we make the following assumptions: (1) There are a finite population of N mobile end users, who can generate data packet to communicate with each other. (2) A common channel is accessed by using the non-persistent CSMA/CA scheme. In this scheme, if the channel is sensed idle, send; if the channel is sensed busy, wait a random time and try again. (3) Every user has his own buffer which can store at most one packet at any time. Once a packet is accommodated at a buffer, it remains there until it is successfully transmitted. (4) If two or more packets are simultaneously transmitted at the same time, a collision of packets occurs. It is assumed that the result, whether success or collision, can be learned immediately after transmission. (5) The time axis is slotted and the slot size is assumed to be fixed and equal to the end-to-end propagation delay denoted by .

(6) A user with a packet ready for transmission randomly sends and receives pulse signals within the collision avoidance period to verify a clear channel before transmitting. The length of collision avoidance period is fixed to be D slots. (7) The transmission time of a packet is denoted by T (slots). T is considered to contain both packet service time and propagation delay for each transmitted packet.

(8) A frame consists of D + T slots that can be considered as system parameters. (9) Each user is in either of the two states: the idle state if he does not have a packet in his own buffer to transmit; or the backlogged state if he has a packet waiting or undergoing transmission. We assume that a user in the idle state generates a packet at the beginning of a frame with probability or remains in the idle state with probability .

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In this system, at the beginning of a frame, all users who have generated a new packet enter the active mode with probability one in the immediatefirst-transmission (IFT) protocol, while they enter the backlogged state once in the delayed-first-transmission (DFT) protocol. Any user in the backlogged state enters the active mode with probability During the collision avoidance period of D slots, users with an outstanding packet can be in either (1) active mode (denoted by a) or (2) backoff mode (denoted by b). Users in the active mode are those who are going to start transmission at the beginning of the transmission period, while users in the backoff mode are those who have already given up transmission in this frame after noticing that there exits another user who is trying to send a packet in the frame. We note that users can enter the active mode only at the beginning of a frame and users in the active mode may transit to the backoff mode in a slot when they listen to pulse signals from other users. During the collision avoidance period, all users in the active mode obey the following algorithm: (1) Each user in the active mode decides whether he sends some pulse signals to the channel with probability or listens to the channel with probability at the beginning of a slot. (2) Users who hear the channel busy are those who decide to listen to the channel, and they transit to the backoff mode at the end of the slot. Otherwise, users who decide to send pulse signals stay in the active mode. (3) When the collision avoidance period ends, all users who are still in the active mode start transmission. If there is more than one user in the active mode, a collision happens and they transit to the backlogged state at the end of the frame. On the other hand, if there is only one user, he succeeds in transmission and becomes empty at the end of the frame. (4) All users in the backoff mode transit to the backlogged state at the end of the frame.

3.

Performance Analysis

We define the imbedded Markov points to be chosen at the end of frames, and define the state of the system by the number of backlogged users at the imbedded Markov point.

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Let x(t) represent the number of users backlogged at the end of frame t. The system state is said to be in state i if and it is denoted by Ei, where i represent the number of backlogged users. The process forms a finite state and discrete-time Markov chain. We assume that a steady-state of the system exists. The imbedded Markov points and channel state transition for both IFT and DFT protocols are shown in Fig. 7.1, where and denote the numbers of users in the backoff mode and active mode at the ends of the current slot and the next slot, and denote the numbers of users in the backoff mode and active mode at the beginning of the packet transmission period, respectively. The total number H of the system states is given by

(7.1)

It is noted that the state transitions in any frame can be divided into three parts as (1) State transition in the first slot (called the first state transition). (2) State transition in a slot between the second slot and the end of the collision avoidance period (called the second state transition).

(3) State transition from the end of the collision avoidance period until the end of the packet transmission period (called the third state transition).

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175

Analysis for Second State Transition

First, we derive the second state transition. The detailed transition diagram from in any slot between the second slot and the end of the collision avoidance period D is shown in Fig. 7.2. From Fig. 7.2, we can define a transition probability for the second state transition between the second slot and the end of the collision avoidance period as follows:

where is Dirac’s delta, that it equals 1 if k = l, and 0 otherwise. We note that if To calculate the transition probability we introduce a function to map the two-dimensional state space into the one-dimensional state space H as follows:

We denote an

3.2

matrix whose elements consist of

by A.

Analysis of IFT for First State Transition

The analyses of the IFT protocol and the DFT protocol are different for the first state transition. In this subsection, we first analyze the first state transition in the first slot of a frame for the IFT protocol.

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Let denote the transition probability such that the system state transits from i to k where The first state transition that is defined as the state transition in the first slot for the IFT protocol is shown in Fig. 7.3. The big arrow on the right in Fig. 7.3 denotes that from the second slot of the frame to the end of the collision avoidance period the users follow the second state transition explained in Fig. 7.2.

Figure 7.3.

First state transition diagram for IFT

The two additional events occurred at the beginning of the first slot for the IFT protocol are (1)

users out of N — i idle users generate new packets with packet generation rate and idle users remain in the idle state with probability respectively where and denote the numbers of backoff users and active users just after the beginning of the first slot, respectively. This probability is given by

(2)

users out of i backlogged users join the active mode with probability µ while the others transit to the backoff mode with probability This probability is given by

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It is noted that from the second slot of the frame to the end of the collision avoidance period the users follow the algorithm explained in Section 2. Because the remaining transition between the second slot and the end of the collision avoidance period D is the same as in Fig. 7.2, once is determined, we can obtain for the IFT protocol by using defined in (7.2) as follows:

where we denote an

3.3

matrix consisting of

Analysis of DFT for First State Transition

In this subsection, we analyze the first state transition for the DFT protocol. The first state transition for the DFT protocol is shown in Fig. 7.4. The big arrow on the right in Fig. 7.4 denotes that from the second slot of the frame to the end of the collision avoidance period the users follow the second state transition explained in Fig. 7.2. Two additional events for the DFT protocol in the first slot are (1)

users out of N – i idle users generate new packets with packet generation rate and users remain in the idle state with probability where and

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denote the numbers of backoff users and active users just after the beginning of the first slot. This probability is given by

(2)

users out of backlogged users join the active mode with probability while the others transit to the backoff mode with probability This probability is given by

Therefore we can obtain in (7.2) as follows:

where we denote an

3.4

for the DFT protocol by using

defined

matrix consisting of

Analysis for Third State Transition

We further proceed to the analysis of the third state transition for both IFT protocol and DFT protocol for the packet transmission period T. During which only one of three cases (successful transmission, no transmission, and unsuccessful transmission) can occur. The transition probabilities for these three cases are denoted by and Fmj as follows, respectively, where and j are the state at the end of the collision avoidance period D and the state at the end of the packet transmission period T, respectively. We note that these three cases are determined only by the number of active users at the end of the collision avoidance period (or equivalently at the beginning of the transmission period), and we obtain that

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(1) For the successful case:

(2) For the idle case:

(3) For the collision case:

We denote by S, G and F matrices with of and

3.5

elements consisting

Stationary Probability Distribution

Finally, we consider the transition of the system state in equilibrium such that there are i backlogged users at the end of the tth frame, and j backlogged users at the end of the (t + 1)st frame where We can obtain the one-frame state transition probability matrix P between consecutive imbedded points having elements

by where D is the number of slots in the collision avoidance period. Let denote an (N + 1)-dimensional row vector of the stationary probability distribution. can be determined by solving P as

3.6

Average Performance Measures

Once we have obtained P and we can proceed to obtain performance measures of the system such as the throughput, channel utilization and average packet delay for the IFT protocol and DFT protocol, respectively, as follows. Let denote throughput in a frame. Then we have

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where e is a column vector with N + 1 elements, all of which equal 1 and à is obtained from (7.4) for the IFT protocol or from (7.5) for the DFT protocol. Let U denote the average channel utilization that the channel is carrying successful transmissions in a packet transmission period. Then we have (7.12) We denote the average number of backlogged users by by

which is given

(7.13) Let E[D] denote the average packet delay (slots). The packet delay is defined as the number of slots elapsed from the generation of a packet to the end of successful transmission. Then it is given by (7.14)

4.

Numerical Results

In this section, we discuss the optimal network design parameters such as the pulse signal transmission probability and ratio of the collision avoidance period D and the transmission period T to maximize the channel utilization and minimize the average packet delay for the IFT protocol. In all numerical examples, the number of users is chosen to be N = 30 and the number of slots for packet transmission in a frame is chosen to be T = 50. Figs. 7.5 and 7.6 show how the channel utilization and average packet delay change with different values of = 0.05,0.1,0.2,0.4,0.5,0.6 for the traffic rate with D = 5 and It should be noted here that small suppresses the start of actual packet transmission too much, which causes reduction of the channel utilization and heavy increase of packet delay. We also observe that when is increased to 0.2 and 0.4, the cases have larger channel utilization and lower packet delay. The maximal channel utilization of the system is the highest for and the minimal packet delay of the system is also the lowest for But when is

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continuously increased over 0.5, the channel utilization of these systems quickly degrades with traffic rate while that of systems with and 0.4 does not. In particular, the system with 0.6 shows the lowest channel utilization gain and the largest packet delay gain. This is because with large pulse transmissions become excessive to the channel when the traffic is heavy and it will rapidly reduce the channel

utilization and increase the packet delay. This result is important because if an appropriate value of is given to the system, the channel utilization increases considerably with little increase of blocking probabilities. has a significant impact on the system performance. Figs. 7.7 and 7.8 show the channel utilization and the average packet delay as a function of the packet generation rate. The length of the collision avoidance period is taken as a parameter where D = 3,5,10,20, respectively, for the same pulse transmission probability and the same retransmission rate as an example. means that during the collision avoidance period an active user sends a pulse signal in every two

slots on average. From Figs. 7.7 and 7.8, we can discuss the values of D to maximize the channel utilization and to minimize the average packet delay by these given parameters. We observe that (1) the difference in the channel utilization of the cases with different values D is smaller when is small, while the cases with smaller D offer a little higher channel utilization values. However, as increases to be larger than 0.01, all cases have a channel utilization gain larger and the difference also becomes larger. Moreover, when increases continuously to be larger than 0.035, the channel utilization with D = 3 sharply decreases to a lower level. On the other hand, the case with D = 10

has the largest value of the average channel utilization and in the case of D = 5, the maximum channel utilization decreases only slightly for large values of The case with D = 20 has a much lower channel utilization than other cases with D = 5,10 all the time; (2) the average packet delay is smaller when is small in all the cases, but the case with D = 3 becomes

very large when increases, while the average packet delay can be kept smaller in the cases D = 5, 10,20 when increases. The case of D = 10 has the smallest values of packet delay among all the cases. The above behaviors are also observed in Fig. 7.9, which shows the average packet delay versus the channel utilization for the same parameters as in Figs. 7.7 and 7.8. When becomes large, in the case of D = 3, for the same pulse transmission probability

the users do not have an enough

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collision avoidance period to verify the clear channel to transmit, so the loss of transmission chances or collisions on the channel are in practice greater than in the other cases. The effect of collision avoidance for D = 3 becomes small when is large. On the other hand, for packet generation rate the case D = 5 offered a higher throughput and lower packet delay than the case D = 10, but for the case D = 10 performs better. This is because for collision avoidance, it is not only necessary to send some pulse signals to let the users in the system know that someone wants to transmit a packet, but also necessary to listen to whether there are some pulse signals sent by other users on the channel when the user does not transmit pulse signals in the collision avoidance period. The smaller D is, the shorter is the time which can be used for collision avoidance to send or listen to pulse signals with each other before the packet transmission. For heavy traffic, a longer period is needed for collision avoidance. But if we increase D too much as in the case of D = 20, the system performance seriously degrades as it wastes the channel resource. These results might be useful in determining an appropriate length of collision avoidance period that maximizes the channel utilization and minimizes the packet delay. From all the above figures, we can conclude that the maximum channel utilization with minimum packet delay can be achieved on the balance of and D.

5.

Conclusion

In this chapter, we presented an exact analysis to numerically evaluate the performance of multiple random access method with CSMA/CA IFT and DFT protocols for high-speed and realizing fully distributed WLANs. The collision avoidance portion of CSMA/CA in this model was performed with a random pulse transmission procedure. In this procedure, a user with a packet ready to transmit initially sends some pulse signals with random intervals before transmitting the packet to verify a clear channel to transmit. The system model consists of a finite number of users to efficiently share a common channel. A discrete-time Markov chain was used to model the system operation. The number of slots in a frame can be arbitrary dependent on the chosen lengths of the collision avoidance period and packet transmission period.

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183

The influence of the possible length of the collision avoidance period and packet transmission period, and pulse transmission probabilities on the network performance was discussed for the IFT protocol, based on the results of the channel utilization and average packet delay for different packet generation rates. The results calculated for our examples show that the choice of collision avoidance period and pulse transmission probability is the most significant issue for better performance.

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185

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Chapter 8

OUTPUT PROCESS ANALYSIS OF WIRELESS CSMA/CA LANS WITH INTEGRATED VOICE AND DATA TRANSMISSION

1.

Introduction

In Chapter 7, we presented an exact analysis to numerically evaluate the system performance of high-speed and fully distributed WLANs with CSMA/CA for single traffic. In Chapter 7, the influence of the possible length of the collision avoidance period and packet transmission period, and pulse transmission probabilities on the network performance were discussed, based on the results of the channel utilization and average packet delay for different packet generation rates. In this chapter, we first present a new detailed system model and a new effective mathematical analysis to numerically evaluate the performance of integrated voice and data traffic in WLAN systems with both non-persistent CSMA/CA IFT and DFT protocols, where the non-persistent CSMA/CA IFT and DFT protocols are the same as in Chapter 7 by IEEE 802.11. The collision avoidance portion of CSMA/ CA is performed by a procedure that let some randomly chosen users talk and the others listen, so that the users who hear somebody talking will refrain from talking. Then we present an exact analysis for the protocol to derive the moment generating function of the packet interdeparture time for the output process. The results obtained in this chapter also include those in Chapter 7 for the single traffic models. In general it is considered that real-time traffic such as voice needs a higher transmission priority than data traffic. In this system, 187

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the larger pulse transmission probability a user employs, the higher priority this user will get. The performance analysis of the CSMA/CA protocol in WLAN systems supporting real-time and data traffic simultaneously with priority for realtime traffic communications presented in this chapter is quite different from those previously known schemes for single traffic systems with CSMA/CD [Klei75a], [Toba80a], [Toba82a], [Coyl83], [Taka85a], [Tasa86], [Mats90a], [Mats90b], [Mats90c] and [Onun91]. The difference is not only in the system model (protocol, traffic types), but also in the performance analytic method and numerical results. The WLANs support multimedia communication of real-time bursty traffic (such as voice or video), which is strict with regard to instantaneous delivery but relatively tolerant of bit errors, and data traffic, which is strict with regard to bit errors but relatively tolerant of instantaneous delivery. The system model consists of a finite number of users to efficiently share a common channel. Each user may generate voice and data traffic. A user with a packet of voice or data ready for transmission randomly senses the channel by pulse signals within a fixed interval (called the collision avoidance period) to verify a clear channel before transmitting. When users do not transmit pulse signals in the collision avoidance period, they can listen continuously to pulse signals transmitted by other users from the channel. The time axis is slotted, and a time frame has a large number of slots and includes two parts: the collision avoidance period and the packet transmission period. The packet transmission period contains both packet service time and propagation delay for each transmitted packet. The number of slots in a frame can be arbitrary and dependent on the chosen lengths of the collision avoidance period and packet transmission period. We define the packet interdeparture time as the interval between two consecutive completions of successful transmissions on the channel. A discrete-time Markov chain is used to model the system operation. This analysis yields a set of equations that can be solved for two important performance measurements of WLAN systems, the channel utilization and average packet delay for both voice traffic and data traffic. And the m.g.f. of the packet interdeparture time is also derived to give the first moment and higher moments. Using the first and second moments of the packet interdeparture time we can determine the parameters in the diffusion approximation to the input process of interconnected WLAN systems with integrated voice and data traffic.

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189

In general it is considered that the real-time traffic such as voice needs a higher transmission priority than data traffic. In this system, the larger the pulse transmission probability a user uses, the higher the priority this user will get. In the numerical results, (1) we discuss the optimal network design parameters such as the pulse signal transmission probability and ratio of the collision avoidance period and transmission period to maximize the channel utilization and minimize the average packet delay of the whole system. (2) We make some comparisons with other previously known schemes for data traffic only using the fixed pulse transmission probability and fixed collision avoidance period length. (3) We compare the average performance measures of CSMA/CA system with the IFT protocol and the DFT protocol. (4) We make comparisons between the single traffic and the multitraffic system models in term of the coefficient of variation of the packet interdeparture time by differentiating the moment generating functions. The chapter is organized as follows. In Section 2 we describe the integrated voice and data WLANs with the CSMA/CA protocol and offers assumptions of the traffic model. In Section 3 we first present performance analysis of the system to obtain the average performance measures of the

system such as the channel utilization and the packet delay for the IFT protocol and the DFT protocol. Then we derive the m.g.f. of the packet interdeparture time for the output process. In Section 4, we offer numerical results to make comparisons between the single traffic and the multi-traffic system models in terms of the average performance measures and coefficient of variation of the packet interdeparture time based on the analysis. Finally, Section 5 concludes the chapter.

2.

System Model

In this section, we present the detailed system model for the system analysis. In the analysis we make the following assumptions: (1) The time axis is slotted and the slot size is assumed to be fixed and equal to the end-to-end propagation delay denoted by . (2) There are a finite population of N mobile end users, which can generate voice and data packet to communicate with each other.

(3) A common channel is accessed by using the non-persistent CSMA/CA scheme. In this scheme, if the channel is sensed idle, send; if the channel is sensed busy, wait a random time and try again.

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(4) Every user has his own buffer which can store at most one packet at any time. Once a packet is accommodated at a buffer, it remains there until it is successfully transmitted. (5) A user with a packet ready for transmission randomly sends and receives pulse signals within the collision avoidance period to verify a clear channel before transmitting. The length of collision avoidance period is fixed to be D slots. (6) The transmission time of a packet is denoted by T (slots). T is considered to contain both packet service time and propagation delay for each transmitted packet. (7) A frame consists of D + T slots that can be considered as system parameters. (8) If more than one packet is simultaneously transmitted at the same time, a collision occurs. Users can know the result of transmission (success or collision) by the end of the frame. (9) Each user is in either of two states: the idle state if he does not have a packet in his own buffer to transmit; or the backlogged state if he has a packet waiting or undergoing transmission. A user in the idle state generates a voice packet with probability or generates a data packet with probability or remains in the idle state with probability at the beginning of a frame. Users with an outstanding voice or data packet can be in either (1) active mode (denoted by a) or (2) backoff mode (denoted by b) during the collision avoidance period of D slots. In the active mode, users are going to start transmission at the beginning of the transmission period T, while in the backoff mode users have already given up transmission in that frame after noticing that there exits another user who is trying to send a packet in the frame. At the beginning of a frame, all users who have generated a new voice or data packet enter the active mode with probability one in the immediatefirst-transmission (IFT) protocol, while they enter the backlogged state once in the delayed-first-transmission (DFT) protocol. Any user having a voice packet or a data packet in the backlogged state enters the active mode with probability or respectively. We note that users can enter the active

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mode only at the beginning of a frame, and users in the active mode may transit to the backoff mode at the end of a slot where they listen to pulse signals from other users. Moreover, the operation of users in the active mode can be described according to the following algorithm: (1) Each user who has a voice packet or a data packet in the active mode decides whether he sends some pulse signals to the channel with probability or or listens to the channel with probability at the beginning of a slot. (2) Those users who hear the channel busy transit to the backoff mode at the end of the slot. Otherwise, they stay in the active mode. (3) When the collision avoidance period ends, all users who are still in the active mode start transmission. If there is more than one user in the active mode, a collision happens and they transit to the backlogged state at the end of the frame. On the other hand, if there is only one user, he succeeds in transmission and becomes empty at the end of the frame. (4) All users in the backoff mode transit to the backlogged state at the end of the frame. We call users having voice or data packets to transmit in

the backlogged state backlogged voice users or backlogged data users.

3.

Performance Analysis

In this section, we present a new effective analytical framework based on the discrete Markov chain model to evaluate the performance of the system. It is noted that the analytical equations in this section are different from those in previous papers [Klei75a], [Toba80a], [Toba82a], [Coyl83], [Taka85a], [Tasa86], [Mats90a], [Mats90b], [Mats90c] and [Onun91], because the multi-traffic systems with a priority to the real-time traffic for CSMA/CA were not analyzed in those papers. The previous analytical methods cannot simply be applied to the present model. Let x(t) represent the numbers of users backlogged of both voice and data traffic at the end of frame t. By defining the system state as the numbers of backlogged voice and data users at the end of time frame t, it is easy to show that the process forms a finite state and discretetime Markov chain. The system state is said to be in state if and it is denoted by where and

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represent the numbers of backlogged voice and data users, respectively. We assume that a steady-state of the system exists. The imbedded Markov points and channel state transition for both IFT and DFT protocols are shown in Fig. 8.1, where denotes the quadruplet of the numbers of users having a voice packet in the backoff mode and active mode, and the numbers of users having a data packet in the backoff mode and active mode, respectively, at the end of the current slot. We also denote by the same quadruplet at the end of the next slot. also denotes the same quadruplet at the beginning of the packet transmission period, respectively. Let H1 and H2 represent the total number of the system states at the end of an arbitrary time frame and the number of the system states at the end of an arbitrary slot during the collision avoidance period, respectively. We note that H1 and H2 can be given by

From Fig. 8.1, we note that the state transitions in any frame can be divided into three parts:

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193

(1) State transition in the first slot (called the first state transition). (2) State transition in a slot between the second slot and the end of the collision avoidance period (called the second state transition). (3) State transition from the end of the collision avoidance period until the end of the packet transmission period (called the third state transition).

3.1

Analysis for Second State Transition

First, we derive the second state transition. In Fig. 8.2, we show the detailed transition diagram from

in any slot between the second slot and the end of the collision avoidance period D. We can define a conditional transition probability for a slot after the first slot in the collision avoidance period as follows:

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where is Dirac’s delta, that it equals 1 if k = l, and 0 otherwise. We note that if The conditional transition probability is the probability that given the numbers of users having a voice packet in the backoff mode and active mode, and the numbers of users having a data packet in the backoff mode and active mode at the end of an arbitrary slot between the second slot and the end of the collision avoidance period D are respectively, the numbers of users having a voice packet in the backoff mode and active mode, and the numbers of users having a data packet in the backoff mode and active mode at the end of the next slot of this period transit to respectively. The following probability of (8.3):

is the probability that and users out of and users in the active mode transmit some pulse signals, so that and users receive pulse signals and they enter the backoff mode. As special cases, if all users sends or listens, whose probabilities are given by or respectively, no change of system states occurs, which results in We denote an matrix whose elements consist of To calculate the transition probability we define a function f(N : a, b, c, d) = to map the four-dimensional state space into the one-dimensional state space H2 as follows. For proof, see Appendix B.

Output Process Analysis of Wireless CSMA/CA LANs with Voice/Data

3.2

195

Analysis of IFT for First State Transition

The analyses of the IFT protocol and the DFT protocol are different for the first state transition. In this subsection, we first analyze the first state transition for the IFT protocol.

The first state transition for both voice users and data users in the first slot for the IFT protocol is shown in Fig. 8.3. The big arrow on the right in Fig. 8.3 denotes that from the second slot of the frame to the end of the collision avoidance period the users follow the second state transition explained in Fig. 8.2. The two additional events occurred at the beginning of the first slot for the IFT protocol are (1)

users out of idle users generate new voice packets with voice packet generation rate and users out of idle users generate new data packets with data packet generation rate and users remain in the idle state with probability respectively, where and denote the numbers of backoff users and active users just after the beginning of the first slot, respectively. This probability is given by

196

(2)

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

voice (data) users out of (data) users join the active mode with probability ers transit to the backoff mode with probability probability is given by

(

backlogged voice ) while the oth( ). This

It is noted that from the second slot of the frame to the end of the collision avoidance period D the users follow the algorithm explained in Section 2. Because the remaining state transition between the second slot and the end of the collision avoidance period D is the same as in Fig. 8.2, once is determined. Let denote the transition probability such that the system state transits from to where and ] are the states at the beginning and at the end of the first slot, respectively. We can obtain for the protocol by using defined in (8.3) as

We denote an

3.3

matrix whose elements consist of

by

Analysis of DFT for First State Transition

In this subsection, we analyze the first state transition for the DFT protocol. The first state transition for the DFT protocol is shown in Fig. 8.4.

The big arrow on the right in Fig. 8.4 denotes that from the second slot of

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197

the frame to the end of the collision avoidance period the users follow the

second state transition explained in Fig. 8.2. Two additional events for the DFT protocol in the first slot are (1)

users out of idle users generate new voice packets with voice packet generation rate and users out of idle users generate new data packets with data packet generation rate and users remain in the idle state with probability respectively where and denote the numbers of backoff users and active users just after the beginning of the first slot. This probability is given by

(2)

voice users out of backlogged voice users join the active mode with probability while the others transit to the backoff mode with probability And data users out of backlogged data

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users join the active mode with probability µ d while the others transit to the backoff mode with probability This probability is given by

Therefore we can obtain fined in (8.3) as follows:

We denote an

3.4

for the DFT protocol by using

matrix whose elements consist of

de-

by

Analysis for Third State Transition

We further proceed to the analysis of the third state transition for both IFT protocol and DFT protocol for the packet transmission period T. During which only one of four cases (successful transmission of a voice packet, successful transmission of a data packet, no transmission, and unsuccessful transmission) can occur. The transition probabilities for these four cases are denoted by as follows, respectively, where and are the state at the end of the collision avoidance period D and the state at the end of the packet transmission period T.

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Because these four cases are determined solely by the number of active users at the end of the collision avoidance period (or equivalently at the beginning of the transmission period), we obtain the following: (1) For the successful cases of voice traffic or data traffic:

(2) For the idle case:

(3) For the collision case:

We denote by of

3.5

,

,

and and

matrices with

elements consisting

Stationary Probability Distribution

Finally, we consider the transition of the system state in equilibrium such that there are backlogged users at the end of the tth frame, and backlogged users at the end of the st frame where Let be the one-frame state transition probability matrix between consecutive imbedded points having elements as

is given by

where D is the number of slots in the collision avoidance period.

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Let denote an -dimensional row vector of the stationary probability distribution. can be determined by solving as or

3.6

Average Performance Measures

Important performance measures used to evaluate the system are the throughput, channel utilization and average packet delay. By using the stationary probability distribution obtained in the previous subsection (see (8.12)), we can calculate these performance measures as follows. Let and denote the throughputs for voice traffic and data traffic per frame, respectively. We have

and

where e is a column vector with elements, all of which equal 1 and is given by (8.5) for the IFT protocol or by (8.6) for the DFT protocol. is given by (8.3) for both IFT and DFT protocols. Then we obtain the total throughput of the system by Let and denote the average channel utilizations for voice traffic and data traffic, per frame, respectively. Using (8.13) and (8.14), we obtain the channel utilizations as follows:

and

Similarly, we have the total channel utilization of the system by

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201

We now consider the average number of backlogged users at the end of any frame, and let and represent the number of backlogged voice users and data users, respectively, which are given by

and

where Finally, let us define the average packet delay as the number of slots elapsed from the generation of a packet to the end of successful transmission. Let and denote the average packet delay for voice traffic and data traffic, respectively. Using and we obtain the average packet delay and (slots), and the total delay of the system (slots) as follows:

3.7

Packet Interdeparture Time Distribution

Now we derive the moment generating function of the probability distribution of packet interdeparture time and give the performance measures to numerically evaluate the system. We define the packet interdeparture time as the time interval between two consecutive successful transmission frames of packets. In this subsection, we concentrate on only the data traffic in the analysis. The analytical method is the same for the voice traffic since the system is symmetrical for voice and data. Let denote the random variable for the data packet interdeparture time, and let denote the m.g.f. of the data packet interdeparture time, we have

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where

is an

-dimensional row vector of the conditional stationary

probability distribution of backlogged data packets at the end of the frame with successful data transmission. is given by

where is an identity matrix. and in (8.21) are matrices of the conditional probabilities for the successful transmissions of data packet and for the other cases, respectively, with elements. and can be calculated by

where of (8.20) is an

and are given in previous subsections. -dimensional column vector. It is given by

where e is a column vector with elements, all of which equal 1. The first term of (8.24) is derived from the following observation: on the condition

that

and

a data packet is successfully transmitted

in this frame and therefore the packet interdeparture time is just one frame. The probability for this successful transmission is given by . The second term of (8.24) expresses that when there is no successful transmission of a data packet in a frame, the distribution of the time interval from the end of the frame to the end of the next successful frame has a moment generating function given by The probability for this case is given by . If we count the packet interdeparture time by slot, we get We can give the exact higher moments of the data packet interdeparture time by differentiating (8.20). Expressions for all moments of can be obtained in the following manner. Let represent the nth factorial moment of the data packet interdeparture time distribution. Then we can obtain as follows:

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where

All moments of the m.g.f. have closed-form expressions. We note that the throughput given by (8.14) for data traffic is equal to the reciprocal of the average data interdeparture time as , where

Using the second moment

that is given by the following:

we can also evaluate, in particular, the coefficient data packet interdeparture time.

4.

of variation of the

Numerical Results

In this section, we first give numerical results in terms of the channel utilization and average packet delay for different total offered traffic to evaluate the network performance. Then we discuss the optimal network design parameters such as the pulse signal transmission probability and ratio of the collision avoidance period D and the transmission period T to maximize the channel utilization and minimize the average packet delay for the IFT protocol. In these numerical examples, we also make comparisons with other previously known schemes for data traffic only using the fixed pulse transmission probability and the fixed length of collision avoidance period. Then we give numerical results in terms of the channel utilization and average packet delay for a variety of offered traffic to compare the average

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performance measures of the integrated voice and data traffic network with the CSMA/CA DFT protocol and the CSMA/CA IFT protocol. We consider all numerical examples to have the total number of users and duration of the packet transmission period with given retransmission probability and pulse transmission probability for data traffic The generation rate of new voice packets equals the generation rate of new data packets, varying from 0.0 to 0.5 in a frame, and we define the total offered rate to be to observe how the system performance can be improved by the pulse signal transmission probability for voice traffic and the collision avoidance period D under the condition of the same offered load. Referring to [Good89], we consider that the mean talkspurt duration is 1.0 s and the mean silent gap duration is 1.35 s. In the numerical results, we consider that 1 frame equals 1.0 ms, including a maximum time length of collision avoidance period of 0.2 ms and a packet transmission time of 0.8 ms, so that the time length of a slot is where 50 (slots) is the length of the packet transmission period T. The unit time of the vertical axis (showing the packet delay) in all figures of this chapter is slot. In general it is considered that real-time traffic such as voice needs a higher priority to transmit than data traffic. In this system, the larger a user employs, the higher priority this user will get. In Fig. 8.5, we show how the average packet delays and for voice and data traffic change with different values of and a fixed length of collision avoidance period for the offered traffic load . From Fig. 8.5, it should be noted here that in the case of small pulse transmission probability for voice traffic such as because no priority or lower priority is given to voice traffic, the actual voice packet transmission is suppressed at the start, which causes a heavy increase of packet delay when the offered load becomes large. We also observe that when is increased to 0.2, the case has lower voice packet delay, and the voice packet delay is the lowest for This is because for heavy offered traffic load, a longer collision avoidance period is necessary. As a result, with large the average voice packet delays decrease considerably with a small increase of the average data packet delays, and the effect of pulse transmission probability is more significant at larger traffic loads. In Fig. 8.6, we show the curves of the total average packet delay E[D] versus the total channel utilization U with the same parameters as in Fig. 8.5

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with an added case of When U becomes large, in the case of the users who have voice packets do not have larger pulse transmission probability, so that the loss of transmission chances or collisions on the channel are in practice greater than in the other cases, and hence the effect of collision avoidance for becomes small. On the other hand, for all channel utilization U, the case of shows lower packet delay than the other cases. This is because for collision avoidance, it is necessary to have a larger pulse transmission probability to let other users in the system know as early as possible that someone wants to transmit a voice packet. When is increased to 0.5, we can observe a further decrease in the total average packet delay for all U, but not so much. This is because for collision avoidance, it is necessary not only to send some pulse signals to let other users in the system know that someone wants to transmit a packet, but also to listen to whether there are any pulse signals sent by other users on the channel. If we raise too much (for example, in the case of it means that an active voice user transmits a pulse signal every two slots during a fixed collision avoidance period of 5 slots), the pulse signal transmissions by voice users become excessive for the channel and

the users who have the data packets lose the chances to transmit their data packets, which causes a decrease in the channel utilization and increase of the packet delay for data traffic even if higher priority is given to the users who have voice packets. As a result, the performance improvement of the whole system becomes very small. For a short collision avoidance period in Fig. 8.6, there is only a very small effect on the improvement of the system performance for larger For the improvement of the system performance, it is necessary to consider not only the effect of priority to increase the channel utilization and to decrease the packet delay for voice traffic, but also the performance for data traffic together. Notice that as we fixed the pulse transmission probability for data traffic to be these numerical examples with the pulse transmission probability for voice traffic plotted in Figs. 8.5 and 8.6 are the same as those for one data traffic only, which have been obtained in previous papers. In the system with it will result in the data traffic’s monopolizing the resource with voice traffic together, and users with voice packets cannot be supported by the system to send their voice packets more quickly.

It is not desirable in a multi-traffic system supporting real-time traffic. This result is important because if an appropriate priority value of is given to the voice traffic, the channel utilization increases considerably with a small

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increase in packet delays. has a significant impact on the network performance. The scheme presented in this chapter can be used to transmit high priority real-time applications such as voice and video traffic. On the other hand, as we observed from Figs. 8.5 and 8.6, the voice packet delay is relatively large – typically more than 50 slots (the length of transmission period T) even in the case of the smallest value of the total offered rate These are due to the fact that in this system, the average packet delay is defined as the number of slots elapsed from the generation of a packet to the end of successful transmission, so that the time of packet delay includes both waiting time and packet transmission time. Regardless of the voice traffic, when is vary small, the successful possibility of packet transmission is very high. Once a packet is successfully transmitted, the packet delay will be at least the time length of a frame that equals D + T (a time length of collision avoidance period and a packet transmission time), that is longer than 50 slots. Notice that the collision avoidance period D is a system parameter and it is variable in different numerical examples. Figs. 8.7 and 8.8 show the average voice packet delay and the average data packet delay respectively, as functions of the offered traffic load with the collision avoidance period for the pulse transmission probabilities and From Figs. 8.7 and 8.8, we can discuss the values of D to minimize the average packet delay for these given parameters. We observe that (1) the difference among

the data packet delays and the voice packet delays with different values of D is smaller when is small. But, as increases, all the voice packet delays have lower levels than the data packet delays. This is because of the fact that in the system having a larger pulse transmission probability as with large there are more voice users sending pulse signals in a fixed collision avoidance period prior to data users, which results in the voice traffic’s monopolizing the resource while most data users remain in the backlogged state; (2) the average packet delay is smaller when is small in all the cases, but the cases with for both voice traffic and data traffic have almost the largest values of the packet delay, while the average packet delay can be kept the smallest in the cases and with all values of among all the cases for the voice traffic and the data traffic, respectively. Fig. 8.9 shows the total average packet delay E[D] versus the total channel utilization U for the same parameters as in Figs. 8.7 and 8.8. We observe that, when becomes large, in the case of the users do not

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have a long enough collision avoidance period to verify whether the channel is clear or not, so that the loss of transmission chances or collisions on the channel are in practice greater than in the other cases. The effect of collision avoidance for becomes small when is large. On the other hand, for all the offered traffic load the case shows a higher channel utilization and lower packet delay than the other cases. The smaller D is, the shorter is the time which can be used for collision avoidance to send or listen to pulse signals before packet transmission. For heavy traffic, a longer period is needed for collision avoidance, as in the case of But if we increase D too much, as in the case of the system performance seriously degrades, since it wastes the channel resource. These results might be useful in determining an appropriate length of collision avoidance period that maximizes the channel utilization and minimizes the packet delay. From all the above figures, we can conclude that the maximum channel utilization with minimum packet delay can be achieved on the balance of and D. In Fig. 8.10, we compare packet delays and with different values of and a fixed versus for the IFT protocol system and the DFT protocol system. We note that the results of in Figs. 8.10 and 8.11 are those of other previously known schemes for data traffic only. From Fig. 8.10, it is observed that for low offered rate , the IFT protocol offers lower packet delay than the DFT protocol, but for large , the DFT protocol performs better than the IFT protocol with for data traffic. It is considered that the rate of the users’ channel access in the IFT protocol case is in practice greater than in the DFT protocol case. When increases over some values, in the IFT protocol case, then more packet collisions occur than in the DFT protocol case, so the data packet delay of the system with the IFT protocol increases heavily, while the system with the DFT protocol has a lower data packet delay. On the other hand, the IFT protocol still offers lower voice packet delay for all offered This is due to the fact that in the case of the system offering larger namely higher priority to voice traffic, a user who generated a new voice immediately transmits pulse signals with probability one without delay to try to insure the transmission chances, so that the IFT protocol can offer shorter voice packet delay.

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In Fig. 8.11, we show how the total packet delay E[D] changes versus total channel utilization U with different values of and 0.4 with a fixed for in both IFT protocol system and DFT protocol system. From Fig. 8.11, it is observed that when is small, the IFT protocol offers higher channel utilization and lower average packet delay than the DFT protocol, but for large , the DFT protocol performs better for the system without voice traffic priority while in the systems having larger (higher priority given to voice traffic), better performance can be obtained by the system with the IFT protocol. The reason for this is the same as that shown in Fig. 8.10, namely, that in the DFT protocol system, upon a new voice or data packet arrival, the user joins the backlogged state at once, then transmits his pulse signals after frames with a geometrically distributed backoff. However, we should note that a packet may be successfully transmitted in the same frame that the packet arrived. Figs. 8.12 and 8.13 show the voice packet delay

and the data

packet delay as functions of with and for and to compare the system performance of the IFT protocol system and the DFT protocol system, respectively. The smaller D is, the shorter the time which can be used for collision avoidance.

From Figs. 8.12 and 8.13 we can see that the average voice packet delays with the DFT protocol are still larger than those with the IFT protocol for all values of and offered rate . However, when increases, the average data packet delay with the DFT protocol becomes small, specially in the case of smaller value of D. This is because of the fact that in the DFT system, when offered rate is low, both voice packet and data packet collisions are fewer, and packet delay is mainly dependent upon the time elapsed from the generation of a packet to getting a chance to transmit

the packet. In this case, the system with the IFT protocol is used properly. However with large more users with packets send the pulse signals in a fixed collision avoidance period, and the rate of the users’ channel access in the DFT protocol case is in practice lower than in the IFT protocol case, which results in the average data packet delay having lower values than the system with the IFT protocol.

Because the value of

is close to the value of

for total

average packet delay versus channel utilization U, in Fig. 8.14 we show

total average packet delay E[D] versus the total channel U for and only. We also notice that the maximum channel utilization of the

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system with the IFT protocol is a bit higher than the system with the DFT protocol. The throughput for the voice or data traffic as a function of is plotted in Fig. 8.15 including four curves with and a fixed length of collision avoidance period We note that as we fixed the pulse transmission probability for data traffic to be the numerical example with corresponds to the case with the single traffic, and since the generation rates of new voice packets and new data packets are equal, the throughputs for voice traffic or data traffic are the same. In Fig. 8.15, we observe that as the pulse transmission probability for voice traffic increases while the pulse transmission probability for data traffic is fixed, the throughput of the system can be significantly improved with . However, because the system consists of a finite population of N users, each one can generate both voice and data packets, the system with larger has more chances available to voice packets. When increases, the relative ratio of users who have data packets in the active mode decreases, and the throughput of the system increases by the majority of voice traffic. In Fig. 8.16, we examine the effects of the pulse transmission probability for voice traffic on the coefficient and of variation of the packet interdeparture time for voice and data traffic, respectively, with the same parameters as in Fig. 8.15. We observe that all coefficients of variation of the packet interdeparture time begin to decrease from 1.0 and reach minimum values depending on . Basically there is not so much difference in the coefficient of variation for different pulse transmission probabilities under a light offered rate. However, as the offered rate further increases, the smaller the pulse transmission probability is, the larger the coefficients and of variation of the packet interdeparture time are. On the other hand, we also observe that the coefficient of variation of the voice packet interdeparture time for the case decreases significantly than as increases while for the other cases of is larger than Fig. 8.17 shows the coefficient for the collision avoidance period as functions of with and . We observe that the difference in between different values of D is smaller when

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is small. However, as increases continuously to be larger than 5, decreases to lower levels in all cases. This is because of the fact that in the system with a larger value of such as more and more voice users send pulse signals in a fixed collision avoidance period D under heavy offered rate, which results in the voice traffic’s monopoly of the resource while most data users remain in the backlogged state. Therefore, when the offered rate is heavy, the larger the collision avoidance period D is, the smaller the variation of the packet interdeparture time is.

5.

Conclusion

In this chapter, we presented a new detailed system model and a new effective analysis for the performance of multiple random access method with CSMA/CA for integrated voice and data WLANs. In the model, a time frame has an arbitrarily large number of slots dependent on the chosen

durations of the collision avoidance period and packet transmission period. A user with a packet ready to transmit initially sends some pulse signals with random intervals within the collision avoidance period to verify a clear channel before transmitting the packet. The system model consists of a finite number of users to efficiently share a common channel. Each user can be a source of both voice traffic and data traffic. We presented a discrete-time Markov chain to model the system operation to yield a set of equations that can be solved for two important performance measurements of the wireless LAN systems, the channel utilization and average packet delay for both voice traffic and data traffic. We also derived the m.g.f. of the packet interdeparture time to give the first moment and higher moments. In the following, our main results are summarized.

(1) We gave the system performance in terms of the channel utilization and average packet delay for different voice and data packet generation rate to discuss the optimal network design parameters such as the pulse signal transmission probability and ratio of the collision avoidance period and transmission period to maximize the channel utilization and minimize the average packet delay of the whole system. (2) We made some comparisons with other previously known schemes for data traffic only using the fixed pulse transmission probability and fixed collision avoidance period length.

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(3) We numerically compared the average performance measures of the systems with the CSMA/CA IFT protocol and the CSMA/CA DFT protocol and showed that for voice traffic, network performance can be improved by the IFT protocol, but for higher traffic, network performance for data traffic can be improved by the DFT protocol. (4) We made comparisons between the single traffic case) and the multi-traffic system models in terms of the coefficient of variation of the packet interdeparture time by differentiating the moment generating functions. Using the first and second moments of the packet interdeparture time we can determine the parameters in the diffusion approximation to the input process of interconnected WLAN systems with integrated voice and data traffic.

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III MULTICHANNEL AND MULTI-TRAFFIC IN WIRELESS COMMUNICATION NETWORKS WITH CDMA PROTOCOL AND HYBRID CHANNEL ASSIGNMENT SCHEME

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Chapter 9

OUTPUT AND DELAY PROCESS ANALYSIS OF SLOTTED CDMA COMMUNICATION NETWORKS WITH INTEGRATED VOICE AND DATA TRANSMISSION 1.

Introduction

Recently, multiple access techniques based on code division multiple access (CDMA) with other protocols and techniques have been positively applied for use in mobile radio and wireless personal communications because the following: (1) CDMA can achieve synchronous communications. (2) The frequency band for each user does not need partitioning. (3) Simultaneous transmissions of more than one user can be captured successfully.

(4) The probability of transmission success increases with the number of simultaneously transmitting users. In [Gilh91] and [Vite93], particularly for terrestrial cellular telephony, it is shown that the interference suppression feature of CDMA can result in a many-fold increase in capacity over analog and even over competing digital techniques. On the other hand, the multiple random access packet switching has already proven to be a very useful technique in environments with high peakto-average traffic ratios. It offers the possibility to efficiently allocate scarce radio communication channels in such a way that users with bursty traffic 221

222 PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

can share a common frequency channel without significant degradation to their throughput. A number of papers have recently appeared in the literature dealing with the analysis of CDMA systems for packet radio environment [Rayc81] and [Abde89]. CDMA with any other contention multiple access protocols (e.g., ALOHA, Slotted ALOHA) has been applied to improve system performance [Yin90], [Liu95] and [Hsu97]. Most existing work on CDMA packet system has been developed for the slotted multiple random access system [Liu95] and [Hsu97]. CDMA with the slotted ALOHA random access protocol has the promise of integrating the properties of both CDMA (i.e., statistical multiplexing) and slotted ALOHA (i.e., random access) to achieve higher spectrum utilization for diverse services. Its multiuser capability enables more than one packet radio user to successfully receive packets from different transmitting users even though the transmissions overlap in time and each of the transmitting radios is within range of each of the receivers. In a CDMA system with slotted ALOHA, the time axis is divided into slots of duration equal to the sum of a packet interval and a guard time. All users are synchronized at the beginning of the time slot for their transmissions. In a CDMA packet radio system, the allocated resource is energy, which must be shared among all users in the system. The system must be able to provide diverse quality of service to each type of users, such as voice, data messages, video, etc. Voice and data users have different traffic characteristics and service requirements. Voice traffic has to be delivered in real time with a negligible and almost constant delay, while data traffic can be enqueued but requires much smaller bit error rates. CDMA systems can offer the flexibility needed to accommodate both voice and data traffic. The nature of human conversation typically yields a natural voice activity period of approximately 0.4 [Srir86]. On average, a voice user is speaking 40% of the time and the remaining time is spent pausing or listening. Therefore, in the design of a CDMA system, it would make sense to use these idle periods in voice activity for queueable data to increase overall capacity of the systems. Some work on integrating voice and data traffic in CDMA systems has been done in [Wils93], [Guo94], [Yang94] and [Capo95]. In [Wils93], simulation models are applied to a comparative performance evaluation of dynamic TDMA and spread-spectrum packet CDMA approaches to multiple access in integrated voice and data personal communication networks en-

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vironment. In [Yang94], optimal admission policies for integrated voice and data traffic in packet radio networks employing CDMA with directsequence spread-spectrum signaling are derived. However, in [Yang94], the CDMA system cannot detect the difference between silent and talk spurt periods of voice calls and thus cannot use the silent periods to transmit data messages or other voice calls. In [Guo94], the voice activity detection is assumed, but a rate reduction policy for data users is used to maintain the quality of service for the voice users. In [Capo95], the effect of data traffic integrating into a CDMA cellular voice system is analyzed. The CDMA system under consideration is a power-controlled cellular architecture in which blocking occurs when the total interference level exceeds the background noise level. Another protocol proposed in [Capo95] admits data traffic into the CDMA cellular system based on the current aggregate voice interference level, and allows for the efficient integration of voice and data without degrading the quality of service for the delay-critical voice traffic. In all the above studies, however, the analyses are based on the simplified models used for the aggregate voice activity. The system models describe only the ON/OFF (call initiation/call termination) behavior of voice traffic whose durations are exponentially distributed or binomially distributed. Other simplified points are in that the number of voice users in the system is fixed and independent of data user population, and the behavior of the silent and talkspurt periods of the voice traffic is not modeled. Also, integrated properties of both CDMA and multiple random access to achieve higher performance utilization for diverse services have not been considered in these analyses. Since the output stream from a network often forms the input stream to an adjacent network, the analysis of the output process and the packet delay distribution is important to predict the performance in relation to the design and development of systems such as the multi-hop integrated voice and data networks or interconnected integrated voice and data networks with local area networks. In particular, to approximate the probability distribution of the interarrival time of internetwork packets at the adjacent network, the higher moments of the packet departure distribution and data packet delay are also required. In order to gain maximum efficiency of system resources and to provide sufficient quality of service to both voice and data users, it is necessary to exploit the statistical characteristics (e.g., voice activity factor) and user requirements (e.g., spatial isolation factor) for the voice and data sources in the network.

224 PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

This chapter presents an exact analysis to numerically evaluate the performance of integrated voice and data slotted CDMA network systems with the multiple random access protocol for packet radio communications. The difficulty in the analysis is mainly due to the fact that the proposed access protocol introduces overlap of packet transmissions in time and interdependency of code utilization. The system model consists of a finite number of users and each user can be a source of both voice traffic and data traffic. The allocation of codes to voice calls is given priority over that to data packets while an admission control, which restricts the maximum number of codes available to voice sources, is considered for voice traffic not to monopolize the resource. Such codes allocated exclusively to voice calls are called voice codes. In addition, the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use the voice codes for transmission if the voice sources are silent. A discretetime Markov chain is used to model the system operation. The output and delay process analysis is presented to evaluate the performance of the systems. In numerical results, we present average performance measures such as the throughput, the average blocking probability of voice calls, and the average delay of data packets, and we demonstrate the correlation coefficient of the voice and data packet departures and the coefficient of variation of the data packet delay for some cases with different numbers of voice codes. The chapter is organized as follows. In Section 2, we introduce the system model. In Section 3, we first derive the stationary distribution of the system. Then we present the exact analysis of the probability distributions of the voice and data packet departures and the data packet delay, and describe the average performance measures and higher moments. Numerical results are shown in Section 4, and concluding remarks are presented in Section 5.

2.

System Model

The slotted CDMA with multiple random access protocols achieves its multiple access property by a random division of the transmissions of different users in codes. These codes accessed are used to transform a users’ signal in a wideband signal (spread-spectrum signal). If a receiver receives multiple wideband signals, it will use the code assigned to a particular user

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to transform the wideband signal received from that user back to the original signal. During this process, the desired signal power is compressed into the original signal bandwidth, while the wideband signals of the other users remain as wideband signals and appear as noise when compared to the desired signal. The structure of CDMA considered in this chapter is such that the entire bandwidth-time space is used simultaneously by all active users: each user employs a different hopping pattern. In this system, interference occurs when two different users land on the same hop frequency. If the codes are synchronized and the hopping patterns are selected in such a way that two users never hop to the same frequency at the same time, the multipleuser interference is eliminated. However, the maximum number of codes available at the same time will be limited due to the limitation of the overall capacity of the system. In this system model, the time axis is slotted into segments of equal length seconds corresponding to the transmission time of a packet. All users are synchronized and all packet transmissions making use of codes are started only at the beginning of a time slot. We assume that the system supports a finite population of N users and they make use of a set of codes M with unique spread-spectrum code sequence, where in general, . It means that the maximum number of the simultaneously available codes is M, and if the number of users transmitting simultaneously is larger than M codes at the same time slot, the base station will still be able to capture M packets successfully. In case there is no code available in a time slot, the packet is assumed to be cleared from the system without retransmission if it is a voice packet, while the packet is assumed to be blocked and retransmitted in the next time slot if it is a data packet. It is assumed that the user will know about his transmission success or failure immediately after the transmission. In general, traffic is classified into two types: real-time bursty traffic (such as voice or video), which is strict with regard to instantaneous delivery but relatively tolerant of bit errors, and data traffic, which is strict with regard to bit errors but relatively tolerant of instantaneous delivery. Our model allows each user to be a source of both voice traffic and data traffic. The system model is assumed to have four operational modes for each of N identical, independently operating users: idle mode (denoted by I), talkspurt mode (denoted by T), silent mode (denoted by S) and data trans-

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mission mode (denoted by D). The talkspurt and silent modes are called the speech mode. A user can only be in one mode at a slot. The system state transition diagram for this model is illustrated in Fig. 9.1.

At the beginning of a slot, each user in the idle mode enters the talkspurt mode by generating a voice call with probability , or enters the data transmission mode by generating a data packet with probability or remains in the idle mode with probability A user with a voice call can be in a talkspurt mode when the user is speaking or in a silent mode when the user just turns off for speech. A user in the talkspurt mode transmits a voice packet in the slot when the user enters the talkspurt mode with probability one (IFT) and in the succeeding slots with probability (typically is set equal to one for talkspurt) until the user either quits the talkspurt mode to return to the idle mode or intermits speech with the silent mode. A user in the silent mode transmits a voice packet in the succeeding slots with probability (typically is set equal to zero for silence) until the user transits to the talkspurt mode. The probabilities that a user in the talkspurt mode returns to the idle mode, that a user in the talkspurt mode transits to the silent mode, and that a user in the silent mode transits to the talkspurt mode are given by and respectively, at any slot. We note that the mean length of a user being in the speech mode is given by slots.

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A user in the data transmission mode transmits a data packet in the slot when the user enters the data transmission mode with probability one. If the data packet is transmitted successfully, the user returns to the idle mode at the end of the slot, while if it is unsuccessful, the user stays in the data transmission mode and the packet enters a first-in first-out queue for retransmission. Therefore, there is no loss of data packets. The allocation of codes to voice calls is given priority over that to data packets. Especially once a user with a new voice call seizes a free code, the user can keep using the code until the end of speech, so that the loss of voice packets (i.e., blocking of voice calls) happens only at the arrival slot. Users whose new voice calls are blocked at the arrival slot are assumed to return to the idle mode at the end of the slot. We apply an admission control for the voice traffic so as not to monopolize the resource. Namely, the admission control restricts the maximum number of voice codes up to where We note that if , there is no admission control for the voice traffic. Since a user with a voice call intermittently turns ON/OFF, it is possible for users with a data packet to find a free code for transmission in some slots where a user with a voice call just turns off for his speech without affecting the quality of the voice users’ service. We assume the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use the voice codes for transmission if the voice sources are silent. For simplicity in analysis, possible imperfections due to the estimation of these periods and switching within these periods are not considered.

3. 3.1

Performance Analysis Stationary Probability Distribution

To analyze the output process and delay process of the system, we first have to obtain the one-step state transition probability and the stationary distribution of the system. The state of the entire system, consisting of N users, M codes and voice codes, can be characterized by a triplet of the numbers of users that are in talkspurt mode, silent mode and the data transmission mode, respectively. We model the system as a finite state and discrete-time Markov chain where the imbedded Markov points are chosen as the end of slots. We define the system state at the imbedded Markov point and denote it by

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0,1,2,...}. Let denote a system state so that there are and users in the talkspurt, silent and data transmission modes at the end of slot t, respectively, where The total number H of the system states is given by

Now let us consider the transition of the system state in equilibrium. Given that there are and users in the talkspurt, silent and data transmission modes, respectively, at the end of the tth slot, the following four events must occur to be and users in the talkspurt, silent and data transmission modes, respectively, at the end of the st slot, where

(1) At the beginning of the slot, users in the idle mode enter the talkspurt (data transmission) mode by generating new voice calls (new data packets), respectively, where and are integer random variables in the range of . This probability is given by

(2) At the beginning of the slot, users out of users in the talkspurt (silent) mode, respectively, transmit voice packets, where and are integer random variables in the range of This probability is given by

(3) Just before the end of the slot, min users and l users in the talkspurt mode transit to the idle mode and to the silent mode, respectively, and users in the silent mode transit to the talkspurt mode, so that there would be users in the talkspurt mode and users in the silent mode at the end of the slot, where l is an integer random variable in the range of max min This probability is given by

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(4) As the number of data packets successfully transmitted in the slot becomes must be equal to We define a conditional transition probability

as follows:

users in the idle mode

generate new voice calls and data packets, users in the talkspurt and silent modes transmit their voice packets at the beginning of st slot where

and

We note that

must be defined in the range of which is denoted by

Considering

the above four events, we can obtain the conditional transition probability as follows:

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where

is Dirac’s delta, so that it equals 1 if and 0 otherwise. We define the one-step state transition probability for the system as follows:

To calculate the one-step state transition probability of the system, we give a function to map the system state into the onedimensional state space H as follows:

Let P be an matrix of the one-step state transition probability where H is the total number of the system states defined in (9.1). Let denote an H-dimensional row vector of the stationary probability distribution of the system state and let denote the stationary probability of the system corresponding to state can be determined by solving the following set of linear equations: or

3.2

Packet Departure Distribution

In this subsection, we derive the joint moment generating function of the probability distribution for packet departures of both voice and data traffic.

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We define the number of packet departures as the number of either voice or data packets successfully transmitted in a slot through all the codes. Denoting the joint moment generating function of voice and data packet departures by and using the conditional transition probability and the stationary distribution given in (9.2) and (9.5), respectively, we have

By differentiating evaluated at we can obtain higher factorial moments of the packet departure distribution as follows:

All moments of the joint m.g.f. of both voice and data packet departures have closed-form expressions. Using these higher factorial moments, in particular, we can evaluate the correlation coefficient of the numbers of voice and data packet departures which is defined as follows:

3.3

Average Performance Measures

In this subsection, we can proceed to obtain average performance measures to numerically evaluate the system performance such as the throughput for both voice and data traffic, the blocking probability of voice calls, the average delay of data packets as follows.

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We define the throughput in this chapter as the average number of either

voice or data packets successfully transmitted in a time slot through all the codes. The throughput can be obtained by differentiating the joint m.g.f. by Let and denote the throughputs of voice traffic and data traffic, respectively. Then we obtain the throughputs as follows: For the voice traffic:

For the data traffic:

Let denote the average number of users per slot that cannot find any free code when they generate new voice calls. It is given by

The blocking probability

of voice calls is given by

where is the average number of users per slot that succeed in transmission of the first packet when they generate a new voice call. It is given as follows:

Finally, we define the average data packet delay E[D] to be the average time from the epoch of a data packet arrival to the epoch of the completion of the data packet transmission. It is presented as follows.

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As all N users are stochastically homogeneous, it is easy to see that the following balance equation holds true:

Solving (9.14) for E[D], we can obtain

3.4

Packet Delay Distribution

In this subsection, we analyze the moment generating function of the probability distribution of the data packet delay. We focus on a data packet called a tagged data packet. Since all data packets are homogeneous in terms of channel access rate and transmission time, we can choose any data packet as the tagged data packet.

We note that the data packet delay can be divided into two parts as follows: (1) The first part is the delay from an arrival point of a new data packet until the end of the first slot where the data packet might be successfully transmitted. (2) The second part is the delay from the end of the first slot until the departure instant of the data packet if the first transmission attempt is unsuccessful. We refer to the former as the initial delay and the latter as the backlog delay. First we derive the conditional m.g.f. of the backlog delay. In order to obtain the conditional m.g.f. of the backlog delay, we first define a conditional transition probability of successful transmissions of the data packets as follows: and i data packets are successfully transmitted in the

st slot

(9.16)

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where

min( ). can be obtained by picking up only when )). Therefore, is given numerically by the following equation:

where is Dirac’s delta and

is defined as

Let denote an matrix of Let represent the conditional m.g.f. of the backlog delay for the tagged data packet that sees i data packets queueing before it, given where We can derive from

the following observations: (1) more than i data packets are transmitted successfully in the st slot, then the delay of the tagged data packet is just one slot, or (2) l data packets are transmitted successfully in the st slot and the tagged data packet finds itself in state at the end of the st slot, then the m.g.f. of the backlog delay is given by Therefore, we have recursive equations as follows:

Express (9.18) in matrix notation and solve it for the column vector as follows:

where is an H-dimensional column vector, e is a column vector with H elements, all of which equal 1 and is an identity matrix. Next we investigate the initial delay. Let represent the probability that the tagged data packet is successfully transmitted in the first slot upon its arrival and let represent the probabilities that the tagged data packet

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fails in transmission in the first slot upon its arrival and finds i data packets queueing before it and itself being in state at the end of the st slot. They are given by normalizing the expectation of the number of data

packets that experience the same situation [Toba82a].

and

where IF

is a function which takes 1 if

holds true or 0 otherwise. K

in (9.20) and (9.21) is a normalizing constant such that

Finally we can obtain the m.g.f.

where

of data packet delay as follows:

is an H-dimensional row vector of

We can give the exact higher moments of the delay by differentiating (9.23). Expressions for all moments of can be obtained in the following manner.

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Let represent the nth factorial moment of the data packet delay distribution. Then we can obtain as follows:

where

We note that all moments of the m.g.f. have closed-form expressions, and the average data packet delay, obtained in (9.24) should equal E[D] in (9.15). Using the second moment we also obtain the coefficient of variation of the data packet delay.

4.

Numerical Results

In order to verify the validity of the analysis and to provide a number of examples of how this analysis can be used to evaluate and compare the performance of the slotted CDMA integrated voice and data network systems with multiple random access protocols, this section presents some numerical results on the throughputs of voice and data traffic, blocking probability of voice calls, average delay of data packets, and correlation coefficient of the voice and data packet departures and coefficient of variation of the data packet delay. In all numerical examples, we fix the number of users at and the total available codes at while we vary the number of voice codes . We note again that in case of there is no admission control for the voice traffic. The system parameters such as the length of a time slot and the parameters for the voice traffic are determined by referring to

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[Good89], where the mean talkspurt duration is 1.00 s, the mean silent gap duration is 1.35 s, and slots have the same duration of 0.8 ms, respectively. In our model, the mean silent gap duration of a voice call corresponds to of 1,688 slots and the mean active duration of a voice call corresponds to of 1,250 slots. Therefore we use and in the numerical examples. and are also fixed. Aiming at the third generation mobile communication systems, IMT2000 is being standardized in ITU (International Telecommunication Union). IMT-2000 can provide 2Mbit/s capacity [Sato98] for multimedia mobile accesses such as voice, data messages, video, etc. In our numerical example, if we consider the data burst length to be segmented into 64 byte-long short packets for transport over networks, the maximum number of slots for transmitting a 64 byte-data burst stream is 0.32. As 2 Mbit/s = 250 byte/ms and 1 slot = 0.8 ms, we can get 200 byte/slot, and so 64/200 = 0.32 slots for each burst stream. It means that in the mean silent gap duration of 1688 slots, the maximum number of data burst streams transmittable is 5.275 K (1688/0.32 5.275 K) and at a maximum 3 burst streams can be transmitted in a slot. In comparison with our example of and it means that the time interval between two consecutive packet arrivals would be 33 slots per code (i.e., packets/slot/code) for the whole system. Therefore, in the mean silent gap duration of 1688 slots, the number of data burst streams would be 153 (1688/33 × 3). In a heavy traffic case of the number of data burst streams would be 5627. It means that 1 packet (i.e., 3 data burst streams) may be transmitted in every slot. The throughputs and for the voice and data traffic as a function of are plotted in Figs. 9.2 and 9.3, respectively. Each of them includes three curves with In the case of all codes are available to voice sources. In Fig. 9.2, we observe the following: (1) When is smaller than 0.2, the throughput is almost constant for all cases, and it is especially the largest for the system with This is because the system with larger has more codes available to voice packets. (2) However, as further increases, all the curves drop suddenly. This is due to the following reason. The present system consists of a finite population of N users and each one can generate both voice and data packets. When increases, the relative ratio of users in the speech mode decreases, so that the throughput of the

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voice traffic decreases. In other words, users in the data transmission mode become the majority of the system. In Fig. 9.3, we observe the following: (1) As the access rate of data packets to free codes increases with while the access rate of voice packets to free codes is fixed, the throughput performance of data traffic can be significantly improved with . (2) As fewer codes are available to voice sources with smaller , the throughput of the data traffic with smaller has a larger value under the same In Fig. 9.4, we show the blocking probability of voice calls for 6,8,10. We observe that when is small, all blocking probabilities are nearly constant, but as increases, the blocking probabilities decrease to almost zero. We also observe that among the three cases, the system with has the largest blocking probability for all . The reason is the same as in Fig. 9.2. These results are important to appropriately control the admission of voice calls. Fig. 9.5 shows the average delay of data packets E[D] versus the offered data rate . We note that in all cases of the difference is quite small for all . It demonstrates that has rather small impact on the average delay of data packets because data sources can find a free code in some slots where voice sources just turn off for speech. Figs. 9.6 and 9.7 display the correlation coefficient of the voice and data packet departures and the coefficient of variation of the data packet delay as a function of the offered data rate for and respectively. In Fig. 9.6, we examine the effects of voice codes on the correlation coefficient of the numbers of voice and data packet departures. We observe the following: (1) The correlation coefficient for all cases starts decreasing from 0.0 and reaches the same minimum value -1.0 as the offered data rate approaches 0.5. This can be explained as follows. First the correlation coefficient is negative because voice packets and data packets compete with each other for the limited number of codes. For the constant offered voice rate, when is small, the interference between voice and data packets is extremely light, so that the maximum throughput of voice packets can be achieved, where we note that the correlation coefficient is almost 0.0. On the other hand, as becomes larger, all codes are fully used and the system reaches such a situation that the increase of the number of successful data packet transmissions by one implies the decrease of the number of successful voice packet transmissions by one, which forces the correlation coefficient down to the minimum value

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-1.0. (2) Among the three cases, the case of has the strongest negative correlation, while the case of has the weakest negative correlation. This is because the less voice codes , the less interference in transmission over each code between voice and data packets, so the output process of voice packets is more independent of the output process of data packets for smaller In Fig. 9.7, we observe that as increases from 0.1, also increases, reaches a peak at about and decreases more weakly. This is because when is very small, many free codes remains and the data packet hardly waits, so that the variation of data packet delay is negligible. As the average number of successful data packet transmissions increases, the variation of data packet delay increases. However, as further increases, data sources become the majority, which contributes toward stabilizing the data packet delay. We also find that for given , the system with a small has a higher throughput of data packets than the system with a large

5.

Conclusion

This chapter presented an exact analysis to numerically evaluate the performance of integrated voice and data slotted CDMA network systems with multiple random access protocols for packet radio communications. The system model consisted of a finite number of users, each of whom can be a source of both voice and data traffic. The allocation of codes to voice calls was given priority over that to data packets while an admission control, which restricts the maximum number

of codes available to voice sources, was considered for voice traffic so as not to monopolize the resource. In addition, the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use voice codes for transmission if the voice sources are in silent periods. A discrete-time Markov chain was used to model the system operation. The throughput, the average delay of data packets, the blocking probability of voice calls, and the moment generating functions of the probability distributions for the packet departure of both voice and data traffic and for the data packet delay were derived to evaluate the performance of the system. In numerical results, we examined the correlation coefficient of the voice and data packet departures and the coefficient of variation of the data packet

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delay for some cases with different numbers of voice codes, as well as average performance measures such as the throughput, average delay of data packets and blocking probability of voice calls. We make the following conclusion: (1) Because system monitoring can distinguish between silent and talkspurt periods of voice sources and hence users with data packets can use the silent gap duration for transmission, the maximum number of voice codes does not significantly affect the average delay of data packets. (2) The larger the maximum number of voice codes, the larger the throughput of voice packets and the lower the blocking probability of voice calls, so that the system can be optimally designed by using the admission control proposed in this chapter.

(3) Because the present system consists of a finite population of users, each of whom can generate both voice and data packets, the effect of admission control is more significant at lower offered rate of data traffic. (4) The system can provide the possibility of achieving higher spectrum utilization for diverse services in packet radio communication network systems using the promise of integrating the properties of both CDMA and multiple random access protocol. (5) The admission control has a significant effect on the correlation coefficient of the voice and data packet departures, but little effect insofar as the offered data rate is smaller or larger. Especially when the offered data rate is large, the correlation coefficient of the voice and data packet departures is close to -1.0. (6) Admission control also has a significant effect on the coefficient of variation of the data packet delay especially when the number of voice codes is large.

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Chapter 10

PERFORMANCE ANALYSIS OF CELLULAR MOBILE TELEPHONE NETWORKS WITH HYBRID CHANNEL ASSIGNMENT SCHEME

1.

Introduction

With regard to mobile telephone communication service, the increasing demand for mobile telecommunication service and the finite spectrum allocated to this service led to the proposal of the cellular structure. The cellular mobile phones first developed by AT & T Bell Laboratories operated in the 800-900 MHz band. Because the spectrum allocated to the mobile radio system is an important resource, it is necessary to use this frequency spectrum efficiently as possible. The concept of the cellular mobile telephone system has been studied intensively in [MacD79]. With the cellular structure approach, the system area is divided into cells and each cell is given its own base station for communication with mobile users in its cell. The frequency reuse concept is the core of the cellular mobile radio system. In this type of frequency reuse system, users in different geographical locations (different cells) may simultaneously use the same frequency channel. To increase the channel utilization, different channel assignment schemes have been studied. A better technique called the hybrid channel assignment (HCA) scheme divides the total number of channels into two groups, one of which is used for fixed allocation to the cells and the other kept at a center station to be shared by all users of the system.

The most important performance measures in the cellular structure mobile telecommunication networks are blocking probability and channel utilization. The blocking probability of a cell in the system using the HCA 245

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scheme is defined as the probability that an arrival call to a cell finds all channels, both fixed and dynamic, are busy. The improvement in the channel utilization is described in terms of the traffic increase which can be serviced by the system with the HCA scheme compared with an ordinary fixed channel assignment (FCA) scheme. In the system using a FCA scheme, the blocking probability of each cell is simply given by the classical Erlang B formula. There, the call arrivals and call services of each cell are dependent on each other. However, when analyzing the performance of the system using an HCA scheme, it is necessary to consider the interference due to the common use of the same channel by cells surrounding the cell of interest. The interference is called cochannel interference. This causes a very complex interaction of the call arrival process in overflow routes to the dynamic channel group. The fact makes exact analytical solutions in practice impossible in the field of queueing theory. Approximation remains the most popular tool for performance evaluation of the systems. Fixed and dynamic channel assignment (DCA) schemes have been extensively studied such as [Cox71], [Cox72a] and [Ande73]. In the DCA scheme, there is no fixed relationship between the channels and the cells [Ande73]. All channels are kept in a central pool, and any channel can be used at any cell if no constraints are violated. Simulation results from these studies showed that under low traffic conditions, the dynamic assignment technique performs better. However, the FCA scheme becomes superior at high offered traffic. Several approximate techniques have been studied to evaluate the performance of the systems using the DCA and HCA schemes such as [Schi70], [Kahw78], [Seng80] and [Bakr82]. The approximate technique that was most used is an approach by simulation. Sin and Georganas [Sin81] have presented a simulation study of the performance of the cellular land mobile radio system using the HCA scheme with Erlang C service discipline. They believed that the hybrid scheme would show its superior performance with nonuniform spatial traffic, since it includes dynamic channels which could move around to serve the random fluctuations in the offered traffic. Kahwa and Georganas [Kahw78] have described the simulation technique of a large cellular system using the DCA scheme and HCA scheme. They provided a comparison between their results and the results presented by Cox and Reudink in an earlier paper [Cox73]. However, simulations

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which attempt to simulate the behavior of an actual system with a large number of channels and cells are time-consuming and expensive. Wilkinson [Wilk56] considered multiple server systems with a no waiting, exponential service, and Poisson arrival process for two channel groups, i.e. the first channel group and repeat channel group. There, both the first moment and the second moment (or variance) of overflow were given to describe adequately the non random character of overflow traffic. However, in that analysis, he assumed that the traffic offered to the repeat channel group was altogether serviced by system, and did not consider the co-channel interference among users. Bakry and Ackroyd [Bakr82] have derived a new teletraffic formula to obtain the blocking probability of calls in mobile radio telephone systems using the FCA, DCA and HCA schemes. In order to simplify the analysis in the HCA system, they approximated the blocking probability in the system as the product of the fixed channel blocking probability and dynamic blocking probability for the HCA scheme. They assumed that the call arrival processes to both channel groups are the same Poisson. As the overflow traffic offered to dynamic channel group is not Poissonian, theoretical results using this approximation significantly disagree with the simulation results for traffic of more than 7 erlangs. Schiff has given some teletraffic formulas for cellular radio systems using dynamic channel assignment scheme but there are some practical difficulties in obtaining numerical results from his formulas [Schi70]. Because exact computation of the performance measures of cellular radio system using the HCA scheme is too complex, in this chapter, we first present two sufficiently accurate approximate techniques to evaluate performance of the large scale mobile radio systems using the HCA scheme and the cellular structure. The two approximate analyses give the expressions for the blocking probability of mobile telephone traffic. In the first method, the blocking probability is obtained by finding the interarrival time probability distribution function of one composite interrupted Poisson process (IPP) stream consisting of several IPP streams overflowing from the cell of interest and its co-channel interference cells. The second method is proposed to solve the blocking probability of the system by regarding each cell as a generalized GI/M/m(m) model, in which, the co-channel interference is considered. Two analytical results are compared with simulation results and good agreements are observed for both fixed and hybrid channel assignment schemes. Then we apply the analysis

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of the second method to present an efficient analysis to evaluate the performance of a hybrid channel assignment algorithm in the cellular mobile networks with hand-off. The methods presented in this chapter will be not only useful for the performance prediction and the optimum design of the cellular mobile radio communication system with an HCA scheme, but also applicable to the study of system with a DCA scheme when the traffic offered to a group of dynamic channels forms a Poisson arrival process. The chapter is organized as follows. In Section 2, we justify the cellular mobile telephone system layout assumptions and give the blocking probability of the FCA scheme which is well known from classical traffic theory. In Section 3, we first present the statistical characteristics of an overflow stream from a fixed channel group that can be viewed as a Poisson process modulated by a random switch (IPP). Then we use those techniques to analyze the performance of the HCA system. Here, we present two techniques (Method I and Method II) to give the blocking probability of the system. Numerical examples for a 49-cell system are presented with two approximate analysis methods. These results are compared with simulation results in Section 4. Finally, a conclusion is offered in Section 5.

2.

System Model

The system considered in this chapter consists of K equal size cells. Fig. 10.1 shows the cellular geometry of a mobile radio system layout. The co-channel interference cells in which the same channel can not be reused are labeled . The number of cells per co-channel interference area is six with single belts of the cell of interest and 18 with double belts. We are considering only the nearest six co-channel interference cells. is called the co-channel reuse ratio, where R is the radius of the cell that is defined as the maximum distance from the center of a cell to the cell boundary and D is the physical distance between the centers of nearest neighboring cells. If the distance separating the cells is more than the same channel can simultaneously be used. C, obtained from [Kahw78], is the least number of channel groups required in order to assign channels to all cells. When at least three different channel groups are required in the system. In the system considered here, the total number of the available radio channels is first divided into three groups. Each cell in the system is allo-

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cated one of the three channel groups, observing the least reuse distance. After the preliminary subdivision of channels, each of the three channel groups is partitioned first into a number of fixed channels and then, into a number of dynamic channels The fixed channels are to be exclusively used in their nominal cells, where as the dynamic channels will be donated to the entire system for a dynamic assignment scheme. In each cell, calls are offered first to the fixed channels. A call that is not serviced by the fixed channels is then offered to the dynamically assigned channels. When an idle dynamic channel is assigned, this channel is said to be borrowed. In most cases, there is more than one, and in fact, many channels which could be borrowed for a given call. The method of selecting one of the available channels makes up most of the variation in the channel assignment algorithms studied. In this system, we use the simplest borrowing algorithm, that is, to borrow the first available channel satisfying the cochannel reuse distance.

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We assume that calls offered to each cell forms an independent Poisson process with arrival rate r and that the holding times of fixed and dynamic channels are random variables having identical negative exponential distributions with means and respectively. The traffic intensity per cell is . The call is blocked only if all and channels are in use. These assumptions are similar to those presented in [MacD79], and this chapter extends the analyses in [MacD79]. Since our objective is to compute the blocking probability using the HCA scheme, we must first ascertain the blocking probability of each cell using the fixed channel. The blocking probability for the FCA scheme is used to calculate the overflow traffic from each cell to the dynamic channel group. Let

denote the blocking probability at each cell in a FCA scheme.

The blocking probability can be provided by the familiar Erlang B formula. The Erlang B formula is used to calculate how much traffic would overflow in the case of a given originating traffic as

This is also the probability of the overflow from the group of fixed channels of each cell to the group of dynamic channels.

3. Performance Analysis 3.1 Interoverflow Time Distribution The overflow traffic from the group of fixed channels of each cell to the group of dynamic channels is not Poisson and a more sophisticated technique has to be applied which is called an interrupted Poisson process (IPP). The IPP provides a simple and approximate description of the overflow traffic and consequently facilitates analytical studies. The overflow traffic from

the group of a fixed channel of each cell can be approximately determined by matching the first three moments of the interarrival time distribution of overflow to dynamic channel group and IPP distribution, since the interoverflow times, i.e., the time between consecutive overflowed calls, are independently identically distributed random variables.

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In order to simplify the analysis, we assume that both the traffic offered to fixed channels and the number of fixed channels in each cell are the same. The overflow stream from each cell is characterized by an IPP, its interarrival time distribution is denoted by is given by [Kucz73]

where

is given by a mixture of two exponential distributions with rate and probability . An overflow stream, which has the first three central moments that are denoted by respectively, is approximated to an IPP with three parameters and which are to be chosen. They are determined as follows:

where is the call arrival rate during the on-state of IPP, and are mean times of the on-time and off-time of the IPP process, respectively. is a factorial moment of IPP, it is given by

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The first three moments of the overflow stream with Poisson input are given by Kosten’s results [Kucz73]:

where

3.2

Analytical Method I

We define the blocking probability of the system in a cellular mobile network environment using the HCA scheme as the probability that an arrival call to a cell finds all channels, both fixed and dynamic, are busy. This is also the blocking probability of the total overflow to the dynamic channel group that an arbitrary overflow call finds all dynamic channels occupied. The overflow traffic from the group of a fixed channel of each cell was given in the previous subsection. In this subsection, we give the analytical method I to compute the blocking probability of dynamic channel group. The analytical method I makes use of the fact that in the system with the HCA scheme, the overflow stream from the cell of interest and co-channel interference cells around it can be combined into one IPP stream under the assumption of independence among cells. The total overflow stream offered to the dynamic channel group is also approximated by the IPP renewal process. The composite IPP stream into the group of dynamic channels will proceed to stage 1 or 2 of arrival with probability and arrival rates 1,2), where and have the meaning explained in (10.3). Let G(x) and denote the probability distribution function of the interarrival time in composite IPP process and its Laplace-Stieltjes transform

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(LST), respectively. They are given by

and

where h is the number of overflow streams offered to the group of dynamic channels. Because each dynamic channel can be reused in the cells satisfying the co-channel reuse distance, when is equal to 3. is the equilibrium state probability of state . The system is said to be in state , when there are overflow streams out of overflow streams in stage 1 and overflow streams in stage 2. If a new call of overflow streams that is in stage 1 arrives, this will convert state to with probability or keep it in state with probability . If a new call of overflow streams that is in stage 2 arrives, this will convert the state to with probability or continue to be in the state j with probability The state transition rate diagram upon arrival of the composite IPP arrival process is shown in Fig. 10.2, where

Using (10.10), we may write down the equilibrium equation as

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Here we use the conservation relation as follows:

and obtain

and

The generalized Erlang loss function of the GI/M/m(m) theory was given by [Taká62]. We apply this blocking probability formula for our analysis, to calculate the blocking probability of the system with the hybrid channel assignment scheme. Let denote the blocking probability of the system with the hybrid channel assignment scheme obtained by the method I, then we have

where

and

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255

Analytical Method II

In this subsection, we give the analytical method II to calculate the blocking probability of the system with the hybrid channel assignment scheme. The definition of the blocking probability of the system is the same as that defined in the method I. Using the analytical method II, the behavior of the system can be studied by characterizing the stationary probability distribution with the binomial moment.

In this method, we are interested in solving for the blocking probability of the system using the HCA scheme by considering the equilibrium probability distribution describing the number of the dynamic channels occupied at the arrival instants of the particular cell taking into account the co-channel interference. 3.3.1

Blocking Probability

Overflowed calls from a particular cell arrive at the instants ... that are defined to be imbedded Markov points. The instants form a sequence of renewal events. The intervals are mutually identically and independently distributed random

variables, with a common distribution function F(x). The expectation of interarrival time is denoted by Let be a random variable, which represents the number of dynamic channels occupied immediately before the arrival of the lth call overflowing from the particular cell. The sequence of random variables forms an irreducible and aperiodic continuous-time Markov chain. We define the state of the system by the number of dynamic channels occupied immediately before the arrival of the lth call overflowing from the particular cell at the imbedded Markov point. The state of the system is said to be in state m if and it is denoted by . Next we define the probability of occupied channels b(m) to be the probability that the channels cannot be offered to service calls by the system because of the influence of co-channel interference cells around it. We first derive the probability of occupied channels b(m) when . Clearly, . The probability of occupied channels b(m) depends on the state of dynamic channel in use, which is determined by considering overflow streams from the co-channel interference cells, channel capacity, etc., so exact computation of b(m) is not possible.

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The probability of occupied channels b(m) is approximately chosen according to the relationship as

where is overflow traffic intensity offered to the group of dynamic channels, given by the following expression as

Moreover,

is overflow traffic carried by an individual channel per cell. is the blocking probability of the system with the hybrid channel assignment scheme which will be calculated by the method II. a and were defined in the Section 2, h was defined in Subsection 3.2, and is the average number of single belt co-channel interference cells around the particular cell and is equal to 6. When just before the st call arrives, the lth call is served with probability , then within an overflowed call interarrival time x, m + 1 – n channels are released with probability and channels continue to be in use until this time with probability or with probability b(m) the lth call is blocked, then by the time channels are released with probability and n channels continue to be in use until this time with probability When , all dynamic channels are occupied. Then the arriving call is blocked and with probability channels are released and with probability channels continue to be in use. To obtain the one-step state transition probability of the system, we first derive the conditional probabilities lows:

and

as fol-

Performance Analysis of Cellular Mobile Networks with HCA Scheme

257

and

Removing the condition on interarrival time x, we then can obtain the one-step state transition probability from the system state m to the system state n. Since has common distribution function F(x), we have

where F(x) is given by (10.2). Let be the stationary probability distribution of the Markov chain imbedded at points just before an overflowed call arrives from the cell of interest. Since this is ergodic, it has an equilibrium state when the time tends to infinity. satisfies the following set of equations subject to the normalization condition:

where P is an matrix of the one-step transition probability The blocking probability of the system with the hybrid channel assignment scheme by the method II is obtained by the following:

3.3.2 Binomial Moment The behavior of the system can be studied by the binomial moment method. We derive the binomial moment in this subsubsection by extending the analytical technique in [Taká62].

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

The steady-state behavior as follows:

can be characterized by binomial moment

where and is the rth binomial moment of Since n depends on m and r, by changing the order of summation, the above equation can be obtained. and are rth binomial moments of and defined in (10.21) and (10.22), which are denoted by and , respectively. By using and , we can write again

But when

into

and

is equal to zero. If we put , respectively, we get

where variables m and r of and Therefore, by using the binomial theorem, we have

and

do not depend on n.

Performance Analysis of Cellular Mobile Networks with HCA Scheme

259

By using the above relation, we can write also that

Let be the rth binomial moment of ing the condition on the interarrival time x as

where

Taking the following:

into (10.32), we obtain

and

,

remov-

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

If we multiply both sides of (10.35) by

and sum over m, then we get

Clearly, the left-hand side of (10.36), is equal to the second term on the right-hand side in the parentheses also equal to (see (10.26)). Therefore, (10.36) can be written as

where

. We define

and

Now let us divide both sides of (10.37) by

then

and is

and by using the relation

can be expressed as

I

We also calculate (10.37) for obtain the expression as follows:

then add them, to finally

Performance Analysis of Cellular Mobile Networks with HCA Scheme

where

and

which is given by

where

and

are given by

261

By substitution of (10.41) and (10.42) in

then { } can directly be obtained. Finally, by using (10.25), the blocking probability of the system will also be obtained by the binomial moment method.

4.

Numerical Results

In this section, numerical results are presented which show system characteristics. The flowchart of the simulation model of the mobile radio communication system using the hybrid channel assignment scheme is shown in Fig. 10.3. The calculated values of the blocking probabilities versus traffic intensity offered to each cell are obtained using two different approximate methods and are shown in Fig. 10.4 and Fig. 10.5.

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

For a particular case of 8 : 2 division where 8 : 2 is the ratio of the number of fixed channels to dynamic channels, we give six dynamic channels to the entire system. For a particular case of 5 : 5 division, we give 15 dynamic channels to the entire system. Fig. 10.4 and Fig. 10.5 show the blocking probabilities as functions of the increase ratio of offered traffic (%). The curves are drawn for different values of traffic intensity per cell, where r is arrival rate and is the holding times of fixed channels for each cell. The case of 10 : 0 division is equal to the fixed channel assignment scheme which is used for comparison. The traffic intensity of each cell is increased from 5 to 10 erlangs. Because transmission probability [1 – q(m)] in the second method depends on blocking probability , we adopt an iteration procedure. The parameter for test of convergence is set to as follows:

The simulation results are also showed in Fig. 10.4 and Fig. 10.5, and good agreement with theoretical results is observed. The simulation is carried out on 49 hexagonal cells having uniform traffic intensity and the statistics are collected from the central cells to overcome the edge effects in a large cellular layout for comparison with the results of Cox and Reudink [Cox72a], Kahwa and Georganas [Kahw78]. The blocking probability of any cell in the system can be calculated by the two methods since there is the same relationship among the cells of the system.

5.

Conclusion

In general, the rigorous solution for the traffic characteristics in a large scale cellular mobile radio communication system using the hybrid channel assignment scheme is not easily obtained, since it depends on the state of co-channel interference cells. We presented in this chapter two approximate methods for analyzing cellular mobile radio communication systems. In the first method, the blocking probability was obtained by finding the interarrival time probability distribution function of one composite interrupted Poisson process (IPP) stream consisting of several IPP streams overflowing from the cell of interest and its co-channel interference cells. The second method was proposed to solve the blocking probability of the system by regarding each call as a GI/M/m(m) model. Two analytical results were

Performance Analysis of Cellular Mobile Networks with HCA Scheme

263

compared with simulation results and good agreements were observed for both fixed and hybrid channel assignment schemes. In the two approximate methods it can be seen that the complexity of the analysis never depends on the number of cells in the system. The methods presented in this chapter will be not only useful for the performance prediction and the optimum design of the cellular mobile radio communication system with a HCA scheme, but also applicable to the study of a system with a DCA scheme when the traffic offered to a group of

dynamic channels forms a Poisson arrival process.

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Performance Analysis of Cellular Mobile Networks with HCA Scheme

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Chapter 11 OUTPUT AND DELAY PROCESS ANALYSIS OF CELLULAR MOBILE TELECOMMUNICATION NETWORKS WITH HAND-OFF

1.

Introduction In high-capacity cellular mobile communication systems, efficient chan-

nel use is achieved by covering the service area with a large number of small

cells. However, in addition to the requirement for more fixed transceiver sites, its bottleneck is in that the small cell system must accommodate an increased number of cell boundary crossings of mobile users who have calls in progress. The calls can include mixed voice and data transmission, image transmission, phone mail, e-mail, etc. Because the hand-off procedure has a significant impact on the system performance, it is very important to evaluate the traffic performance of the cellular system with hand-off.

To enhance the quality of cellular service, hand-off priority schemes must be introduced to reduce the probability of hand-off failure. The systems with hand-off priority schemes have been studied in [Hong86] and [McMi95]. Hong and Rappaport [Hong86] presented two prioritized hand-

off procedures to the system. They assumed that the traffic is of pure chance with different arrival rates at one cell for new and hand-off calls and the holding times obey a negative exponential distribution with the same mean. In priority scheme I, a number of channels are used exclusively for hand-off calls while the remaining channels are used for both new calls and hand-off calls. Blocked calls are cleared from the system immediately. In priority scheme II, channels are shared in the same way as in priority scheme I, but hand-off calls can be queued in an unlimited buffer. We note that in micro267

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

cell systems, the assumption of the unlimited buffer size and the same mean holding time for both new and hand-off calls is unrealistic. A priority queueing model analyzed in [McMi95] is a combination of a nonpreemptive priority queueing system with a channel reservation policy and a hysteresis mechanism. In the system it is also assumed that the channel holding time of new calls in the originating cell and the residing time of calls in the hand-off cell obey the exponential distribution with the same mean. As for other related studies, loss systems with two classes of calls and three groups of servers have been studied in [Sato83] and [Kawa85]. Sato and Mori [Sato83] considered a telephone-based ticket reservation system for an airline company. In the system, two streams of reservation calls are received and answered by servers. Each input stream obeys a Poisson process. The total number of servers is divided into three groups. Two groups can only deal with one or other input stream and what they call a commonly usable group is able to serve both classes of input streams. They assumed that commonly usable servers deal with streams overflowed from their own assigned servers. If servers are all busy, arriving calls should be lost. Kawashima [Kawa85] analyzed a similar model by approximating the overflow process to an interrupted Poisson process (IPP). He proposed an efficient and fairly accurate approximation method to compute loss probabilities for both classes of calls. It requires much shorter time than the exact method employing lumping. The models analyzed by Sato and Mori [Sato83] and by Kawashima [Kawa85] correspond to our model without waiting room as special cases. In this chapter, we present an exact analysis to evaluate the performance of mobile radio networks with two streams of calls (referred to as new call and hand-off call) and finite buffer-capacity of N calls for hand-off. We apply a matrix-analytic method to describe the stochastic behavior of the system and we efficiently calculate the stationary probabilities with a large state space to obtain the blocking probability, channel utilization for both streams, average queue length and average waiting time for hand-off stream. Numerical results show how priority can be given to hand-off calls according to the number of reserved channels and buffer size. Using the matrix-analytic solution for the stationary state probability distribution, we further derive the Laplace-Stieltjes transform (LST) of the probability distribution of the waiting time of a hand-off call.

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 269

The analysis presented in this chapter will not only be useful for the performance evaluation of cellular mobile systems with hand-off and the optimum design of the cellular system but also applicable to evaluating other performance measures in interconnected networks, multichannel networks or integrated networks of voice and data. The chapter is organized as follows. In Section 2, the cellular system model with hand-off is presented. Traffic performance and the waiting time distribution of a hand-off call are analyzed in Section 3 by the matrixanalytic method. In Section 4, we show numerical results and compare them with results in [Kawa85]. A conclusion is drawn in Section 5.

2.

System Model

In cellular mobile environments (see Fig. 10.1), the co-channel interference cells are taken into consideration where the same channel cannot be reused [Kawa85]. is called the co-channel reuse ratio, where R is the radius of the cell, that is defined as the maximum distance from the center of a cell to the cell boundary, and D is the physical distance between the centers of nearest neighboring cells that can use the same channel simultaneously, is related to the number of cells per group by The cell radius should be small for a high capacity system since this allows more channel reuse in a given service area. We consider only the nearest six co-channel interference cells and three different channels sets in the whole system. In fixed channel assignment schemes, a specific set of channels is permanently allocated to each cell. Because these channels can only be used to serve the call requests within the cell, we can analyze the performance of an arbitrary cell independently of other cells. We consider two Poisson streams offered to a cell and denote the arrival rates of new call traffic and hand-off call traffic by and respectively. We note that in the cellular systems, can be given as a system parameter, while is related to other system parameters such as arrival rate of new calls, mean call holding time, cell size and users’ mobility. Under memoryless assumptions, the average hand-off rate into an adjacent cell can be given by the ratio of the average call holding time to the average cell sojourn time, which is what one would intuitively expect [Nand93]. To apply this fundamental result on mean hand-off rate in [Nand93] to our model, we assume an independent, identically distributed cell sojourn time with mean for new calls and for hand-off calls. The holding time of

270

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

a channel is assumed to be independent and exponentially distributed with different mean for new calls and for hand-off calls. If we consider the case of a uniform traffic distribution and at most one hand-off per call, the arrival rate of hand-off calls into a cell can be given by

Fig. 11.1 shows the flow diagram of the system model with hand-off. The total number of channels in a cell is denoted by . Among channels, channels are reserved for new (hand-off) calls only and the remaining channels are allowed to be used by both types of calls.

and in Fig. 11.1 are defined to be the offered new traffic load, (erlangs/cell) and the offered hand-off traffic load, (erlangs/cell), respectively. When a call arrives, the reservation channels of its type are checked first for transmission. If there is no reservation channel available, then the common channels are checked for transmission. If there is no common channel available, the call is assumed to be blocked and cleared from the system if it is a new call, while the call is assumed to enter a buffer if it is a hand-off call. The buffer is assumed to have finite capacity of calls, which are served

in the first-in first-out discipline. If the buffer is full, however, the hand-off call will be blocked and cleared from the system. Once a call seizes a channel in a cell, it holds the channel until either the transmission is completed

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 271

within the cell or it moves out of the cell. Whenever a call releases a channel besides the one reserved for new calls, it is assumed that the first hand-off call waiting in the buffer, if any, immediately seizes the channel.

3. Performance Analysis 3.1 Stationary Probability Distribution We define the system state of a cell by the pair of the numbers of channels being used by new calls and by hand-off calls as well as the number of hand-off calls in the buffer. Let and denote the numbers of channels being used by new calls and by hand-off calls, respectively. Then the system state is defined as and the number of hand-off calls in the buffer. We formulate the model

as a continuous-time Markov chain and observe the system at an arbitrary instant. Let denote the row vector of the stationary probability distribution at an arbitrary instant, where is the row vector of the stationary state probabilities for the case that there are hand-off calls in the buffer. We can write

Namely, for

as a function of

and

for

where denotes the stationary probability of the state that there are and channels being used by new calls and by hand-off calls, respectively, and hand-off calls in the buffer at an arbitrary instant. Let H (H') denote the total number of the system states for the case that the buffer is empty (not empty), respectively. Because all accessible channels must be occupied if and H' are given by

As the size of is given by H, while the size of is given by H' , we note that the total size of x becomes H + NH', which is too large to solve it directly. In Appendix C, we will show an efficient iterative procedure to solve x.

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

To express the state transition probabilities easily, we define a function to map the system state into the one-dimensional state space H as follows:

For the case that there are hand-off calls waiting in the buffer, we define a function which maps the system state into the onedimensional state space as follows:

We note that using the above functions and and for

, we can express that

. The state transition diagram for this system is shown in Fig. 11.2. Let be an

infinitesimal matrix as follows:

The elements of are infinitesimal generators that we define as follows: is an infinitesimal submatrix that corresponds to the case that there is no hand-off call in the buffer. Each element of is given as follows:

(i) Transition rates from

and

are given by

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 273

(ii) Transition rates from

and

(iii) Transition rates from

(iv) Transition rates from

and

and

are given by

are given by

are given by

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

(v) Transition rates from

where in the case of

and

are given by

and

or

then

(vi) Transition rates from

and

are given by

(vii) Transition rates from

(viii) Transition rates from by

and

are given by

and

are given

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 275

where in the case of

and

or

(ix) Transition rates from by

where in the case of

then

and

and

are given

or

then

is an infinitesimal submatrix which corresponds to the transitions from the case of having just one hand-off call in the buffer to the case that the call is severed by an available channel. Each element of is given as follows:

(i) Transition rates from

and

(ii) Transition rates from given by

are given by

and

are

is an infinitesimal submatrix for the case that a hand-off call enters an empty buffer as no channel is available for the call upon arrival. Each element of is given by where and and are all infinitesimal submatrices. corresponds to the case that the number of hand-off calls in the buffer does not change. is given as follows:

(i) Transition rates from

and

are given by

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

where in the case of

and

(ii) Transition rates from

where in the case of

(iii) Transition rates from

where in the case of

then

and

and

are given by

or

and

and

(iv) Transition rates from given by

where in the case of

or

then

are given by

or

then

and

and

are

or

then

and represent infinitesimal submatrices that the number of handoff calls decreases and increases by one in the buffer, respectively. is given as follows: (i) Transition rates from

(ii) Transition rates from given by

are given by

and

are

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off

277

and

is given by where E is an identity matrix. includes the case that a hand-off call is blocked upon arrival because the buffer is full. It can be obtain by the following: (i) Transition rates from

and

(ii) Transition rates from

(iii) Transition rates from

are given by

and

and

(iv) Transition rates from given by

3.2

are given by

are given by

and

are

Average Performance Measures

Let and denote the average channel utilization, which is defined as the average ratio of channels occupied by new calls (hand-off calls) to the number of channels accessible to new calls (hand-off calls), and blocking probability of new calls (hand-off calls), respectively. Using the above infinitesimal generators, we can calculate the steady-state probability, through which we can obtain the channel utilizations as follows: For N = 0:

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

For

Using the PASTA (Poisson Arrivals See Time Averages) property presented in [Wolf82], we can derive blocking probabilities as follows: For

For

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 279

where e is a column vector with elements, all of which equal 1. The blocking probability of an arbitrary call is given by

We define the average queue length E[Q] of hand-off calls to be the average number of hand-off calls in the system. It is given by

We define the average waiting time as the average time elapsed from the arrival of a call to the beginning of its transmission. The average waiting time E[W] of a hand-off call is given by

3.3 Waiting Time Distribution In this subsection we analyze the waiting time distribution of a hand-off call. Let W and denote a random variable for the waiting time of a hand-off call and the LST of the waiting time distribution, respectively. We can obtain by the following equation:

We note that formulates five independent events. The first two terms on the right-hand side of (11.39) correspond to the case that the buffer

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

is empty and the cell has several channels available to hand-off calls. Then an arriving hand-off call can immediately seize a channel for transmission. In this case the waiting time is zero. The following two terms correspond to the case that the buffer is empty but there is no channel available to the hand-off call upon arrival. As soon as one of the channels becomes free, the call seizes the channel. In this case, the time this call spent in the buffer equals the time interval until one of the busy channels becomes available to the call, which depends on the number of new calls on transmission. We note that normalized by gives the steady-state probability at arrival instant of a hand-off call because Poisson arrivals see the time average. The last term of (11.39) corresponds to the case that when a hand-off call enters the buffer, some hand-off calls are already waiting. Here we note that if the number of new calls on transmission is less than or equal to

it does not influence the waiting time of the hand-off call. Therefore we may collect the states to reduce the total number of the system states. Then we have the follows:

The matrix represents the LST of the distribution of the time interval until the system state changes due to the arrival of a call or the completion of service. has elements

and it is defined as

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 281

We denote, by Var the variance of the random variable W, and by the standard deviation of the random variable W. By differentiating we can obtain the average waiting time E[W ], E[W 2] and other higher moments. The average waiting time E[W ] is explicitly given by

where e is defined as in the previous subsection. We note that E[W] obtained in (11.42) should equal E[W] in (11.38). and E[T] are both matrices and they are given by

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

We can give the second moment of the random variable W by differentiating twice at as follows:

where

4.

is given by

Numerical Results

In this section, we present numerical results to show system performance characteristics with hand-off priority and compare them with the case without hand-off priority under the same assumptions. We define the following: the total offered traffic load to a cell is (erlangs/cell) where is given as a parameter and is given by (11.1). In Fig. 11.3, the average channel utilization is depicted as a function of the total offered traffic load ranging from 0.0 to 25.0 (erlangs/cell), and in Fig. 11.4, the blocking probability is also depicted as a function of ranging from 0.0 to 10.0, where the mean of cell sojourn time is adjusted at

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 283

to make the rates of and equal to [Kawa85] (a special case of our model without buffer). The total number of channels is As the value of 3% is often used for the performance criterion of blocking probability, we plot the vertical scale of Fig. 11.4 by the maximum value of 0.1 with log scale. We consider the cases with (i.e., and without (i.e., their own reservation channels for to examine how change the channel utilization and blocking probability with the offered traffic load. We first calculate the channel utilizations and blocking probabilities with a given set of parameters for various combinations of channel assignment as in [Kawa85]. Then we choose the sets of and in such a way that the blocking probabilities and become the closest. In Figs. 11.3 and 11.4, we show the average channel utilization and blocking probability for three sets of and (5,3)). We observe that the difference in the channel utilization among all sets of channel assignment is smaller when is small, but as increases, the difference between with and without channel assignment becomes larger and especially the difference in the case of and is larger than that in the case of and The effect of channel assignment on new calls and hand-off calls is more significant at larger traffic loads. It is also observed that is smaller than in all cases. This is due to the fact that in this numerical example, the ratio of if the other conditions are the same, so that new calls compete for all empty channels more often than hand-off calls. As can be seen from Fig. 11.4, the blocking probability for and has the largest value with all and the differences between and for the cases of and are very small. We obtain the offered load at which blocking probabilities become about 3%: at for at and at for and at and at for This result is important because if additional channels are given to hand-off calls, the channel utilization increases considerably with a little increase of blocking probabilities. Figs. 11.5 and 11.6 show the average queue length and average waiting time E[W] of a hand-off call, respectively, versus the total offered

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

traffic load between 0.0 and 25.0, where the ratio of new call load to hand-

off call load is 1.5 for the case we consider

and and in the

In case we

consider These figures exhibit: (1) with the offered traffic load, how the average queue length and waiting time change for different combinations between channels and buffer size, and (2) points where the average queue length and average waiting time increase rapidly (at about offered traffic load and 7.5) depend on channel size M , but not on buffer size N. Fig. 11.7 shows how the average waiting time E[W] of hand-off calls

changes with and without reserved channels for new calls (i.e. and We give the total offered traffic load from 0.0 to 25.0, where the load ratio of new calls to hand-off calls is the same as in Figs. 11.5 and 11.6,

and the total channel number is

We consider

six cases: and and and for respectively. It should be noted here that in all cases the average waiting time increases rapidly between and 15.0. We also observe that: (l)When all channels can be used to the maximum by hand-off calls, so the average waiting time is the smallest among all cases. On the other hand, in this case the blocking probability for new calls is the largest. (2) When all channels are used to the maximum by new calls, so the average waiting time is the largest. (3) The difference in the average waiting time between buffer sizes and 8 are very large when is large. If we increase the buffer size of hand-off calls too much, the expense of new calls will not be negligible. We can conclude that the proposed hand-off priority scheme can decrease the average waiting time of hand-off calls by properly reserving channels for the new call stream and the hand-off call stream, and the assumption of the unlimited buffer size for hand-off priority would be unrealistic. In Fig. 11.8, we show the standard deviation of waiting time of hand-off calls with the total offered traffic load for and We notice that for any buffer size N , the standard deviation of waiting time monotonously increases with This is because

when

is small, most calls are served immediately upon arrival while only

a small number of hand-off calls happen to wait in the buffer, which causes the standard deviation of the waiting time to be small. In Fig. 11.8, we also observe that the standard deviation of waiting time for is larger than that for in each case of This is because in the

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 285

case the waiting time of a hand-off call can be vary small, so that it is less

likely that some of hand-off calls happen to wait in the buffer. It is also observed that the increase of buffer size has a significant impact on the system performance. Fig. 11.8 clearly shows that in our model the waiting time of hand-off calls is heavily dependent on the offered traffic load, the number of channels and buffer size.

5.

Conclusion

In this chapter, we have presented techniques for analyzing the behavior of hand-off priority schemes. We allocated the total channels to two types of streams (new calls and hand-off calls), and gave hand-off calls a finite buffer-capacity. We presented a matrix-analytic solution to numerically evaluate the performance of the mobile radio network with hand-off stream. A non-priority scheme and a priority scheme were also compared in several cases and trade-offs between performance measures of new calls and hand-off calls were clarified in numerical examples. Using the matrixanalytic solution for the stationary state probability distribution, we further derived the higher moments of waiting time for the hand-off stream. From the numerical results and discussions we can conclude that to reduce the forced termination of calls in progress, one of the most promising treatments is to make channel reservation and buffer space for hand-off calls. The priority scheme we proposed in this chapter is useful to improve the system performance with hand-off calls. The analysis presented in this chapter will be not only useful for performance evaluation of the cellular mobile systems with hand-off and the optimum design of the system, but also applicable to performance evaluation of interconnected networks, multichannel networks or integrated networks of voice and data.

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PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Output and Delay Process Analysis of Cellular Mobile Networks with Hand-off 287

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Chapter 12

CONCLUDING REMARKS

1.

Summary of the Book Because the dependency among the hops, channels, users or cells and

multiple random access protocols complicate performance analysis, more

attention was paid to these queueing networks than to classical queueing theory. Furthermore, in multi-hop and multichannel packet radio systems, and cellular mobile wireless communication systems, many problems are

not yet modeled and analyzed to evaluate their performance. The purpose of this book was to offer detailed exact and approximate analytical solution methods and techniques using queueing theory to model the complex multimedia traffic, multichannel and multi-hop systems with procedures of multiple random access schemes and reliably evaluate the performance of the systems. In this book, we applied queueing theory to present some new efficient analytic methods for analysis and evaluation of the performances of the multi-hop and multichannel packet radio network systems and the cellular mobile telephone wireless communication system.

2.

Topics for Future Research

Wireless communications with the demand for voice, Internet, and multimedia services for fixed and mobile user have a bright future. We would like to suggest several important future research topics for wireless communication networks. We offer some discussion as to the kinds of wireless communication systems that will emerge over the next few years 289

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and our reflections on the model and methodology for these wireless communication systems as follows: (1) Ad Hoc Wireless Network Mobile computing systems based on IEEE 802.11 and HIPERLAN protocols are classified into the following two types: infrastructured networks and ad-hoc networks. An ad hoc wireless network is a collection of wireless mobile hosts forming a temporary network without the aid of any established infrastructure or centralized control. Ad hoc wireless networking is experiencing a resurgence of interest because of new applications and improved technologies. These networks are now being considered for many commercial applications, including in-home networking, wireless LANs, nomadic computing, and short-term networking for disaster relief public events, and temporary offices. Both the IEEE 802.11 and HIPERLAN wireless LAN standards support ad hoc wireless networking within a small area. It is a significant technical challenge to provide reliable high-speed end-to-end communications in ad hoc wireless networks given their dynamic network topology, decentralized control, and multi-hop connections. It is necessary to analyze the system performance and the departure processes of successful packet or message transmissions and the probability distribution of their delays employing some random multiple protocols in ad hoc network environments by application of queueing theory where we need to take into account the interdependence in the transmissions among users, and multi-hop transmissions with user mobility and user’s random multi-traffic demand. The main characteristics of ad hoc networks relevant to the performance analysis can be considered as follows: (i) Dynamic Network Topology: We all want any-where any-time communication. Ubiquitous communication has been made possible in the recent years with the advent of mobile ad hoc networks. Ad hoc networks require a peerto-peer architecture, and the dynamic topology of the network depends on the location of different mobile users, which changes randomly and rapidly over unpredictable times. Network topological changes can occur due to the breakdown of a mobile user in a hostile environment, and the failure of a connected link due to signal in-

Concluding Remarks

291

terference and changes in signal propagation conditions. Therefore, an ad hoc routing protocol must be able to dynamically update the status of its links and reconfigure itself in order to maintain strong connectivity to support communications among the users. In wireless networks, given a channel access protocol and a set of source-to-destination paths, the performance evaluations such as end-to-end throughput and delay are widely used. However, since the network topology is dynamically changing, the bandwidth and battery power are important factors in wireless ad hoc networks. In the performance analysis, we should also consider these factors such as the design of routing protocols, trade-offs in using different performance measures to make the maximum end-to-end throughput, the minimum end-to-end delay, total power, bandwidth, and the shortest path/minimum hop, and so on. (ii) Bandwidth Constraints and Variable Link Capacity: Due to the effects of multiple access, multipath fading, noise, and signal interference, the capacity of a wireless link can be degraded over time and the effective throughput may be less than the radio’s maximum transmission capacity. We should consider efficient use of the limited available bandwidth. In order to reduce interference among mobile users occurring over the same channel, a multichannel assignment algorithm should be provided, thus decreasing the possibility of affecting an existing transmission and wasting. No detailed numerical analysis and evaluation seem to have appeared on the effects of multipath fading, noise, and signal interference on the multichannel network capacity with user mobility. We have not determined what features involved in the multipath fading, noise, signal interference and user mobility are decisive to the network performance. (iii) Multi-Hop Communications: Since the propagation range of a given mobile user is limited,

the mobile user may need to enlist the aid of other mobile users in forwarding a packet to his final destination. Thus the end-to-end connection between any two mobile users may consist of multiple wireless hops. That is, mobile users that cannot reach the destination user directly will need to relay their packets through other users.

292

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Basically, there are two approaches in providing ad hoc network connectivity. One is to employ a flat-routed network architecture, and the other is to use hierarchical network architecture. In the former type of network, all the users are equal and packet routing is done based on peer-to-peer connections. On the other hand, in the latter type of network, at least one user in each lower layer is designated to serve as a gateway or coordinator to higher layers. We can model the former to be a general multi-hop queueing system and the latter to be a priority multi-hop queueing system but with some peculiar conditions. For example, since ad hoc mobile users need to relay their messages through other users toward their intended destinations, a decrease (i.e., users (hosts) can stop transmitting and/or receiving for arbitrary periods of time) in the number of mobile users can also

degrade network performance. As the number of available users (hosts) decreases, the network may also be partitioned into smaller networks. When we analyze the system, we need to obtain not only average performance measures but also higher moments of packet interdeparture times and packet delay, etc., because the output stream from one hop often forms the input stream to another taking dynamic number of available users into consideration. Moreover, in an ad hoc network the traffic generated by different sources (which are associated with several applications such as voice, video and data) must be transmitted between any two mobile users. In general, depending on the application, each type of traffic imposes different requirements on the system (such as, for example, response time). Under such a situation, a priority mechanism must be employed in order to provide a solution which satisfies the particular requirements for each source. We should consider prioritized multiple access, broadcast communication channels employing message-based and/or user-based priorities, to compare the performance of systems with different multiple access protocols.

(iv) Hidden User Problem and Exposed User Problem: More specifically, the hidden user problem still exists in multihop networks, although some standards have paid much attention

Concluding Remarks

293

to this problem such as CSMA/CA with Request-To-Send (RTS) and Clear-To-Send (CTS) in an 802.11 basic service set. But these schemes do not work well to prevent the hidden user problem in an ad hoc wireless network, because there exists simultaneously an exposed user problem. The larger interfering range, where all users do not sense each other’s transmissions, makes the hidden user problem worse. On the other hand, the larger sensing range, where any user that can possibly interfere with the reception of a packet from user A to B is within the sensing range of A, intensifies the exposed user problem. The larger interfering and sensing ranges will degrade the network performance severely in the multi-hop case. Note, however, these performance measures are not completely independent, because hidden users and exposed users behave independently, ignoring the ongoing transmissions. Which protocol will be more harmful in a multi-hop ad hoc wireless network to solve simultaneously the hidden user problem and the exposed node problem? There is no performance analysis in this standard to deal with the exposed node problem, which will be more harmful in a multi-hop ad hoc wireless network. We have not identified what features involved in the hidden user problem and the exposed node problem are decisive to the network performance. We must address the issues of the two problems together in multi-hop ad hoc wireless networks and focus on the performance analysis of hidden user configurations with the exposed node problem by use of some approach. (2) International Mobile Telecommunications 2000 (IMT-2000) IMT-2000 is a worldwide standard sponsored by the ITU to provide wireless access to the global telecommunications infrastructure through both satellite and land-based systems. IMT-2000 is designed to support a wide range of services including voice, high-rate and variable-rate data, and multimedia in both indoor and outdoor environments. A family of radio protocols suitable for a range of environments and applications is being developed to support these requirements. The goal for the protocol family is to maximize commonality within the family while maintaining flexibility to adapt to different environments and applications.

294

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

A majority of the proposals for IMT-2000 radio transmission technology is based on wideband CDMA (W-CDMA) technology. As the wireless communication and computing become more and more ubiquitous, future W-CDMA systems are required to efficiently utilize the limited wireless spectrum due to the rapidly growing demands for multimedia services. We can find some very successful applications of queueing theory to carry out performance analysis and design of the W-CDMA protocol for IMT-2000 system. (i) One of the theoretical performance studies for the W-CDMA protocol in IMT-2000 system is to consider integrated properties of both CDMA and multiple random access to achieve higher performance utilization such as we considered in Chapter 9. CDMA is being widely accepted as a system solution for next-generation mobile cellular systems, but it extends to other system aspects as well. Moreover, over the last decade, several wireless medium access control (MAC) protocols have been proposed. For example, a packet scheduling algorithm based on bit error rate (BER) requirements for CDMA systems was proposed in [Akyi99]. The algorithm is limited to only one received power level for each slot. A hybrid time-/code-division MAC protocol for a time division duplex (TDD) W-CDMA system was considered in [Huan00]. The basic idea behind the hybrid technique is to control interference in the CDMA system using TDMA-type multiplexing. A complicated

control function is required for the system. Study of performance analysis of IMT-2000 system with such protocols is still in the early stages. Almost the important characteristics of network operation have yet to be solved, and effective and practical channel access protocol in general have not yet been determined. We need clear answers including which network protocol gives the best system performance (not only averages but higher moments) under specified conditions.

(ii) Several contributions address critical issues regarding multimedia services for third generation mobile radio networks ranging from high rate data transmission with CDMA technology to resource allocation for integrated voice and data traffic. Network architectures are proposed for efficient mobility management in an Internet based

Concluding Remarks

295

mobile wireless network and for route optimization for hand-off connections in wireless ATM. User mobility, the hostile wireless propagation environment, and heterogeneous characteristics of multimedia traffic pose significant challenges in resource allocation. It is necessary to take into account to efficiently accommodate real-time voice and video traffic and non-real-time bursty data traffic between the system and end mobile users or among the end mobile users. Normally, a video source has a much higher packet generation rate than a voice source. Voice and video are real-time traffic and thus have strict transmission delay requirements. However, they can tolerate a certain level of transmission errors. On the other hand, data traffic is non real-time in nature but requires high transmission accuracy. The transmission delay requirement depends on each particular data application. Packetized transmission over wireless links can make it possible to achieve a high statistical multiplexing gain. But to simultaneously maximize wireless channel utilization and guarantee quality of service (QoS) satisfaction, the development and performance evaluation of efficient

protocols for future wireless multimedia traffic W-CDMA systems remain an open research area. (3) High Speed Digital Cellular Network A high speed digital cellular network provides voice mail, paging, and e-mail services in addition to voice. All digital cellular standards are undergoing enhancements to support high rate packet data transmission. The great popularity of high speed digital cellular network systems indicates that users are willing to tolerate inferior voice communications in exchange for mobility. Digital cellular systems can use any of the multiple access techniques described above to divide up the signal bandwidth in a given cell. Almost all the important aspects of high speed digital cellular network operation are virtually untouched. Different access systems are organized in a layered structure, which can be compared to hierarchical cell structures in cellular mobile radio systems. This concept facilitates optimum system design for different application areas, cell ranges, and radio environments, since a variety of access technologies complement each other on a common platform.

296

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

While multiple channel access in high speed digital cellular networks still remains an active research area due to the limited availability of wireless bandwidth, the absence of infrastructure makes the problem more challenging. Mobility, being one of the inherent properties of high speed digital cellular networks, results in frequent changes in the network topology, making routing in such dynamic environments more complex. Providing services in such networks while guaranteeing the performance requirements specified by the users remains an interesting and active research area. There are many more difficult problems in performance analysis of high speed digital cellular network systems related to multi-traffic, user mobility, multichannel transmissions, hand-off and so on. System performance will certainly improve as the technology and networks mature. We suggest that theoretical performance study of

high speed digital cellular network systems should be carried out much more extensively. We should show how to achieve high performance in these networks while maintaining fair sharing of network resources and demonstrate the trade-off between fairness and performance in terms of throughput achieved. On the other hand, a cellular network and an ad-hoc network will be combined into one wireless network system for the efficient use of network resources, where communication between two users in a cell is guaranteed by relaying capability of the base station in the system, and two users can directly communicate with each other while they are close to each other. The performance of such network depends on various system parameters such as the size of a cell, location of users, communication range of users, channel assignment schemes, traffic behavior and so on.

Future wireless communications systems beyond the third generation will be characterized by horizontal communication between different access technologies such as cellular, cordless, WLAN-type systems, shortrange connectivity, and wired systems (e.g., see [Mohr00]). They will be combined on a common platform to complement each other in an optimum way and satisfy different service requirements in a variety of radio environments. These access systems will be connected to a common, flexible, and seamless IP-based core network.

Concluding Remarks

297

Users will have a single number for all access technologies. A new media access system (generalized access network) connects the core network to the appropriate access technology. It also contains the mobility management. Global roaming is required for all access technologies. Key requirements include the interworking between these different access systems in terms of horizontal (intrasystem) and vertical (intersystem) handover as well as seamless services with service negotiations including mobility, security, and quality of service. They will be handled in the newly developed media access system and in the core network. The media access system connects each access system to the common core network. Rapidly growing wireless communications continue to raise new research and development problems that require unprecedented interactions among communication engineers. In particular, specialists in transmission and networks must often cross each other’s boundaries. Major challenges lie ahead, from the design of physical and radio access to network architecture, resource management, mobility management, and capacity and performance aspects. We hope that this book will be helpful for further theoretical performance study in this growing wireless communications field.

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Appendix A Derivation of Equation (5.16)

Theorem: For

the equation

has a unique solution in the unit circle

The solution

is

given by (5.17) as

Proof: Since for some

depends on

then there exists Therefore for and

299

we have

300

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

Hence, applying Rouche’s theorem [Cops48], has exactly one zero within

Appendix B Derivation of Equation (8.4)

To calculate the one-step state transition probability and the stationary probability distribution, we must map a K-multiple dimensional state space into a one-dimensional state space. In this appendix, we explain

in detail how to map the four-dimensional state space as [a, b, c, d] into the one-dimensional state space. For generality, we consider the case that we classify N users into K + 1 types. We note that in this case the state space can be expressed by K dimensions as the total number of users is fixed, so that the remaining is automatically classified into the (K + l)st type. We also note that the total number of different states is given by

Let denote the function that numbers all different

states in the K-dimensional space from 0 to

The order of

numbering is ascending, namely, [0, . . . , 0,0] is numbered 0, [0, . . . , 0,1] is numbered 1, [0, . . . , 0, N] is numbered N, [0, . . . , 1,0] is numbered N + 1, and [N, . . . , 0,0] is numbered To derive

we observe that 301

302

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

(1) the number of different states in the K-dimensional space for is given by

(2) For given be numbered as

the state of

Therefore the following recursive equation holds true:

From (B.1), we obtain

should

APPENDIX B: Derivation of Equation (8.4)

303

By calculating (B.4), we can obtain f(N : a, b, c, d) in Section 3 of Chapter 8, Performance Analysis.

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Appendix C Numerical Calculation Method to Solve Row Vector x in Chapter 11

In this appendix we present a numerical calculation method to solve x in

Chapter 11. As

and

we have

where e is a column vector with N elements, all of which equal 1. From (C.4), we obtain

305

306

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

where From (C.3), we have

where

where

where

where According to (C.1), we have

where We plug:

into (C.2) and after some calculations, we have

In summary the calculation algorithm is given as follows: [Algorithm] [Step 1] Start calculation from G1, then G2,..., GN – 2 and GN –1

APPENDIX C: Numerical Calculation Method to Solve Row Vector x in Chapter 11307

[Step 2] Solve

with a condition

[Step 3] Solve to get

[Step 4] Normalize

as

to get

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Index

Ad hoc wireless network, 290 bandwidth constraint, 291 dynamic network topology, 290 end-to-end connection, 291 flat-routed network, 292 hierarchical network, 292

multi-hop ad hoc, 293

multiuser capability, 222 spread-spectrum signal, 224 voice code, 224 wideband signal, 225 Cellular mobile networks, 7, 249, 269 call holding time, 269 cell radius, 248, 269

variable link capacity, 291

cell sojourn time, 269

ARPANET, 2

channel assignment algorithm, 249

Average number of backlogged users, 180, 201

co-channel interference, 248, 269 co-channel reuse distance, 26 co-channel reuse ratio, 248, 269 DCA scheme, 27, 263 dynamic channel, 249 FCA scheme, 26, 250, 269 fixed channel, 249 frequency reuse, 7 hand-off rate, 269 hand-off, 29, 270 HCA scheme, 27, 252, 255

Average number of blocking calls, 232 Average number of collision packets, 40,96 Average performance measures, 51, 76, 94, 132, 155, 179, 200, 231, 277 Balance equation, 52, 76, 97, 233

Bernoulli arrival, 118 Binary random variable, 118 Binomial moment, 257 Blocking probability, 40, 232, 252, 278 Boundary function, 120 Branching probability, 118 Buffer capacity, 116 Buffer size, 34 Burst stream, 237 CDMA, xxii admission control, 224 allocated resource, 222

bandwidth-time space, 225 call initiation, 223 call termination, 223 desired signal power, 225 frequency band, 221 hop frequency, 225 hopping pattern, 225 integrated voice and data, 224

macrocell, 29 microcell, 29 overflow traffic, 250 picocell, 29 probability of occupied channels, 255 Channel state transition, 50, 89, 128, 174, 192

Channel utilization, 40, 52, 76, 96, 155, 180, 200, 277

Closed-form expression, 46, 94, 102, 152, 203, 231, 236

Coefficient of variation, 41, 66, 103, 163, 203, 236 Composite IPP stream, 252 Continuous-time Markov chain, 31, 119, 129, 255, 271 Correlation coefficient, 94, 231

321

322

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

CSMA/BT-CD, 25 CSMA/CA

clear-to-send, 293 collision avoidance period, 171, 188

collision avoidance portion, 170, 187 priority based CSMA/CA, 188 request-to-send, 293 CSMA/CD collision detection time, 153 CSMA/CD-IC, 151 CSMA/CD-RC, 151 end-to-end propagation delay, 15–16

propagation delay, 162 DFT, 42, 48, 72, 173, 190

Dirac’s delta, 175, 230 Discrete-time Markov chain, 31, 48, 74, 87, 153, 174, 191, 227 DQDB, 4

ITU, 237 Joint moment generating function, 92, 231 Joint probability distribution, 55, 157 average, 59, 159 moment, 59, 159

packet departure, 56, 157 packet interdeparture time, 56, 157 LAN, 4 Laplace-Stieltjes transform, 40, 252, 279 Little’s result, 123 Loss probability, 40, 97 M/G/1, 115 MAN, 4

Marginal generating function, 120 Matrix-analytic method, 268 Method of decomposition, 127 Mobile packet radio networks, 8 capture, 20

DSSS, 9

channel noise, 8

End-to-end propagation delay, 153, 172, 189

CSMA, 19 CSMA/CD, 20 error rate, 8 transmission error, 8

Ergodic, 119, 257

Erlang B formula, 27, 250 Exposed user problem, 293 Factorial moment, 41, 59, 94, 102, 202, 231, 236 FDDI, 4

Moment generating function, 40, 53, 60, 98, 155–156, 160, 201, 233

FH/FSK CSMA/CD, 152

Moments, 40

FHSS, 9 Finite population, 48, 72, 85, 152, 172, 189, 225

Multi-hop networks, 1 CDMA, 25

Fully connected networks, 1 Generalized access network, 297

CSMA, 25

Generalized GI/M/m(m), 247

emission rate, 121

Generalized M/G/1, 127 Geographic service area, 7 Hidden user problem, 292

CSMA/CD, 25 error rate, 118

fully connected three-hop network, 119 idle period, 129 internal arrival, 127

High speed digital cellular network, 295 HIPERLAN, 9, 290 IEEE 802.11, 9, 18, 169, 187, 290 IFT, 42, 48, 72, 86, 173, 190, 226 Imbedded Markov point, 49, 74, 89, 128–129, 153, 173, 192, 227, 255

routing polynomial, 119 slotted ALOHA, 24, 116

IMT-2000, 11, 237, 293

tandem N-hop network, 126

Independently identically distributed, 117 Infinitesimal submatrix, 275 Interarrival time distribution, 251 Interdeparture time distribution, 33, 52, 155, 201

average, 55, 157, 203 moment, 55, 157, 203 Internet, 289

Interrupted Poisson process, 250 factorial moment, 251 first three moments, 252 off-time, 251 on-time, 251

parallel network, 139 priority based slotted ALOHA, 115 routing policy, 123

tandem three-hop network, 123 transmission error, 117, 123 transmission period, 129 Multi-traffic

active duration, 237 data traffic, 85, 188, 225 data transmission mode, 86, 226 idle mode, 86, 225 real-time bursty traffic, 85, 188, 225

silent gap duration, 103, 204, 237 silent mode, 225 spatial isolation factor, 223

323

INDEX

speech mode, 86

talkspurt duration, 103, 204, 237 talkspurt mode, 225 voice activity factor, 223 Multichannel networks backlogged state, 48, 72 capture ratio, 73 capture restriction parameter, 74 capture, 71 channel bandwidth, 5, 10, 65, 77, 103, 162 CSMA, 22 CSMA/CD, 22, 152

FM receiver, 73 idle state, 48, 72 integrated voice and data, 84 LAN, 152 noise level, 73 power level, 73

slotted ALOHA, 22, 46, 71, 84 system configuration, 47 Multimedia service, 289 Multiple access network, 1 Multiple access protocols, 9 fixed assignment protocols, 10 CDMA, 11 FDMA, 10 TDMA, 10

random access protocols, 11 CSMA, 14,,, CSMA/CA, 18 CSMA/CD, 14

pure ALOHA, 13 reservation multiple access, 18 slotted ALOHA, 14 Non-persistent CSMA/CA, 172, 189 Nonreservation system, 86 Normalization condition, 131 Packet communication networks, 3

multi-hop, 6 multichannel, 5 one-hop, 6 single channel, 4 Packet delay distribution, 33, 60, 98, 160, 233 average, 64, 103, 236

backlog delay, 60, 98, 160, 233 initial delay, 60, 98, 160, 233 moment, 64, 103, 236

Peak-to-average, 221

Performance measures, 39 Personal communication networks, 7, 221 CDMA slotted ALOHA, 222 priority based CDMA, 224 PN code, 11 Poisson arrival, 118 Poisson arrivals see time averages, 278, 280

Poisson process, 126, 133, 250, 268 Poisson stream, 115, 134, 269 hand-off call traffic, 269

new call traffic, 269 Population size, 34

PRMA, 18 Probability density function, 40 Probability distribution function, 252 Probability distributions, 33 generating functions, 33

input process, 33 output process, 33 Probability generating function, 117 Pseudo-random noise, 11 QoS, 295 Queue length, 123, 132, 279 Renewal cycle, 48, 72 Reservation system, 87

reservation channel, 87

selection probability, 87

Retransmission probability, 48, 73 Rouche’s theorem, 121, 300 SSMA, 20 Standard deviation, 281 State transition diagram, 86, 175–177, 193, 195, 197, 226, 272 State transition probability, 51, 74, 89, 154, 179, 199, 230, 257

State transition rate diagram, 122, 253 Stationary probability distribution, 51, 76, 90, 92, 155, 179, 200, 230, 257, 271

Store-and-forward packet switching, 149 Throughput, 40, 51, 76, 94, 155, 179, 200, 232

Topographical database, 74 Variance, 41 W-CDMA, 294 bit error rate, 294 MAC, 294 TDD, 294 Waiting time distribution, 279

normalizing constant, 63, 102, 235 tagged packet, 60, 233 tagged user, 60, 98, 160 Packet delay, 40, 52, 76, 97, 123, 180, 201, 232

average, 281 moment, 282 Waiting time, 279

Packet departure distribution, 92, 155, 230 Packet departure, 92, 155, 231

Wireless communication networks, 7 engineered resources, 30

324

PERFORMANCE ANALYSIS OF MULTICHANNEL AND MULTI-TRAFFIC

external traffic load, 30 multi-traffic, 30

collision avoidance portion, 18 CSMA/CA, 170, 187

multichannel, 30

first state transition, 174, 193

multimedia service, 30 WLANs, 8, 169 active mode, 173, 190 backlogged state, 172, 190

idle state, 172, 190

integrated voice and data, 187 second state transition, 174, 193 third state transition, 174, 193 transmission time, 172, 190

backoff mode, 173, 190

WWW, xix

collision avoidance period, 172

Z-transform, 132, 135