MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS

JEREMIAH F. HAYES

THIMMA V. J. GANESH BABU

A JOHN WILEY & SONS, INC., PUBLICATION

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MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS

MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS

JEREMIAH F. HAYES

THIMMA V. J. GANESH BABU

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright # 2004 by John Wiley & Sons, Inc. All rights reserved

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Library of Congress Cataloging-in-Publication Data:

Hayes, Jeremiah F., 1934-

Modeling and analysis of telecommunications networks/Jeremiah F.

Hayes & Thimma V. J. Ganesh Babu.

p. cm.

A Wiley-Interscience Publication.

ISBN 0-471-34845-7 (Cloth)

1. Telecommunication systems. I. Babu, Thimma V. J. Ganesh. II. Title.

TK5101 .H39 2004

621.38201dc222003020806

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should

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To the Older Generation

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CONTENTS

Preface xiii

Retrieving Files from the Wiley FTP and Internet Sites xix

1 Performance Evaluation in Telecommunications 1

1.1 Introduction: The Telephone Network, 1

1.1.1 Customer Premises Equipment, 1

1.1.2 The Local Network, 2

1.1.3 Long-Haul Network, 4

1.1.4 Switching, 4

1.1.5 The Functional Organization of Network Protocols, 6

1.2 Approaches to Performance Evaluation, 8

1.3 Queueing Models, 9

1.3.1 Basic Form, 9

1.3.2 A Brief Historical Sketch, 10

1.4 Computational Tools, 13

Further Reading, 14

2 Probability and Random Processes Review 17

2.1 Basic Relations, 17

2.1.1 Set Functions and the Axioms of Probability, 17

2.1.2 Conditional Probability and Independence, 20

2.1.3 The Law of Total Probability and Bayes Rule, 21

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2.2 Random VariablesProbability Distributions and Densities, 22

2.2.1 The Cumulative Distribution Function, 22

2.2.2 Discrete Random Variables, 23

2.2.3 Continuous Random Variables, 31

2.3 Joint Distributions of Random Variables, 38

2.3.1 Probability Distributions, 38

2.3.2 Joint Moments, 40

2.3.3 Autocorrelation and Autocovariance Functions, 41

2.4 Linear Transformations, 42

2.4.1 Single Variable, 42

2.4.2 Sums of Random Variables, 42

2.5 Transformed Distributions, 46

2.6 Inequalities and Bounds, 47

2.7 Markov Chains, 52

2.7.1 The Memoryless Property, 52

2.7.2 State Transition Matrix, 53

2.7.3 Steady-State Distribution, 56

2.8 Random Processes, 61

2.8.1 Defintion: Ensemble of Functions, 61

2.8.2 Stationarity and Ergodicity, 61

2.8.3 Markov Processes, 63

References, 64

Exercises, 64

3 Application of Birth and Death Processes to Queueing Theory 67

3.1 Elements of the Queueing Model, 67

3.2 Littles Formula, 69

3.2.1 A Heuristic, 69

3.2.2 Graphical Proof, 70

3.2.3 Basic Relationship for the Single-Server Queue, 73

3.3 The Poisson Process, 74

3.3.1 Basic Properties, 74

3.3.2 Alternative Characterizations of the Poisson Process, 75

3.3.3 Adding and Splitting Poisson Processes, 78

3.3.4 Pure Birth Processes, 79

3.3.5 Poisson Arrivals See Time Averages (PASTA), 81

3.4 Birth and Death Processes: Application to Queueing, 82

3.4.1 Steady-State Solution, 82

3.4.2 Queueing Models, 85

3.4.3 The M/M/1 QueueInfinite Waiting Room, 863.4.4 The M/M/1/L QueueFinite Waiting Room, 893.4.5 The M/M/S QueueInfinite Waiting Room, 913.4.6 The M/M/S/L QueueFinite Waiting Room, 953.4.7 Finite Sources, 97

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3.5 Method of Stages, 98

3.5.1 Laplace Transform and Averages, 98

3.5.2 Insensitivity Property of Erlang B, 100

3.5.3 The Erlang B Blocking Formula: N Lines, Homogeneous Traffic, 103

References, 106

Exercises, 106

4 Networks of Queues: Product Form Solution 113

4.1 Introduction: Jackson Networks, 113

4.2 Reversibility: Burkes Theorem, 114

4.2.1 Reversibility Defined, 114

4.2.2 Reversibility and Birth and Death Processes, 116

4.2.3 Departure Process from the M/M/S Queue: Burkes Theorem, 118

4.3 Feedforward Networks, 119

4.3.1 A Two-Node Example, 119

4.3.2 Feedforward Networks: Application of Burkes Theorem, 120

4.3.3 The Traffic Equation, 121

4.4 Product Form Solution for Open Networks, 123

4.4.1 Flows Within Feedback Paths, 123

4.4.2 Detailed Derivation for a Two-Node Network, 124

4.4.3 N-Node Open Jackson Networks, 127

4.4.4 Average Message Delay in Open Networks, 132

4.4.5 Store-and-Forward Message-Switched Networks, 134

4.4.6 Capacity Allocation, 138

4.5 Closed Jackson Networks, 139

4.5.1 Traffic Equation, 139

4.5.2 Global Balance EquationSolution, 141

4.5.3 Normalization ConstantConvolution Algorithm, 142

4.5.4 Extension to the Infinite Server Case, 146

4.5.5 Mean Value Analysis of Closed Chains, 147

4.5.6 Application to General Networks, 149

4.6 BCMP Networks, 150

4.6.1 Overview of BCMP Networks, 150

4.6.2 Single NodeExponential Server, 151

4.6.3 Single NodeInfinite Server, 152

4.6.4 Single NodeProcessor Sharing, 156

4.6.5 Single NodeLast Come First Served (LCFS), 158

4.7 Networks of BCMP Queues, 161

4.7.1 Store-and-Forward Message-Switched Nodes, 163

4.7.2 Example: Window Flow ControlA Closed Network Model, 170

4.7.3 Cellular Radio, 175

References, 178

Exercises, 179

CONTENTS ix

5 Markov Chains: Application to Multiplexing and Access 187

5.1 Time-Division Multiplexing, 187

5.2 The Arrival Process, 188

5.2.1 Packetization, 188

5.2.2 Compound Arrivals, 189

5.3 Asynchronous Time-Division Multiplexing, 190

5.3.1 Finite Buffer, 192

5.3.2 Infinite Buffer, 195

5.4 Synchronous Time-Division Multiplexing, 197

5.4.1 Application of Rouches Theorem, 199

5.4.2 Calculations Involving Rouches Theorem, 201

5.4.3 Message Delay, 203

5.5 Random Access Techniques, 207

5.5.1 Introduction to ALOHA, 207

5.5.2 Analysis of Delay, 210

References, 215

Exercises, 216

6 The M/G/1 Queue: Imbedded Markov Chains 2196.1 The M/G/1 Queue, 219

6.1.1 Imbedded Markov Chains, 220

6.1.2 Distribution of Message Delay: FCFS, 222

6.1.3 Residual Life Distribution: Alternate Derivation of

the PollaczekKhinchin Formula, 231

6.1.4 Variation for the Initiator of a Busy Period, 234

6.1.5 Busy Period of the M/G/1 Queue, 2376.2 The G/M/1 Queue, 2416.3 Priority Queues, 244

6.3.1 Preemptive Resume Discipline, 245

6.3.2 L-Priority Classes, 252

6.3.3 Nonpreemptive Priorities, 256

6.4 Polling, 265

6.4.1 Basic Model: Applications, 265

6.4.2 Average Cycle Time, 267

6.4.3 Average Delay: Exhaustive, Gated, and Limited Service, 267

References, 274

Exercises, 275

7 Fluid Flow Analysis 281

7.1 OnOff Sources, 281

7.1.1 Single Source, 281

7.1.2 Multiple Sources, 284

7.2 Infinite Buffers, 286

7.2.1 The Differential Equation for Buffer Occupancy, 286

x CONTENTS

7.2.2 Derivation of Eigenvalues, 289

7.2.3 Derivation of the Eigenvectors, 292

7.2.4 Derivation of Coefficients, 295

7.3 Finite Buffers, 298

7.4 More General Sources, 300

7.5 Analysis: Leaky Bucket, 300

7.6 Equivalent Bandwidth, 303

7.7 Long-Range-Dependent Traffic, 304

7.7.1 Definitions, 304

7.7.2 AMatching Technique for LRD Traffic Using the Fluid FlowModel, 306

References, 309

Exercises, 310

8 The Matrix Geometric Techniques 313

8.1 Introduction, 313

8.2 Arrival Processes, 313

8.2.1 The Markov Modulated Poisson Process (MMPP), 314

8.2.2 The Batch Markov Arrival Process, 316

8.2.3 Further Extensions, 319

8.2.4 Solutions of Forward Equation for the Arrival Process, 319

8.3 Imbedded Markov Chain Analysis, 321

8.3.1 Revisiting the M/G/1 Queue, 3218.3.2 The Multidimensional Case, 323

8.3.3 Application of Renewal Theory, 328

8.3.4 Moments at Message Departure, 334

8.3.5 Steady-State Queue Length at Arbitrary Points in Time, 335

8.3.6 Moments of the Queue Length at Arbitrary Points in Time, 336

8.3.7 Virtual Waiting Time, 336

8.4 A Matching Technique for LRD Traffic, 337

8.4.1 d MMPPs and Equivalents, 337

8.4.2 A Fitting Algorithm, 339

Appendix 8A: Derivation of Several Basic Equations Used in Text, 343

Appendix 8B: Derivation of Variance and Covariance Functions of Two-State

MMPP, 347

References, 355

Exercises, 355

9 Monte Carlo Simulation 359

9.1 Simulation and Statistics, 359

9.1.1 Introduction, 359

9.1.2 Sample Mean and Sample Variance, 359

9.1.3 Confidence Intervals, 361

9.1.4 Sample Sizes and Run Times, 362

9.1.5 Histograms, 364

CONTENTS xi

9.1.6 Hypothesis Testing and the Chi-Square Test, 368

9.2 Random-Number Generation, 370

9.2.1 Pseudorandom Numbers, 370

9.2.2 Generation of Continuous Random Variables, 371

9.2.3 Discrete Random VariablesGeneral Case, 375

9.2.4 Generating Specific Discrete Random Variables, 377

9.2.5 The Chi-Square Test Revisited, 379

9.3 Discrete-Event Simulation, 380

9.3.1 Time-Driven Simulation, 380

9.3.2 Event-Driven Simulation, 381

9.4 Variance Reduction Techniques, 382

9.4.1 Common Random-Number Technique, 383

9.4.2 Antithetic Variates, 384

9.4.3 Control Variates, 385

9.4.4 Importance Sampling, 386

References, 387

Exercises, 387

Index 389

xii CONTENTS

PREFACE

BACKGROUND

The insinuation of telecommunications into the daily fabric of our lives has been

arguably the most important and surprising development of the last 25 years. Before

this revolution, telephone service and its place in our lives had been largely stable

for more than a generation. The growth was, so to speak, lateral, as the global reach

of telecommunications extended and more people got telephone service. The

distinction between oversea and domestic calls blurred with the advances in

switching and transmission, undersea cable, and communication satellites. Traffic

on the network remained overwhelmingly voice, largely in analog format with

facsimile (Fax) beginning to make inroads. A relatively small amount of data traffic

was carried by modems operating at rates up to 9600 bits per second over voice

connections. Multiplexing of signals was rudimentarymost connections were

point-to-point business applications.

The contrast with todays network is overwhelming. The conversion from analog

to digital has long since been completed. A wide range of services, each with its

unique set of traffic characteristics and performance requirements, are available. At

the core of the change is the Internet, which is becoming accepted to handle all

telecommunications traffic and functions.

In order to effect such a far-reaching change, many streams converged. At the

most basic level was the explosive growth of the technology. The digital switching

and processing that are intrinsic to the modern network are possible only through

integrated-circuit technology. There are any number of examples of this technology

at work. For one who has worked in data communications, the most striking is the

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