Fundamentals of Queueing Theory

Preface ix

Acknowledgments xi

About the Companion Website xiii

1 Introduction 1

1.1 Measures of System Performance 2

1.2 Characteristics of Queueing Systems 4

1.3 The Experience of Waiting 9

1.4 Little's Law 10

1.5 General Results 19

1.6 Simple Bookkeeping for Queues 22

1.7 Introduction to the QtsPlus Software 26

Problems 27

2 Review of Stochastic Processes 35

2.1 The Exponential Distribution 35

2.2 The Poisson Process 39

2.3 Discrete-Time Markov Chains 49

2.4 Continuous-Time Markov Chains 62

Problems 69

3 Simple Markovian Queueing Models 73

3.1 Birth-Death Processes 73

3.2 Single-Server Queues (M=M=1) 77

3.3 Multiserver Queues (M=M=c) 90

3.4 Choosing the Number of Servers 97

3.5 Queues with Truncation (M=M=c=K) 100

3.6 Erlang's Loss Formula (M=M=c=c) 105

3.7 Queues with Unlimited Service (M=M=1) 108

3.8 Finite-Source Queues 109

3.9 State-Dependent Service 115

3.10 Queues with Impatience 119

3.11 Transient Behavior 121

3.12 Busy-Period Analysis 126

Problems 127

4 Advanced Markovian Queueing Models 147

4.1 Bulk Input (M[X]=M=1) 147

4.2 Bulk Service (M=M[Y ]=1) 153

4.3 Erlang Models 158

4.4 Priority Queue Disciplines 172

4.5 Retrial Queues 191

Problems 204

5 Networks, Series, and Cyclic Queues 213

5.1 Series Queues 215

5.2 Open Jackson Networks 221

5.3 Closed Jackson Networks 229

5.4 Cyclic Queues 243

5.5 Extensions of Jackson Networks 244

5.6 NonJackson Networks 246

Problems 248

6 General Arrival or Service Patterns 255

6.1 General Service, Single Server (M=G=1) 255

6.2 General Service, Multiserver (M=G=c=_,M=G=1) 290

6.3 General Input (G=M=1, G=M=c) 295

Problems 306

7 General Models and Theoretical Topics 313

7.1 G=Ek=1, G[k]=M=1, and G=PHk=1 313

7.2 General Input, General Service (G=G=1) 320

7.3 Poisson Input, Constant Service, Multiserver (M=D=c) 330

7.4 Semi-Markov and Markov Renewal Processes in Queueing 332

7.5 Other Queue Disciplines 337

7.6 Design and Control of Queues 342

7.7 Statistical Inference in Queueing 353

Problems 361

8 Bounds and Approximations 365

8.1 Bounds 366

8.2 Approximations 378

8.3 Deterministic Fluid Queues 392

8.4 Network Approximations 400

Problems 411

9 Numerical Techniques and Simulation 417

9.1 Numerical Techniques 417

9.2 Numerical Inversion of Transforms 433

9.3 Discrete-Event Stochastic Simulation 446

Problems 469

References 475

Appendix A: Symbols and Abbreviations 487

Appendix B: Tables 495

Appendix C: Transforms and Generating Functions 503

C.1 Laplace Transforms 503

C.2 Generating Functions 510

Appendix D: Differential and Difference Equations 515

D.1 Ordinary Differential Equations 515

D.2 Difference Equations 531

Appendix E: QtsPlus Software 537

E.1 Instructions for Downloading 540

Index 541

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