Fundamentals of Queueing Theory, Fifth Edition
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The definitive guide to queueing theory and its practical applications—features numerous real-world examples of scientific, engineering, and business applications

Thoroughly updated and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fifth Edition presents the statistical principles and processes involved in the analysis of the probabilistic nature of queues. Rather than focus narrowly on a particular application area, the authors illustrate the theory in practice across a range of fields, from computer science and various engineering disciplines to business and operations research. Critically, the text also provides a numerical approach to understanding and making estimations with queueing theory and provides comprehensive coverage of both simple and advanced queueing models. As with all preceding editions, this latest update of the classic text features a unique blend of the theoretical and timely real-world applications. The introductory section has been reorganized with expanded coverage of qualitative/non-mathematical approaches to queueing theory, including a high-level description of queues in everyday life. New sections on non-stationary fluid queues, fairness in queueing, and Little’s Law have been added, as has expanded coverage of stochastic processes, including the Poisson process and Markov chains.

• Each chapter provides a self-contained presentation of key concepts and formulas, to allow readers to focus independently on topics relevant to their interests

• A summary table at the end of the book outlines the queues that have been discussed and the types of results that have been obtained for each queue

• Examples from a range of disciplines highlight practical issues often encountered when applying the theory to real-world problems

• A companion website features QtsPlus, an Excel-based software platform that provides computer-based solutions for most queueing models presented in the book.

Featuring chapter-end exercises and problems—all of which have been classroom-tested and refined by the authors in advanced undergraduate and graduate-level courses—Fundamentals of Queueing Theory, Fifth Edition is an ideal textbook for courses in applied mathematics, queueing theory, probability and statistics, and stochastic processes. This book is also a valuable reference for practitioners in applied mathematics, operations research, engineering, and industrial engineering.


John F. Shortle, PhD, is Professor in the Department of Systems Engineering and Operations Research at George Mason University.

James M. Thompson is an Enterprise Architect at the Federal Home Loan Mortgage Corporation.

Donald Gross, PhD, is Professor emeritus, The George Washington University, and was Distinguished Research Professor of Operations Research and Engineering at George Mason University.

The late Carl M. Harris, PhD, was BDM International Professor and the founding chair of the Systems Engineering and Operations Research Department at George Mason University.


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 Qts Plus 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 SemiMarkov 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