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JOHN · Statistical Methods in Engineering and Quality Assurance

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JOHNSON and BALAKRISHNAN · Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz

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JOHNSON and KOTZ · Distributions in Statistics

JOHNSON and KOTZ (editors) · Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present

JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 1, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 2, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN · Discrete Multivariate Distributions

JOHNSON, KEMP, and KOTZ · Univariate Discrete Distributions, Third Edition

JUDGE, GRIFFITHS, HILL, LÜTKEPOHL, and LEE · The Theory and Practice of Econometrics, Second Edition

JUREČKOVÁ and SEN · Robust Statistical Procedures: Aymptotics and Interrelations

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KADANE AND SCHUM · A Probabilistic Analysis of the Sacco and Vanzetti Evidence

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KARIYA and KURATA · Generalized Least Squares KASS and VOS · Geometrical Foundations of Asymptotic Inference

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† SEARLE · Linear Models for Unbalanced Data

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Library of Congress Cataloging-in-Publication Data

Gross, Donald.
Fundamentals of queueing theory/Donald Gross, John F. Shortle, Carl M. Harris. — 4th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-471-79127-0 (cloth)
1. Queuing theory. I. Shortle, John F., 1969- II. Harris, Carl M., 1940- III. Title. T57.9.G76 2008
519.8′2—dc22
2008003734

Carl Harris, 1940–2000

This book is dedicated to the memory of Carl M. Harris. Carl and I first entertained the idea of a queueing text back in 1968 and collaborated on the first three editions. We were friends and colleagues from that time until his untimely death from a heart attack while exercising at a local gym, one month after his 60th birthday.

Carl was the BDM International Professor of Operations Research and the founding chair of the Systems Engineering and Operations Research Department for the Volgenau School of Information Technology and Engineering at George Mason University, Fairfax, Virginia. In 1999, he was awarded the Institute for Operations Research and the Management Sciences (INFORMS) Kimball Medal in recognition of distinguished service to the Operations Research profession and the Operations Research Society of America (INFORMS’ predecessor). He was the society’s 39th president. Carl’s research interests were in the areas of applied probability and statistics, particularly queueing theory and stochastic processes. He authored or co-authored about 80 scholarly papers (on many of which I was fortunate enough to have worked with him.) In addition to the first 3 editions of this book, he co-authored with Saul I. Gass, The Encyclopedia of Operations Research and Management Science.

His warmth, collegiality, and friendship are sorely missed.

Donald Gross

PREFACE

The changes in this fourth edition reflect the feedback from numerous students, teachers, and colleagues since the third edition came out ten years ago. Almost all the material from the third edition is kept in the fourth, however, with a fair amount of editing and reorganization.

Chapter 2 contains a new section on choosing the number of servers and a new subsection on computational issues of the Erlang B formula. The chapter now begins with a section on birth-death processes, which was the old Section 1.10 from the previous edition. Chapter 3 is substantially edited and contains a new section on retrial queues. Chapter 5 contains an expanded discussion of the level crossing method developed by Percy Brill. Chapter 7 is now split into two separate chapters: Chapter 7, Bounds and Approximations, and Chapter 8, Numerical Techniques and Simulation. Chapter 7 includes a new section on network approximations, and Chapter 8 includes a new section on numerical inversion of transforms.

Two appendices are added back to this edition, one on transforms and generating functions and the other on differential and difference equations (by popular request). The appendix on the QtsPlus software is completely rewritten to reflect the changes and expansion made to the software. Also, a subsection on how to use the software is added to Chapter 1. Finally, many more examples and problems are added. In this edition we do not denote which problems are solvable on the computer — we leave that up to the discretion of the student and/or instructor. We also do not include here a table on suggested text material for various course lengths (quarter, semester, etc.). Again, we believe that the instructor is the best one to decide.

For errata, updates, and other information about the text and associated QtsPlus software, see the text website:

http://mason.gmu.edu/jshortle/fqt4th.xhtml.

Donald Gross
John F. Shortle
James M. Thompson

Fairfax, Virginia
March 2008

ACKNOWLEDGMENTS

We are grateful for the assistance given to us over the years by many professional colleagues and students, whose numerous comments and suggestions have been so helpful in improving our work. We particularly thank Prof. Andrew Ross, Eastern Michigan University, Prof. Percy Brill, University of Windsor, and Prof. John Mullen, New Mexico State University, for their very detailed comments, many of which are reflected in this edition. We also appreciate the help and encouragement of our research colleagues, Dr. Martin Fischer and Dr. Denise Masi of Noblis (formerly Mitretek Systems) and Prof. Brian Mark of George Mason University.

With heartfelt thanks, we extend special appreciation once more to our families for their unlimited and continuing encouragement and to all the people at John Wiley & Sons who have been wonderfully supportive of us through these four editions. We also appreciate the support of the Volgenau School of Information Technology and Engineering and the Department of Systems Engineering and Operations Research at George Mason University.

D. G.
J. F. S.
J. M. T.

CHAPTER 1

INTRODUCTION

All of us have experienced the annoyance of having to wait in line. Unfortunately, this phenomenon continues to be common in congested, urbanized, “high-tech” societies. We wait in line in our cars in traffic jams or at toll booths; we wait on hold for an operator to pick up our telephone calls; we wait in line at supermarkets to check out; we wait in line at fast-food restaurants; and we wait in line at banks and post offices. We, as customers, do not generally like these waits, and the managers of the establishments at which we wait also do not like us to wait, since it may cost them business. Why then is there waiting?

The answer is simple: There is more demand for service than there is facility for service available. Why is this so? There may be many reasons; for example, there may be a shortage of available servers, it may be infeasible economically for a business to provide the level of service necessary to prevent waiting, or there may be a space limit to the amount of service that can be provided. Generally these limitations can be removed with the expenditure of capital, and to know how much service should then be made available, one would need to know answers to such questions as, “How long must a customer wait?” and “How many people will form in the line?” Queueing theory attempts (and in many cases succeeds) to answer these questions through detailed mathematical analysis. The word “queue” is in more common usage in Great Britain and other countries than in the United States, but it is rapidly gaining acceptance in this country, although it must be admitted that it is just as displeasing to spend time in a queue as in a waiting line.

1.1 Description of the Queueing Problem

A queueing system can be described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served. The term “customer” is used in a general sense and does not imply necessarily a human customer. For example, a customer could be a ball bearing waiting to be polished, an airplane waiting in line to take off, or a computer program waiting to be run. Such a basic system can be schematically shown as in Figure 1.1. Although any queueing system may be diagrammed in this manner, it should be clear that a reasonably accurate representation of such a system would require a detailed characterization of the underlying processes.

Figure 1.1 A typical queueing process.

Queueing theory was developed to provide models to predict the behavior of systems that attempt to provide service for randomly arising demands; not unnaturally, then, the earliest problems studied where those of telephone traffic congestion. The pioneer investigator was the Danish mathematician A. K. Erlang, who, in 1909, published “The Theory of Probabilities and Telephone Conversations.” In later works he observed that a telephone system was generally characterized by either (1) Poisson input, exponential holding (service) times, and multiple channels (servers), or (2) Poisson input, constant holding times, and a single channel. Erlang was also responsible for the notion of stationary equilibrium, for the introduction of the so-called balance-of-state equations, and for the first consideration of the optimization of a queueing system.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!