Introduction_to_Probability_Models_11th_edition.pdf

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Preface

1 Introduction to Probability Theory

1.1 Introduction

1.2 Sample Space and Events

1.3 Probabilities Defined on Events

1.4 Conditional Probabilities

1.5 Independent Events

Exercises

References

2 Random Variables

3 Conditional Probability and Conditional Expectation

4 Markov Chains

5 The Exponential Distribution and the Poisson Process

6 Continuous-Time Markov Chains

7 Renewal Theory and Its Applications

8 Queueing Theory

9 Reliability Theory

10 Brownian Motion and Stationary Processes

11 Simulation

17 Solutions to Starred Exercises

Introduction to Probability Models Eleventh Edition

Introduction to Probability Models Eleventh Edition Sheldon M. Ross University of Southern California Los Angeles, California AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE SYDNEY • TOKYO Academic Press is an Imprint of Elsevier

Academic Press is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA Eleventh edition 2014 Tenth Edition: 2010 Ninth Edition: 2007 Eighth Edition: 2003, 2000, 1997, 1993, 1989, 1985, 1980, 1972 Copyright © 2014 Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/ locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data Ross, Sheldon M., author. Introduction to probability models / by Sheldon Ross. – Eleventh edition. pages cm Includes bibliographical references and index. ISBN 978-0-12-407948-9 1. Probabilities. I. Title. QA273.R84 2014 519.2–dc23 2013035819 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-407948-9 For information on all Academic Press publications visit our web site at store.elsevier.com Printed and bound in USA 14 15 16 17 18 10 9 8 7 6 5 4 3 2 1

Preface This text is intended as an introduction to elementary probability theory and stochastic processes. It is particularly well suited for those wanting to see how probability theory can be applied to the study of phenomena in fields such as engineering, computer sci- ence, management science, the physical and social sciences, and operations research. It is generally felt that there are two approaches to the study of probability theory. One approach is heuristic and nonrigorous and attempts to develop in the student an intuitive feel for the subject that enables him or her to “think probabilistically.” The other approach attempts a rigorous development of probability by using the tools of measure theory. It is the first approach that is employed in this text. However, because it is extremely important in both understanding and applying probability theory to be able to “think probabilistically,” this text should also be useful to students interested primarily in the second approach. New to This Edition The tenth edition includes new text material, examples, and exercises chosen not only for their inherent interest and applicability but also for their usefulness in strengthen- ing the reader’s probabilistic knowledge and intuition. The new text material includes Section 2.7, which builds on the inclusion/exclusion identity to find the distribution of the number of events that occur; and Section 3.6.6 on left skip free random walks, which can be used to model the fortunes of an investor (or gambler) who always invests 1 and then receives a nonnegative integral return. Section 4.2 has additional material on Markov chains that shows how to modify a given chain when trying to determine such things as the probability that the chain ever enters a given class of states by some time, or the conditional distribution of the state at some time given that the class has never been entered. A new remark in Section 7.2 shows that results from the classical insurance ruin model also hold in other important ruin models. There is new material on exponential queueing models, including, in Section 2.2, a determina- tion of the mean and variance of the number of lost customers in a busy period of a finite capacity queue, as well as the new Section 8.3.3 on birth and death queueing models. Section 11.8.2 gives a new approach that can be used to simulate the exact stationary distribution of a Markov chain that satisfies a certain property. Among the newly added examples are 1.11, which is concerned with a multiple player gambling problem; 3.20, which finds the variance in the matching rounds problem; 3.30, which deals with the characteristics of a random selection from a population; and 4.25, which deals with the stationary distribution of a Markov chain.

xii Preface Course Ideally, this text would be used in a one-year course in probability models. Other possible courses would be a one-semester course in introductory probability theory (involving Chapters 1–3 and parts of others) or a course in elementary stochastic processes. The textbook is designed to be flexible enough to be used in a variety of possible courses. For example, I have used Chapters 5 and 8, with smatterings from Chapters 4 and 6, as the basis of an introductory course in queueing theory. Examples and Exercises Many examples are worked out throughout the text, and there are also a large number of exercises to be solved by students. More than 100 of these exercises have been starred and their solutions provided at the end of the text. These starred problems can be used for independent study and test preparation. An Instructor’s Manual, contain- ing solutions to all exercises, is available free to instructors who adopt the book for class. Organization Chapters 1 and 2 deal with basic ideas of probability theory. In Chapter 1 an axiomatic framework is presented, while in Chapter 2 the important concept of a random vari- able is introduced. Section 2.6.1 gives a simple derivation of the joint distribution of the sample mean and sample variance of a normal data sample. Chapter 3 is concerned with the subject matter of conditional probability and con- ditional expectation. “Conditioning” is one of the key tools of probability theory, and it is stressed throughout the book. When properly used, conditioning often enables us to easily solve problems that at first glance seem quite difficult. The final section of this chapter presents applications to (1) a computer list problem, (2) a random graph, and (3) the Polya urn model and its relation to the Bose-Einstein distribution. Section 3.6.5 presents k-record values and the surprising Ignatov’s theorem. In Chapter 4 we come into contact with our first random, or stochastic, process, known as a Markov chain, which is widely applicable to the study of many real-world phenomena. Applications to genetics and production processes are presented. The concept of time reversibility is introduced and its usefulness illustrated. Section 4.5.3 presents an analysis, based on random walk theory, of a probabilistic algorithm for the satisfiability problem. Section 4.6 deals with the mean times spent in transient states by a Markov chain. Section 4.9 introduces Markov chain Monte Carlo methods. In the final section we consider a model for optimally making decisions known as a Markovian decision process. In Chapter 5 we are concerned with a type of stochastic process known as a count- ing process. In particular, we study a kind of counting process known as a Poisson process. The intimate relationship between this process and the exponential distribu- tion is discussed. New derivations for the Poisson and nonhomogeneous Poisson processes are discussed. Examples relating to analyzing greedy algorithms, minimiz- ing highway encounters, collecting coupons, and tracking the AIDS virus, as well as

Preface xiii material on compound Poisson processes, are included in this chapter. Section 5.2.4 gives a simple derivation of the convolution of exponential random variables. Chapter 6 considers Markov chains in continuous time with an emphasis on birth and death models. Time reversibility is shown to be a useful concept, as it is in the study of discrete-time Markov chains. Section 6.7 presents the computationally important technique of uniformization. Chapter 7, the renewal theory chapter, is concerned with a type of counting pro- cess more general than the Poisson. By making use of renewal reward processes, limiting results are obtained and applied to various fields. Section 7.9 presents new results concerning the distribution of time until a certain pattern occurs when a sequence of independent and identically distributed random variables is observed. In Section 7.9.1, we show how renewal theory can be used to derive both the mean and the variance of the length of time until a specified pattern appears, as well as the mean time until one of a finite number of specified patterns appears. In Section 7.9.2, we suppose that the random variables are equally likely to take on any of m possible values, and compute an expression for the mean time until a run of m dis- tinct values occurs. In Section 7.9.3, we suppose the random variables are continuous and derive an expression for the mean time until a run of m consecutive increasing values occurs. Chapter 8 deals with queueing, or waiting line, theory. After some preliminaries dealing with basic cost identities and types of limiting probabilities, we consider exponential queueing models and show how such models can be analyzed. Included in the models we study is the important class known as a network of queues. We then study models in which some of the distributions are allowed to be arbitrary. Included are Section 8.6.3 dealing with an optimization problem concerning a single server, general service time queue, and Section 8.8, concerned with a single server, general service time queue in which the arrival source is a finite number of potential users. Chapter 9 is concerned with reliability theory. This chapter will probably be of greatest interest to the engineer and operations researcher. Section 9.6.1 illustrates a method for determining an upper bound for the expected life of a parallel system of not necessarily independent components and Section 9.7.1 analyzes a series structure reliability model in which components enter a state of suspended animation when one of their cohorts fails. Chapter 10 is concerned with Brownian motion and its applications. The theory of options pricing is discussed. Also, the arbitrage theorem is presented and its rela- tionship to the duality theorem of linear programming is indicated. We show how the arbitrage theorem leads to the Black–Scholes option pricing formula. Chapter 11 deals with simulation, a powerful tool for analyzing stochastic mod- els that are analytically intractable. Methods for generating the values of arbitrarily distributed random variables are discussed, as are variance reduction methods for increasing the efficiency of the simulation. Section 11.6.4 introduces the valuable simulation technique of importance sampling, and indicates the usefulness of tilted distributions when applying this method.

xiv Preface Acknowledgments We would like to acknowledge with thanks the helpful suggestions made by the many reviewers of the text. These comments have been essential in our attempt to continue to improve the book and we owe these reviewers, and others who wish to remain anonymous, many thanks: Mark Brown, City University of New York Zhiqin Ginny Chen, University of Southern California Tapas Das, University of South Florida Israel David, Ben-Gurion University Jay Devore, California Polytechnic Institute Eugene Feinberg, State University of New York, Stony Brook Ramesh Gupta, University of Maine Marianne Huebner, Michigan State University Garth Isaak, Lehigh University Jonathan Kane, University of Wisconsin Whitewater Amarjot Kaur, Pennsylvania State University Zohel Khalil, Concordia University Eric Kolaczyk, Boston University Melvin Lax, California State University, Long Beach Jean Lemaire, University of Pennsylvania Andrew Lim, University of California, Berkeley George Michailidis, University of Michigan Donald Minassian, Butler University Joseph Mitchell, State University of New York, Stony Brook Krzysztof Osfaszewski, University of Illinois Erol Pekoz, Boston University Evgeny Poletsky, Syracuse University James Propp, University of Massachusetts, Lowell Anthony Quas, University of Victoria Charles H. Roumeliotis, Proofreader David Scollnik, University of Calgary

Preface xv Mary Shepherd, Northwest Missouri State University Galen Shorack, University of Washington, Seattle Marcus Sommereder, Vienna University of Technology Osnat Stramer, University of Iowa Gabor Szekeley, Bowling Green State University Marlin Thomas, Purdue University Henk Tijms, Vrije University Zhenyuan Wang, University of Binghamton Ward Whitt, Columbia University Bo Xhang, Georgia University of Technology Julie Zhou, University of Victoria

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