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Introduction to Probability Models 10th Edition.pdf

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    Title Page
    Copyright Page
    Contents
    Preface
    Chapter 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
    1.6 Bayes’ Formula
    Exercises
    References
    Chapter 2. Random Variables
    2.1 Random Variables
    2.2 Discrete Random Variables
    2.2.1 The Bernoulli Random Variable
    2.2.2 The Binomial Random Variable
    2.2.3 The Geometric Random Variable
    2.2.4 The Poisson Random Variable
    2.3 Continuous Random Variables
    2.3.1 The Uniform Random Variable
    2.3.2 Exponential Random Variables
    2.3.3 Gamma Random Variables
    2.3.4 Normal Random Variables
    2.4 Expectation of a Random Variable
    2.4.1 The Discrete Case
    2.4.2 The Continuous Case
    2.4.3 Expectation of a Function of a Random Variable
    2.5 Jointly Distributed Random Variables
    2.5.1 Joint Distribution Functions
    2.5.2 Independent Random Variables
    2.5.3 Covariance and Variance of Sums of Random Variables
    2.5.4 Joint Probability Distribution of Functions of Random Variables
    2.6 Moment Generating Functions
    2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population
    2.7 The Distribution of the Number of Events that Occur
    2.8 Limit Theorems
    2.9 Stochastic Processes
    Exercises
    References
    Chapter 3. Conditional Probability and Conditional Expectation
    3.1 Introduction
    3.2 The Discrete Case
    3.3 The Continuous Case
    3.4 Computing Expectations by Conditioning
    3.4.1 Computing Variances by Conditioning
    3.5 Computing Probabilities by Conditioning
    3.6 Some Applications
    3.6.1 A List Model
    3.6.2 A Random Graph
    3.6.3 Uniform Priors, Polya’s Urn Model, and Bose–Einstein Statistics
    3.6.4 Mean Time for Patterns
    3.6.5 The k-Record Values of Discrete Random Variables
    3.6.6 Left Skip Free Random Walks
    3.7 An Identity for Compound Random Variables
    3.7.1 Poisson Compounding Distribution
    3.7.2 Binomial Compounding Distribution
    3.7.3 A Compounding Distribution Related to the Negative Binomial
    Exercises
    Chapter 4. Markov Chains
    4.1 Introduction
    4.2 Chapman–Kolmogorov Equations
    4.3 Classification of States
    4.4 Limiting Probabilities
    4.5 Some Applications
    4.5.1 The Gambler’s Ruin Problem
    4.5.2 A Model for Algorithmic Efficiency
    4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem
    4.6 Mean Time Spent in Transient States
    4.7 Branching Processes
    4.8 Time Reversible Markov Chains
    4.9 Markov Chain Monte Carlo Methods
    4.10 Markov Decision Processes
    4.11 Hidden Markov Chains
    4.11.1 Predicting the States
    Exercises
    References
    Chapter 5. The Exponential Distribution and the Poisson Process
    5.1 Introduction
    5.2 The Exponential Distribution
    5.2.1 Definition
    5.2.2 Properties of the Exponential Distribution
    5.2.3 Further Properties of the Exponential Distribution
    5.2.4 Convolutions of Exponential Random Variables
    5.3 The Poisson Process
    5.3.1 Counting Processes
    5.3.2 Definition of the Poisson Process
    5.3.3 Interarrival and Waiting Time Distributions
    5.3.4 Further Properties of Poisson Processes
    5.3.5 Conditional Distribution of the Arrival Times
    5.3.6 Estimating Software Reliability
    5.4 Generalizations of the Poisson Process
    5.4.1 Nonhomogeneous Poisson Process
    5.4.2 Compound Poisson Process
    5.4.3 Conditional or Mixed Poisson Processes
    Exercises
    References
    Chapter 6. Continuous-Time Markov Chains
    6.1 Introduction
    6.2 Continuous-Time Markov Chains
    6.3 Birth and Death Processes
    6.4 The Transition Probability Function Pij(t)
    6.5 Limiting Probabilities
    6.6 Time Reversibility
    6.7 Uniformization
    6.8 Computing the Transition Probabilities
    Exercises
    References
    Chapter 7. Renewal Theory and Its Applications
    7.1 Introduction
    7.2 Distribution of N(t)
    7.3 Limit Theorems and Their Applications
    7.4 Renewal Reward Processes
    7.5 Regenerative Processes
    7.5.1 Alternating Renewal Processes
    7.6 Semi-Markov Processes
    7.7 The Inspection Paradox
    7.8 Computing the Renewal Function
    7.9 Applications to Patterns
    7.9.1 Patterns of Discrete Random Variables
    7.9.2 The Expected Time to a Maximal Run of Distinct Values
    7.9.3 Increasing Runs of Continuous Random Variables
    7.10 The Insurance Ruin Problem
    Exercises
    References
    Chapter 8. Queueing Theory
    8.1 Introduction
    8.2 Preliminaries
    8.2.1 Cost Equations
    8.2.2 Steady-State Probabilities
    8.3 Exponential Models
    8.3.1 A Single-Server Exponential Queueing System
    8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity
    8.3.3 Birth and Death Queueing Models
    8.3.4 A Shoe Shine Shop
    8.3.5 A Queueing System with Bulk Service
    8.4 Network of Queues
    8.4.1 Open Systems
    8.4.2 Closed Systems
    8.5 The System M/G/1
    8.5.1 Preliminaries: Work and Another Cost Identity
    8.5.2 Application of Work to M/G/1
    8.5.3 Busy Periods
    8.6 Variations on the M/G/1
    8.6.1 The M/G/1 with Random-Sized Batch Arrivals
    8.6.2 Priority Queues
    8.6.3 An M/G/1 Optimization Example
    8.6.4 The M/G/1 Queue with Server Breakdown
    8.7 The Model G/M/1
    8.7.1 The G/M/1 Busy and Idle Periods
    8.8 A Finite Source Model
    8.9 Multiserver Queues
    8.9.1 Erlang’s Loss System
    8.9.2 The M/M/k Queue
    8.9.3 The G/M/k Queue
    8.9.4 The M/G/k Queue
    Exercises
    References
    Chapter 9. Reliability Theory
    9.1 Introduction
    9.2 Structure Functions
    9.2.1 Minimal Path and Minimal Cut Sets
    9.3 Reliability of Systems of Independent Components
    9.4 Bounds on the Reliability Function
    9.4.1 Method of Inclusion and Exclusion
    9.4.2 Second Method for Obtaining Bounds on r(p)
    9.5 System Life as a Function of Component Lives
    9.6 Expected System Lifetime
    9.6.1 An Upper Bound on the Expected Life of a Parallel System
    9.7 Systems with Repair
    9.7.1 A Series Model with Suspended Animation
    Exercises
    References
    Chapter 10. Brownian Motion and Stationary Processes
    10.1 Brownian Motion
    10.2 Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem
    10.3 Variations on Brownian Motion
    10.3.1 Brownian Motion with Drift
    10.3.2 Geometric Brownian Motion
    10.4 Pricing Stock Options
    10.4.1 An Example in Options Pricing
    10.4.2 The Arbitrage Theorem
    10.4.3 The Black-Scholes Option Pricing Formula
    10.5 White Noise
    10.6 Gaussian Processes
    10.7 Stationary and Weakly Stationary Processes
    10.8 Harmonic Analysis of Weakly Stationary Processes
    Exercises
    References
    Chapter 11. Simulation
    11.1 Introduction
    11.2 General Techniques for Simulating Continuous Random Variables
    11.2.1 The Inverse Transformation Method
    11.2.2 The Rejection Method
    11.2.3 The Hazard Rate Method
    11.3 Special Techniques for Simulating Continuous Random Variables
    11.3.1 The Normal Distribution
    11.3.2 The Gamma Distribution
    11.3.3 The Chi-Squared Distribution
    11.3.4 The Beta (n, m) Distribution
    11.3.5 The Exponential Distribution—The Von Neumann Algorithm
    11.4 Simulating from Discrete Distributions
    11.4.1 The Alias Method
    11.5 Stochastic Processes
    11.5.1 Simulating a Nonhomogeneous Poisson Process
    11.5.2 Simulating a Two-Dimensional Poisson Process
    11.6 Variance Reduction Techniques
    11.6.1 Use of Antithetic Variables
    11.6.2 Variance Reduction by Conditioning
    11.6.3 Control Variates
    11.6.4 Importance Sampling
    11.7 Determining the Number of Runs
    11.8 Generating from the Stationary Distribution of a Markov Chain
    11.8.1 Coupling from the Past
    11.8.2 Another Approach
    Exercises
    References
    Appendix: Solutions to Starred Exercises
    Index
    Introduction to Probability Models Tenth Edition
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    Introduction to Probability Models Tenth Edition Sheldon M. Ross University of Southern California Los Angeles, California AMSTERDAM • BOSTON • HEIDELBERG • LONDON SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO NEW YORK • OXFORD • PARIS • SAN DIEGO Academic Press is an Imprint of Elsevier
    Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA Elsevier, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK Copyright © 2010 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability 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. Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. Introduction to probability models / Sheldon M. Ross. – 10th ed. p. cm. Includes bibliographical references and index. ISBN 978-0-12-375686-2 (hardcover : alk. paper) 1. Probabilities. I. Title. QA273.R84 2010 519.2–dc22 2009040399 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-375686-2 For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Typeset by: diacriTech, India Printed in the United States of America 09 10 11 9 8 7 6 5 4 3 2 1
    Contents Preface 1 2 Introduction Sample Space and Events Probabilities Defined on Events Introduction to Probability Theory 1.1 1.2 1.3 1.4 Conditional Probabilities 1.5 1.6 Exercises References Independent Events Bayes’ Formula Random Variables 2.1 Random Variables 2.2 Discrete Random Variables xi 1 1 1 4 7 10 12 15 20 2.3 Continuous Random Variables 2.2.1 The Bernoulli Random Variable 2.2.2 The Binomial Random Variable 2.2.3 The Geometric Random Variable 2.2.4 The Poisson Random Variable 21 21 25 26 27 29 30 31 32 2.3.1 The Uniform Random Variable 34 2.3.2 Exponential Random Variables 34 2.3.3 Gamma Random Variables 34 2.3.4 Normal Random Variables 36 Expectation of a Random Variable 36 2.4.1 The Discrete Case 38 2.4.2 The Continuous Case 40 2.4.3 44 Jointly Distributed Random Variables 44 2.5.1 2.5.2 48 2.5.3 Covariance and Variance of Sums of Random Variables 50 Joint Distribution Functions Independent Random Variables Expectation of a Function of a Random Variable 2.4 2.5
    vi 3 Contents 59 62 71 74 77 84 86 95 97 97 97 102 106 117 122 140 140 141 149 153 157 160 166 169 171 172 173 191 191 195 204 214 230 230 234 237 243 245 2.5.4 Joint Probability Distribution of Functions of Random Variables 2.6 Moment Generating Functions 2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population The Distribution of the Number of Events that Occur Limit Theorems Stochastic Processes 2.7 2.8 2.9 Exercises References Conditional Probability and Conditional Expectation 3.1 3.2 3.3 3.4 Computing Expectations by Conditioning Introduction The Discrete Case The Continuous Case 3.4.1 Computing Variances by Conditioning 3.5 Computing Probabilities by Conditioning 3.6 Some Applications 3.6.1 A List Model 3.6.2 A Random Graph 3.6.3 Uniform Priors, Polya’s Urn Model, and Bose–Einstein Statistics 3.6.4 Mean Time for Patterns 3.6.5 The k-Record Values of Discrete Random Variables 3.6.6 Left Skip Free Random Walks 3.7 An Identity for Compound Random Variables Poisson Compounding Distribution Binomial Compounding Distribution 3.7.1 3.7.2 3.7.3 A Compounding Distribution Related to the Negative Binomial Exercises 4 Markov Chains Introduction 4.1 4.2 Chapman–Kolmogorov Equations 4.3 Classification of States Limiting Probabilities 4.4 Some Applications 4.5 4.5.1 The Gambler’s Ruin Problem 4.5.2 A Model for Algorithmic Efficiency 4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 4.6 Mean Time Spent in Transient States 4.7 Branching Processes
    Contents Time Reversible Markov Chains 4.8 4.9 Markov Chain Monte Carlo Methods 4.10 Markov Decision Processes 4.11 Hidden Markov Chains 4.11.1 Predicting the States Exercises References 5.3 5 6 7 The Exponential Distribution and the Poisson Process 5.1 5.2 Properties of the Exponential Distribution Further Properties of the Exponential Distribution Introduction The Exponential Distribution 5.2.1 Definition 5.2.2 5.2.3 5.2.4 Convolutions of Exponential Random Variables The Poisson Process 5.3.1 Counting Processes 5.3.2 Definition of the Poisson Process 5.3.3 5.3.4 5.3.5 Conditional Distribution of the Arrival Times 5.3.6 Interarrival and Waiting Time Distributions Further Properties of Poisson Processes Estimating Software Reliability 5.4 Generalizations of the Poisson Process 5.4.1 Nonhomogeneous Poisson Process 5.4.2 Compound Poisson Process 5.4.3 Conditional or Mixed Poisson Processes Exercises References Introduction Birth and Death Processes The Transition Probability Function Pij(t) Limiting Probabilities Time Reversibility Continuous-Time Markov Chains 6.1 6.2 Continuous-Time Markov Chains 6.3 6.4 6.5 6.6 6.7 Uniformization 6.8 Computing the Transition Probabilities Exercises References Introduction Renewal Theory and Its Applications 7.1 7.2 Distribution of N(t) 7.3 Limit Theorems and Their Applications vii 249 260 265 269 273 275 290 291 291 292 292 294 301 308 312 312 313 316 319 325 336 339 339 346 351 354 370 371 371 372 374 381 390 397 406 409 412 419 421 421 423 427
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