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Introduction to Probability Models

Tenth Edition

Introduction to Probability

Models

Tenth Edition

Sheldon M. Ross

University of Southern California

Los Angeles, California

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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–dc 2009040399

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ISBN: 978-0-12-375686-

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vi Contents

• 1 Introduction to Probability Theory Preface xi
  • 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
• 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
      • Variables 2.5.4 Joint Probability Distribution of Functions of Random
  • 2.6 Moment Generating Functions - Variance from a Normal Population 2.6.1 The Joint Distribution of the Sample Mean and Sample
  • 2.7 The Distribution of the Number of Events that Occur
  • 2.8 Limit Theorems
  • 2.9 Stochastic Processes
  • Exercises
  • References
• 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
      • Statistics 3.6.3 Uniform Priors, Polya’s Urn Model, and Bose–Einstein
    • 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
      • Binomial 3.7.3 A Compounding Distribution Related to the Negative
  • Exercises
• 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
      • Algorithm for the Satisfiability Problem 4.5.3 Using a Random Walk to Analyze a Probabilistic
  • 4.6 Mean Time Spent in Transient States
  • 4.7 Branching Processes
  • 4.8 Time Reversible Markov Chains Contents vii
  • 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 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
• 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 P ij ( t )
  • 6.5 Limiting Probabilities
  • 6.6 Time Reversibility
  • 6.7 Uniformization
  • 6.8 Computing the Transition Probabilities
  • Exercises
  • References
• 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 viii Contents
  • 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
      • Distinct Values 7.9.2 The Expected Time to a Maximal Run of
    • 7.9.3 Increasing Runs of Continuous Random Variables
  • 7.10 The Insurance Ruin Problem
  • Exercises
  • References
• 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
      • Finite Capacity 8.3.2 A Single-Server Exponential Queueing System Having
    • 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 /
    • 8.5.1 Preliminaries: Work and Another Cost Identity
    • 8.5.2 Application of Work to M / G /
    • 8.5.3 Busy Periods
  • 8.6 Variations on the M / G /
    • 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 /
    • 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 Contents ix
    • 8.9.3 The G / M / k Queue
    • 8.9.4 The M / G / k Queue
  • Exercises
  • References
• 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 - Parallel System 9.6.1 An Upper Bound on the Expected Life of a
  • 9.7 Systems with Repair
    • 9.7.1 A Series Model with Suspended Animation
  • Exercises
  • References
• 10 Brownian Motion and Stationary Processes
  • 10.1 Brownian Motion - Ruin Problem 10.2 Hitting Times, Maximum Variable, and the Gambler’s
  • 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
• 11 Simulation
  • 11.1 Introduction - Variables 11.2 General Techniques for Simulating Continuous Random
    • 11.2.1 The Inverse Transformation Method x Contents
    • 11.2.2 The Rejection Method
    • 11.2.3 The Hazard Rate Method
    • Variables 11.3 Special Techniques for Simulating Continuous Random
    • 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
      • Algorithm 11.3.5 The Exponential Distribution—The Von Neumann
  • 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
    • Markov Chain 11.8 Generating from the Stationary Distribution of a
    • 11.8.1 Coupling from the Past
    • 11.8.2 Another Approach
  • Exercises
  • References
• Appendix: Solutions to Starred Exercises
• Index

xii Preface

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.

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, containing 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 variable is introduced. Subsection 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 conditional 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 diffi- cult. 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. Subsection 3.6.5 presents k -record values and the surprising Ignatov’s theorem.

Preface xiii

In Chapter 4 we come into contact with our first random, or stochastic, pro- cess, 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 illus- trated. Subsection 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 counting process. In particular, we study a kind of counting process known as a Poisson process. The intimate relationship between this process and the expo- nential distribution is discussed. New derivations for the Poisson and nonhomo- geneous Poisson processes are discussed. Examples relating to analyzing greedy algorithms, minimizing highway encounters, collecting coupons, and tracking the AIDS virus, as well as material on compound Poisson processes, are included in this chapter. Subsection 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 compu- tationally important technique of uniformization. Chapter 7, the renewal theory chapter, is concerned with a type of counting process more general than the Poisson. By making use of renewal reward pro- cesses, limiting results are obtained and applied to various fields. Section 7. presents new results concerning the distribution of time until a certain pattern occurs when a sequence of independent and identically distributed random vari- ables is observed. In Subsection 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 Subsection 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 distinct values occurs. In Subsection 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 prelimi- naries dealing with basic cost identities and types of limiting probabilities, we consider exponential queueing models and show how such models can be ana- lyzed. 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 Subsection 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.

Preface xv

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 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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2 Introduction to Probability Theory

Some examples are the following.

1. If the experiment consists of the flipping of a coin, then

S = { H , T }

where H means that the outcome of the toss is a head and T that it is a tail.

2. If the experiment consists of rolling a die, then the sample space is

S = {1, 2, 3, 4, 5, 6}

where the outcome i means that i appeared on the die, i = 1, 2, 3, 4, 5, 6.

3. If the experiments consists of flipping two coins, then the sample space consists of the following four points:

S = {( H , H ), ( H , T ), ( T , H ), ( T , T )}

The outcome will be ( H , H ) if both coins come up heads; it will be ( H , T ) if the first coin comes up heads and the second comes up tails; it will be ( T , H ) if the first comes up tails and the second heads; and it will be ( T , T ) if both coins come up tails.

4. If the experiment consists of rolling two dice, then the sample space consists of the following 36 points:

S =

⎧ ⎪⎪⎪ ⎪⎪⎪ ⎨ ⎪⎪ ⎪⎪⎪ ⎪⎩

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

⎫ ⎪⎪⎪ ⎪⎪⎪ ⎬ ⎪⎪ ⎪⎪⎪ ⎪⎭

where the outcome ( i , j ) is said to occur if i appears on the first die and j on the second die.

5. If the experiment consists of measuring the lifetime of a car, then the sample space consists of all nonnegative real numbers. That is,

S = [0, ∞)∗^ 

Any subset E of the sample space S is known as an event. Some examples of events are the following.

1 ′. In Example (1), if E = { H }, then E is the event that a head appears on the flip of the coin. Similarly, if E = { T }, then E would be the event that a tail appears. 2 ′. In Example (2), if E = { 1 }, then E is the event that one appears on the roll of the die. If E = {2, 4, 6}, then E would be the event that an even number appears on the roll.

∗ (^) The set ( a , b ) is defined to consist of all points x such that a < x < b. The set [ a , b ] is defined to consist of all points x such that a  x  b. The sets ( a , b ] and [ a , b ) are defined, respectively, to consist of all points x such that a < x  b and all points x such that a  x < b.

1.2 Sample Space and Events 3

3 ′. In Example (3), if E = {( H , H ), ( H , T )}, then E is the event that a head appears on the first coin. 4 ′. In Example (4), if E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}, then E is the event that the sum of the dice equals seven. 5 ′. In Example (5), if E = (2, 6), then E is the event that the car lasts between two and six years.  We say that the event E occurs when the outcome of the experiment lies in E. For any two events E and F of a sample space S we define the new event E ∪ F to consist of all outcomes that are either in E or in F or in both E and F. That is, the event E ∪ F will occur if either E or F occurs. For example, in (1) if E = { H } and F = { T }, then

E ∪ F = { H , T }

That is, E ∪ F would be the whole sample space S. In (2) if E = {1, 3, 5} and F = {1, 2, 3}, then

E ∪ F = {1, 2, 3, 5}

and thus E ∪ F would occur if the outcome of the die is 1 or 2 or 3 or 5. The event E ∪ F is often referred to as the union of the event E and the event F. For any two events E and F , we may also define the new event EF , sometimes written E ∩ F , and referred to as the intersection of E and F , as follows. EF consists of all outcomes which are both in E and in F. That is, the event EF will occur only if both E and F occur. For example, in (2) if E = {1, 3, 5} and F = {1, 2, 3}, then

EF = {1, 3}

and thus EF would occur if the outcome of the die is either 1 or 3. In Exam- ple (1) if E = { H } and F = { T }, then the event EF would not consist of any outcomes and hence could not occur. To give such an event a name, we shall refer to it as the null event and denote it by Ø. (That is, Ø refers to the event consisting of no outcomes.) If EF = Ø, then E and F are said to be mutually exclusive. We also define unions and intersections of more than two events in a simi- lar manner. If⋃ E 1 , E 2 ,... are events, then the union of these events, denoted by ∞ n = 1 E^ n , is defined to be the event that consists of all outcomes that are in^ En for at least one value of n = 1, 2,.... Similarly, the intersection of the events E (^) n , denoted by

n = 1 E^ n , is defined to be the event consisting of those outcomes that are in all of the events En , n = 1, 2,.... Finally, for any event E we define the new event Ec , referred to as the complement of E , to consist of all outcomes in the sample space S that are not in E. That is, E c^ will occur if and only if E does not occur. In Example (4)