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Introduction to Probability Models (10th Edition) Ross.pdf
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
Sheldon M. Ross
University of Southern California
Los Angeles, California
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
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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–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
viii
Contents
7.4 Renewal Reward Processes
7.5 Regenerative Processes
7.5.1 Alternating Renewal Processes
Semi-Markov Processes
The Inspection Paradox
7.6
7.7
7.8 Computing the Renewal Function
7.9 Applications to Patterns
Patterns of Discrete Random Variables
7.9.1
7.9.2 The Expected Time to a Maximal Run of
Distinct Values
Increasing Runs of Continuous Random Variables
7.9.3
7.10 The Insurance Ruin Problem
Exercises
References
8.1
8.2
8 Queueing Theory
Introduction
Preliminaries
8.2.1 Cost Equations
8.2.2
Exponential Models
8.3.1 A Single-Server Exponential Queueing System
8.3.2 A Single-Server Exponential Queueing System Having
Steady-State Probabilities
8.3
Finite Capacity
Birth and Death Queueing Models
8.3.3
8.3.4 A Shoe Shine Shop
8.3.5 A Queueing System with Bulk Service
8.4 Network of Queues
8.5
8.4.1 Open Systems
8.4.2 Closed Systems
The System M/G/1
8.5.1
8.5.2 Application of Work to M/G/1
8.5.3
Busy Periods
Preliminaries: Work and Another Cost Identity
8.6 Variations on the M/G/1
Priority Queues
8.6.1 The M/G/1 with Random-Sized Batch Arrivals
8.6.2
8.6.3 An M/G/1 Optimization Example
8.6.4 The M/G/1 Queue with Server Breakdown
The Model G/M/1
8.7.1 The G/M/1 Busy and Idle Periods
8.7
8.8 A Finite Source Model
8.9 Multiserver Queues
8.9.1
Erlang’s Loss System
439
447
450
457
460
463
466
467
474
476
478
484
495
497
497
498
499
500
502
502
511
517
522
524
527
527
532
538
538
539
540
541
541
543
546
550
553
558
559
562
563
Contents
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
9
Reliability Theory
Introduction
9.1
Structure Functions
9.2
9.2.1 Minimal Path and Minimal Cut Sets
9.3 Reliability of Systems of Independent Components
9.4
Second Method for Obtaining Bounds on r(p)
Bounds on the Reliability Function
9.4.1 Method of Inclusion and Exclusion
9.4.2
System Life as a Function of Component Lives
Expected System Lifetime
9.6.1 An Upper Bound on the Expected Life of a
9.5
9.6
9.7
Parallel System
Systems with Repair
9.7.1 A Series Model with Suspended Animation
Exercises
References
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
11 Simulation
11.1 Introduction
11.2 General Techniques for Simulating Continuous Random
Variables
ix
564
565
567
568
578
579
579
580
582
586
590
591
600
602
610
614
616
620
623
629
631
631
635
636
636
636
638
638
640
644
649
651
654
659
661
665
667
667
672
x
Contents
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
672
673
677
680
680
684
684
685
686
688
691
696
697
703
706
707
710
715
717
722
723
723
725
726
734
735
775
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