Pub. Date:
Elsevier Science
Introduction to Probability Models / Edition 6

Introduction to Probability Models / Edition 6

by Sheldon M. Ross


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Product Details

ISBN-13: 9780125984706
Publisher: Elsevier Science
Publication date: 02/28/1997
Edition description: Older Edition
Pages: 669
Product dimensions: 6.24(w) x 9.27(h) x 1.39(d)

About the Author

Dr. Sheldon M. Ross is a professor in the Department of Industrial and Systems Engineering at the University of Southern California. He received his PhD in statistics at Stanford University in 1968. He has published many technical articles and textbooks in the areas of statistics and applied probability. Among his texts are A First Course in Probability, Introduction to Probability Models, Stochastic Processes, and Introductory Statistics. Professor Ross is the founding and continuing editor of the journal Probability in the Engineering and Informational Sciences. He is a Fellow of the Institute of Mathematical Statistics, a Fellow of INFORMS, and a recipient of the Humboldt US Senior Scientist Award.

Table of Contents

Preface to the Fifth Editionxi
Preface to the Sixth Editionxiii
Preface to the Seventh Editionxv
1.Introduction to Probability Theory1
1.2.Sample Space and Events1
1.3.Probabilities Defined on Events4
1.4.Conditional Probabilities6
1.5.Independent Events10
1.6.Bayes' Formula12
2.Random Variables23
2.1.Random Variables23
2.2.Discrete Random Variables27
2.2.1.The Bernoulli Random Variable27
2.2.2.The Binomial Random Variable28
2.2.3.The Geometric Random Variable31
2.2.4.The Poisson Random Variable31
2.3.Continuous Random Variables33
2.3.1.The Uniform Random Variable34
2.3.2.Exponential Random Variables35
2.3.3.Gamma Random Variables35
2.3.4.Normal Random Variables36
2.4.Expectation of a Random Variable37
2.4.1.The Discrete Case37
2.4.2.The Continuous Case40
2.4.3.Expectation of a Function of a Random Variable42
2.5.Jointly Distributed Random Variables46
2.5.1.Joint Distribution Functions46
2.5.2.Independent Random Variables50
2.5.3.Covariance and Variance of Sums of Random Variables51
2.5.4.Joint Probability Distribution of Functions of Random Variables59
2.6.Moment Generating Functions62
2.6.1.The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population70
2.7.Limit Theorems73
2.8.Stochastic Processes79
3.Conditional Probability and Conditional Expectation93
3.2.The Discrete Case93
3.3.The Continuous Case98
3.4.Computing Expectations by Conditioning101
3.5.Computing Probabilities by Conditioning114
3.6.Some Applications128
3.6.1.A List Model128
3.6.2.A Random Graph129
3.6.3.Uniform Priors, Polya's Urn Model, and Bose-Einstein Statistics137
3.6.4.The k-Record Values of Discrete Random Variables141
4.Markov Chains163
4.2.Chapman-Kolmogorov Equations166
4.3.Classification of States168
4.4.Limiting Probabilities178
4.5.Some Applications188
4.5.1.The Gambler's Ruin Problem188
4.5.2.A Model for Algorithmic Efficiency192
4.5.3.Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem194
4.6.Mean Time Spent in Transient States200
4.7.Branching Processes202
4.8.Time Reversible Markov Chains205
4.9.Markov Chain Monte Carlo Methods216
4.10.Markov Decision Processes222
5.The Exponential Distribution and the Poisson Process241
5.2.The Exponential Distribution242
5.2.2.Properties of the Exponential Distribution243
5.2.3.Further Properties of the Exponential Distribution248
5.2.4.Convolutions of Exponential Random Variables253
5.3.The Poisson Process256
5.3.1.Counting Processes256
5.3.2.Definition of the Poisson Process258
5.3.3.Interarrival and Waiting Time Distributions261
5.3.4.Further Properties of Poisson Processes264
5.3.5.Conditional Distribution of the Arrival Times270
5.3.6.Estimating Software Reliability281
5.4.Generalizations of the Poisson Process284
5.4.1.Nonhomogeneous Poisson Process284
5.4.2.Compound Poisson Process289
6.Continuous-Time Markov Chains313
6.2.Continuous-Time Markov Chains314
6.3.Birth and Death Processes316
6.4.The Transition Probability Function P[subscript ij](t)323
6.5.Limiting Probabilities331
6.6.Time Reversibility338
6.8.Computing the Transition Probabilities349
7.Renewal Theory and Its Applications363
7.2.Distribution of N(t)365
7.3.Limit Theorems and Their Applications368
7.4.Renewal Reward Processes377
7.5.Regenerative Processes386
7.5.1.Alternating Renewal Processes389
7.6.Semi-Markov Processes395
7.7.The Inspection Paradox398
7.8.Computing the Renewal Function400
7.9.Applications to Patterns403
7.9.1.Patterns of Discrete Random Variables404
7.9.2.The Expected Time to a Maximal Run of Distinct Values410
7.9.3.Increasing Runs of Continuous Random Variables412
8.Queueing Theory427
8.2.1.Cost Equations429
8.2.2.Steady-State Probabilities430
8.3.Exponential Models432
8.3.1.A Single-Server Exponential Queueing System432
8.3.2.A Single-Server Exponential Queueing System Having Finite Capacity438
8.3.3.A Shoeshine Shop442
8.3.4.A Queueing System with Bulk Service444
8.4.Network of Queues447
8.4.1.Open Systems447
8.4.2.Closed Systems452
8.5.The System M/G/1458
8.5.1.Preliminaries: Work and Another Cost Identity458
8.5.2.Application of Work to M/G/1459
8.5.3.Busy Periods460
8.6.Variations on the M/G/1461
8.6.1.The M/G/1 with Random-Sized Batch Arrivals461
8.6.2.Priority Queues463
8.6.3.An M/G/1 Optimization Example466
8.7.The Model G/M/1470
8.7.1.The G/M/1 Busy and Idle Periods475
8.8.A Finite Source Model475
8.9.Multiserver Queues479
8.9.1.Erlang's Loss System479
8.9.2.The M/M/k Queue481
8.9.3.The G/M/k Queue481
8.9.4.The M/G/k Queue483
9.Reliability Theory499
9.2.Structure Functions500
9.2.1.Minimal Path and Minimal Cut Sets502
9.3.Reliability of Systems of Independent Components506
9.4.Bounds on the Reliability Function510
9.4.1.Method of Inclusion and Exclusion511
9.4.2.Second Method for Obtaining Bounds on r(p)519
9.5.System Life as a Function of Component Lives521
9.6.Expected System Lifetime529
9.6.1.An Upper Bound on the Expected Life of a Parallel System533
9.7.Systems with Repair535
9.7.1.A Series Model with Suspended Animation539
10.Brownian Motion and Stationary Processes549
10.1.Brownian Motion549
10.2.Hitting Times, Maximum Variable, and the Gambler's Ruin Problem553
10.3.Variations on Brownian Motion554
10.3.1.Brownian Motion with Drift554
10.3.2.Geometric Brownian Motion555
10.4.Pricing Stock Options556
10.4.1.An Example in Options Pricing556
10.4.2.The Arbitrage Theorem558
10.4.3.The Black-Scholes Option Pricing Formula561
10.5.White Noise567
10.6.Gaussian Processes569
10.7.Stationary and Weakly Stationary Processes572
10.8.Harmonic Analysis of Weakly Stationary Processes577
11.2.General Techniques for Simulating Continuous Random Variables590
11.2.1.The Inverse Transformation Method590
11.2.2.The Rejection Method591
11.2.3.The Hazard Rate Method595
11.3.Special Techniques for Simulating Continuous Random Variables598
11.3.1.The Normal Distribution598
11.3.2.The Gamma Distribution602
11.3.3.The Chi-Squared Distribution602
11.3.4.The Beta (n, m) Distribution603
11.3.5.The Exponential Distribution--The Von Neumann Algorithm604
11.4.Simulating from Discrete Distributions606
11.4.1.The Alias Method610
11.5.Stochastic Processes613
11.5.1.Simulating a Nonhomogeneous Poisson Process615
11.5.2.Simulating a Two-Dimensional Poisson Process621
11.6.Variance Reduction Techniques624
11.6.1.Use of Antithetic Variables625
11.6.2.Variance Reduction by Conditioning629
11.6.3.Control Variates633
11.6.4.Importance Sampling634
11.7.Determining the Number of Runs639
AppendixSolutions to Starred Exercises649

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