Probability and Statistics with Reliability, Queuing, and Computer ...

Probability and Statistics with Reliability, Queuing, and Computer Science Applications Kishor Shridharbhai Trivedi Duke University Durham, North Carolina

A Wiley-Interscience Publication JOHN WILEY & SONS, INC.

Contents

Preface to the Second Edition

ix

Preface to the First Edition

xi

Acronyms

xiii

1 Introduction 1.1 Motivation 1.2 Probability Models 1.3 Sample Space 1.4 Events 1.5 Algebra of Events 1.6 Graphical Methods of Representing Events 1.7 Probability Axioms 1.8 Combinatorial Problems 1.9 Conditional Probability 1.10 Independence of Events 1.11 Bayes' Rule 1.12 Bernoulli Trials

1 1 2 3 6 7 11 13 19 23 25 38 45

2 Discrete R a n d o m Variables 2.1 Introduction 2.2 Random Variables and Their Event Spaces 2.3 The Probability Mass Function 2.4 Distribution Functions 2.5 Special Discrete Distributions 2.6 Analysis of Program MAX 2.7 The Probability Generating Function 2.8 Discrete Random Vectors 2.9 Independent Random Variables

61 61 62 64 66 68 92 96 99 104

v

vi

CONTENTS

3

Continuous Random Variables 3.1 Introduction 3.2 The Exponential Distribution 3.3 The Reliability and Failure Rate 3.4 Some Important Distributions 3.5 Functions of a Random Variable 3.6 Jointly Distributed Random Variables 3.7 Order Statistics 3.8 Distribution of Sums 3.9 Functions of Normal Random Variables

115 115 119 124 129 148 153 157 167 182

4

Expectation 4.1 Introduction 4.2 Moments 4.3 Expectation Based on Multiple Random Variables 4.4 Transform Methods 4.5 Moments and Transforms of Some Distributions 4.6 Computation of Mean Time to Failure 4.7 Inequalities and Limit Theorems

193 193 197 200 208 217 228 237

5

Conditional Distribution and Expectation 5.1 Introduction 5.2 Mixture Distributions 5.3 Conditional Expectation 5.4 Impefect Fault Coverage and Reliability 5.5 Random Sums

247 247 255 262 268 279

6

Stochastic Processes 6.1 Introduction 6.2 Classification of Stochastic Processes 6.3 The Bernoulli Process 6.4 The Poisson Process 6.5 Renewal Processes 6.6 Availability Analysis 6.7 Random Incidence 6.8 Renewal Model of Program Behavior

289 289 294 300 304 314 319 328 332

7

Discrete-Time Markov Chains 7.1 Introduction 7.2 Computation of n-step Transition Probabilities 7.3 State Classification and Limiting Probabilities 7.4 Distribution of Times Between State Changes 7.5 Markov Modulated Bernoulli Process 7.6 Irreducible Finite Chains with Aperiodic States 7.7 * The M/Q/l Queuing System

337 337 341 347 356 358 361 377

CONTENTS

vii

7.8 7.9

385 392

Discrete-Time Birth-Death Processes Finite Markov Chains with Absorbing States

8

Continuous-Time Markov Chains 8.1 Introduction 8.2 The Birth-Death Process 8.3 Other Special Cases of the Birth-Death Model 8.4 Non-Birth-Death Processes 8.5 Markov Chains with Absorbing States 8.6 Solution Techniques 8.7 Automated Generation

9

Networks of Queues 9.1 Introduction 9.2 Open Queuing Networks 9.3 Closed Queuing Networks 9.4 General Service Distribution and Multiple Job Types 9.5 Non-product-form Networks 9.6 Computing Response Time Distribution 9.7 Summary

405 405 412 446 454 496 520 530 555 555 560 . 568 596 604 617 630

10 Statistical Inference 10.1 Introduction 10.2 Parameter Estimation 10.3 Hypothesis Testing

637 637 639 692

11 Regression and Analysis of Variance 11.1 Introduction 11.2 Least-squares Curve Fitting 11.3 The Coefficients of Determination 11.4 Confidence Intervals in Linear Regression 11.5 Trend Detection and Slope Estimation 11.6 Correlation Analysis 11.7 Simple Nonlinear Regression 11.8 Higher-dimensional Least-squares Fit 11.9 Analysis of Variance

727 727 732 735 738 742 745 748 749 751

A Bibliography A.l Theory A.2 Applications

765 765 770

B Properties of Distributions

777

C Statistical Tables

780

D Laplace Transforms

801

viii

CONTENTS

E Program Performance Analysis

808

Author Index

811

Subject Index

819