Table of Contents
Preface xi
Authors xiii
1 Probability Modeling Fundamentals 1
1.1 Random Experiments and Events 2
1.2 Probability 8
1.3 Conditional Probability 11
Homework Problems 15
1.1 Random Experiments and Events 15
1.2 Probability 16
1.3 Conditional Probability 17
Application: Basic Reliability Theory 19
2 Analysis of Random Variables 23
2.1 Introduction to Random Variables 23
2.2 Discrete Random Variables 25
2.3 Continuous Random Variables 26
2.4 Expectation 29
2.5 Generating Functions 33
2.6 Common Applications of Random Variables 35
2.6.1 Equally Likely Alternatives 35
2.6.2 Random Sampling 39
2.6.3 Normal Random Variables 41
Homework Problems 42
2.2 Discrete Random Variables 42
2.3 Continuous Random Variables 43
2.4 Expectation 43
2.5 Generating Functions 44
2.6 Common Applications of Random Variables 45
Application: Basic Warranty Modeling 45
3 Analysis of Multiple Random Variables 49
3.1 Two Random Variables 49
3.1.1 Two Discrete Random Variables 50
3.1.2 Two Continuous Random Variables 52
3.1.3 Expectation 55
3.2 Common Applications of Multiple Random Variables 61
3.2.1 The Multinomial Distribution 61
3.2.2 The Bivariate Normal Distribution 62
3.3 Analyzing Discrete Random Variables Using Conditional Probability 62
3.4 Analyzing Continuous Random Variables Using Conditional Probability 67
3.5 Computing Expectations by Conditioning 70
3.6 Computing Probabilities by Conditioning 75
Homework Problems 77
3.1 Two Random Variables 77
3.2 Common Applications of Multiple Random Variables 79
3.3 Analyzing Discrete Random Variables Using Conditional Probability 79
3.4 Analyzing Continuous Random Variables Using Conditional Probability 80
3.5 Computing Expectations by Conditioning 81
3.6 Computing Probabilities by Conditioning 83
Application: Bivariate Warranty Modeling 84
4 Introduction to Stochastic Processes 89
4.1 Introduction to Stochastic Processes 89
4.2 Introduction to Counting Processes 90
4.3 Introduction to Renewal Processes 91
4.3.1 Renewal-Reward Processes 94
4.3.2 Alternating Renewal Processes 95
4.4 Bernoulli Processes 97
Homework Problems 102
4.1 Introduction to Stochastic Processes 102
4.2 Introduction to Counting Processes 102
4.3 Introduction to Renewal Processes 102
4.4 Bernoulli Processes 104
Application: Acceptance Sampling 105
5 Poisson Processes 111
5.1 Introduction to Poisson Processes 111
5.2 Interarrival Times 114
5.3 Arrival Times 118
5.4 Decomposition and Superposition of Poisson Processes 121
5.5 Competing Poisson Processes 124
5.6 Nonhomogeneous Poisson Processes 125
Homework Problems 126
5.1 Introduction to Poisson Processes 126
5.2 Interarrival Times 128
5.3 Arrival Times 130
5.4 Decomposition and Superposition of Poisson Processes 131
5.5 Competing Poisson Processes 133
5.6 Nonhomogeneous Poisson Processes 133
Application: Repairable Equipment 134
6 Discrete-Time Markov Chains 137
6.1 Introduction 137
6.2 Manipulating the Transition Probability Matrix 141
6.3 Classification of States 147
6.4 Limiting Behavior 149
6.5 Absorbing States 152
Homework Problems 157
6.1 Introduction 157
6.2 Manipulating the Transition Probability Matrix 159
6.3 Classification of States 161
6.4 Limiting Behavior 161
6.5 Absorbing States 162
Application: Inventory Management 163
7 Continuous-Time Markov Chains 165
7.1 Introduction 165
7.2 Birth and Death Processes 168
7.3 Limiting Probabilities 170
7.4 Time-Dependent Behavior 173
7.5 Semi-Markov Processes 176
Homework Problems 177
7.2 Birth and Death Processes 177
7.3 Limiting Probabilities 177
7.5 Semi-Markov Processes 179
8 Markovian Queueing Systems 181
8.1 Queueing Basics 181
8.2 The M/M/1 Queue 184
8.3 The M/M/1/c Queue 186
8.4 The M/M/s Queue 188
8.5 The M/U/s/c Queue 191
8.6 The M/G/1 Queue 193
8.7 Networks of Queues 194
Homework Problems 196
8.1 Queueing Basics 196
8.2 The M/M/1 Queue 197
8.3 The M/M/1/c Queue 197
8.4 The M/M/s Queue 198
8.5 The M/M/s/c Queue 198
8.6 The M/G/1 Queue 198
8.7 Networks of Queues 199
Bibliography 201
Index 203
