Simulation

Ross's Simulation, Fourth Edition introduces aspiring and practicing actuaries, engineers, computer scientists and others to the practical aspects of constructing computerized simulation studies to analyze and interpret real phenomena. Readers learn to apply results of these analyses to problems in a wide variety of fields to obtain effective, accurate solutions and make predictions about future outcomes. This text explains how a computer can be used to generate random numbers, and how to use these random numbers to generate the behavior of a stochastic model over time. It presents the statistics needed to analyze simulated data as well as that needed for validating the simulation model.

Author Notes

Sheldon M. Ross is a professor in the Department of Industrial Engineering and Operations Research at the University of Southern California. He received his Ph.D. 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, and a recipient of the Humboldt US Senior Scientist Award.

Preface	p. ix
1 Introduction	p. 1
Exercises	p. 3
2 Elements of Probability	p. 5
2.1 Sample Space and Events	p. 5
2.2 Axioms of Probability	p. 6
2.3 Conditional Probability and Independence	p. 7
2.4 Random Variables	p. 9
2.5 Expectation	p. 11
2.6 Variance	p. 14
2.7 Chebyshev's Inequality and the Laws of Large Numbers	p. 16
2.8 Some Discrete Random Variables	p. 18
Binomial Random Variables	p. 18
Poisson Random Variables	p. 20
Geometric Random Variables	p. 22
The Negative Binomial Random Variable	p. 23
Hypergeometric Random Variables	p. 24
2.9 Continuous Random Variables	p. 24
Uniformly Distributed Random Variables	p. 25
Normal Random Variables	p. 26
Exponential Random Variables	p. 27
The Poisson Process and Gamma Random Variables	p. 29
The Nonhomogeneous Poisson Process	p. 32
2.10 Conditional Expectation and Conditional Variance	p. 33
The Conditional Variance Formula	p. 34
Exercises	p. 35
References	p. 39
3 Random Numbers	p. 41
Introduction	p. 41
3.1 Pseudorandom Number Generation	p. 41
3.2 Using Random Numbers to Evaluate Integrals	p. 42
Exercises	p. 46
References	p. 48
4 Generating Discrete Random Variables	p. 49
4.1 The Inverse Transform Method	p. 49
4.2 Generating a Poisson Random Variable	p. 55
4.3 Generating Binomial Random Variables	p. 57
4.4 The Acceptance-Rejection Technique	p. 58
4.5 The Composition Approach	p. 60
4.6 Generating Random Vectors	p. 61
Exercises	p. 62
5 Generating Continuous Random Variables	p. 67
Introduction	p. 67
5.1 The Inverse Transform Algorithm	p. 67
5.2 The Rejection Method	p. 71
5.3 The Polar Method for Generating Normal Random Variables	p. 78
5.4 Generating a Poisson Process	p. 82
5.5 Generating a Nonhomogeneous Poisson Process	p. 83
Exercises	p. 87
References	p. 91
6 The Discrete Event Simulation Approach	p. 93
Introduction	p. 93
6.1 Simulation via Discrete Events	p. 93
6.2 A Single-Server Queueing System	p. 94
6.3 A Queueing System with Two Servers in Series	p. 97
6.4 A Queueing System with Two Parallel Servers	p. 99
6.5 An Inventory Model	p. 102
6.6 An Insurance Risk Model	p. 103
6.7 A Repair Problem	p. 105
6.8 Exercising a Stock Option	p. 108
6.9 Verification of the Simulation Model	p. 110
Exercises	p. 111
References	p. 115
7 Statistical Analysis of Simulated Data	p. 117
Introduction	p. 117
7.1 The Sample Mean and Sample Variance	p. 117
7.2 Interval Estimates of a Population Mean	p. 123
7.3 The Bootstrapping Technique for Estimating Mean Square Errors	p. 126
Exercises	p. 133
References	p. 135
8 Variance Reduction Techniques	p. 137
Introduction	p. 137
8.1 The Use of Antithetic Variables	p. 139
8.2 The Use of Control Variates	p. 147
8.3 Variance Reduction by Conditioning	p. 154
Estimating the Expected Number of Renewals by Time t	p. 164
8.4 Stratified Sampling	p. 166
8.5 Applications of Stratified Sampling	p. 175
Analyzing Systems Having Poisson Arrivals	p. 176
Computing Multidimensional Integrals of Monotone Functions	p. 180
Compound Random Vectors	p. 182
8.6 Importance Sampling	p. 184
8.7 Using Common Random Numbers	p. 197
8.8 Evaluating an Exotic Option	p. 198
8.9 Estimating Functions of Random Permutations and Random Subsets	p. 203
Random Permutations	p. 203
Random Subsets	p. 206
8.10 Appendix: Verification of Antithetic Variable Approach When Estimating the Expected Value of Monotone Functions	p. 207
Exercises	p. 209
References	p. 217
9 Statistical Validation Techniques	p. 219
Introduction	p. 219
9.1 Goodness of Fit Tests	p. 219
The Chi-Square Goodness of Fit Test for Discrete Data	p. 220
The Kolmogorov-Smirnov Test for Continuous Data	p. 222
9.2 Goodness of Fit Tests When Some Parameters Are Unspecified	p. 227
The Discrete Data Case	p. 227
The Continuous Data Case	p. 230
9.3 The Two-Sample Problem	p. 230
9.4 Validating the Assumption of a Nonhomogeneous Poisson Process	p. 237
Exercises	p. 241
References	p. 244
10 Markov Chain Monte Carlo Methods	p. 245
Introduction	p. 245
10.1 Markov Chains	p. 245
10.2 The Hastings-Metropolis Algorithm	p. 248
10.3 The Gibbs Sampler	p. 251
10.4 Simulated Annealing	p. 262
10.5 The Sampling Importance Resampling Algorithm	p. 264
Exercises	p. 269
References	p. 272
11 Some Additional Topics	p. 273
Introduction	p. 273
11.1 The Alias Method for Generating Discrete Random Variables	p. 273
11.2 Simulating a Two-Dimensional Poisson Process	p. 277
11.3 Simulation Applications of an Identity for Sums of Bernoulli Random Variables	p. 280
11.4 Estimating the Distribution and the Mean of the First Passage Time of a Markov Chain	p. 285
11.5 Coupling from the Past	p. 289
Exercises	p. 291
References	p. 293
Index	p. 294

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