Table of Contents
Preface vii
1 Basic Simulation Programming 1
1.1 Introduction 1
1.2 Random Numbers Generators 1
1.2.1 Basic Generators 4
1.2.2 The Need for Multiple Substreams 5
1.2.3 Computing (a × s) mod m 6
1.2.4 Computing the Jumping Matrices 6
1.2.5 A Random Number Package 7
1.2.6 Jumping Backward 9
1.3 Examples of Using Random Number Generator 10
1.4 Nonuniform Random Variates 13
1.4.1 Random Variates of Various Distributions 14
1.4.2 Correlated Random Variates 18
1.5 Utilities 20
1.5.1 Numerical Approximation of Normal Distribution 20
1.5.2 Quantile of Normal Distribution 21
1.5.3 Quantile of t Distribution 22
1.5.4 Quantile of Chi-square Distribution 23
1.5.5 Standard Deviation 25
1.6 Summary 26
2 Sample Sizes and Stopping Rules 27
2.1 Definitions 30
2.2 Batch-Means Method 32
2.3 Determining the Simulation Run Length 33
2.3.1 The von Neumann Test of Independence 33
2.3.2 A Source Code of the von Neumann Test 33
2.3.3 The Runs Test of Independence 34
2.3.4 A Source Code of Runs Up Test 36
2.3.5 An Implementation of Determining the Simulation Run Length 38
2.4 Constructing the Confidence Interval 38
2.5 A Correlation Adjustment 39
2.6 An Implementation of Batch-Means Method 39
2.7 An Illustration of Allocated Sample Sizes 43
2.8 Empirical Experiments 43
2.8.1 Experiment 1 46
2.8.2 Experiment 2 47
2.8.3 Experiment 3 48
2.9 Summary 49
3 Generating Independent and Identically Distributed Batch Means 51
3.1 Discussion of Batch-Means Method 51
3.2 Generating Independent and Normally Distributed Batch Means 52
3.2.1 Validation of Normality 52
3.2.2 A Source Code of Normality test 53
3.2.3 Batch Means Variance Estimator 54
3.2.4 The Implementation 56
3.2.5 Discussions of Batch-Means Procedures 58
3.3 Empirical Experiments 59
3.3.1 Experiment 1: Independence and Normality Tests 60
3.3.2 Experiment 2: Batch Sizes Determination 62
3.3.3 Experiment 3: Coverages of Confidence Interval 64
3.4 Summary 66
4 Distributions of Order Statistics 69
4.1 Joint and Conditional Distributions of Order Statistics 72
4.2 Using Range Statistics to Perform Equivalence Tests 74
4.2.1 Indifference-Zone Selection 74
4.2.2 Variance of Weighted Sample Means 75
4.2.3 Effects of the Indifference Amount and Sample Size 76
4.2.4 Equivalence Tests 77
4.2.5 Confidence Interval Half Width of Interest 79
4.3 Statistical Analysis of the Range 80
4.3.1 Simulating the Sample Range 81
4.3.2 Estimating Quantiles of the Range 82
4.4 Empirical Experiments 84
4.5 Summary 85
5 Order Statistics from Correlated Normal Random Variables 87
5.1 Order Statistics of Correlated Random Variables 88
5.1.1 Method of Evaluation of the Percentage Points 89
5.2 Applications of Correlated Order Statistics 90
5.2.1 Multiple Comparisons with a Control 90
5.2.2 Multiple Decision (Ranking and Selection) 92
5.2.3 Multiple Comparisons with a Control: Unknown Equal Variances 92
5.2.4 Multiple Comparisons with a Control: Unknown Unequal Variances 93
5.3 Empirical Experiments 94
5.3.1 Experiment 1: Known Equal Variances 94
5.3.2 Experiment 2: Unknown Equal Variances 95
5.3.3 Experiment 3: Unknown Unequal Variances 95
5.4 Summary 97
6 Histogram and Quasi-Independent Procedure 99
6.1 Introduction and Definitions 101
6.1.1 The Natural Estimators 103
6.1.2 Proportion Estimation 104
6.1.3 Quantile Estimation 105
6.2 Methodologies 106
6.2.1 Determining the Simulation Run Length 106
6.2.2 Histogram Approximation 106
6.2.3 Two-Phase Quantile Estimation 110
6.2.4 A Source Code of Quantile Estimation 113
6.3 Empirical Experiments 120
6.3.1 Independent Sequences 120
6.3.2 Correlated Sequences 122
6.3.3 A Practical Application 125
6.4 Summary 127
7 Metamodels 129
7.1 Introduction 129
7.2 Constructing Metamodels of Quantiles 131
7.3 Constructing Quantile Confidence Interval 133
7.4 Empirical experiments 133
7.4.1 Choosing the Design Points 133
7.4.2 Estimating Quantiles of Moving-Average and Autoregressive Processes via Non-functional-form Metamodels 134
7.4.3 Estimating Quantiles of Queuing Systems via Nonfunctional-form Metamodels 136
7.5 Summary 138
8 Density Estimation 141
8.1 Theoretical Basis 142
8.1.1 Empirical Distribution Functions 142
8.1.2 The Density Estimator 144
8.1.3 The Complication of Lack of Independence 146
8.2 An Implementation 146
8.2.1 Determine the Bandwidth 147
8.2.1 Determine the Sample Size 148
8.2.2 Density Confidence Interval 149
8.2.3 The Density-Estimation Procedure 150
8.3 Empirical Experiments 151
8.4 Summary 157
9 Comparing Two Alternatives 161
9.1 Background 161
9.1.1 Inference Procedures of Two Means 161
9.1.2 Null Hypothesis Tests of Equivalence 163
9.2 Methodology 164
9.2.1 A Weighted-Sample-Means Approach 164
9.2.2 Fix the Value of β = α/2 of Null Hypothesis Tests 165
9.3 Empirical Experiments 167
9.3.1 Experiment 1: Difference of Means 167
9.3.2 Experiment 2: Null Hypothesis of Equal Means 169
9.4 Summary 170
10 Ranking and Selection 171
10.1 Introduction 172
10.1.1 Generalized Subset Selection 172
10.1.2 Order Statistics of Continuous Distributions 173
10.1.3 A Review of Confidence Interval Half Width 175
10.1.4 Confidence Interval Half Width of Interest 176
10.1.5 Adjustment of the Difference of Sample Means 177
10.1.6 The Source Code of Computing Additional Sample Size 179
10.1.7 A Sequential Ranking and Selection Procedure (SRS) 180
10.2 Some Extensions of Selection of Continuous Distributions 182
10.2.1 Restricted Subset Selection 182
10.2.2 An Indifference-Zone Procedure to Select Only and/or All The Best Systems 183
10.2.3 Ratio Statistics of Variance of Normally Distributed Variables 186
10.2.4 Multiple Comparisons with the Best 190
10.3 Lognormally Distributed Samples 191
10.3.1 The Property of the Constant hL 192
10.4 Other Approach of Selection Procedures 195
10.5 Empirical Experiments 196
10.5.1 Experiment 1: Normal Populations 196
10.5.2 Experiment 2: Exponential Populations 197
10.5.3 Experiment 3: Lognormal Populations 197
10.6 Summary 199
11 Computing Budget Allocation of Selection Procedures 201
11.1 Problem Statement 202
11.2 A Heuristic Computing Budget Allocation Rule 203
11.2.1 Confidence Interval Half-Width and Computing Budget 207
11.2.2 Maximizing Probability of Correction Selection with a Given Computing Budget 209
11.2.3 Optimal Computing Budget Allocation (OCBA) 211
11.3 Empirical Experiments 212
11.3.1 Experiment 1 Equal Variances 213
11.3.2 Experiment 2 Increasing Variances 214
11.3.3 Experiment 3 Decreasing Variances 216
11.4 Summary 216
12 Using Common Random Numbers with Selection Procedures 219
12.1 Common Random Numbers 219
12.2 The Basis of Correlated Order Statistics 220
12.2.1 Using CRNs with Dudewicz and Dalal's Procedure 220
12.2.2 Subset Selection with CRN 222
12.3 Empirical Experiments 225
12.3.1 Experiment 1: All Systems are Correlated 225
12.3.2 Experiment 2: Best System is Independent with Others 226
12.3.3 Experiment 3: Best System is Negatively Correlated with Others 229
12.3.4 Experiment 4: Unequal Variances 229
12.3.5 Experiment 5: Subset Selection - All Systems are Correlated 231
12.3.6 Experiment 6: Subset Selection - Independence Between Groups 1 233
12.3.7 Experiment 7: Subset Selection - Independence Between Groups 2 233
12.3.8 Experiment 8: Subset Selection - Unequal Variances 233
12.4 Summary 236
13 Parallel and Distributed Simulation 237
13.1 Introduction 237
13.2 Parallel and Distributed Selection 238
13.3 The Framework 240
13.4 Selection with All Pairwise Comparisons 241
13.5 Empirical Experiments 243
13.6 Summary 245
14 Multi-Objective Selection 247
14.1 Introduction 248
14.1.1 A Multi-Objective Selection Procedure 249 I
14.2 Methodologies 250
14.2.1 Prolog 250
14.2.2 The Strategy 251
14.2.3 The Incomplete Pareto Set Selection Procedure 252
14.2.4 The Two-Stage Pareto Set Selection Procedure 253
14.2.5 Incorporating Indifference-Zone 254
14.3 Empirical Experiments 255
14.3.1 Experiment 1: The Parameter mp = 2 255
14.3.2 Experiment 2: The Parameter mp = 3 256
14.3.3 Experiment 3: The Parameter mp = 3 257
14.4 Summary 257
15 Generic Selection with Constraints 261
15.1 Methodologies 262
15.1.1 Multi-Objective Selection 262
15.1.2 A Generic Selection-With-Constraints Procedure 263
15.1.3 Variance as the Constraint 266
15.1.4 Variance as the Primary Performance Measure 267
15.2 Empirical Experiments 268
15.2.1 Selection With Constraints 268
15.2.2 Variance as the Constraint 270
15.2.3 Variance as the Primary Performance Measure 271
15.3 Summary 274
Appendix A Tables of Critical Constants 275
Bibliography 277
Index 285
