Simulation and the Monte Carlo Method
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Persons
Reuven Y. Rubinstein, DSc, was Professor Emeritus in the Faculty of Industrial Engineering and Management at Technion-Israel Institute of Technology. He served as a consultant at numerous large-scale organizations, such as IBM, Motorola, and NEC. The author of over 100 articles and six books, Dr. Rubinstein was also the inventor of the popular score-function method in simulation analysis and generic cross-entropy methods for combinatorial optimization and counting.
Dirk P. Kroese, PhD, is a Professor of Mathematics and Statistics in the School of Mathematics and Physics of The University of Queensland, Australia. He has published over 100 articles and four books in a wide range of areas in applied probability and statistics, including Monte Carlo methods, cross-entropy, randomized algorithms, tele-traffic c theory, reliability, computational statistics, applied probability, and stochastic modeling.
Content
Preface xiii
Acknowledgments xvii
1 Preliminaries 1
1.1 Introduction 1
1.2 Random Experiments 1
1.3 Conditional Probability and Independence 2
1.4 Random Variables and Probability Distributions 4
1.5 Some Important Distributions 5
1.6 Expectation 6
1.7 Joint Distributions 7
1.8 Functions of Random Variables 11
1.8.1 Linear Transformations 12
1.8.2 General Transformations 13
1.9 Transforms 14
1.10 Jointly Normal Random Variables 15
1.11 Limit Theorems 16
1.12 Poisson Processes 17
1.13 Markov Processes 19
1.13.1 Markov Chains 19
1.13.2 Classification of States 21
1.13.3 Limiting Behavior 22
1.13.4 Reversibility 24
1.13.5 Markov Jump Processes 25
1.14 Gaussian Processes 27
1.15 Information 28
1.15.1 Shannon Entropy 29
1.15.2 Kullback-Leibler Cross-Entropy 31
1.15.3 Maximum Likelihood Estimator and Score Function 32
1.15.4 Fisher Information 33
1.16 Convex Optimization and Duality 34
1.16.1 Lagrangian Method 35
1.16.2 Duality 37
Problems 41
References 46
2 Random Number, Random Variable, and Stochastic Process Generation 49
2.1 Introduction 49
2.2 Random Number Generation 49
2.2.1 Multiple Recursive Generators 51
2.2.2 Modulo 2 Linear Generators 52
2.3 Random Variable Generation 55
2.3.1 Inverse-Transform Method 55
2.3.2 Alias Method 57
2.3.3 Composition Method 58
2.3.4 Acceptance-Rejection Method 59
2.4 Generating from Commonly Used Distributions 62
2.4.1 Generating Continuous Random Variables 62
2.4.2 Generating Discrete Random Variables 67
2.5 Random Vector Generation 70
2.5.1 Vector Acceptance-Rejection Method 71
2.5.2 Generating Variables from a Multinormal Distribution 72
2.5.3 Generating Uniform Random Vectors over a Simplex 73
2.5.4 Generating Random Vectors Uniformly Distributed over a Unit Hyperball and Hypersphere 74
2.5.5 Generating Random Vectors Uniformly Distributed inside a Hyperellipsoid 75
2.6 Generating Poisson Processes 75
2.7 Generating Markov Chains and Markov Jump Processes 77
2.7.1 Random Walk on a Graph 78
2.7.2 Generating Markov Jump Processes 79
2.8 Generating Gaussian Processes 80
2.9 Generating Diffusion Processes 81
2.10 Generating Random Permutations 83
Problems 85
References 89
3 Simulation of Discrete-Event Systems 91
3.1 Introduction 91
3.2 Simulation Models 92
3.2.1 Classification of Simulation Models 94
3.3 Simulation Clock and Event List for DEDS 95
3.4 Discrete-Event Simulation 97
3.4.1 Tandem Queue 97
3.4.2 Repairman Problem 101
Problems 103
References 106
4 Statistical Analysis of Discrete-Event Systems 107
4.1 Introduction 107
4.2 Estimators and Confidence Intervals 108
4.3 Static Simulation Models 110
4.4 Dynamic Simulation Models 112
4.4.1 Finite-Horizon Simulation 114
4.4.2 Steady-State Simulation 114
4.5 Bootstrap Method 126
Problems 127
References 130
5 Controlling the Variance 133
5.1 Introduction 133
5.2 Common and Antithetic Random Variables 134
5.3 Control Variables 137
5.4 Conditional Monte Carlo 139
5.4.1 Variance Reduction for Reliability Models 141
5.5 Stratified Sampling 144
5.6 Multilevel Monte Carlo 146
5.7 Importance Sampling 149
5.7.1 Weighted Samples 149
5.7.2 Variance Minimization Method 150
5.7.3 Cross-Entropy Method 154
5.8 Sequential Importance Sampling 159
5.9 Sequential Importance Resampling 165
5.10 Nonlinear Filtering for Hidden Markov Models 167
5.11 Transform Likelihood Ratio Method 171
5.12 Preventing the Degeneracy of Importance Sampling 174
Problems 179
References 184
6 Markov Chain Monte Carlo 187
6.1 Introduction 187
6.2 Metropolis-Hastings Algorithm 188
6.3 Hit-and-Run Sampler 193
6.4 Gibbs Sampler 194
6.5 Ising and Potts Models 197
6.5.1 Ising Model 197
6.5.2 Potts Model 198
6.6 Bayesian Statistics 200
6.7 Other Markov Samplers 202
6.7.1 Slice Sampler 204
6.7.2 Reversible Jump Sampler 205
6.8 Simulated Annealing 208
6.9 Perfect Sampling 212
Problems 214
References 219
7 Sensitivity Analysis and Monte Carlo Optimization 221
7.1 Introduction 221
7.2 Score Function Method for Sensitivity Analysis of DESS 224
7.3 Simulation-Based Optimization of DESS 231
7.3.1 Stochastic Approximation 232
7.3.2 Stochastic Counterpart Method 237
7.4 Sensitivity Analysis of DEDS 246
Problems 252
References 255
8 Cross-Entropy Method 257
8.1 Introduction 257
8.2 Estimation of Rare-Event Probabilities 258
8.2.1 Root-Finding Problem 267
8.2.2 Screening Method for Rare Events 268
8.2.3 CE Method Combined with Sampling from the Zero-Variance Distribution 271
8.3 CE Method for Optimization 272
8.4 Max-Cut Problem 276
8.5 Partition Problem 282
8.5.1 Empirical Computational Complexity 283
8.6 Traveling Salesman Problem 283
8.6.1 Incomplete Graphs 288
8.6.2 Node Placement 289
8.6.3 Case Studies 290
8.7 Continuous Optimization 291
8.8 Noisy Optimization 292
8.9 MinxEnt Method 294
Problems 298
References 303
9 Splitting Method 307
9.1 Introduction 307
9.2 Counting Self-Avoiding Walks via Splitting 308
9.3 Splitting with a Fixed Splitting Factor 310
9.4 Splitting with a Fixed Effort 313
9.5 Generalized Splitting 314
9.6 Adaptive Splitting 318
9.7 Application of Splitting to Network Reliability 321
9.8 Applications to Counting 322
9.9 Case Studies for Counting with Splitting 325
9.9.1 Satisfiability (SAT) Problem 325
9.9.2 Independent Sets 330
9.9.3 Permanent and Counting Perfect Matchings 332
9.9.4 Binary Contingency Tables 334
9.9.5 Vertex Coloring 336
9.10 Splitting as a Sampling Method 337
9.11 Splitting for Optimization 340
9.11.1 Continuous Optimization 343
Problems 344
References 348
10 Stochastic Enumeration Method 351
10.1 Introduction 351
10.2 Tree Search and Tree Counting 352
10.3 Knuth's Algorithm for Estimating the Cost of a Tree 355
10.4 Stochastic Enumeration 357
10.4.1 Combining SE with Oracles 359
10.5 Application of SE to Counting 360
10.5.1 Counting the Number of Paths in a Network 360
10.5.2 Counting SATs 363
10.5.3 Counting the Number of Perfect Matchings in a Bipartite Graph 366
10.6 Application of SE to Network Reliability 368
10.6.1 Numerical Results 370
Problems 373
References 375
Appendix 377
A.1 Cholesky Square Root Method 377
A.2 Exact Sampling from a Conditional Bernoulli Distribution 378
A.3 Exponential Families 379
A.4 Sensitivity Analysis 382
A.4.1 Convexity Results 383
A.4.2 Monotonicity Results 384
A.5 A Simple CE Algorithm for Optimizing the Peaks Function 385
A.6 Discrete-Time Kalman Filter 385
A.7 Bernoulli Disruption Problem 387
A.8 Complexity 389
A.8.1 Complexity of Rare-Event Algorithms 389
A.8.2 Complexity of Randomized Algorithms: FPRAS and FPAUS 390
A.8.3 SATs in CNF 394
A.8.4 Complexity of Stochastic Programming Problems 395
Problems 402
References 403
Abbreviations and Acronyms 405
List of Symbols 407
Index 409
PREFACE
Since the publication in 2008 of the second edition of Simulation and the Monte Carlo Method, significant changes have taken place in the field of Monte Carlo simulation. This third edition gives a fully updated and comprehensive account of the major topics in Monte Carlo simulation.
The book is based on an undergraduate course on Monte Carlo methods given at the Israel Institute of Technology (Technion) and the University of Queensland for the last five years. It is aimed at a broad audience of students in engineering, physical and life sciences, statistics, computer science, mathematics, and simply anyone interested in using Monte Carlo simulation in their study or work. Our aim is to provide an accessible introduction to modern Monte Carlo methods, focusing on the main concepts, while providing a sound foundation for problem solving. For this reason most ideas are introduced and explained via concrete examples, algorithms, and experiments.
Although we assume that the reader has some basic mathematical background, such as an elementary course in probability and statistics, we nevertheless review the basic concepts of probability, stochastic processes, information theory, and convex optimization in Chapter 1.
In a typical stochastic simulation, randomness is introduced into simulation models via independent uniformly distributed random variables. These random variables are then used as building blocks to simulate more general stochastic systems. Chapter 2 deals with the generation of such random numbers, random variables, and stochastic processes.
Many real-world complex systems can be modeled as discrete-event systems. Examples of discrete-event systems include traffic systems, flexible manufacturing systems, computer-communications systems, inventory systems, production lines, coherent lifetime systems, PERT networks, and flow networks. The behavior of such systems is identified via a sequence of discrete "events" that causes the system to change from one "state" to another. We discuss how to model such systems on a computer in Chapter 3.
Chapter 4 treats the statistical analysis of the output data from static and dynamic simulation models. The main difference is that the former do not evolve in time whereas the latter do. For dynamic models, we distinguish between finite-horizon and steady-state simulations. Two popular methods for estimating steady-state performance measures - the batch means and regenerative methods - are discussed as well.
Chapter 5 deals with variance reduction techniques in Monte Carlo simulation, such as antithetic and common random numbers, control random variables, conditional Monte Carlo, stratified sampling, and importance sampling. Using importance sampling, one can often achieve substantial (sometimes dramatic) variance reduction, in particular when estimating rare-event probabilities. While dealing with importance sampling, we present two alternative approaches, called the variance minimization and the cross-entropy methods. Special attention is paid to importance sampling algorithms in which paths are generated in a sequential manner. Further improvements of such algorithms are obtained by resampling successful paths, giving rise to sequential importance resampling algorithms. We illustrate their use via a nonlinear filtering example. In addition, this chapter contains two new importance sampling based methods, called the transform likelihood ratio method and the screening method for variance reduction. The former presents a simple, convenient and unifying way of constructing efficient importance sampling estimators, whereas the latter ensures lowering of the dimensionality of the importance sampling density. This is accomplished by identifying (screening out) the most important (bottleneck) parameters to be used in the importance sampling distribution. As results, the accuracy of the importance sampling estimator increases substantially.
Chapter 6 gives a concise treatment of the generic Markov chain Monte Carlo (MCMC) method for approximately generating samples from an arbitrary distribution. We discuss the classic Metropolis-Hastings algorithm and the Gibbs sampler. In the former, one simulates a Markov chain such that its stationary distribution coincides with the target distribution, while in the latter the underlying Markov chain is constructed on the basis of a sequence of conditional distributions. We also deal with applications of MCMC in Bayesian statistics, and explain how MCMC is used to sample from the Boltzmann distribution for the Ising and Potts models, which are extensively used in statistical mechanics. Moreover, we show how MCMC is used in the simulated annealing method to find the global minimum of a multi-extremal function. We also show that both the Metropolis-Hastings and Gibbs samplers can be viewed as special cases of a general MCMC algorithm and then present two more modifications, namely the slice and the reversible jump samplers.
Chapter 7 is on sensitivity analysis and Monte Carlo optimization of simulated systems. Because of their complexity, the performance evaluation of discrete-event systems is usually studied by simulation, and the simulation is often associated with the estimation of the performance function with respect to some controllable parameters. Sensitivity analysis is concerned with evaluating sensitivities (gradients, Hessians, etc.) of the performance function with respect to system parameters. This provides guidance to operational decisions and to selecting system parameters that optimize the performance measures. Monte Carlo optimization deals with solving stochastic programs, that is, optimization problems where the objective function and some of the constraints are unknown and need to be obtained via simulation. We deal with sensitivity analysis and optimization of both static and dynamic models. We introduce the celebrated score function method for sensitivity analysis, and two alternative methods for Monte Carlo optimization, the so-called stochastic approximation and stochastic counterpart methods. In particular, in the latter method, we show how using a single simulation experiment one can approximate quite accurately the true unknown optimal solution of the original deterministic program.
Chapter 8 deals with the cross-entropy (CE) method, which was introduced by the first author in 1997 as an adaptive algorithm for rare-event estimation using a cross-entropy minimization technique. It was soon realized that the underlying ideas had a much wider range of application than just in rare-event simulation: they could be readily adapted to tackle quite general combinatorial and multi-extremal optimization problems, including many problems associated with learning algorithms and neural computation. We provide a gradual introduction to the CE method, and show its elegance and versatility. In particular, we present a general CE algorithm for the estimation of rare-event probabilities and then slightly modify it for solving combinatorial optimization problems. We discuss applications of the CE method to several combinatorial optimization problems, such as the max-cut problem and the traveling salesman problem, and provide supportive numerical results on its effectiveness. Due to its versatility, tractability, and simplicity, the CE method has potentially a diverse range of applications, for example, in computational biology, DNA sequence alignment, graph theory, and scheduling. Over the last 10 years many hundreds of papers have been written on the theory and applications of CE. For more details see the site www.cemethod.org, our book The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning (Springer, 2004), and search in the wikipedia under "cross-entropy method". The chapter concludes with a discussion of the minimum cross-entropy (MinxEnt) optimization program.
Chapter 9 introduces the splitting method, which uses a sequential sampling plan to decompose a "difficult" problem into a sequence of "easy" problems. The method was originally designed for rare-event simulation, but it has developed into a highly versatile "particle MCMC" algorithm that can be used for rare-event estimation, optimization, and sampling. The chapter presents various splitting algorithms for dynamic and static simulation models, and demonstrates how they can be used to (1) estimate rare-event probabilities, (2) solve hard counting problems, (3) find solutions to challenging optimization problems, and (4) sample from complicated probability distributions. The chapter features a wide variety of case studies and numerical experiments, demonstrating the effectiveness of the method.
Many combinatorial problems can be formulated in terms of searching or counting the total cost of a tree. Chapter 10 presents a new Monte Carlo called stochastic enumeration (SE) that is well suited to solve such problems by generating random paths through the tree in a parallel fashion. The SE algorithm can be viewed as a sequential importance sampling method on a "hyper-tree" whose vertices are sets of vertices of the original tree. By combining SE with fast polynomial decision algorithms, we show how it can be used for counting #P-complete problems, such as the number of satisfiability assignments, number of paths in a general network, and the number of perfect matchings in a graph. The usefulness...
