Simulation and the Monte Carlo Method

Preface. Acknowledgments. 1. Preliminaries 1. 1.1 Random Experiments. 1.2 Conditional Probability and Independence. 1.3 Random Variables and Probability Distributions. 1.4 Some Important Distributions. 1.5 Expectation. 1.6 Joint Distributions. 1.7 Functions of Random Variables. 1.8 Transforms. 1.9 Jointly Normal Random Variables. 1.10 Limit Theorems. 1.11 Poisson Processes. 1.12 Markov Processes. 1.13 Efficiency of Estimators. 1.14 Information. 1.15 Convex Optimization and Duality. Problems. References. 2. Random Number, Random Variable and Stochastic Process Generation. 2.1 Introduction. 2.2 Random Number Generation. 2.3 Random Variable Generation. 2.4 Generating From Commonly Used Distributions. 2.5 Random Vector Generation. 2.6 Generating Poisson Processes. 2.7 Generating Markov Chains and Markov Jump Processes. 2.8 Generating Random Permutations. Problems. References. 3. Simulation of Discrete Event Systems. 3.1 Simulation Models. 3.2 Simulation Clock and Event List for DEDS. 3.3 Discrete Event Simulation. Problems. References. 4. Statistical Analysis of Discrete Event Systems. 4.1 Introduction. 4.2 Static Simulation Models. 4.3 Dynamic Simulation Models. 4.4 The Bootstrap Method. Problems. References. 5. Controlling the Variance. 5.1 Introduction. 5.2 Common and Antithetic Random Variables. 5.3 Control Variables. 5.4 Conditional Monte Carlo. 5.5 Stratified Sampling. 5.6 Importance Sampling. 5.7 Sequential Importance Sampling. 5.8 The Transform Likelihood Ratio Method. 5.9 Preventing the Degeneracy of Importance Sampling. Problems. References. 6. Markov Chain Monte Carlo. 6.1 Introduction. 6.2 The Metropolis-Hastings Algorithm. 6.3 The Hit-and-Run Sampler. 6.4 The Gibbs Sampler. 6.5 Ising and Potts Models. 6.6 Bayesian Statistics. 6.7 Other Markov Samplers. 6.8 Simulated Annealing. 6.9 Perfect Sampling. Problems. References. 7. Sensitivity Analysis and Monte Carlo Optimization. 7.1 Introduction. 7.2 The Score Function Method for Sensitivity Analysis of DESS. 7.3 Simulation-Based Optimization of DESS. 7.4 Sensitivity Analysis of DEDS. Problems. References. 8. The Cross-Entropy Method. 8.1 Introduction. 8.2 Estimation of Rare Event Probabilities. 8.3 The CE-Method for Optimization. 8.4 The Max-cut Problem. 8.5 The Partition Problem. 8.6 The Travelling Salesman Problem. 8.7 Continuous Optimization. 8.8 Noisy Optimization. Problems. References. 9. Counting via Monte Carlo. 9.1 Counting Problems. 9.2 Satisfiability Problem. 9.3 The Rare-Event Framework for Counting. 9.4 Other Randomized Algorithms for Counting. 9.5 MinxEnt and Parametric MinxEnt. 9.6 PME for COPs and Decision Making. 9.7 Numerical Results. Problems. References. Appendix A. A.1 Cholesky Square Root Method. A.2 Exact Sampling from a Conditional Bernoulli Distribution. A.3 Exponential Families. A.4 Sensitivity Analysis. A.5 A simple implementation of the CE algorithm for optimizing the 'peaks' function. A.6 Discrete-time Kalman Filter. A.7 Bernoulli Disruption Problem. A.8 Complexity of Stochastic Programming Problems. Problems. References. Acronyms. List of Symbols. Index.