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.

1.12.1 Markov Chains.

1.12.2 Markov Jump Processes.

1.13 Efficiency of Estimators.

1.14 Information.

1.15 Convex Optimization and Duality.

1.15.1 Lagrangian Method.

1.15.2 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.

2.3.1 Inverse-Transform Method.

2.3.2 Alias Method.

2.3.3 Composition Method.

2.3.4 Acceptance-Rejection Method.

2.4 Generating From Commonly Used Distributions.

2.4.1 Generating Continuous Random Variables.

2.4.2 Generating Discrete Random Variables.

2.5 Random Vector Generation.

2.5.1 Vector Acceptance-Rejection Method.

2.5.2 Generating Variables From a Multinormal Distribution.

2.5.3 Generating Uniform Random Vectors Over a Simplex.

2.5.4 Generating Random Vectors, Uniformly Distributed Over a Unit Hyper-Ball and Hyper-Sphere.

2.5.5 Generating Random Vectors, Uniformly Distributed Over a Hyper-Ellipsoid.

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.

3.3.1 Tandem Queue.

3.3.2 Repairman Problem.

Problems.

References.

**4. Statistical Analysis of Discrete Event Systems.**

4.1 Introduction.

4.2 Static Simulation Models.

4.3 Dynamic Simulation Models.

4.3.1 Finite-Horizon Simulation.

4.3.2 Steady-State Simulation.

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.

5.4.1 Variance Reduction for Reliability Models.

5.5 Stratified Sampling.

5.6 Importance Sampling.

5.6.1 The Variance Minimization Method.

5.6.2 The Cross-Entropy Method.

5.7 Sequential Importance Sampling.

5.7.1 Non-linear Filtering for Hidden Markov Models.

5.8 The Transform Likelihood Ratio Method.

5.9 Preventing the Degeneracy of Importance Sampling.

5.9.1 The Two-Stage Screening Algorithm.

5.9.2 Case Study.

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.

7.3.1 Stochastic Approximation.

7.3.2 The Stochastic Counterpart Method.

7.4 Sensitivity Analysis of DEDS.

Problems.

References.

**8. The Cross-Entropy Method.**

8.1 Introduction.

8.2 Estimation of Rare Event Probabilities.

8.2.1 The Root-Finding Problem.

8.2.2 The Screening Method for Rare Events.

8.3 The CE-Method for Optimization.

8.4 The Max-cut Problem.

8.5 The Partition Problem.

8.6 The Travelling Salesman Problem.

8.6.1 Incomplete Graphs.

8.6.2 Node Placement.

8.6.3 Case Studies.

8.7 Continuous Optimization.

8.8 Noisy Optimization.

Problems.

References.

**9. Counting via Monte Carlo.**

9.1 Counting Problems.

9.2 Satisfiability Problem.

9.2.1 Random K-SAT (K-RSAT).

9.3 The Rare-Event Framework for Counting.

9.3.1 Rare-Events for the Satisfiability Problem.

9.4 Other Randomized Algorithms for Counting.

9.4.1 Complexity of Randomized Algorithms: FPRAS and FPAUS.

9.5 MinxEnt and Parametric MinxEnt.

9.5.1 The MinxEnt Method.

9.5.2 Rare-Event Probability Estimation Using PME.

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.4.1 Convexity Results.

A.4.2 Monotonicity Results.

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.