Simulation and the Monte Carlo Method

Chapter 1: Preliminaries

• • 1.1 Introduction
• • 1.2 Random Experiments
• • 1.3 Conditional Probability and Independence
• • 1.4 Random Variables and Probability Distributions
• • 1.5 Some Important Distributions
• • 1.6 Expectation
• • 1.7 Joint Distributions
• • 1.8 Functions of Random Variables
• • 1.9 Transforms
• • 1.10 Jointly Normal Random Variables
• • 1.11 Limit Theorems
• • 1.12 Poisson Processes
• • 1.13 Markov Processes
• • 1.14 Gaussian Processes
• • 1.15 Information
• • 1.16 Convex Optimization and Duality
• • Problems
• • References

Chapter 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 Gaussian Processes
• • 2.9 Generating Diffusion Processes
• • 2.10 Generating Random Permutations
• • Problems
• • References

Chapter 3: Simulation of Discrete-Event Systems

• • 3.1 Introduction
• • 3.2 Simulation Models
• • 3.3 Simulation Clock and Event List for DEDS
• • 3.4 Discrete-Event Simulation
• • Problems
• • References

Chapter 4: Statistical Analysis of Discrete-Event Systems

• • 4.1 Introduction
• • 4.2 Estimators and Confidence Intervals
• • 4.3 Static Simulation Models
• • 4.4 Dynamic Simulation Models
• • 4.5 Bootstrap Method
• • Problems
• • References

Chapter 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 Multilevel Monte Carlo
• • 5.7 Importance Sampling
• • 5.8 Sequential Importance Sampling
• • 5.9 Sequential Importance Resampling
• • 5.10 Nonlinear Filtering for Hidden Markov Models
• • 5.11 Transform Likelihood Ratio Method
• • 5.12 Preventing the Degeneracy of Importance Sampling
• • Problems
• • References

Chapter 6: Markov Chain Monte Carlo

• • 6.1 Introduction
• • 6.2 Metropolis–Hastings Algorithm
• • 6.3 Hit-and-Run Sampler
• • 6.4 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

Chapter 7: Sensitivity Analysis and Monte Carlo Optimization

• • 7.1 Introduction
• • 7.2 Score Function Method for Sensitivity Analysis of DESS
• • 7.3 Simulation-Based Optimization of DESS
• • 7.4 Sensitivity Analysis of DEDS
• • Problems
• • References

Chapter 8: Cross-Entropy Method

• • 8.1 Introduction
• • 8.2 Estimation of Rare-Event Probabilities
• • 8.3 CE Method for Optimization
• • 8.4 Max-Cut Problem
• • 8.5 Partition Problem
• • 8.6 Traveling Salesman Problem
• • 8.7 Continuous Optimization
• • 8.8 Noisy Optimization
• • 8.9 MinxEnt Method
• • Problems
• • References

Chapter 9: Splitting Method

• • 9.1 Introduction
• • 9.2 Counting Self-Avoiding Walks via Splitting
• • 9.3 Splitting with a Fixed Splitting Factor
• • 9.4 Splitting with a Fixed Effort
• • 9.5 Generalized Splitting
• • 9.6 Adaptive Splitting
• • 9.7 Application of Splitting to Network Reliability
• • 9.8 Applications to Counting
• • 9.9 Case Studies for Counting with Splitting
• • 9.10 Splitting as a Sampling Method
• • 9.11 Splitting for Optimization
• • Problems
• • References

Chapter 10: Stochastic Enumeration Method

• • 10.1 Introduction
• • 10.2 Tree Search and Tree Counting
• • 10.3 Knuth’s Algorithm for Estimating the Cost of a Tree
• • 10.4 Stochastic Enumeration
• • 10.5 Application of SE to Counting
• • 10.6 Application of SE to Network Reliability
• • Problems
• • References

Chapter Appendix: Appendix

• • 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 CE Algorithm for Optimizing the Peaks Function
• • A.6 Discrete-Time Kalman Filter
• • A.7 Bernoulli Disruption Problem
• • A.8 Complexity
• • Problems
• • References