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Simulation and the Monte Carlo Method, 2nd Edition

Simulation and the Monte Carlo Method, 2nd Edition

Reuven Y. Rubinstein, Dirk P. Kroese

ISBN: 978-0-470-23038-1

Jun 2007

384 pages


This accessible new edition explores the major topics in Monte Carlo simulation

Simulation and the Monte Carlo Method, Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences.

The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including:

  • Markov Chain Monte Carlo
  • Variance reduction techniques such as the transform likelihood ratio method and the screening method
  • The score function method for sensitivity analysis
  • The stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization
  • The cross-entropy method to rare events estimation and combinatorial optimization
  • Application of Monte Carlo techniques for counting problems, with an emphasis on the parametric minimum cross-entropy method

An extensive range of exercises is provided at the end of each chapter, with more difficult sections and exercises marked accordingly for advanced readers. A generous sampling of applied examples is positioned throughout the book, emphasizing various areas of application, and a detailed appendix presents an introduction to exponential families, a discussion of the computational complexity of stochastic programming problems, and sample MATLAB programs.

Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method.

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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.



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.



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.



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.



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.



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.



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.



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.



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.



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.




List of Symbols.


  • This long awaited second edition gives a fully updated and comprehensive account of the major topics in Monte Carlo simulation since the early 1980's.
  • The basic concepts of probability, Markov processes, and convex optimization are now carefully reviewed in a completely revised Chapter.
  • A new co-author has been added to enliven the writing style and to provide modern day expertise .
  • The authors aim to provide an accessible introduction to modern MCM, focusing on the main concepts, while providing a sound foundation for problem solving.
  • Most ideas are introduced and explained by way of concrete examples, algorithms, and practical experiments.
""The book is clearly written and easy to read for people with mathematical background.... The material of the book is useful in most areas of the nowadays research work."" (International Statistical Review, April 2009)

""I enjoyed reading the book, and found the individual examples quite interesting."" (Biometrics, December 2008)

""I enjoyed reading the book, and found the individual examples quite interesting."" (Biometrics, December 2008)

""..if you need to learn how to use Monte Carlo in your simulations, this is probably the best single document I have ever read. ""(Computing Reviews, September 2008)

""Rubinstein and Kroese did an exemplary job of addressing major issues and providing much needed updated information in this area."" (CHOICE, June 2008)

""the book is nicely written and the additional to the book from the 1st edition certainly make it more attractive to a wider audience. I would recommend it to students and practioners with appropriate background."" (MAA Review March 2008)

  • An extensive range of exercises is provided at the end of each chapter. More difficult sections and exercises are marked accordingly.
  • Examples of cross-entropy programs, written in Matlab, are given in an appendix.
  • A generous sampling of applied examples are positioned throughout the book emphasizing areas such as engineering, computer science, finance, statistics, and the physical and life sciences.