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Essentials of Monte Carlo Simulation

Statistical Methods for Building Simulation Models

verfasst von: Nick T. Thomopoulos

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Essentials of Monte Carlo Simulation focuses on the fundamentals of Monte Carlo methods using basic computer simulation techniques. The theories presented in this text deal with systems that are too complex to solve analytically. As a result, readers are given a system of interest and constructs using computer code, as well as algorithmic models to emulate how the system works internally. After the models are run several times, in a random sample way, the data for each output variable(s) of interest is analyzed by ordinary statistical methods. This book features 11 comprehensive chapters, and discusses such key topics as random number generators, multivariate random variates, and continuous random variates. Over 100 numerical examples are presented as part of the appendix to illustrate useful real world applications. The text also contains an easy to read presentation with minimal use of difficult mathematical concepts. Very little has been published in the area of computer Monte Carlo simulation methods, and this book will appeal to students and researchers in the fields of Mathematics and Statistics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The Monte Carlo method originated in the 1940s, just when computer technology was getting off the ground. It was a tool that allowed researches to solve very complicated problems that they could not solve analytically. The tool required a large stream of random numbers to have available on call, a requirement that previously was prohibitive because of the lack of high-speed data processing. The 1940s was the early days of data processing and the beginning of computers. As computer technology improved, mathematicians developed algorithms to generate random numbers in large quantities as needed. At the same time, practitioners of all sorts discovered the power of this new tool in problem solving. Special simulation languages and simulation software systems became available to allow the user quicker and easier ways to perform simulations on a computer. The chapter describes the basic fundamentals of the Monte Carlo method. Included here is a review on each of the remaining Chaps. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11, of the book.

Nick T. Thomopoulos

Chapter 2. Random Number Generators

Abstract

The integrity of computer simulation models is only as good as the reliability of the random number generator that produces the stream of random numbers one after the other. The chapter describes the difficult task of developing an algorithm to generate random numbers that are statistically valid and have a large cycle length. The linear congruent method is currently the common way to generate the random numbers for a computer. The parameters of this method include the multiplier and the seed. Only a few multipliers are statistically recommended, and two popular ones in use for 32-bit word length computers are presented. Another parameter is the seed and this allows the analyst the choice of altering the sequence of random numbers with each run, and also when necessary, offers the choice of using the same sequence of random numbers from one run to another.

Nick T. Thomopoulos

Chapter 3. Generating Random Variates

Abstract

This chapter shows how the continuous uniform u~U(0,1) random variates are used to generate random variates for random variables from defined probability distributions. To accomplish in a computer simulation model, a random number generator algorithm isapplied whenever a random uniform u~U(0,1) variate is needed. The random number generator is the catalyst that delivers the uniform,u~U(0,1), random variates, one after another, as they are needed in the simulation model. This is essential since it allows the analyst the opportunity to create simulation models that use any probability distribution that pertains and gives flexibility to emulate the actual system under study.

Nick T. Thomopoulos

Chapter 4. Generating Continuous Random Variates

Abstract

This chapter shows how to transform the continuous uniform random variates, u∼U(0,1), to random variates for a variable that comes from one of the common continuous probability distributions. The probability distributions described here are the following: the continuous uniform, exponential, Erlang, gamma, beta, Weibull, normal, lognormal, chi-square, student’s t, and Fishers F. The chapter also shows how to use the (Hasting’s) approximation formulas for the standard normal distribution.

Nick T. Thomopoulos

Chapter 5. Generating Discrete Random Variates

Abstract

This chapter shows how to transform continuous uniform random variates, u∼U(0,1), to random discrete variates for a variable that comes from one of the more common discrete probability distributions. The probability distributions described here are the following: discrete arbitrary, discrete uniform, Bernoulli, binomial, hyper-geometric, geometric, Pascal and Poisson.

Nick T. Thomopoulos

Chapter 6. Generating Multivariate Random Variates

Abstract

This chapter considers some of the more popular multivariate distributions and shows how to generate random variates for each. The probability distributions described are the following: multivariate discrete arbitrary, multinomial, multivariate hyper geometric, bivariate normal, bivariate lognormal, multivariate normal and multivariate lognormal. The Cholesky decomposition method is also presented because of its important role in generating random variates from the multivariate normal and multivariate lognormal distributions.

Nick T. Thomopoulos

Chapter 7. Special Applications

Abstract

This chapter concerns applications that are not from the common probability distributions, continuous or discrete. The applications are instructive since they show some popular deviations in generating random variates as is often needed in building computer simulation models. The applications presented are the Poisson process, constant Poisson process, batch arrivals, active redundancy, standby redundancy, random integers without replacement and poker.

Nick T. Thomopoulos

Chapter 8. Output from Simulation Runs

Abstract

Computer simulation models are mainly developed to emulate actual systems that are too complex to analyze mathematically. The systems often fall into the terminating or the nonterminating type. The chapter describes how output data is collected for either type of system. Terminating systems have a defined beginning and ending event, and nonterminating systems include a combination of transient, equilibrium and cyclical stages. The output data from these systems are needed subsequently to statistically analyze the performance of the system that is under study. Another simulation model presented is one that creates the database to be used as test data for software applications like forecasting and inventory replenishments.

Nick T. Thomopoulos

Chapter 9. Analysis of Output Data

Abstract

This chapter describes the common statistical methods that are used to analyze the output data from computer models that are based on terminating and nonterminating systems. The statistical methods are essentially the same that are described in the common statistical textbooks. They include measuring the average, standard deviation, confidence interval from output data, some of the variable type and some of the proportion type. The methods described also pertain when the two or more variables are in review.

Nick T. Thomopoulos

Chapter 10. Choosing the Probability Distribution from Data

Abstract

Computer simulation models often include one or more variables that play important roles in the model. Some of the random variables are of the continuous type and others are discrete. The analyst is confronted with choosing the proper probability distribution for each variable, and also with estimating the associated parameter(s) value. The chapter describes some of the common ways to select the distribution and to estimate the associated parameter values when some empirical or sample data is available from the real system.

Nick T. Thomopoulos

Chapter 11. Choosing the Probability Distribution When No Data

Abstract

Sometimes the analyst may need to develop a computer simulation model that includes one or more variables where no empirical or sample data is available. This is where he/she seeks opinions from one or more experts who give some estimates on the characteristics of the variable. The chapter pertains to these situations and shows some of the common ways to select the probability distribution and estimate the associated parameters.

Nick T. Thomopoulos

Backmatter

Titel: Essentials of Monte Carlo Simulation
verfasst von: Nick T. Thomopoulos
Verlag: Springer New York
Electronic ISBN: 978-1-4614-6022-0
Print ISBN: 978-1-4614-6021-3
DOI: https://doi.org/10.1007/978-1-4614-6022-0