Simulation - · PDF filexii Preface in either case, the usual approach to estimating θ...
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Sheldon M. RossEpstein Department of Industrial
and Systems Engineering
University of Southern California
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Fifth edition 2013
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Library of Congress Cataloging-in-Publication DataRoss, Sheldon M.Simulation / Sheldon M. Ross, Epstein Department of Industrial and Systems
Engineering, University of Southern California. Fifth edition.pages cm
Includes bibliographical references and index.ISBN 978-0-12-415825-2 (hardback)1. Random variables. 2. Probabilities. 3. Computer simulation. I. Title.QA273.R82 2012519.2dc23
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In formulating a stochastic model to describe a real phenomenon, it used to be thatone compromised between choosing a model that is a realistic replica of the actualsituation and choosing one whose mathematical analysis is tractable. That is, theredid not seem to be any payoff in choosing a model that faithfully conformed tothe phenomenon under study if it were not possible to mathematically analyzethat model. Similar considerations have led to the concentration on asymptotic orsteady-state results as opposed to the more useful ones on transient time. However,the advent of fast and inexpensive computational power has opened up anotherapproachnamely, to try to model the phenomenon as faithfully as possible andthen to rely on a simulation study to analyze it.
In this text we show how to analyze a model by use of a simulation study. Inparticular, we first show how a computer can be utilized to generate random (moreprecisely, pseudorandom) numbers, and then how these random numbers can beused to generate the values of random variables from arbitrary distributions. Usingthe concept of discrete events we show how to use random variables to generate thebehavior of a stochastic model over time. By continually generating the behavior ofthe system we show how to obtain estimators of desired quantities of interest. Thestatistical questions of when to stop a simulation and what confidence to place inthe resulting estimators are considered. A variety of ways in which one can improveon the usual simulation estimators are presented. In addition, we show how to usesimulation to determine whether the stochastic model chosen is consistent with aset of actual data.
New to This Edition
New exercises in most chapters. A new Chapter 6, dealing both with the multivariate normal distribution, and
with copulas, which are useful for modeling the joint distribution of randomvariables.
Chapter 9, dealing with variance reduction, includes additional material onstratification. For instance, it is shown that stratifying on a variable alwaysresults in an estimator having smaller variance than would be obtaind byusing that variable as a control. There is also a new subsection on the use ofpost stratification.
There is a new chapter dealing with additional variance reduction methodsbeyond those previously covered. Chapter 10 introduces the conditionalBernoulli sampling method, normalized importance sampling, and LatinHypercube sampling.
The chapter on Markov chain Monte Carlo methods has an new sectionentitled Continuous time Markov chains and a Queueing Loss Model.
The successive chapters in this text are as follows. Chapter 1 is an introductorychapter which presents a typical phenomenon that is of interest to study. Chapter 2is a review of probability. Whereas this chapter is self-contained and does notassume the reader is familiar with probability, we imagine that it will indeed be areview for most readers. Chapter 3 deals with random numbers and how a variantof them (the so-called pseudorandom numbers) can be generated on a computer.The use of random numbers to generate discrete and then continuous randomvariables is considered in Chapters 4 and 5.
Chapter 6 studies the multivariate normal distribution, and introduces copulaswhich are useful for modeling the joint distribution of random variables. Chapter 7presents the discrete event approach to track an arbitrary system as it evolvesover time. A variety of examplesrelating to both single and multiple serverqueueing systems, to an insurance risk model, to an inventory system, to a machinerepair model, and to the exercising of a stock optionare presented. Chapter 8introduces the subject matter of statistics. Assuming that our average reader has notpreviously studied this subject, the chapter starts with very basic concepts and endsby introducing the bootstrap statistical method, which is quite useful in analyzingthe results of a simulation.
Chapter 9 deals with the important subject of variance reduction. This is anattempt to improve on the usual simulation estimators by finding ones havingthe same mean and smaller variances. The chapter begins by introducing thetechnique of using antithetic variables. We note (with a proof deferred to thechapters appendix) that this always results in a variance reduction along with
a computational savings when we are trying to estimate the expected value ofa function that is monotone in each of its variables. We then introduce controlvariables and illustrate their usefulness in variance reduction. For instance, we showhow control variables can be effectively utilized in analyzing queueing systems,reliability systems, a list reordering problem, and blackjack. We also indicate howto use regression packages to facilitate the resulting computations when usingcontrol variables. Variance reduction by use of conditional expectations is thenconsidered, and its use is indicated in examples dealing with estimating , andin analyzing finite capacity queueing systems. Also, in conjunction with a controlvariate, conditional expectation is used to estimate the expected number of eventsof a renewal process by some fixed time. The use of stratified sampling as a variancereduction tool is indicated in examples dealing with queues with varying arrivalrates and evaluating integrals. The relationship between the variance reductiontechniques of conditional expectation and stratified sampling is explained andillustrated in the estimation of the expected return in video poker. Applications ofstratified sampling to queueing systems having Poisson arrivals, to computation ofmultidimensional integrals, and to compound random vectors are also given. Thetechnique of importance sampling is next considered. We indicate and explain howthis can be an extremely powerful variance reduction technique when estimatingsmall probabilities. In doing so, we introduce the concept of tilted distributionsand show how they can be utilized in an importance sampling estimation ofa small convolution tail probability. Applications of importance sampling toqueueing, random walks, and random permutations, and to computing conditionalexpectations when one is conditioning on a rare event are presented. The finalvariance reduction technique of Chapter 9 relates to the use of a common stream ofrandom numbers. Chapter 10 introduces additional variance reduction techniques.
Chapter 11 is concerned with statistical validation techniques, which arestatistical procedures that can be used to validate the stochastic model when somereal data are available. Goodness of fit tests such as the chi-square test and theKolmogorovSmirnov test are presented. Other sections in this chapter deal withthe two-sample and the n-sample problems and with ways of statistically testingthe hypothesis that a given process is a Poisson process.
Chapter 12 is concerned with Markov chain Monte Carlo methods. These aretechniques that have greatly expanded the use of simulation in recent years. Thestandard simulation paradigm for estimating = E[h(X)], where X is a randomvector, is to simulate independent and identically distributed copies of X andthen use the average value of h(X) as the estimator. This is the so-called rawsimulation estimator, which c