IE406, I & IEUTKSimulationStatistical Models in SimulationDr. Xueping LiUniversity of TennesseeBased onBanks, Carson, Nelson & NicolDiscrete-Event System Simulation2Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model might well describe the variations. An appropriate model can be developed by sampling the phenomenon of interest: Select a known distribution through educated guesses Make estimate of the parameter(s) Test for goodness of fit We will: Review several important probability distributions Present some typical application of these models3Review of Terminology and Concepts In this section, we will review the following concepts: Discrete random variables Continuous random variables Cumulative distribution function Expectation4Discrete Random Variables [Probability Review] X is a discrete random variable if the number of possible values of X is finite, or countably infinite. Example: Consider jobs arriving at a job shop. Let X be the number of jobs arriving each week at a job shop. Rx= possible values of X (range space of X) = {0,1,2,…} p(xi) = probability the random variable is xi= P(X = xi) p(xi), i = 1,2, … must satisfy: The collection of pairs [xi, p(xi)], i = 1,2,…, is called the probability distribution of X, and p(xi) is called the probability mass function (pmf) of X.∑∞==≥11)( 2. allfor ,0)( 1.iiixpixp5Continuous Random Variables [Probability Review] X is a continuous random variable if its range space Rxis an interval or a collection of intervals. The probability that X lies in the interval [a,b] is given by: f(x), denoted as the pdf of X, satisfies: PropertiesXRXRxxfdxxfRxxfXin not is if ,0)( 3.1)( 2.in allfor , 0)( 1.==≥∫∫=≤≤badxxfbXaP )()(0)( because ,0)( 1.000dxxfxXPxx===∫)()()()( .2 bXaPbXaPbXaPbXaP ≺≺≺≺ =≤=≤=≤≤6Continuous Random Variables [Probability Review] Example: Life of an inspection device is given by X, a continuous random variable with pdf: X has an exponential distribution with mean 2 years Probability that the device’s life is between 2 and 3 years is:⎪⎩⎪⎨⎧≥=−otherwise ,00 x,21)(2/xexf14.021)32(322/==≤≤∫−dxexPx7Cumulative Distribution Function [Probability Review] Cumulative Distribution Function (cdf) is denoted by F(x), where F(x) = P(X <= x) If X is discrete, then If X is continuous, then Properties All probability question about X can be answered in terms of the cdf, e.g.:∑≤=xxiixpxF all)()(∫∞−=xdttfxF )()(0)(lim 3.1)(lim 2.)()( then , If function. ingnondecreas is 1.==≤−∞→∞→xFxFbFaFbaFxx≺baaFbFbXaP ≺≺ allfor ,)()()(−=≤8Cumulative Distribution Function [Probability Review] Example: An inspection device has cdf: The probability that the device lasts for less than 2 years: The probability that it lasts between 2 and 3 years:2/02/121)(xxtedtexF−−−==∫632.01)2()0()2()20(1=−==−=≤≤−eFFFXP145.0)1()1()2()3()32(1)2/3(=−−−=−=≤≤−−eeFFXP9Expectation [Probability Review] The expected value of X is denoted by E(X) If X is discrete If X is continuous a.k.a the mean, m, or the 1stmoment of X A measure of the central tendency The variance of X is denoted by V(X) or var(X) or σ2 Definition: V(X) = E[(X – E[X]2] Also, V(X) = E(X2) – [E(x)]2 A measure of the spread or variation of the possible values of X around the mean The standard deviation of X is denoted by σ Definition: Square root of V(X) Expressed in the same units as the mean∑=iiixpxxE all)()(∫∞∞−= dxxxfxE )()(10Expectations [Probability Review] Example: The mean of life of the previous inspection device is: To compute variance of X, we first compute E(X2): Hence, the variance and standard deviation of the device’s life are:22/21)(02/002/=+−==∫−∫∞−∞∞−dxexdxxeXExxxe82/221)(02/002/22=+−==∫−∫∞−∞∞−dxexdxexXExxex428)(2=−=XV2)( == XVσ11Useful Statistical Models In this section, statistical models appropriate to some application areas are presented. The areas include: Queueing systems Inventory and supply-chain systems Reliability and maintainability Limited data12Queueing Systems [Useful Models] In a queueing system, interarrival and service-time patterns can be probablistic. Sample statistical models for interarrival or service time distribution: Exponential distribution: if service times are completely random Normal distribution: fairly constant but with some random variability (either positive or negative) Truncated normal distribution: similar to normal distribution but with restricted value. Gamma and Weibull distribution: more general than exponential (involving location of the modes of pdf’s and the shapes of tails.)13Inventory and supply chain [Useful Models] In realistic inventory and supply-chain systems, there are at least three random variables: The number of units demanded per order or per time period The time between demands The lead time Sample statistical models for lead time distribution: Gamma Sample statistical models for demand distribution: Poisson: simple and extensively tabulated. Negative binomial distribution: longer tail than Poisson (more large demands). Geometric: special case of negative binomial given at least one demand has occurred.14Reliability and maintainability [Useful Models] Time to failure (TTF) Exponential: failures are random Gamma: for standby redundancy where each component has an exponential TTF Weibull: failure is due to the most serious of a large number of defects in a system of components Normal: failures are due to wear15Other areas [Useful Models] For cases with limited data, some useful distributions are: Uniform, triangular and beta Other distribution: Bernoulli, binomial and hyperexponential.16Discrete Distributions Discrete random variables are used to describe random phenomena in which only integer values can occur. In this section, we will learn about: Bernoulli trials and Bernoulli distribution Binomial distribution Geometric and negative binomial distribution Poisson distribution17Bernoulli Trials and Bernoulli Distribution[Discrete Dist’n] Bernoulli Trials: Consider an experiment consisting of n trials, each can be a
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