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# Class Note for EMSE 273 with Professor Dorp at GW (11)

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Date Created: 02/07/15

SIMULATION MODELING AND ANALYSIS WITH ARENA T Altiok and B Melamed Chapter 4 Random Number and Variate Generation The Inverse Transform Method a Every cdf FX 13 is nondecreasing so the inverse cdf u is always well de ned b Applying the distribution function FX to the underlying variate X results in a variate FXX N Unif01 uniform between 0 and l 0 Conversely applying an inverse distribution function to a Unif0 l variate U results in a variate X F391u with cdf FX X 1 The Inverse Transform method uses a c in a twostep algorithm 1 Use your favorite RNG to generate a realization u from a variate U N Unif0 1 2 Compute 13 F u as a realization of X Altiok Melam ed Simulation Modeling and Analysis with Arena Chapter 4 The Inverse Transform Method Cont F Xm Altiok Melam ed Simulation Modeling and Analysis with Arena 3 Chapter 4 Generation of Uniform Variates Suppose we wish to generate a realization x om the uniform distribution Unif2lO for a realization u 065 of the underlying RNG Recall that the uniform cdf can be rewritten as 1 a b a ru where u is given and x is unknown Solving the above for x to obtain the inverse cdf readily yields the formula 1 F1 b au a and substituting a 2 b 10 and u 065 into the above results in the requisite value 1 10 2 065 2 72 Altiok Melam ed Simulation Modeling and Analysis with Arena Chapter 4 Generation of Exponential Variates 0 Suppose we wish to generate a realization x from the exponential distribution Exp005 of rate 05 mean 2 for a realization u 045 of the underlying RNG 0 Recall that the cdf of the exponential distribution can be written as u 1 vs Aw where u is given and x is unknown 0 Solving the above for x readily yields the formula 1 2 F121 lnl u and substituting 0 5 and u 045 above results in the requisite value 1 21nl 045 11957 0 Since U Unif0 1 implies 1 U W Unif01 it follows that the equation above may be simplified into the equivalent formula 1 2 F121 lnu Altiok Melam ed Simulation Modeling and Analysis with Arena 5 Chapter 4 Generation of Discrete Variates Suppose we code the state space by integers say S 12 and we wish to generate a realization x from a discrete pdf over S as in the table below m PXW 1 03 2 05 3 02 Recall that the discrete cdf can be rewritten as d ifwltl k Epjifk wltkl Altiok Melam ed Simulation Modeling and Analysis with Arena 6 Chapter 4 Generation of Discrete Variates Cont Utilizing the general formula for an inverse cdf we deduce that the inverse cdf can be written as 1 1 FX kildsk 19 1 k h 139 2 W 6 6 4 n21pnan21pn or equivalently 1 F u k when El lt lt nzlpn u nzlpn l for 0 S u lt 03 In our case we have 1 F u 2 for 03 g ult 08 3 for 08 S u lt l Altiok Melam ed Simulation Modeling and Analysis with Arena 7 Chapter 4 Generation of Discrete Variates Cont A 0 I I I I I 0 45 030 l 39 gt 0 1 2 3 4 The Inverse Transform method for generating a discrete variate Altiok Melam ed Simulation Modeling and Analysis with Arena 8 Chapter 4 Generation of Step Variates Suppose we wish to generate a realization x from a step distribution with J 4 steps whose pdf is speci ed in the table below i 1 7 pi l O 2 01 2 2 4 03 3 4 6 04 4 6 8 02 0 Recall that the step cdf can be rewritten as 0 if 1 lt l J j 1 pj 1 u 1 wzpw l nf lwltr j1 Z Z r l j j 7 7 1f 1 r 1 gt J Altiok Melam ed Simulation Modeling and Analysis with Arena 9 Chapter 4 Generation of Step Variates Cont 0 Solving the cdf for X yields the inverse step cdf r l 1 J j J J 1FXuZ11 julju 122315 p J h j 1 j W ere j nzz pnnZ1pn 0 It follows that the requisite variate realization x corresponding to u 05 is 2 I l sz 105l 05 p 3 3 X 3 11 1 p 3 4 05 0103 45 Altiok Melam ed Simulation Modeling and Analysis with Arena 10 Chapter 4 Generation of Step Variates Cont FXOC 45 The Inverse Transform method for generating a step variate Altiok Melam ed Simulation Modeling and Analysis with Arena 1 1 Chapter 4 Generation of Step Variates Cont Instead of using the Inverse Transform method for generating a step variate we could utilize the following properties of a step variate Property 1 PrX E Interval 2 p1 Property 2 The conditional distribution of X given that X E Inlte r ual i is a uniform distribution on l1 7 Algorithm Step 1 Sample Interval indea i from discrete distribution as before plapza apn Step 2 Given Interval indea i from Step 1 sample X from a uniform distribution on 13 as before Altiok Melam ed Simulation Modeling and Analysis with Arena 12 Chapter 4

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