Chapter7 - 1 Chapter 7 Random-Number Generation Banks,...

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Unformatted text preview: 1 Chapter 7 Random-Number Generation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation 2 Purpose & Overview ¡ Discuss the generation of random numbers. ¡ Introduce the subsequent testing for randomness: ¢ Frequency test ¢ Autocorrelation test. 2 3 Properties of Random Numbers ¡ Two important statistical properties: ¢ Uniformity ¢ Independence. ¡ Random Number, R i , must be independently drawn from a uniform distribution with pdf: Figure: pdf for random numbers ≤ ≤ = otherwise , 1 , 1 ) ( x x f 2 1 2 ) ( 1 2 1 = = = ∫ x xdx R E 4 Generation of Pseudo-Random Numbers ¡ “Pseudo”, because generating numbers using a known method removes the potential for true randomness. ¡ Goal: To produce a sequence of numbers in [ 0,1 ] that simulates, or imitates, the ideal properties of random numbers (RN). ¡ Important considerations in RN routines: ¢ Fast ¢ Portable to different computers ¢ Have sufficiently long cycle ¢ Replicable ¢ Closely approximate the ideal statistical properties of uniformity and independence. 3 5 Techniques for Generating Random Numbers ¡ Linear Congruential Method (LCM). ¡ Combined Linear Congruential Generators (CLCG). ¡ Random-Number Streams. 6 Linear Congruential Method [Techniques] ¡ To produce a sequence of integers, X 1 , X 2 , … between and m-1 by following a recursive relationship: ¡ The selection of the values for a , c , m , and X drastically affects the statistical properties and the cycle length. ¡ The random integers are being generated [ 0,m-1 ], and to convert the integers to random numbers: ,... 2 , 1 , , mod ) ( 1 = + = + i m c aX X i i The multiplier The increment The modulus ,... 2 , 1 , = = i m X R i i 4 7 Example [LCM] ¡ Use X = 27 , a = 17 , c = 43 , and m = 100 . ¡ The X i and R i values are: X 1 = (17*27+43) mod 100 = 502 mod 100 = 2, R 1 = 0.02; X 2 = (17*2+ 32 ) mod 100 = 77, R 2 = 0.77 ; X 3 = (17*77+ 32 ) mod 100 = 52, R 3 = 0.52; … 8 Characteristics of a Good Generator [LCM] ¡ Maximum Density ¢ Such that he values assumed by R i , i = 1,2,… , leave no large gaps on [0,1] ¢ Problem: Instead of continuous, each R i is discrete ¢ Solution: a very large integer for modulus m ¡ Approximation appears to be of little consequence ¡ Maximum Period ¢ To achieve maximum density and avoid cycling....
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Chapter7 - 1 Chapter 7 Random-Number Generation Banks,...

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