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# Chapter7 - Chapter 7 Random-Number Generation Banks Carson...

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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.

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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 , 0 1 0 , 1 ) ( x x f 2 1 2 ) ( 1 0 2 1 0 = = = 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 0 and m-1 by following a recursive relationship: The selection of the values for a , c , m , and X 0 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 , 0 , mod ) ( 1 = + = + i m c aX X i i The multiplier The increment The modulus ,... 2 , 1 , = = i m X R i i

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4 7 Example [LCM] Use X 0 = 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 - Chapter 7 Random-Number Generation Banks Carson...

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