Simulation Modeling and Analysis

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Simulation Modeling and Analysis

• Pseudo-Random Numbers

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Outline

• Properties of Random Numbers
• Generating Random Numbers
• Testing Random Numbers

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Properties of Random Numbers

• Key Properties
• Uniformity
• Independence
• Density function (continuous!)
• f(x) 1 for 0 lt x lt 1, 0 otherwise
• Moments
• E(R) 1/2 V(R) 1/12

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Generating Random Numbers

• Random Numbers vs Pseudo-random Numbers
• Requirements of a RNG routine
• Speed
• Portable
• Long Cycle
• Replicable RN
• Uniform and Independent RNs

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Random Number Generation

• Linear Congruential Method
• X i1 (a Xi c) mod m
• Ri Xi/m
• Note Only values from the set
• I 0,1/m,2/m,,(m-1)/m
• are obtained

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Random Number Generation -contd

• Longest Possible Period (P)
• If m 2b and c gt 0 , P m
• If m 2b and c 0 , P m/4
• If m prime and c 0 , P m-1
• Example
• X i1 (75 Xi ) mod (231-1)

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Random Number Generation -contd

• Combined Congruential Generators. Two distinct
  congruential generators can be combined to obtain
  PRNs with longer periods.
• X i1 (?(-1)j-1 Xi,j ) mod (m1 - 1)
• Ri Xi/m1, Xi gt 0 Ri (m1-1)/m1, Xi gt 0

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Testing Random Numbers

• Null Hypotheses
• H0 Ri U0,1 H0 Ri independent
• Tests
• Frequency test
• Runs test
• Autocorrelation test
• Gap test
• Poker test

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Kolmogorov-Smirnov Frequency Test

• 1.- Arrange data in increasing value
• 2.- Compute D, D- and D
• 3.- Find critical Dc (Handout) for given a
• 4.- Accept or reject the null hypothesis.
• 5.- Example StatFit

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Chi-Square Frequency Test

• The Chi static compares observed frequencies of
  occurrence of PRNs in selected subdomains
  against expected frequencies derived from the U
  distribution function. See StatFit
• X02 ?n (Oi - Ei)2/Ei

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Runs Testing

• Run sequence of similar events
• Runs up and runs down (independence)
• Maximum number of runs (N numbers) N-1
• mean (2N-1)/3 variance (16N-29)/90
• Test hypothesis against normal distribution.

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Runs Testing -contd

• Runs above and below the mean
• Maximum number of runs (N numbers, n1 above and
  n2 below the mean) n1n2
• mean 2 n1 n2/N 1/2
• variance 2 n1 n2 (2 n1 n2 - N)/N2 (N-1)
• Test hypothesis against normal distribution.

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Runs Testing -contd

• Runs length
• Test hypothesis against Chi square distribution

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Autocorrelation Testing

• Seek the autocorrelation between every m numbers
  (I.e. dependence)
• Null Hypothesis H0 ?im 0
• Note If values are uncorrelated, ?im has normal
  distribution. So, test hypothesis against normal
  distribution.

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Gap Testing

• Gap Interval of recurrence of same digit.
• Monitor Frequency of gaps and test
• 1.- Specify the cdf F(x) 1-0.9 x1
• 2.- Arrange the observed gaps into S(x)
• 3.- Find D and Dc
• 4.- Accept or reject the null hypothesis.

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Poker Test

• Frequency of repetition of certain digits in a
  series
• Null hypothesis is tested againts the Chi-square
  distribution.

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