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Random Number GeneratorsPowerPoint Presentation

Random Number Generators

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Random Number Generators

xn = f ( xn-1, xn-2) where x0 is seed

- Pseudo-random since, given the same seed, the sequence is repeatable and deterministic
- Cycle length – length of repeating sequence
- Example: xn = a xn-1 + b mod m

seed

cycle

period

D = S

(oi – ei)2

ei

i=1

Testing Random NumbersChi-Square test

- Discreet distributions, large sample sizes, general
- Does an observed data set satisfy a specified distribution?
- Prepare a histogram of observed data – k cells
- D has chi-square distribution with k-1 degrees of freedom
- Null hypothesis that observations come from distribution can not be rejected at significance a if computed D is less than C 2[1-a;k-1]
- Works best with equiprobable cells – cell sizes so the frequencies are equal.

Chi-Square Example

(observed-expected)2

expected

6.25

0.49

0.25

0.09

0.0

0.16

0.04

0.49

0.36

2.25

Sum = 10.38 whereas X 2[0.9;9] = 14.68

K+ = n max [Fo(x) – Fe(x)]

x

K- = n max [Fe(x) – Fo(x)]

x

Testing Random NumbersKolmogorov-Smirnov test

- Continuous distributions, small sample sizes, general
- Based on differences between observed and expected CDFs
- If K+ and K- are smaller than K[1-a;n] the observations are said to come from the distribution with level of significance a.

1

Rk = S

(Ui – ½)(Ui+k – ½)

n-k

i=1

Rk! z1-a/2 /(12 n-k)

Testing Random NumbersSerial-Correlation test

- For a sequence of numbers, compute covariance between numbers that are k apart: xi and xi+k
- Autocovariance at lag k, do for range of lags.
- If the C.I. includes zero, not significant correlation

100(1-a)% CI:

Simulation Techniques Overview

Simulation environments

emulation

Workloadparameters

exec-drivensim

SystemConfigparameters

Result Data

trace-drivensim

-> discussion of timing-firstpaper

Factorlevels

stochasticsim

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