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Simulation Modeling and AnalysisPowerPoint Presentation

Simulation Modeling and Analysis

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Simulation Modeling and Analysis. Sampling from Probability Distributions. 1. Outline. Inverse Transforms for Random Variate Generation Direct Transform and Convolution Acceptance-Rejection Technique. 2. Inverse Transforms for Random Variate Generation.

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Outline

- Inverse Transforms for Random Variate Generation
- Direct Transform and Convolution
- Acceptance-Rejection Technique

2

Inverse Transforms for Random Variate Generation

- Random variates are required to simulate the vagaries of arrivals, service, processing and the like which take place in the real world.
- Once a reliable RNG is available, how does one use it to obtain random variates with selected statistical distributions?

3

Steps in Inverse Transform Method

1.- Determine the cdf for the desired RV X.

2.- Set F(X) = R

3.- Solve F(X) = R for X in terms of R. I.e. X = F-1(R)

4.- Generate the necessary RN sequence Ri and use it to compute corresponding values Xi for i = 1,2,…, n

4

Inverse Transform Method for the Exponential Distribution

1.- F(x) = 1 - e - x , x > 0

2.- F(X) = 1 - e - x = R

3.- X = - (1/) ln (1 - R)

4.- For i = 1,2,…, n

Xi = - (1/) ln (1 - Ri)

- Example: Use RAND in Excel to create 1000 PRN’s and transform them to E. Examine your results in Stat::Fit.

5

Inverse Transform Method for the Uniform Distribution

1.- F(x) = {(x-a)/(b-a) ; 0 for a < x < b

2.- F(X) = (X -a)/(b-a) = R

3.- X = a + (b-a) R

4.- For i = 1,2,…, n

Xi = a + (b-a) Ri

- Example: Use RAND in Excel to create 1000 PRN’s and transform them to U. Examine your results in Stat::Fit.

6

Inverse Transform Method for the Weibull Distribution

1.- F(x) = 1 - e -x/) , x > 0

2.- F(X) = 1 - e -X/) = R

3.- X = [ ln (1 - R)]1/

4.- For i = 1,2,…, n

Xi = [ ln (1 - Ri )]1/

- Example: Use RAND in Excel to create 1000 PRN’s and transform them to W. Examine your results in Stat::Fit.

7

Inverse Transform Method for the Triangular Distribution

1.- F(x) = {0, for x <0; x2/2 for 0<x<1; 1-(2-x)2/2, for 1 <x<2; 1, for x > 2.

2.- R=X2/2 (0<X<1) and R = 1- (2-X)2/2 (1<X<2)

3.- X = 2R1/2 , (0<R<1/2); X = 2 - (2(1-R))1/2 , (1/2<R<1)

- Example: Use RAND in Excel to create 1000 PRN’s and transform them to W. Examine your results in Stat::Fit.

8

Inverse Transform Method for Empirical Distributions

- Similar procedure applies
- 1.- Determine the empirical cdf F(x)
- 2.- Generate R
- 3.- Using the F(X) curve determine the corresponding value of X.

9

Transform Method for Distributions without Closed Form Inverse

- Normal, gamma and beta distributions have no closed form inverses. Inverse transform method is applicable only approximately.
- For example, for the standard normal distribution
X ~ (R0.135 - (1-R) 0.135)/0.1975

- Example: Use RAND in Excel to create 1000 PRN’s and transform them to N. Examine your results in Stat::Fit.

10

Inverse Transform Method for Discrete Distributions Inverse

- Similar method applies.
- Lookup table
- Algebraic

11

Direct Transform for the Normal Inverse

- For the normal distribution
F(x) = -x (1/(2)1/2) e -(t2/2) dt

- Box-Muller method (pp. 341-343)
Z1 = (-2 ln R1)1/2 cos(2 R2)

Z2 = (-2 ln R1)1/2 sin(2 R2)

X1 = + Z1

X2 = + Z2

12

Convolution Method Inverse

- Convolution: the probability distribution of two or more independent RV
- Useful for Erlang and binomial variates
- Erlang w/parameters (K,
- Sum of K independent exponential RV each with mean 1/K
X = -(1/ K ) ln Ri

- Sum of K independent exponential RV each with mean 1/K

13

Acceptance-Rejection Technique Inverse

- Say random variates X are needed uniformly distributed between 1/4 and 1
- Steps
1.- Generate R

2a.- If R > 1/4 make X = R and go to 3

2b.- If R < 1/4 go to 1 until X is obtained

3.- Repeat from 1 for another X

14

Acceptance-Rejection Technique for Poisson Distribution Inverse

- Recall Poisson and exponential are closely related.
P(N=n) = e n/n!

- Steps
1.- Set n =0, P=1

2.- Generate R and replace P by P R

3.- If P < e , accept N = n, otherwise go to 2

15

Acceptance-Rejection Technique for Gamma Distribution Inverse

- Steps
1.- Compute a = (2-1)1/2 ; b = 2 - ln 4 + 1/a

2.- Generate R1, R2

3.- Compute X = [R1/(1-R1)]a

4a.- If X > b-ln(R12R2) reject and go to 2

4b.- If X < b-ln(R12R2) use X as is or as X/

16

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