Computer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.

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1 Simulation Discrete-Event System Simulation

2 Chapter 6 andom-variate Generation

3 Purpose & Overview Develop understanding of generating samples from a specified distribution as input to a simulation model. Illustrate some widely-used techniques for generating random variates. Inverse-transform technique Acceptance-rejection technique Special properties 3

4 Preparation It is assumed that a source of uniform [0,] random numbers eists. Linear Congruential Method andom numbers,,, with PDF f 0 ( ) 0 otherwise CDF F ( ) 0 0 < 0 > 4

5 Inverse-transform Technique The concept: For CDF function: r F() Generate r from uniform (0,), a.k.a U(0,) Find, F - (r) F() F() r F() r F() r r 5

6 Inverse-transform Technique The inverse-transform technique can be used in principle for any distribution. Most useful when the CDF F() has an inverse F - () which is easy to compute. equired steps. Compute the CDF of the desired random variable.. Set F() on the range of. 3. Solve the equation F() for in terms of. 4. Generate uniform random numbers,, 3, and compute the desired random variate by i F - ( i ) 6

7 Eponential Distribution Eponential Distribution PDF f ( ) CDF F( ) λe λ e Simplification ln() λ λ Since and (-) are uniformly distributed on [0,] To generate,, 3 e e λ λ λ ln( ) ln( ) λ ln( ) λ F ( ) 7

8 Eponential Distribution Figure: Inverse-transform technique for ep(λ ) 8

9 Eponential Distribution Eample: Generate 00 variates i with distribution ep(λ ) Generate 00 s with U(0,), the histogram of s become: 0,7 0,6 0,5 0,4 0,3 0, 0, 0 0,5,5,5 3 3,5 4 4,5 5 5,5 6 6,5 7 Empirical Histogram Theor. PDF Check: Does the random variable have the desired distribution? P ( 0) P( F( 0)) F( 0) 9

10 Other Distributions Eamples of other distributions for which inverse CDF works are: Uniform distribution Weibull distribution Triangular distribution 0

11 Computer Science, Informatik 4 Uniform Distribution andom variable uniformly distributed over [a, b] ) ( ) ( ) ( a b a a b a a b a F +

12 Weibull Distribution The Weibull Distribution is described by PDF f ( ) CDF F( ) β β α β e e ( ) β α ( ) β α The variate is F( ) β ( ) α e β ( ) α e ( ) α α β β β β ln( ln( α β ) ) ln( ) β β α ln( ) β α ln( )

13 Computer Science, Informatik 4 3 Triangular Distribution The CDF of a Triangular Distribution with endpoints (0, ) is given by is generated by ) ( and for For 0 > < < ) ( ) ( F < ) ( 0

14 Empirical Continuous Distributions When theoretical distribution is not applicable To collect empirical data: esample the observed data Interpolate between observed data points to fill in the gaps For a small sample set (size n): Arrange the data from smallest to largest () () (n) Set (0) 0 Assign the probability /n to each interval The slope of each line segment is defined as ( i) ( i) ( i) ( i) ai / n ( i ) / n / n The inverse CDf is given by Fˆ ( ) ( i ) + a i ( i ) n (i-) (i) 4

15 Empirical Continuous Distributions i Interval Probability Cumulative Probability Slope a i 0.0 < < < < < (4) + a 4 ( (4 ) / ( ).66 n) 5

16 Empirical Continuous Distributions What happens for large samples of data Several hundreds or tens of thousand First summarize the data into a frequency distribution with smaller number of intervals Afterwards, fit continuous empirical CDF to the frequency distribution Slight modifications Slope a i ( i) c i c ( i) i The inverse CDf is given by ˆ F ( ) + a c ( ) ( i) i i c i cumulative probability of the first i intervals 6

17 Empirical Continuous Distributions Eample: Suppose the data collected for 00 broken-widget repair times are: i Interval (Hours) Frequency elative Frequency Cumulative Frequency, c i Slope, a i Consider 0.83: c < < c 4.00 (4-) + a 4 ( c (4-) ) ( ).75 7

18 Empirical Continuous Distributions Problems with empirical distributions The data in the previous eample is restricted in the range The underlying distribution might have a wider range Thus, try to find a theoretical distribution Hints for building empirical distribution based on frequency tables It is recommended to use relatively short intervals - Number of bin increase This will result in a more accurate estimate 8

19 Discrete Distribution All discrete distributions can be generated via inverse-transform technique Method: numerically, table-lookup procedure, algebraically, or a formula Eamples of application: Empirical Discrete uniform Gamma 9

20 Discrete Distribution Eample: Suppose the number of shipments,, on the loading dock of a company is either 0,, or Data - Probability distribution: p() F() The inverse-transform technique as table-lookup procedure F ( i ) ri < ri F( i ) Set i 0

21 Discrete Distribution Method - Given, the generation scheme becomes: 0,,, 0.5 < 0.8 < i 3 Table for generating the discrete variate Input r i Output i 0 Consider 0.73: F( i- ) < < F( i ) F( 0 ) < 0.73 < F( ) Hence,

22 Acceptance-ejection technique Useful particularly when inverse cdf does not eist in closed form, a.k.a. thinning Illustration: To generate random variates, ~ U(/4, ) Procedures: Step. Generate ~ U[0,] Step a. If > ¼, accept. Step b. If < ¼, reject, return to Step no Generate Condition yes Output does not have the desired distribution, but conditioned ( ) on the event { ¼} does. Efficiency: Depends heavily on the ability to minimize the number of rejections.

23 Poisson Distribution PMF of a Poisson Distribution n α α P( N n) e n! Eactly n arrivals during one time unit A + A + L+ An < A + A + L+ An + An + Since interarrival times are eponentially distributed we can set A i ln( i ) α Well known, we derived this generator in the beginning of the class 3

24 Poisson Distribution Substitute the sum by Simplify by n i n+ ln( ) i ln( ) < i α α i multiply by -α, which reverses the inequality sign sum of logs is the log of a product ln n i i n i ln( i ) α > n+ i ln( ) i ln n i i Simplify by e ln() n i i e α > n+ i i 4

25 Poisson Distribution Procedure of generating a Poisson random variate N is as follows. Set n0, P. Generate a random number n+, and replace P by P n+ 3. If P < ep(-α), then accept Nn - Otherwise, reject the current n, increase n by one, and return to step. 5

26 Poisson Distribution Eample: Generate three Poisson variates with mean α0. ep(-0.) Variate Step : Set n0, P Step : , P Step 3: Since P < ep(- 0.), accept N 0 Variate Step : Set n0, P Step : 0.446, P Step 3: Since P < ep(-0.), accept N 0 Variate 3 Step : Set n0, P Step : , P Step 3: Since P > ep(-0.), reject n0 and return to Step with n Step : 0.995, P Step 3: Since P > ep(-0.), reject n and return to Step with n Step : , P Step 3: Since P < ep(-0.), accept N 6

27 Poisson Distribution It took five random numbers to generate three Poisson variates In long run, the generation of Poisson variates requires some overhead! N n+ P Accept/eject esult P < ep(- α) Accept N P < ep(- α) Accept N P ep(- α) eject P ep(- α) eject P < ep(- α) Accept N 7

28 Special Properties Based on features of particular family of probability distributions For eample: Direct Transformation for normal and lognormal distributions Convolution Beta distribution (from gamma distribution) 8

29 Direct Transformation Approach for N(0,): Consider two standard normal random variables, Z and Z, plotted as a point in the plane: In polar coordinates: Z B cos(φ) Z B sin(φ) B Z + Z ~ χ distribution with degrees of freedom Ep(λ ). / Hence, B ( ln ) The radius B and angle φ are mutually independent. Z Z ( ln ) ( ln ) / / cos(π sin(π ) ) 9

30 Direct Transformation Approach for N(μ,σ ): Generate Z i ~ N(0,) i μ + σ Z i Approach for Lognormal(μ,σ ): Generate ~ N((μ,σ ) Y i e i 30

31 Summary Principles of random-variate generation via Inverse-transform technique Acceptance-rejection technique Special properties Important for generating continuous and discrete distributions 3