1. 28-1 ©2006 Raj Jain www.rajjain.com Random Variate Generation

2. 28-2 ©2006 Raj Jain www.rajjain.com Overview 1.Inverse transformation 2.Rejection 3.Composition 4.Convolution 5.Characterization

3. 28-3 ©2006 Raj Jain www.rajjain.com Random-Variate Generation  General Techniques  Only a few techniques may apply to a particular distribution  Look up the distribution in Chapter 29

4. 28-4 ©2006 Raj Jain www.rajjain.com Inverse Transformation  Used when F -1 can be determined either analytically or empirically. 0 0.5 u 1.0 x CDF F(x)

5. 28-5 ©2006 Raj Jain www.rajjain.com Proof

6. 28-6 ©2006 Raj Jain www.rajjain.com Example 28.1  For exponential variates:  If u is U(0,1), 1-u is also U(0,1)  Thus, exponential variables can be generated by:

7. 28-7 ©2006 Raj Jain www.rajjain.com Example 28.2  The packet sizes (trimodal) probabilities:  The CDF for this distribution is:

8. 28-8 ©2006 Raj Jain www.rajjain.com Example 28.2 (Cont)  The inverse function is:  Note: CDF is continuous from the right  the value on the right of the discontinuity is used  The inverse function is continuous from the left  u=0.7  x=64

9. 28-9 ©2006 Raj Jain www.rajjain.com Applications of the Inverse-Transformation Technique

10. 28-10 ©2006 Raj Jain www.rajjain.com Rejection  Can be used if a pdf g(x) exists such that c g(x) majorizes the pdf f(x)  c g(x) > f(x) 8 x  Steps: 1. Generate x with pdf g(x). 2. Generate y uniform on [0, cg(x)]. 3. If y f(x)  Efficiency = how closely c g(x) envelopes f(x) Large area between c g(x) and f(x)  Large percentage of (x, y) generated in steps 1 and 2 are rejected  If generation of g(x) is complex, this method may not be efficient.

11. 28-11 ©2006 Raj Jain www.rajjain.com Example 28.2  Beta(2,4) density function:  Bounded inside a rectangle of height 2.11  Steps:  Generate x uniform on [0, 1].  Generate y uniform on [0, 2.11].  If y < 20 x(1-x) 3, then output x and return. Otherwise repeat from step 1.

12. 28-12 ©2006 Raj Jain www.rajjain.com Composition  Can be used if CDF F(x) = Weighted sum of n other CDFs.  Here,, and F i 's are distribution functions.  n CDFs are composed together to form the desired CDF Hence, the name of the technique.  The desired CDF is decomposed into several other CDFs  Also called decomposition.  Can also be used if the pdf f(x) is a weighted sum of n other pdfs:

13. 28-13 ©2006 Raj Jain www.rajjain.com Steps:  Generate a random integer I such that:  This can easily be done using the inverse- transformation method.  Generate x with the ith pdf f i (x) and return.

14. 28-14 ©2006 Raj Jain www.rajjain.com Example 28.4  pdf:  Composition of two exponential pdf's  Generate  If u 1 <0.5, return; otherwise return x=a ln u 2.  Inverse transformation better for Laplace

15. 28-15 ©2006 Raj Jain www.rajjain.com Convolution  Sum of n variables:  Generate n random variate y i 's and sum  For sums of two variables, pdf of x = convolution of pdfs of y 1 and y 2. Hence the name  Although no convolution in generation  If pdf or CDF = Sum  Composition  Variable x = Sum  Convolution

16. 28-16 ©2006 Raj Jain www.rajjain.com Convolution: Examples  Erlang-k =  i=1 k Exponential i  Binomial(n, p) =  i=1 n Bernoulli(p)  Generated n U(0,1), return the number of RNs less than p    ( ) =  i=1 N(0,1) 2   a, b  )+  (a,b 2 )=  (a,b 1 +b 2 )  Non-integer value of b = integer + fraction    n Any = Normal   U(0,1)  Normal    m Geometric = Pascal    2 Uniform = Triangular

17. 28-17 ©2006 Raj Jain www.rajjain.com Characterization  Use special characteristics of distributions  characterization  Exponential inter-arrival times  Poisson number of arrivals  Continuously generate exponential variates until their sum exceeds T and return the number of variates generated as the Poisson variate.  The a th smallest number in a sequence of a+b+1 U(0,1) uniform variates has a  (a, b) distribution.  The ratio of two unit normal variates is a Cauchy(0, 1) variate.  A chi-square variate with even degrees of freedom  2 ( ) is the same as a gamma variate  (2, /2).  If x 1 and x 2 are two gamma variates  (a,b) and  (a,c), respectively, the ratio x 1 /(x 1 +x 2 ) is a beta variate  (b,c).  If x is a unit normal variate, e  +  x is a lognormal( ,  ) variate.

18. 28-18 ©2006 Raj Jain www.rajjain.com Summary Is pdf a sum of other pdfs? Use Composition Yes Is CDF a sum of other CDFs? Use composition Yes Is CDF invertible? Use inversion Yes

19. 28-19 ©2006 Raj Jain www.rajjain.com Summary (Cont) Does a majorizing function exist? Use rejection Yes Is the variate related to other variates? Use characterization Yes Is the variate a sum of other variates Use convolution Yes Use empirical inversion No

20. 28-20 ©2006 Raj Jain www.rajjain.com Exercise 28.1  A random variate has the following triangular density:  Develop algorithms to generate this variate using each of the following methods: a.Inverse-transformation b.Rejection c.Composition d.Convolution

21. 28-21 ©2006 Raj Jain www.rajjain.com Homework  A random variate has the following triangular density:  Develop algorithms to generate this variate using each of the following methods: a.Inverse-transformation b.Rejection c.Composition d.Convolution

22. 28-22 ©2006 Raj Jain www.rajjain.com Answer to Homework  A random variate has the following triangular density:  Develop algorithms to generate this variate using each of the following methods: a.Inverse-transformation F(x) = a.Rejection b.Composition c.Convolution

23. 28-23 ©2006 Raj Jain www.rajjain.com Inverse Transformation 048 x x F(x) 048 x f(x) x/16(8-x)/16 0.25

24. 28-24 ©2006 Raj Jain www.rajjain.com Rejection  Find a majoring function: f(x) < cg(x)  Generate x with g(x) and generate y uniform on (0,cg(x))  If y < f(x), then output x and return. Otherwise, repeat from step 1.  g(x) = 1/8 0<x<8, c=2  Generate x with U(0,8) and y with U(0,0.25) if y<(1/16)min(x,8-x) output x, otherwise repeat 048 x f(x) x/16(8-x)/16 0.25

25. 28-25 ©2006 Raj Jain www.rajjain.com Composition  Can be used if the pdf f(x) is a weighted sum of n other pdfs: 048 x f(x) x/16 1/4 048 x f(x) (8-x)/16 1/4  Generate u1 = U(0,1), if u1 0.5 use right triangle to generate x  Left Triangle: f(x)=x/16 => F(x)=x 2 /32=u =>  Right Triangle: f(x)=(8-x)/16  F(x)=x-x 2 /32=u 

26. 28-26 ©2006 Raj Jain www.rajjain.com Convolution  Sum of n variables:  Generate n random variate y i 's and sum  Triangle is a convolution of two squares 04u1 f(x) 1/4 04u2 f(x) 1/4 048 x f(x) 0.25  Generate u1=U(0,4)  Generate u2=U(0,4)  Return x=u1+u2

27. 28-27 ©2006 Raj Jain www.rajjain.com Thank You!

28. 28-28 ©2006 Raj Jain www.rajjain.com Copyright Notice  These slides have been provided to instructors using “The Art of Computer Systems Performance Analysis” as the main textbook in their course or tutorial.  Any other use of these slides is prohibited.  Instructors are allowed to modify the content or templates of the slides to suite their audience.  The copyright notice on every slide and this copyright slide should not be removed when these slides’ content or templates are modified.  These slides or their modified versions are not transferable to other instructors without their agreeing with these conditions directly with the author.