Random Number Generation

1. Random Number Generation

2. Random Number Generators • Without random numbers, we cannot do Stochastic Simulation • Most computer languages have a subroutine, object or function generating random numbers (uniformly distributed) • Simulation languages provide more than that (you can get random samples from many distributions) • How do they generate it? • How can we test their randomness?

3. Properties of Random Numbers • A sequence of random numbers, R1, R2, . . ., must have have two important statistical properties, uniformity and independence • Each random number Ri, should be an independent sample from the continuous uniform distribution between 0 and 1

4. Properties of Random Numbers • Some consequences of the uniformity and independence properties are • If the interval (0,1) is divided into n subintervals of equal length, the expected number of observations in each interval is N/n, where N is the total number of observations • The probability of observing a value in a particular interval is independent of the previous values drawn

5. Pseudo-Random Numbers • pseu·doadj. False or counterfeit; fake. [American Heritage Dictionary] • Because these numbers are produced using a deterministic algorithm • Given the method, the set of random numbers produced can be replicated • Thus they are not truly random • They simply imitate the properties of uniform distribution and independence • A statistical test should conclude they are indistinguishable from true random numbers • Thus, they can be used for all practical purposes instead of true random numbers

6. Considerations in Generating Pseudo-Random Numbers • The routine should be fast ( A simulation may require billions of RNs) • The routine should be portable to different environments • The routine should have a sufficiently long cycle • The random numbers generated should be replicable (Useful in debugging and comparing systems) • The random numbers generated should imitate the statistical properties of uniformity and independence

7. Linear Congruential Method • This method produces a sequence of integers between 0 and m-1 according to the following recursive relationship: • The initial value “X0” is called the seed, “a” the constant multiplier, “c” the increment, and “m” the modulus • If c  0, the form is called the mixed congruential method • if c = 0, the form is called the multiplicative congruential method • The choice of the parameters affect the statistical properties and the cycle length • To convert the integers to random numbers use Ri = Xi /m

8. Example • X0 = 27, a = 17, c=43, and m = 100 (Ri = Xi /100) • X0 = 27 • X1 = (17*27+43) mod 100 = 502 mod 100 = 2 (P1 = .02) • X2 = (17*2+43) mod 100 = 77 mod 100 = 77 (P2 = .77) • X3 = (17*77+43) mod 100 = 1352 mod 100 = 52 (P3 = .52) • . . .

9. Linear Congruential Method • Maximum density (no gaps in the distribution): • The numbers generated with this method can only assume values from the set I = {0, 1/m, 2/m, . . ., (m-1)/m} • Thus Ri’s are actually distributed with a discrete distribution defined over I • This can be accepted as an approximation given that m is large enough • Maximum period (avoid cycling, autocorrelation) • Cycle length (P) depends on the choice of parameters (always less than m) • For m = 2b, c  0 and relatively prime to m and a = 1+4k, the longest possible period P = m. • For m = 2b, c = 0, X0 (seed) odd, and a = 3+8k or a = 5+8k, the longest possible period P = m/4 • For m a prime number, etc.. P = m-1

10. Example • Use multiplicative congruential method with a = 13, m = 26 = 64 and X0 = 1, 2, 3, 4 X0 = 1, I = {1, 5, 9 ,13, …, 53, 57, 61} Gap = 4/64 = 0.0625

11. Combined Linear Congruential Generators • To simulate more complex systems, the simulation runs need to go through larger numbers of elementary events • This means that kind of simulation runs have to use more random numbers • In order to have healthy runs, pseudo-random generators with longer periods are needed (So that cycles can be avoided during the run) • It is possible to combine two or more multiplicative congruential generators in such a way that the combined generator has good statistical properties and a longer period

12. Random-Numbers Streams and Seeds • The seed for a random-number generator: • Is the integer value X0 that initializes the random-number sequence. • Any value in the sequence can be used to “seed” the generator. • A random-number stream: • Refers to random numbers obtained by using a starting seed • If the streams are b values apart, then stream i could defined by starting seed: • Older generators: b = 105; Newer generators: b = 1037. • A single random-number generator with k streams can act like k distinct virtual random-number generators • To compare two or more alternative systems. • Advantageous to dedicate portions of the pseudo-random number sequence to the same purpose in each of the simulated systems.

13. Pseudo-Random Number Generation in SIMAN • SIMAN employs the multiplicative congruential method with a = 16807, m = 231-1 = 2,147,483,647 • This is an almost full-period generator • For any initial seed between 1 and 231-2, all unnormalized random numbers between 1 and 231-2, are generated exactly once before the generator cycles again (P = 231-2) • A SIMAN model may employ several different random-number streams • Each stream is generated by using the same generator with different initial seed values

14. Pseudo-Random Number Generation in ARENA • With today’s computing power, the cycling can occur in minutes of simulation with with a cycle length of 2 billion (2.1x109) • ARENA thus uses a combined multiple recursive generator which combines two separate generators

15. Pseudo-Random Number Generation in ARENA • The cycle length of this generator is 3.1x1057 • This is inexhaustible with the current computing speeds • Just to generate them would take 1040 millennia (thousand years) on a 2GHz PC • The Arena generator has facility to split this cycle into 1.8x109 separate streams, each of length 1.7x1038 • Each stream is further subdivided into 2.3x1015 separate substreams of length 7.6x1022 apiece

16. Tests for Random Numbers • Two categories: • Testing for uniformity: H0: Ri ~ U[0,1] H1: Ri ~ U[0,1] • Failure to reject the null hypothesis, H0, means that evidence of non-uniformity has not been detected. • Testing for independence: H0: Ri ~ independently H1: Ri ~ independently • Failure to reject the null hypothesis, H0, means that evidence of dependence has not been detected. • Level of significance a, the probability of rejecting H0 when it is true:a = P(reject H0|H0 is true) / /

17. Tests for Random Numbers • When to use these tests: • If a well-known simulation languages or random-number generators is used, it is probably unnecessary to test • If the generator is not explicitly known or documented, e.g., spreadsheet programs, symbolic/numerical calculators, tests should be applied to many sample numbers. • Types of tests: • Theoretical tests: evaluate the choices of m, a, and c without actually generating any numbers • Empirical tests: applied to actual sequences of numbers produced.