Christine M. Mitchell Center for Human-Machine Systems Research (chmsr)

1. ISYE 7210--Simulation of Real-Time Systems Fall 2005Random Number Generators & Probability Distributions www.chmsr.gatech.edu/ISyE7210/www.horstmann.com/corejava.html Christine M. Mitchell Center for Human-Machine Systems Research (chmsr) School of Industrial and Systems Engineering (chmsr lab) ISyE Main, Room 426, 404 385-0363 (office) ISyE Groseclose, Room 334, 404 894-4321 {cm}@chmsr.gatech.edu

2. “Lite” Introduction to Discrete-Event Simulation • In discrete-event simulation, time hops from one event to the next • Time is captured in discrete units • Time recorded at • Change of state, e.g., new customer arrives, tn • An event occurse.g., new server joins set of servers, tn + i • In discrete-event simulation, time can be • Deterministic, e.g., time for one customer to depart and another customer to step up to the sever is 1 min. • Stochastic, e.g., time defined by sampling from a probability distribution, such as transaction time is exponentially distributed with mean 5 min. • Example • Entities • Server(s)—such as bank teller(s) • Customers —such as people depositing paycheck

3. “Lite” Introduction to Discrete-Event Simulation (cont’d) • Example (cont’d) • Discrete events • Customer arrivals at queue (assume one waiting line) • Customer service time (assume one type of transaction) • Time to switch from completed customer (departing) to customer waiting in queue to step up to server • New server arrival • H0: Assume, one server & at least three customers: A at server, B & C in waiting in queue t0 simulation time begins t1 ~ (t1 - t0) A completes transaction (random time) t2 = t1+ 10 s. A departs & B moves to the server (deterministic time) t3 ~ t2 + (t1 - t0) B completes transaction t4 = t3 + 10 s. B departs & C moves to the server t5 customer, D, arrives (random time) t6 ~ t4 + (t1 - t0) C completes transaction t7 secondserver arrives * tn tn + 1 * *

4. Random Number Generator (RNG) &Pseudo-Random Number Generator (P-RNG) • Random Number Generator (RNG) • An algorithm that generates a sequence of numbers from 0 to MAX_Value (often the largest number a computer can hold) • Sequence is uncorrelated, that is, any number from [0, MAX_VALUE] is equally likely • Numbers are often scaled on the unit interview, [0, 1] • In practice, there is no such thing as a statistically random sequence • For mathematical purposes, e.g., discrete-event simulation timing, use a pseudo-random number generator (P-RNG) • A working definition of random (in the context of a computer-generated sequences) • Deterministic algorithm that produces a sequence of numbers • Sequence is statistically uncorrelated

5. t3+ .3 t0 t1 t1+ .3 t3 t5 t5 t5 + .3 t6 t7 Pseudo-Random Number Generator (P-RNG) (con’td) • Popular and preferred class of P-RNG is a mixed-congruential type • A mixed-congruential RNG • Sequence of random numbers calculated by adding a random number to the last number obtained • R2 = R1 + Rnew1 • Rn = Rn - 1 + Rnew2 • Allows a modeler to regenerate the same random number stream, given knowledge of the constants and random number seed, xo • Random number, xn+1 , is generated by the nth random number, Xn, by using a recursive formula

6. Pseudo-Random Number Generator (con’td)A Mixed-Congruential Generator (cont’d) A mixed-congruential P-RNG looks as follows • a, c, and m are positive integers (a < m, c < m) •  Xn+1 is the remainder when (aXn+ c) is dived by m • Xn+1= (aXn + c) (modulo m) • Modulo: m—a large prime integer • Xn has a value in the range 0 to m-1 • Using a computer, the typical choice for m is m = 2*b, where b is the “word size” of the computer, e.g., 16 bit, 32 bit, 64 bit • The multiplier, a, is an integer in the range 1 to m-1

7. Pseudo-Random Number Generator (con’td)A Mixed-Congruential Generator (cont’d) • A mixed-congruential algorithm looks as follows • f(xn) = xn+1where the generating function f(..) is defined for all x in the range of 1 to m-1 • Sequence of x’s must be initialized by using an initial seed, x0, from the sequence 1, 2, 3, ….., m-1 • Sequence of x’s is conventionally normalized to the unit interval via division by the modulus to produce the sequence, u1, u2, u3, u4, contained in [0, 1] •  un = xn/ m for n = 1, 2, 3, 4, ….

8. Pseudo-Random Number Generators • Good random number generators are to find!!!! • Always useyour own random number generator • You control the random numbers, not the computer hardware or language (implementation) • Languages, etc., will often start generating random sequences with a different seed (and unknown starting point) for each run • E.g., seconds since 1970 • Even if you can set the seed, may not have control, test this! • Potentially, sad consequences may occur if you need to replicate runs with exactly the same random numbers for each type of event • For example, experiments that include human decision makers

9. Distribution Class Hierarchy for ISyE 7210

10. Distribution Class Hierarchy for ISyE 7210

11. Uniform01: P-RNG & Distribution (cont’d) //Uniform01.java /** Class: Uniform01 Generates uniformly distributed random numbers in the interval [0,1]. I also provides the basis for the pseudo-random number generator for all other probability distribution subclasses. The pseud0-random number generator uses the recursion: IX=16087 * IX mod (2 (31 – 1) ) <br> using only 32 bits including the sign. <p> Code modified from UNIF in Bratley, Fox, and Schrage (1987), <i>A Guide to Simulation (2nd ed)</i>, Springer-Verlag, New York, p. 331. <p> <pre> input: seed...an integer random number from a stream of numbers greater than 0 and less than 2147483647 ~ (2 (31 – 1) ) <p> output: seed = new seed value </pre> //(cont’d)

12. Uniform01: P-RNG & Distribution (cont’d) //Uniform01.java @author Christine M. Mitchell<BR> * Center for Human-Machine Systems Research<BR> * School of Industrial and Systems Engineering<BR> * Georgia Institute of Technology<BR> * * Last Updated: September 2005 */ public class Uniform01 { private long seed = 0; /** * Sets the internal seed in this class for the * the random number generator */ public Uniform01(long newSeed) //constructor taking one argument, a (long) seed value { seed = newSeed; } /** * Returns the next value from a Uniform distribution on * [0, 1] using recursion above */ //Uniform)1 (cont’d)

13. Uniform01: P-RNG & Distribution (cont’d) //Uniform01.java public double getNextRandom() { long k1 = 0; double value = 0.0; k1 = seed /127773; seed = 16807 * (seed - k1 * 127773) - k1 *2836; if (seed < 0) seed = seed + 2147483647; //twos-complement if negative value = seed * 4.656612875E-10; //scale to [0, 1] interval return value; } /** * Returns current value of the random number seed */ public long getSeed() { return seed; } //Uniform01 (cont’d)

14. Uniform01: P-RNG & Distribution (cont’d) //Uniform01.java /** * Resets seed value */ public void setSeed(long newSeed) { seed = newSeed; } /** * Returns string representation of Uniform01 */ public String toString() { return new String("Seed: " + seed + "\t" + "\t" + "RV: " + getNextRandom()); } /** * main tests the class Uniform01 */ //Uniform01 (cont’d)

15. Uniform01: P-RNG & Distribution (cont’d) //Uniform01.java public static void main (String argv []) { Uniform01 rng1 = new Uniform01(123456789); Uniform01 rng2 = new Uniform01(256712345); System.out.println("Printing from Uniform01 main( )"); System.out.println("Two Uniform [0, 1] distributions"); for (int i = 0; i < 8; i++) { System.out.println(rng1.toString() + "\t" + rng2.toString()); }//end for } //end main }// end class Uniform01 Printing from Uniform01 main( ) Two Uniform [0, 1] distributions Seed: 123456789 RV: 0.2184182969823758 Seed: 256712345 RV: 0.1256054230552947 Seed: 469049721 RV: 0.9563175765089297 Seed: 269735592 RV: 0.05034540176607601 Seed: 2053676357 RV: 0.8295092339327006 Seed: 108115927 RV: 0.15516752709521853 Seed: 1781357515 RV: 0.5616954427668942 Seed: 333219727 RV: 0.9006280269466985 Seed: 1206231778 RV: 0.41530708147596124 Seed: 1934083960 RV: 0.8552496921076923 Seed: 891865166 RV: 0.06611873491603133 Seed: 1836634728 RV: 0.18157601270870943 Seed: 141988902 RV: 0.2575777923820449 Seed: 389931518 RV: 0.7480457563247773 Seed: 553144097 RV: 0.1099567935324706 Seed: 1606416029 RV: 0.4050272141392882

16. Distributions for ISyE 7210 • Comments on set of probability distributions for 7210 • Each distribution is a Java class • Each distribution generates a pseudo-random stream of random variables from the specified probability distributions • Set of distribution are a good example of the utility of subclassing