Modesim Report

8/3/2019 Modesim Report

http://slidepdf.com/reader/full/modesim-report 1/4

STI College San Fernando Modeling and Simulation

Generating Random Numbers and Variables Page 1 of 4

GENERATING RANDOM NUMBERS

WHAT ARE RANDOM NUMBERS?

Random numbers are a necessary basic ingredient in the simulation of almost all discrete systems.

Most computer languages have a subroutine, object, or function that will generate a random number.Similarly simulation languages generate random numbers that are used to generate event limes and other

random variables.

It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely,

it is also very difficult for humans to produce a string of random digits, and computer programs can be written

which, on average, actually predict some of the digits humans will write down based on previous ones.

Intuitively, we can list a number of criteria that a sequence of numbers must fulfill to pass as a random

number sequence:

y unpredictability,

y independence,

y without pattern.

These criteria appear to be the minimum request for an algorithm to produce random numbers. More

precisely we can formulate:

y uniform distribution,

y uncorrelated,

y passes every test of randomness,

y large period before the sequence repeats,

y sequence repeatable and possibility to vary starting values,

y fast algorithm.

PROPERTIES OF RANDOM NUMBERS

Two important statistical properties:

y Uniformity

y Independence

PSEUDORANDOM NUMBERS

Pseudo means false, so false random numbers are being generated. The goal of any generation scheme is

to produce a sequence of numbers between zero(0) and one(1) which simulates, or initiates, the ideal

properties of uniform distribution and independence as closely as possible.





When generating pseudo-random numbers, certain problems or errors can occur. These errors, or

departures from ideal randomness, are all related to the properties stated previously. Some examples

include the following:

1. The generated numbers may not be uniformly distributed.

2. The generated numbers may be discrete - valued instead continuous valued

3. The mean of the generated numbers may be too high or too low.

4. The variance of the generated numbers may be too high or low

5. There may be dependence. The following are examples:

(a) Autocorrelation between numbers.

(b) Numbers successively higher or lower than adjacent numbers.

(c) Several numbers above the mean followed by several numbers below the mean.

Usually, random numbers are generated by a digital computer as part of the simulation. Numerous

methods can be used to generate the values. In selecting among these methods, or routines, there are a

number of important considerations.

y Fast

y Portable to different computers

y Have sufficiently long cycle.

y Replicable.

y Closely approximate the ideal statistical properties of uniformity and independences.

TECHNIQUES FOR GENERATING RANDOM NUMBERS

y Linear Congruential Method (LCM)

y Combined Linear Congruential Generators (CLCG)

y Random-Number Streams

Linear Congruential Method (LCM)

The linear congruential method, initially proposed by Lehmer [1951], produces a sequence of integers,

X 1, X2,... between zero and m 1 according to the following recursive relationship:

Xi+1 = (a Xi + c) mod m, i = 0,1, 2,...

multiplier modulusincrement





The initial value X0 is called the seed, a is called the constant multiplier, c is the increment, and m

is the modulus.

The selection of the values for a, c, m, and X 0 drastically affects the statistical properties and the cycle

length.

If c 0, the form is called the mixed congruential method . When c = 0, the form is known as the

multiplicative congruential method. The selection of the values for a, c , m and X0 drastically affects the

statistical properties and the cycle length.

The random integers are being generated [0, m-1], and to convert the integers to random numbers:

X

Ri = ---- , i=1,2,3

m

Example 1:

Use the linear congruential method to generate a sequence of random numbers with X 0 = 27, a = 17, c= 43, and m = 100. Here, the integer values generated will all be between zero and 99 because of

the value of the modulus. These random integers should appear to be uniformly distributed the integers

zero to 99.Random numbers between zero and 1 can be generated by:

Ri = Xi / m, i=1,2,3.

The sequence of Xi and subsequent Ri values is computed as follows:

X0 = 27

X1 = (17.

27 + 43) mod 100 = 502 mod 100 = 2 R1 = 2 100 = 0.02

X 2 = (17 2 + 43) mod 100 = 77 mod 100 = 77

R2 = 77 100 = 0.77

X3 = (1777+ 43) mod 100 = 1352 mod 100 = 52

R3 = 52 100 = 0.52

Example 2:

Find the sequence of random numbers given a = 13, c = 0, m = 31, X0 = 1.

X0 = 1

X1 = (13 * 1 + 0) mod 31 = 13 mod 31 = 13

X2 = (13 * 13 + 0) mod 31 = 169 mod 31 = 14

X3 = (13 * 14 + 0) mod 31 = 182 mod 31 = 27

X4 = (13 * 27 + 0) mod 31 = 351 mod 31 = 10

X5 = (13 * 10 + 0) mod 31 = 130 mod 31 = 6

X6 = (13 * 6 + 0) mod 31 = 78 mod 31 = 16





X7 = (13 * 16 + 0) mod 31 = 208 mod 31 = 22

X8 = (13 * 22 + 0) mod 31 = 286 mod 31 = 7

X9 = (13 * 7 + 0) mod 31 = 91 mod 31 = 29

X10 = (13 * 29 + 0) mod 31 = 377 mod 31 = 5

X11 = (13 * 5 + 0) mod 31 = 65 mod 31 = 3

The sequence begins with: 1, 13, 14, 27, 10, 6, 16, 22, 7, 29, 5, 3,

What's the next value? Well, it looks pretty unpredictable, but you've been initiated. So you can

compute. The first 30 terms in the sequence are a permutation of the integers from 1 to 30 and then the

sequence repeats itself. It has a period equal to m - 1.

If a pseudorandom integer sequence with values between 0 and m is scaled by dividing by m, the result

is floating-point numbers uniformly distributed in the interval [0 , 1]. Our simple example begins with:

0.0323 , 0.4194 , 0.4516 , 0.8710 , 0.3226 , 0.1935 , 0.5161 , .

There are only a finite number of values, 30 in this case. The smallest value is 1 / 31 and the largest is30 / 31.

Seatwork:

Use the linear congruential method to generate a sequence of random numbers with X 0 = 9, a = 5, c =

11, and m = 16. Find the list of random numbers that will be generated by the given values before an

occurrence of any random number in the list will occur and determine the pseudorandom integer

sequence with values between 0 and m. Show your solution.

Documents

Modesim Report