4 - Generating Random Variables for Simulation

Summary

In Chapter 3 we reviewed the most common discrete and continuous random variables. This chapter shows how we can use the density function or cumulative distribution function for a distribution to generate instances of that distribution. For example, we might have a system in which the interarrival times of jobs are well modeled by an Exponential distribution and the job sizes (service requirements) are well modeled by a Normal distribution. To simulate the system, we need to be able to generate instances of Exponential and Normal random variables. This chapter reviews the two basic methods used in generating random variables. Both these methods assume that we already have a generator of Uniform(0,1) random variables, as is provided by most operating systems.

Inverse-Transform Method

This method assumes that (i) we know the c.d.f. (cumulative distribution function), F_x (x) = P{X ± x}, of the random variable X that we are trying to generate, and (ii) that this distribution is easily invertible, namely that we can get x from F_X (x).

The Continuous Case

Idea: We would like to map each instance of a uniform r.v. generated by our operating system – that is, u ∈ U (0, 1) – to some x, which is an instance of the random variable X, where X has c.d.f. F_X. We assume WLOG that X ranges from 0 to ∞. Let's suppose there is some mapping that takes each u and assigns it a unique x. Such a mapping is illustrated by g^-1(·) in Figure 4.1.