Page 199 - DCAP601_SIMULATION_AND_MODELING

Unit 11: Design and Evaluation of Simulation Experiments (I)

These examples show that it can be very misleading to do an exploratory simulation with one Notes
set of parameter values to determine appropriate simulation run lengths, because the appropriate
run lengths are very different for different traffic intensities.

Did u know? Full form of RBM
Reflecting Brownian Motion

11.3 Variance Reduction Techniques

In a simulation study, we are interested in one or more performance measures for some stochastic
model. For example, we want to determine the long-run average waiting time, E(W ), of customers
th
in a G/G/1 queue. To estimate E(W ) we can do a number of, say K , independent runs, the i one
yielding the output random variable Wi with E(Wi ) = E(W ). After K runs have been performed,
an estimator of E(W ) is given by W   K i 1 W /K. However, if we were able to obtain a different
i

unbiased estimator of E(W ) having a smaller variance than W , we would obtain an estimator
with a smaller confidence interval. In this section we will present a number of different methods
that one can use to reduce the variance of the estimator W . We will successively describe the
following techniques:
1. Common Random Numbers
2. Antithetic Variables

3. Control Variates
4. Conditioning
The first method is typically used in a simulation study in which we want to compare performance
measures of two different systems. All other methods are also useful in the case that we want to
simulate a performance measure of only a single system.

11.3.1 Common Random Numbers

The idea of the method of common random numbers is that if we compare two different systems
with some random components it is in general better to evaluate both systems with the same
realizations of the random components. Key idea in the method is that if X and Y are two
random variables, then
Var(X – Y) = Var(X) + Var(Y) – 2Cov(X, Y).
Hence, in the case that X and Y are positively correlated, i.e. Cov(X, Y) > 0, the variance of X – Y
will be smaller than in the case that X and Y are independent. In general, the use of common
random numbers leads to positive correlation of the outcomes of a simulation of two systems.
As a consequence, it is better to use common random numbers instead of independent random
numbers when we compare two different systems.
Let us illustrate the method using the following scheduling example. Suppose that a finite
number of N jobs has to be processed on two identical machines. The processing times of the
jobs are random variables with some common distribution function F. We want to compare the
completion time of the last job, C , under two different policies. In the LPTF policy, we always
max
choose the remaining job with the longest processing time first, in the SPTF policy we choose
the remaining job with the shortest processing time first.

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