10.3 Variance Reduction Methods

As is evident from Equation 10.5, the length of a confidence interval is inversely proportional to where n is the number of simulation runs or experiments, and proportional to the standard deviation of the random variable under study. Note that the standard deviation that is used in calculating the confidence interval is itself in practice an estimate obtained from the simulation data and may therefore vary slightly with n. The brute force way to shrink the confidence interval of an estimate is obviously to increase n. However, in the interest of efficiency, we should also consider the option of somehow reducing the variance (and, consequently, the standard deviation) of the estimate. In this section, we consider several schemes for doing so.

The first two approaches rely on the following facts from elementary statistics:

where Cov( X, Y) = E([X E(X)][Y E(Y)]) is called the covariance of X and Y.

10.3.1 Antithetic Variables

Suppose we run simulations to estimate some parameter (for example, Mean Time to Data Loss [MTDL] in a RAID system). In traditional simulation, we would run n independent simulations and use the results. If Z ₁,Z ₂ are the outputs from two independent runs, we can expect that

so that

When the method of antithetic variables is used, we try to run simulations in pairs, coupled together in such a way that their results (any parameter that is estimated by the simulation, be it reliability,...