Variance Reduction Techniques (VRT)

(Law and Kelton)

VRTs• Increase the simulation efficiency -- statistical

efficiency.• Reducing the variances of the random variables’

outputs from a simulation without disturbing its expectation, we can obtain greater precision with less simulation time.

• These techniques can make the difference of making a very expensive simulation versus a frugal one.

• All VRTs require some effort of understanding the model and the effects of VRT on it.

• Developed mainly for Monte Carlo simulation. But can also be used in independent sampling and correlated sampling.

Common Random Numbers (CRN)• Used when comparing two or more alternative

system configurations instead of investigating a single configuration.

• Sometimes called correlated sampling, matched streams, matched pairs.

• Random numbers are used to generate random variates from other distributions which are used to drive the simulation models. Thus it is necessary to ensure that generated variates react monotonically to U.

• Be aware of backfiring.• It is important to dedicate a stream of random

number to a specific random variate generator.

Antithetic Variates (AV)• Used in simulating single systems.• Induced correlation between separate runs, but

seeks negative correlation.• To make pairs of small observation and a large

observation. These observations should offset each other. These average of the two will be closer to the expected compared to when the two outputs are independent.

• AV uses complimentary random numbers to drive the two replication in pairs.

• -It is important to point out to use U in one replication and U -1 in the succeeding.

Control Variates (CV)• Attempts to take advantage of correlation between

certain random variables to obtain a variance reduction.

• These correlations may arise during simulation or induced through an auxiliary simulation.

• Given X and Y random variables, if Y affects X, and if Y > v, then X > and vice-versa.

Indirect Estimation

• applied usually for queueing-type simulation models where the measures to be estimated are steady-state measures

• we can substitute actual expectations which can be arrived at through an exact solution.

Conditioning

• Similar to indirect estimation.

• Exploits some special property of a model to replace an estimate of a quantity by its exact analytical value.