Concomitant Control Variables Applied to the Regenerative Simulation of Queuing Systems

We investigate using multiple concomitant control variables to reduce the width of confidence intervals when estimating steady-state response variables via the regenerative method of simulation. A concomitant control variable is an estimator of a known quantity, defined with respect to the system being simulated, that is believed to be correlated with the response variable estimator. We give examples of such control variables for two regenerative queuing systems, the stable GI / G /1 queue and a closed queuing system. An estimator that uses multiple control variables involves unknown coefficients; one wishes to choose these coefficients to minimize the variance of the estimator. We establish the asymptotic validity of confidence intervals constructed using control variables when the coefficients themselves are estimated via regenerative simulation. For the response variables and control variables in our examples we compute the variance reduction that could be obtained if the optimum coefficients were known and empirically investigate the variance reduction obtained in practice using various methods for estimating the coefficients. We find that when the coefficients are estimated from a small fraction of the simulated tours the controlled estimators yield confidence intervals whose width is less than and whose coverage is comparable to that obtained without using control variables.