Comparison of gradient estimation techniques for queues with non-identical servers

Gradient estimation techniques for stochastic discrete-event simulation have been a major topic of research over the past decade. In this paper, we apply two of the techniques—perturbation analysis and the likelihood ratio method—to a single-queue system with non-identical multiple servers. We derive estimates for derivatives of mean steady-state system time with respect to parameters of the underlying timing distributions. In terms of perturbation analysis, we consider both an infinitesimal perturbation analysis estimator, which is biased for this problem, and two smoothed perturbation analysis estimators, one unbiased but not very practical and one approximate but easily implementable. For two servers, we provide an analytical proof of unbiasedness in steady state for the Markovian case. For the likelihood ratio method, we apply the regenerative likelihood ratio estimator. We provide simulation results for both Markovian and non-Markovian examples, and compare the performance of the various estimators. We conclude that no one method performs universally well, and provide recommendations as to when one is likely to be preferred to the others.