A New Estimator of Sensitivity Measures for Simulations Based on Light Traffic Theory

Abstract

Consider an open queueing system that is stable for arrival rates, 0 ≤ λ < c. Let f(λ) be some function such as the mean sojourn time of a customer. Normally, light traffic theory is concerned with limits as λ → 0, e.g., f(n)(0), n = 0, 1, …. Here we use light traffic theory to obtain derivatives of f(·) at any value of λ for which the system is stable. The derivatives, f(n)(λ), n = 0, 1, …, are obtained by perturbing a system with arrival rate, λ, with a small auxiliary arrival process. The nth derivative is obtained by looking at sample paths of the original system with k auxiliary customers thrown in, k = 0, 1, …, n. In practice, the only way to obtain f(n)(λ) for λ > 0 is by observing the effect of additional customers on the sample path of a simulation. We make this technique rigorous by showing that a certain function of the sample paths is “admissible.” The analysis suggests a method for analytically bounding the derivatives, and yields a proof that a large class of queueing functions are infinitely differentiable with respect to λ. The calculations are reminiscent of other sensitivity analyses of simulations such as (infinitesimal) perturbation analysis and the likelihood ratio method. The method described here (the “light traffic perturbation method”) shares the rigorous foundation of the likelihood ratio method and numerical evidence suggests that (at least for simple systems) it has convergence properties similar to perturbation analysis. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

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