Infinitesimal perturbation analysis (IPA) derivative estimation with unknown parameters

Abstract

We consider the problem of infinitesimal perturbation analysis (IPA) derivative estimation in a setting where some parameters associated with input processes are unknown. In general, IPA estimates depend explicitly on these parameters. Therefore they need to be estimated in order to compute IPA estimates. A “brute-force” way to solve this problem is to estimate the parameters first based on historical input data and then use the estimates in place of the parameters in calculating IPA estimates. However, this method does not utilize the most recent data and thus may lose some accuracy, particularly in an environment where data are received continuously and sequentially We propose an adaptive method in which IPA estimates are computed based on the latest parameter estimates that are continuously updated as more input data are obtained. We prove that our IPA estimators are strongly consistent and have a convergence rate of $1/\sqrt{n}$, which is the same as the traditional IPA estimators. We use the $G/G/1$ queue as an illustrative example to show how our method works in detail. Simulation experiments on several queueing systems are also provided to support our theoretical conclusions.