Gradient estimation for a class of systems with bulk services

Abstract

This article presents a comparison of different gradient estimators for the sensitivity of waiting times in a bulk server system. Inspired by a transportation network, our model is that of a bursty arrival process that waits at a “platform” until the server is available (representing a train or bus ready for departure). At the departure epochs, all waiting passengers leave at once. The departure process is assumed to be a renewal process and, based on a limiting result, the interdeparture times are approximated by truncated normal random variables. The interarrival times are assumed to be identically and independently distributed (i.i.d.), with a general distribution of bounded density. We are interested in calculating the sensitivities of the total cumulative waiting time of all passengers with respect to the interdeparture times. For this general model where neither the interarrival times nor the interdeparture times are exponential, there is no analytical formula available. However, the estimation of such sensitivities is an important problem for flow control in such networks. We establish a Smoothed Perturbation Analysis (SPA), a Measure-Valued Differentiation (MVD), and a Score Function (SF) estimator, including numerical experiments.