Gradient Estimation for a Class of Systems with Bulk Services: A Problem in Public Transportation

Abstract

This paper deals with a system where batch arrivals wait in a station until a server (a train) is available, at which moment it services all customers in waiting. This is an example of a bulk server, which has many applications in public transportation, telecommunications, computer resource allocation, and multiple access telecommuncation networks, among others. We consider a subway model and focus on a metro line serving a particular metro station. Denote the planned inter-departure time of this line by x. The metro station is served by several other lines and passengers change trains at the station. Traveling times of trains are assumed to be given by fixed times and an additional stochastic noise. We perform a sensitivity analysis of the total delay of passengers waiting for the x line with respect to x. We establish a smoothed perturbation analysis (SPA), a measure-valued differentiation (MVD), and a score function (SF) estimator. Numerical experiments are performed to compare the ensuing estimators. It turns out that the SPA and MVD estimators are intrinsically different and the model presented in this paper may serve as a counter-example to the widespread belief that SPA and MVD yield similar estimators.