Gradient estimation for discrete-event systems by measure-valued differentiation

Abstract

In simulation of complex stochastic systems, such as Discrete-Event Systems (DES), statistical distributions are used to model the underlying randomness in the system. A sensitivity analysis of the simulation output with respect to parameters of the input distributions, such as the mean and the variance, is therefore of great value. The focus of this article is to provide a practical guide for robust sensitivity, respectively, gradient estimation that can be easily implemented along the simulation of a DES. We study the Measure-Valued Differentiation (MVD) approach to sensitivity estimation. Specifically, we will exploit the “modular” structure of the MVD approach, by firstly providing measure-valued derivatives for input distributions that are of importance in practice, and subsequently, by showing that if an input distribution possesses a measure-valued derivative, then so does the overall Markov kernel modeling the system transitions. This simplifies the complexity of applying MVD drastically: one only has to study the measure-valued derivative of the input distribution, a measure-valued derivative of the associated Markov kernel is then given through a simple formula in canonical form. The derivative representations of the underlying simple distributions derived in this article can be stored in a computer library. Combined with the generic MVD estimator, this yields an automated gradient estimation procedure. The challenge in automating MVD so that it can be included into a simulation package is the verification of the integrability condition to guarantee that the estimators are unbiased. The key contribution of the article is that we establish a general condition for unbiasedness which is easily checked in applications. Gradient estimators obtained by MVD are typically phantom estimators and we discuss the numerical efficiency of phantom estimators with the example of waiting times in the G/G/1 queue.