Variance and Derivative Estimation of Virtual Performance

Abstract

Virtual performance is a class of time-dependent performance measures conditional on a particular event occurring at time τ 0 for a (possibly) nonstationary stochastic process; virtual waiting time of a customer arriving to a queue at time τ 0 is one example. Virtual statistics are estimators of the virtual performance. In this article, we go beyond the mean to propose estimators for the variance, and for the derivative of the mean with respect to time, of virtual performance, examining both their small-sample and asymptotic properties. We also provide a modified K -fold cross validation method for tuning the parameter k for the difference-based variance estimator, and we evaluate the performance of both variance and derivative estimators via controlled studies and a realistic illustration. The variance and derivative provide useful information that is not apparent in the mean of virtual performance.