Functional Estimation with Respect to a Threshold Parameter via Dynamic Split-and-Merge

We consider a class of stochastic models for which the performance measure is deened as a mathematical expectation that depends on a parameter , say (), and we are interested in constructing estimators of in functional form (i.e., entire functions of), which can be computed from a single simulation experiment. We focus on the case where is a continuous parameter, and also consider estimation of the derivative 0 (). One approach for doing that, when is a parameter of the probability law that governs the system, is based on the use of likelihood ratios and score functions. In this paper, we study a diierent approach, called split-and-merge, for the case where is a threshold parameter. This approach can be viewed as a practical way of running parallel simulations at an innnite number of values of , with common random numbers. We give several examples showing how diierent kinds of parameters such as the arrival rate in a queue, the probability that an arriving customer be of a given type, a scale parameter of a service time distribution, and so on, can be turned into threshold parameters. We also discuss implementation issues. 1. Functional Estimation Let f((; ; P); 2 g be a family of probability spaces deened over the same measurable space, where = a; b] is a bounded interval of the real line. In general, the probability law P may depend on a parameter. Consider a nite-horizon discrete event model deened over that family of probability spaces and let h(; !) be some random variable of interest (e.g. total sojourn time of all the customers served during a given day in a queueing system, or the total number of rejected customers in a nite-buuer system, etc). Suppose that we are interested in the function () = E h(; !)] = Z h(; !)P (d!): Normally, a simulation performed at = 0 permits one to estimate (0) and perhaps 0 (0) or higher order derivatives. Techniques for doing that include the likelihood ratio or score function method, as well as perturbation analysis and its numerous variants (see 2], 5], 9], 12], 14], 15], 18] and the several references given there). To obtain estimations at diierent values of , one would usually