EXPLOITING REGENERATIVE STRUCTURE TO ESTIMATE FINITE TIME AVERAGES VIA SIMULATION: SUPPLEMENTARY MATERIAL

Abstract

We propose nonstandard simulation estimators of expected time averages over ¯nite intervals [0; t], seeking to enhance estimation e±ciency. We make three key assumptions: (i) the underlying stochastic process has regenerative structure, (ii) the time average approaches a known limit as time t increases and (iii) time 0 is a regeneration time. To exploit those properties, we propose a residual-cycle estimator, based on data from the regenera- tive cycle in progress at time t, using only the data after time t. We prove that the residual-cycle estimator is unbiased and more e±cient than the standard estimator for all su±ciently large t. Since the relative e±ciency increases in t, the method is ideally suited to use when applying simu- lation to study the rate of convergence to the known limit. We also consider two other simulation techniques to be used with the residual-cycle estimator. The ¯rst involves overlapping cycles, par- alleling the technique of overlapping batch means in steady-state estimation; multiple observations are taken from each replication, starting a new observation each time the initial regenerative state is revisited. The other technique is splitting, which involves independent replications of the termi- nal period after time t, for each simulation up to time t. We demonstrate that these alternative estimators provide e±ciency improvement by conducting simulations of queueing models.