Reliable estimation via simulation

Leta ands denote the inter arrival times and service times in aGI/GI/1 queue. Let a⁽ⁿ⁾, s⁽ⁿ⁾ be the r.v.s. with distributions as the estimated distributions ofa ands from iid samples ofa ands of sizesn. Letw be a r.v. with the stationary distribution of the waiting times of the queue with input(a,s). We consider the problem of estimatingEw ], > 0 and via simulations when (a ⁽ⁿ⁾,s⁽ⁿ⁾) are used as input. Conditions for the accuracy of the asymptotic estimate, continuity of the asymptotic variance and uniformity in the rate of convergence to the estimate are obtained. We also obtain rates of convergence for sample moments, the empirical process and the quantile process for the regenerative processes. Robust estimates are also obtained when an outlier contaminated sample ofa ands is provided. In the process we obtain consistency, continuity and asymptotic normality of M-estimators for stationary sequences. Some robustness results for Markov processes are included.