- Published 2002 in SIGMETRICS Performance Evaluation Review

0 f ( X(u) ) du, where f is a real-valued function and {X(t) : t ≥ 0 } is the underlying stochastic process that records the state of the system as it evolves over continuous time. In this paper we assume that {X(t) : t ≥ 0 } can be represented as a generalized semi-Markov process (gsmp) and consider simulation-based methods for obtaining point estimates and confidence intervals for time-average limits. We also consider time-average limits of the form r̃(f̃) = limn→∞(1/n) ∑n−1 j=0 f̃(Sj , Cj), where { (Sn, Cn) : n ≥ 0 } is the general state space Markov chain used to define the gsmp (see below). When the output process { f ( X(t) ) : t ≥ 0} or { f̃(Sn, Cn) : n ≥ 0 } obeys a central limit theorem (clt), there are two main approaches to obtaining an asymptotic confidence interval for the time-average limit. The first approach is to derive the confidence interval using a limit theorem in which the variance constant that appears in the clt is “cancelled out” [4] and hence need not be estimated. “Cancellation” procedures of this type include the original “standardized time series” (sts) area and maximum methods, the original methods of batch means and spaced batch means (where the number of batches is independent of the simulation run length), and the sts-weighted-area method. The second approach is to consistently estimate the variance constant. Procedures of this type include the regenerative method, the method of “variable” batch means (where the number of batches increases as the run length increases), and spectral methods. It is usually nontrivial to determine for a specified gsmp model whether time-average limits are well defined. It is even harder to determine whether the output process obeys a clt and, if so, whether a specified estimation method is applicable. Most conditions in the literature involve unrealistic assumptions (such as stationarity of the output process) or are difficult to verify. In this paper we provide new conditions on the building blocks of a gsmp under which long-run estimation problems are well defined and a variety of cancellation and consistent estimation methods are provably valid. Our first set of results provides building-block conditions under which time-average limits exist and the output process { f ( X(t) ) : t ≥ 0} or { f̃(Sn, Cn) : n ≥ 0 } obeys a functional central limit theorem (fclt). When an fclt holds, the output process obeys an ordinary clt. Moreover, the validity of a broad class of cancellation methods— including all of those mentioned above—follows directly from results in [4]. Our moment conditions are significantly weaker than those in [6] and, in fact, appear to be the weakest conditions possible. Our remaining results provide building-block conditions under which various estimators of the variance constant in the clt for the output process are (weakly) consistent, so that confidence intervals based on these variance estimators are asymptotically valid. We use a coupling approach to extend consistency results for variance estimators from a stationary to a non-stationary setting. By combining this approach with known results for stationary processes, we obtain sufficient conditions under which a class of “quadratic form” variance estimators are consistent. This class includes batch means and spectral estimators. Our results complement those of [3], which establish strong consistency for variance estimators under the harder-to-verify assumption that the output process obeys a strong invariance principle. For example, it appears difficult to establish strong consistency for the popular version of variable batch means in which the number of batches grows as the 2/3 power of the run length—our results can be used to establish weak consistency for this method.

@article{Haas2002OnTV,
title={On the validity of long-run estimation methods for discrete-event systems},
author={Peter J. Haas and Peter W. Glynn},
journal={SIGMETRICS Performance Evaluation Review},
year={2002},
volume={30},
pages={35-37}
}