This paper develops central limit theorems (CLT's) and large deviations results for additive functionals associated with reflecting diffusions in which the functional may include a term associated with the cumulative amount of boundary reflection that has occurred. Extending the known central limit and large deviations theory for Markov processes to include additive functionals that incorporate boundary reflection is important in many applications settings in which reflecting diffusions arise, including queueing theory and economics. In particular, the paper establishes the partial differential equations that must be solved in order to explicitly compute the mean and variance for the CLT, as well as the associated rate function for the large deviations principle.

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In the planning of steady-state simulations, a central issue is the initial transient problem, in which an initial segment of the simulation output is adversely contaminated by initialization bias. Our article makes several contributions toward the analysis of this computational challenge. To begin, we introduce useful ways for measuring the magnitude of the initial transient effect in the single replication setting. We then analyze the marginal standard error rule (MSER) and prove that MSER’s deletion point is determined, as the simulation time horizon tends to infinity, by the minimizer of a certain random walk. We use this insight, together with fluid limit intuition associated with queueing models, to generate two nonpathological examples in which at least one variant of MSER fails to accurately predict the duration of the initial transient. Our results suggest that the efficacy of a deletion procedure is sensitive to the choice of performance measure, and that the set of standard test problems on which initial transient procedures are tested should be significantly broadened.

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