Most likely paths to error when estimating the mean of a reflected random walk

TL;DR

This paper investigates the large deviation behavior of the mean position in a reflected random walk, revealing conditions under which simulation estimates tend to overestimate the true steady-state mean, impacting queueing system evaluations.

Contribution

It provides a detailed sample path large deviation analysis for RRWs, identifying the most likely paths leading to estimation errors and highlighting when naive simulation estimates are conservative.

Findings

Most likely paths involve the process being zero outside a specific interval.

When the rate function is coercive, overestimation of the mean is highly probable.

Results have significant implications for simulation-based performance evaluation of queueing systems.

Abstract

It is known that simulation of the mean position of a Reflected Random Walk (RRW) $\{W_n\}$ exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let $I$ denote the rate function for the one dimensional increment process. If $I$ is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, $\psi^*$, of the paths $n^{-1} W_{\lfloor tn\rfloor}$ is to be zero apart from on an interval $[T_0,T_1]\subset[0,1]$ and to satisfy the functional equation \begin{align*} \nabla I\left(\ddt\psi^*(t)\right)=\lambda^*(T_1-t) \quad \text{whenever } \psi(t)\neq 0. \end{align*} If $I$ is non-coercive, a similar, but slightly more involved, result holds. These results prove, in broad generality, that Monte Carlo estimates of the steady-state mean position of a RRW have a high likelihood of over-estimation. This has serious implications for the performance evaluation of queueing systems by simulation techniques where steady state expected queue-length and waiting time are key performance metrics. The results show that na\"ive estimates of these quantities from simulation are highly likely to be conservative.