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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 incrementdistributions with negative drift. The Large Deviation Principle (LDP) holdsfor deviations below the mean, but for deviations at the usual speed above themean the rate function is null. This paper takes a deeper look at thisphenomenon. Conditional on a large sample mean, a complete sample path LDPanalysis is obtained. Let $I$ denote the rate function for the one dimensionalincrement process. If $I$ is coercive, then given a large simulated meanposition, under general conditions our results imply that the most likelyasymptotic behavior, $\psi^*$, of the paths $n^{-1} W_{\lfloor tn\rfloor}$ isto be zero apart from on an interval $[T_0,T_1]\subset[0,1]$ and to satisfy thefunctional equation \begin{align*} \nablaI\left(\ddt\psi^*(t)\right)=\lambda^*(T_1-t) \quad \text{whenever } \psi(t)\neq0. \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 thesteady-state mean position of a RRW have a high likelihood of over-estimation.This has serious implications for the performance evaluation of queueingsystems by simulation techniques where steady state expected queue-length andwaiting time are key performance metrics. The results show that na\"iveestimates of these quantities from simulation are highly likely to beconservative.

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