  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.
