It is known that simulation of the mean position of a Reflected Random Walk (RRW) {Wn} 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, ∗, of the paths n−1W⌊tn⌋ is to be zero apart from on an interval [T0, T1] ⊂ [0, 1] and to satisfy the functional equation ∇I