New analytical bounds on the average undershoot in an infinite horizon (s,S) inventory system

We provide new bounds on the average undershoot in an infinite horizon $(s,S)$ inventory system with i.i.d demands with known moments up to order three. The two moment bound improves on Lorden’s bound [G. Lorden, On Excess over the Boundary, Annals of Mathematical Statistics 41 (2) (1970) 520–527] for small values of $S$ – $s$ and low values of the coefficient of variation. The three moment bound provides further improvement for negatively skewed demands. The bounds are sharp and shown to be useful when the asymptotically tight two moment approximation fails.