Structural properties of the lost sales problem with fractional lead times

We consider an infinite horizon periodic review base-stock inventory system with discrete demand and lost sales. Every T periods the inventory is reviewed and an order, with arbitrary lead time L, is placed to bring the inventory position to R. Using a recursive Markovian representation, we develop the operating characteristics for any system with positive integer-valued base-stock R, review period T, and lead time L. Then, we develop an algorithm to find the optimal stationary R. Surprisingly, computational results yield the unintuitive findings that the optimal base-stock and cost are not monotone in L given T.