AbstractsMathematics

In a discrete review inventory process, when the demand forms a stochastically convergent sequence of random variables, it seems reasonable that the optimal stationary (s, S) inventory policy will be a function of the limiting demand and cost structure only. The intent of this paper is to provide a rigorous justification of this conjecture under suitable restrictions. Assuming linear costs and integer valued demand, the problem is essentially reduced to showing the existence and finding an expression for the stationary inventory distribution. The stationary inventory distribution, with an (s,S) policy in effect, is derived by applying renewal theory to the inventory process with renewals defined as those periods in which a positive amount is ordered. For this purpose a version of the key renewal theorem for stochastically convergent sequences is proved and formulated in terms of integrals. The integral formulation is used to derive the stationary distribution of the excess variable and the stationary probability that a renewal will occur, or equivalently, that an order will be placed.