A Queuing Model for an Inventory Problem

The relation between lost sales and inventory level is an important problem in inventory control. An explicit mathematical solution is obtained by methods of general interest for a probabilistic model that arose in connection with consulting work for an industrial client. Customer demand for a given commodity is a Poisson process with mean rate λ, and replenishment time for restocking is random. At any moment, the constant inventory n is divided between in-stock amount n 0 , and inreplenishment process amount n − n 0 . Customer arrival when n 0 > 0 results in a unit sale and the initiation of replenishment of that unit. Successive replenishment times are independent. Customer arrival when n 0 = 0, results in a lost sale. The unique stationary probabilities p(n 0 ∣n) of the states n 0 (fixed n), are obtained, they are given by the Erlang congestion formula, and depend upon the replenishment time only to the extent of its mean value. A generalization is obtained where λ may be a function of the state of the system. The ratio of lost sales to total demand, given by p(0∣n), is shown to be convex decreasing in n. The problem of allocation of inventory dollars among various competing commodities, so as to minimize over-all lost sales dollars, is treated.