A Queuing Model for an Inventory Problem

William Karush
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William Karush: The Ramo-Wooldridge Corporation, Los Angeles, California

Operations Research, 1957, vol. 5, issue 5, 693-703

Abstract: 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 (lambda), 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 (lambda) 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.

Date: 1957
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