Clearing Systems and (s, S) Inventory Systems with Nonlinear Costs and Positive Lead Times

Stochastic clearing systems model many different applications, including certain bulk server queues, shuttle buses, and buffers in communication systems. We extend the characterization of the optimal parameters of a generalized stochastic clearing system (Stidham, S., Jr. 1977. Cost models for stochastic clearing systems. Opns. Res.25 100–127.) from piece-wise linear to general convex cost-rate functions, and discuss applications to continuous-review and periodic-review (s, S) inventory models with nonlinear holding and storage costs and/or nonzero lead times. As in Stidham, we show that log-concavity of the sojourn function associated with the input process guarantees that the necessary conditions are also sufficient for optimality of the clearing parameters, which correspond to s and S in inventory examples. Our conditions are weaker and our model more general than those found in the previous literature on (s, S) inventory models.