The M/M/1 queue with a production-inventory system and lost sales

In this paper, we study an M/M/1 queue with an attached production-inventory system. Customers arrive in the system according to a Poisson process, and a single server serves the customers. The service times are assumed to be i.i.d. exponential random variables. The customers leave the system with exactly one item from the inventory at his service completion epoch. If there is no inventory item, all arriving customers are lost. The stocks are replenished by (1) an external order under (r,Q)-policy, or (2) an internal production. The internal production process is assumed to be a Poisson process. We first derive the stationary joint distribution of the queue length and the on-hand inventory in product form. Using the joint distribution, we introduce long-run performance measures and a cost model. Then, we show numerical examples, which minimize the long-run cost per unit time.