Inventory system with renewal demands at service facilities

Inventory systems where customers join waiting line to receive their demanded items are recently considered by many researchers, as these models allow the study of both queue length and size of the inventory. This article considers a continuous review inventory system at a service facility, wherein an item demanded by a customer is issued to him/her only after performing service of random duration on the item. The service facility is assumed to have waiting hall of infinite size. The arrival time points of customers form a renewal process. The service times are assumed to be distributed as negative exponential. The operating policy is (s,S) policy with instantaneous supply of ordered items. We consider two models which differ in the way that ordered items, when received, is brought into the stock. In the first model, the ordered items are brought into the stock immediately. In the second model, the supplied items are brought into the stock only at the next demand epoch. The stationary distribution of the underlying Markov chain is obtained in matrix-geometric form. The joint probability distribution of the number of customers in the system and the inventory level is obtained in the steady-state case.