A production-inventory system with a Markovian service queue and lost sales

We study an (s, S) production-inventory system with an attached Markovian service queue. A production facility gradually replenishes items in the inventory based on the (s, S) scheme, and the production process is assumed to be a Poisson process. In addition to the production-inventory system, c servers process customers that arrive in the system according to the Poisson process. The service times are assumed to be independent and identically distributed exponential random variables. The customers leave the system with exactly one item at the service completion epochs. If an item is unavailable, the customers cannot be served and must wait in the system. During this out-of-stock period, all newly arriving customers are lost. A regenerative process is used to analyze the proposed model. We show that the queue size and inventory level processes are independent in steady-state, and we derive an explicit stationary joint probability in product form. Probabilistic interpretations are presented for the inventory process. Finally, using mean performance measures, we develop cost models and show numerical examples.