Continuous-type (s,Q)-inventory model with an attached M/M/1 queue and lost sales

In this study, we consider an M/M/1 queuing model with an attached continuous-type inventory. Customers arrive in the system according to a Poisson process and are served individually on a first-come, first-served basis. The service times of customers are assumed to be independent and identically distributed exponential random variables. Along with the queue, there is an internal finite storage for the inventory and each service requires an exponentially distributed random amount H of inventory from the storage. Therefore, a customer leaves the system with H amount of item at his/her service completion time. The inventory is replenished by an outside supplier with a random lead time under an (s,Q) inventory control policy. We assume that the customers who arrive during stock-out periods are lost (lost sales). For this queuing-inventory system, we derive the stationary joint probability distribution of queue length and inventory level in explicit product form. Numerical examples followed by a cost model are also presented.