Optimal inventory replenishment policy for a queueing system with finite waiting room capacity

We treat an inventory control problem in a facility that provides a single type of service for customers. Items used in service are supplied by an outside supplier. To incorporate lost sales due to service delay into the inventory control, we model a queueing system with finite waiting room and non-instantaneous replenishment process and examine the impact of finite buffer on replenishment policies. Employing a Markov decision process theory, we characterize the optimal replenishment policy as a monotonic threshold function of reorder point under the discounted cost criterion. We present a simple procedure that jointly finds optimal buffer size and order quantity.