Product Form Solution of a Queuing-Inventory System with Lost Sales and Server Vacation

In this study, the authors consider an M/M/1 queuing system with attached inventory under an (s, S) control policy. The server takes multiple vacations whenever the inventory is depleted. It is assumed that the lead time and the vacation time follow exponential distributions. The authors formulate the model as a quasi-birth-and-dearth (QBD) process and derive the stability condition of the system. Then, the stationary distribution in product form for the joint process of the queue length, the inventory level, and the server’s status is obtained. Furthermore, the conditional distributions of the inventory level when the server is on and operational, and when it is off due to a vacation, are derived. Using the stationary distribution, the authors obtain some performance measures of the system. The authors investigate analytically the effect of the server’s vacation on the performance measures. Finally, several numerical examples are presented to investigate the effects of some parameters on the performance measures, the optimal policy, and the optimal cost.