Product form solution for a queueing-inventory system with lost sales and phase type distribution of the vacation time

In this paper, a continuous review (s,S),S queueing-inventory system with lost sales is investigated. Customers arrive according to a Poisson process, and each arriving customer needs exactly one item from inventory for service. The server takes a single vacation once the inventory is depleted. It is assumed that the vacation time follows a phase type distribution, and the service time and the lead time follow exponential distributions. We obtain the product form solution for the steady-state distribution of the joint process of the queue length, the on-hand inventory level and the server's status. Furthermore, the conditional distributions of the on-hand inventory level when the server is turned off due to a vacation and when it is turned on are derived, respectively. Some performance measures of the system are computed by using the steady-state distribution of system. Based on some performance measures, we develop a cost model and determine a total expected cost under an appropriate cost structure by using the genetic algorithm. The effects of various parameters on the optimal inventory policy and the optimal expected cost are investigated by numerical examples.