Analysis of a production inventory system with MAP arrivals, phase-type services and vacation to production facility

In this paper, we discuss a production inventory system with service times. Customers arrive in the system according to a Markovian arrival process. The service times follow a phase-type distribution. We assume that there is an infinite waiting space for customers. Arriving customers demand only one unit of item from the inventory. The production facility produces items according to an (s, S)-policy. Once the inventory level becomes the maximum level S, the production facility goes on a vacation of random duration. When the production facility returns from the vacation, if the inventory level depletes to the fixed level s, it is immediately switched on and starts production until the inventory level becomes S. Otherwise, if the inventory level is greater than s on return from the vacation, it takes another vacation. The vacation times are exponentially distributed. The production inventory system in the steady-state is analyzed by using the matrix-geometric method. A numerical study is performed on the system performance measures. Besides, an optimization study is discussed for the inventory policy.