On an (s, S) Inventory Policy with Service Time, Vacation to Server and Correlated Lead Time

Abstract

We consider an inventory with random positive service time. The control policy is of (s, S) type. Customer demands constitute a Markovian arrival process (MAP). Service times have phase type distribution. Lead time for replenishment of inventory follows a correlated process similar to the customer arrival process. At a service completion epoch if no customer is waiting or the inventory level is zero then the server goes on a vacation of random duration. The vacation time is also phase type distributed. On return from vacation if the server finds no customer waiting or inventory level is zero, the server goes on another vacation of random duration having the same phase type distribution as the earlier one; we assume that the successive vacation times are independent of each other. This goes on until on return he finds at least one customer and at least one item in the inventory at which he starts serving. The customer arrival process is subject to balking. While waiting for service, customers may become impatient and leave the system (renege). Under these assumptions the underlying level dependent quasi birth-death process (LDQBD) is analysed. Several particular cases are considered in which the system is exhaustively studied. Numerical illustrations are provided. A cost function is introduced to numerically evaluate the optimal values of s and S.