This paper considers a production inventory with positive service time. The time for producing each item is assumed to follow a ‘Markovian production scheme’. Because of this, correlation automatically gets into the production process. The customer arrival process follows a Markovian arrival process. When inventory falls to s, the production process is switched on, and is switched off when the on-hand inventory reaches S. The service time to each customer has a phase-type distribution. We investigate the system stability. Under the condition of stability, we investigate the system state distribution. Next, several performance measures are computed, such as the fraction of time server is on vacation; fraction of time the inventory level is zero; fraction of time the production process is On; expected number of customers in the system while server is on vacation and also while it is busy. Numerical results indicate, among other things, the effect of the control variables s and S on the fraction of time the system goes out of inventory and on expected loss rate of customers.

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.

This article introduces an additional control policy—the N-policy–into (s, S) inventory system with positive service time. Under specified interarrival and service time distributions, which are independent of each other, we obtain the necessary and sufficient condition for the system to be stable. We also obtain the optimal values of the control variables s, S, and N; it is seen that the cost function attains the minimum value at s = 0. It is also shown that the cost function is separately convex in the variables S and N. Numerical illustrations are provided. Several measures of performance of the system are evaluated.

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.