Stochastic Analysis of a Production Inventory System with Deteriorating Items, Unreliable Server and Retrial of Customers

This paper considers a production inventory system with perishable items where the customers arrive according to a homogeneous Poisson process. The time interval between the production of each item is exponentially distributed. This is a single server system, where the service time follows exponential distribution. The server may become breakdown according to a Poisson process in which case the service restarts after an exponentially distributed time. On arrival, a customer leads to service if the server is available and the inventory is non-empty. The arriving customer on finding the server busy or breakdown goes to a waiting place(orbit) of infinite capacity with pre-determined probability or exits the system with complementary probability. Each customer in the orbit retries to access the service facility in an exponentially distributed time interval. If the retrial is unsuccessful, the customer returns to the orbit with a pre-allotted probability or is lost forever with probability equal to its complement. An algorithmic solution to the problem is obtained using Matrix Analytic Method. Average inventory level in the system, mean number of customers in the orbit, expected rate at which production is switched ON, the mean number of customer loss before and after entering the system, the rate of successful retrials among overall retrials, average breakdown rate and repair rate and some other performance measures of the system are derived. The impacts of system parameters on different measures are numerically studied. A suitable cost function is constructed to find the optimum cost.