Markov Process for Service Facility systems with perishable inventory and analysis of a single server queue with reneging Stochastic Model

In this paper, we develop a supply network model for a service facility system with perishable inventory (on hand) by considering a two dimensional stochastic process of the form (L, X) = 0 )); ( ), ( ( t t X t L , where L (t) is the level of the on hand inventory and X (t) is the number of customers at time t. The inter-arrival time to the service station is assumed to be exponentially distributed with mean 1/λ. The service time for each customer is exponentially distributed with mean 1/ μ. The maximum inventory level is S and the maximum capacity of the waiting space is N. The replenishment process is assumed to be (S-1, S) with a replenishment of only one unit at any level of the inventory. Lead time is exponentially distributed with parameter β. The items are replenished at a rate of β whose mean replenishment time is 1/β. Item in inventory is perishable when it’s utility drops to zero or the inventory item become worthless while in storage. Perishable of any item occurs at a rate of γ. Once entered a queue, the customer may choose to leave the queue at a rate of α if they have not been served after a certain time (reneging). The steady state probability distributions for the system states are obtained. A numerical example is provided to illustrate the method described in the model.