Stochastic Inventory System with Multi Optional Service and Finite Source

Abstract

In this article, we consider a continuous review perishable inventory system with a finite number of homogeneous sources of demands. The maximum storage capacity is fixed as $S$. The operating policy is $(s, S)$ policy, that is, whenever the inventory level drops to $s$, an order for $Q(= S- s)$ items is placed. The ordered items are received after a random time which is distributed as exponential. The life time of each items is assumed to be exponential. All arriving customers demand first the essential service (regular service) and some of them may further demand one of other optional services: Type $1$, Type $2$, $\ldots$, and Type $N$ service. The service times of the essential service and of the Type $j$ $(j = 1, 2, \ldots, N)$ service are assumed to be exponentially distributed. The joint probability distribution of the number of customers in the waiting hall and the inventory level is obtained for the steady state case. We have derived the Laplace-Stieljes transforms of waiting time distribution of customers in the waiting hall. Some important system performance measures in the steady state are derived, and the long-run total expected cost rate is also derived.