Inventory with Positive Service Time and Retrial of Demands: An Approach through Multiserver Queues

We analyze an inventory with positive service time and retrial of demands by considering the inventory as servers of a multiserver queuing system. Demands arrive according to a Poisson process and service time distribution is exponential. On each service completion, the number of demands in the system as well as the number of inventories (servers) is reduced by one. When all servers are busy, new arrivals join an orbit from which they try to access the service at an exponential rate. Using matrix geometric methods the steady state joint distribution of the demands and inventory has been analyzed and a numerical illustration is given. 1. Introduction Inventory models with positive service time were first investigated by Sigman and Simchi-Levi [1]. Thereafter a number of researchers analyzed inventory systems where a queue of demands could be formed. We refer to the survey article of inventory with positive service time by Krishnamoorthy et al. [2] for further details. Many of the results are based on the idea that an inventory is served at a service facility. But in certain cases an arriving demand (customer) itself serves the inventory as in the case of a self-serviceable inventory. To analyze such a model, we consider the inventory as the server which is quite different from the usual multiserver queuing models. When a customer leaves the system after service completion, one server (inventory) also leaves the system which results in the decrement of the total number of servers (inventory) by one. As soon as the inventory reaches a lower level , an order is placed instantaneously and the inventory is restocked to . When the inventory is not available, an arriving customer leaves the system and he tries later for the inventory. To address such a model, the idea of retrial of customers in queuing theory can be used. A retrial queue is similar to an ordinary queue which has the following additional characteristics. When an arriving customer finds that all servers accessible to him are busy, he joins an infinite buffer called orbit from which he tries to access the server with a given probabilistic or deterministic policy. Thus the formation of a primary queue is not mandatory. The orbit is not an actual physical waiting area, but it simply represents a pool of returning customers. This type of situation arises in many real situations such as telecommunication and computer networks. A retrial queuing system with servers has been first investigated by Wilkinson [3]. In a multiserver retrial queue with servers, when an arriving customer finds a free server,