An Inventory System with Multi-server Service Facility with Negative Demands

In this paper, we consider a continuous review perishable inventory system with multi-server service facility. In such systems the demanded item is delivered to the customer only after performing some service, such as assembly of parts or installation, etc. Compared to most inventory models in which inventory is depleted at the demand rate, it is depleted, in this model, at the rate that the service occurs. We assume that the arrivals of customers are according to a Markovian arrival process (MAP) and that service time has exponential distribution. The ordering policy is based on (s, S) policy. The lead time is assumed to have exponential distribution. The customer who finds either all servers are busy or no item (excluding those in service) is in the stock, enters into an orbit of infinite size. These orbiting customers send requests at random time points for possible selection of their demands. The interval time between two successive request-time points is assumed to have exponential distribution. In addition to the regular customers, a second flow of negative customers following an independent MAP is also considered which will remove one of the customers in the orbit. The joint probability distribution of the number of busy servers, the inventory level and the number of customers in the orbit is obtained in the steady state. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated.