A Production Inventory System with Two Types of Customers and Machine Breakdowns

We consider a production inventory system with machine breakdowns and two types of demands (type - 1 and type - 2). The arrivals of customers are according to two independent Poisson processes and each customer demands unit item. The maximum storage capacity of the system is fixed as S. If the inventory level falls to a prefixed level, says(< S), a production machine is started to produce an item. The production time is assumed to follow an exponential distribution. We assume that the production is stopped when the on hand inventory level reaches. The machine, during production may fail and the time to fail is assumed to have an exponential distribution. The failed machine can be repaired and the repair time is also assumed to have an exponential distribution. The demands of type - 2 customers are satisfied whenever there is a positive inventory level, otherwise they are assumed to be lost. The demands of type - 1 customers are satisfied if the inventory level is above or the inventory level lies between 1 and s and the production machine is on. In all other cases, the type - 1 customers are sent to an orbit of infinite size from which they retry at times so that the interval times between two successive retrials are assumed to have iid exponential distributions. The joint probability distribution of the number of customers in the orbit, inventory level and status of machine is obtained in the steady state case. Various system performance measures in the steady state are derived and the long-run total expected cost per unit time is calculated. A numerical illustration is also provided.