An inventory system with replacement and refurbishment of failed items

We consider a continuous review inventory system with two types of inventory, namely type-I and type-II. Type-I and type-II inventory, respectively, contain fresh and refurbished items. Two streams of customers, called type-A and type-B customers arrive to the system in which type-A customer demands for an item either from type-I or type-II inventory according to a Bernoulli trial and type-B customer occurs to replace the failed item with new one from type-I inventory. The arrival time points of type-A and type-B customers follow independent Poisson Processes. Failed items are stored in a place, called pool, of infinite size for refurbishment. After refurbishment, these items are kept in type-II inventory. The refurbishment of pooled item is stopped once type-II inventory reaches its maximum level S2 and is again commenced only when type-II inventory level falls to s2. The refurbishment time follows exponential distribution. Type-I inventory is replenished according to a (s, S) policy with exponential lead time. The joint pdf of number of items in the pool, type-II inventory level, status of refurbishment and type-I inventory level is obtained in the steady state. Various system performance measures are derived and total expected cost rate is calculated. Results are illustrated numerically.