A perishable inventory system with service facility and feedback customers

In this article, we consider a continuous review perishable inventory system with service facility consisting of finite waiting hall and a single server. The primary customers arrive according to a Markovian arrival process. An arriving customer, who finds the waiting hall is full, is considered to be lost. The individual customer's unit demand is satisfied after a random time of service which is distributed as exponential. The life time of each item is assumed to be exponentially distributed. The items are replenished based on variable ordering policy. The lead time is assumed to have phase type distribution. After the service completion, the primary customer may decide either to join the secondary (feedback) queue, which is of infinite size, or leave the system according to a Bernoulli trial and the server decides to serve either for primary or feedback customer according to a Bernoulli trail. The primary and secondary service are at different counters. The service time for feedback customers is assumed to be independent exponential distribution. After the service completion for feedback customer, the server starts immediately for primary customer's service, whenever the inventory level and the primary customer level is positive, otherwise the server becomes idle for an exponential duration. If the primary customer level and inventory level becomes positive during the server idle period then he starts service for primary customer immediately. After completing his idle period, the server goes to secondary counter to serve for feedback customer, if any. The joint probability distribution of the system is obtained in the steady state. Important system performance measures are derived and the long-run total expected cost rate is also calculated. The results are illustrated numerically.