An inventory system with service facility and feedback customers

We consider a continuous review inventory system with service facility and a single server. The customers arrive according to a Poisson process. Any arriving customer, who finds the waiting room 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. After a customer is served, he/she will decide either to join the orbit, for another service or leave the system according to a Bernoulli trial. The service time for these feedback customers are assumed to be exponential distribution. The inventory is replenished according to an (s, S) inventory policy and the replenishing times are assumed to be exponential. We have carried out the busy period analysis of the system and the waiting time distribution for both primary and feedback customers. The sensitivity analysis has been carried out the effect of different parameters and costs rates.