Queueing Inventory System with Multiple Service Nodes and Addressed Retrials from a Common Orbit

In this paper, we consider a queueing inventory model with K service nodes located apart making it impossible to know the status of the other service nodes. The primary arrival of customers follows Marked Markovian Arrival Process and the service times are exponentially distributed. If a customer arriving at a node finds the server busy or the inventory level to be zero, he joins a common orbit with infinite capacity. An orbital customer shall choose a service node at random according to some predetermined probability distribution dependent on the orbit size. Each service node is assigned with a continuous review inventory replenished according to an (s, S) policy with lead time. This scenario is modeled as a level dependent quasi birth and death process which belongs to the class of asymptotically quasi-Teoplitz Markov chains. Steady-state probabilities and some important performance measures are obtained. A cost function is introduced and employed for computing the optimal values of reorder levels and replenishment rates.