A queueing-inventory system with two classes of customers

We consider a queueing-inventory system with two classes of customers. Customers arrive at a service facility according to Poisson processes. Service times follow exponential distributions. Each service uses one item in the attached inventory supplied by an outside supplier with exponentially distributed lead time. We find a priority service rule to minimize the long-run expected waiting cost by dynamic programming method and obtain the necessary and sufficient condition for the priority queueing-inventory system being stable. Formulating the model as a level-dependent quasi-birth-and-death (QBD) process, we can compute the steady state probability distribution by Bright-Taylor algorithm. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters to the system performance measures.