Scheduling services in a queuing system with impatience and setup costs

We consider a single server queue in discrete time, in which customers must be served before some limit sojourn time of geometrical distribution. A customer who is not served before this limit leaves the system. The fact of serving customers, holding them in queue or losing them induce costs. The purpose is to decide when to serve the customers so as to minimize these costs. We use a Markov Decision Process with infinite horizon and discounted criterion. We establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type, and we compute the threshold explicitly.