Scheduling 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: it is impatient. The fact of serving customers and the fact of losing them due to impatience induce costs. The purpose is to decide when to serve the customers so as to minimize costs. We use a Markov Decision Process with infinite horizon and discounted cost. We establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type. In addition, thanks to a pathwise comparison analysis of two threshold policies, we are able to compute explicitly the optimal value of this threshold according to the parameters of problem.