Abstract
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 service of customers, the loss due to impatience and the holding of customers in the queue induce costs. The purpose is to decide when to serve the customers so as to minimize them. 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, we are able to compute explicitly the optimal value of this threshold in terms of the parameters of problem.