Abstract
We consider a single server queue in continuous time, in which customers must beserved before some limit sojourn time of exponential distribution. A customer who is not servedbefore this limit leaves the system: it is impatient. The fact of serving customers and the fact oflosing them due to impatience induce costs. The fact of holding them in the queue also induces aconstant cost per customer and per unit time. The purpose is to decide when to serve the customersso as to minimize costs. We use a Markov Decision Process with infinite horizon and discountedcost. Since the standard uniformization approach is not applicable here, we introduce a familyof approximated uniformizable models, for which we establish the structural properties of thestochastic dynamic programming operator, and we deduce that the optimal policy is of thresholdtype. The threshold is computed explicitly. We then pass to the limit to show that this thresholdpolicy is also optimal in the original model. A particular care is given to the completeness of theproof. We also illustrate the difficulties involved in the proof with numerical examples.