Optimal control of admission in service in a queue with impatience and setup costs

International audienceWe consider a single server queue in continuous time, in which customers must be served before some limit sojourn time of exponential distribution. Customers who are not served before this limit leave the system: they are impatient. The fact of serving customers and the fact of losing them due to impatience induce costs. The fact of holding them in the queue also induces a constant cost per customer and per unit time. The purpose is to decide whether to serve customers or to keep the server idle, so as to minimize costs. We use a Markov Decision Process with infinite horizon and discounted cost. Since the standard uniformization approach is not applicable here, we introduce a family of approximated uniformizable models, for which we establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type. The threshold is computed explicitly. We then pass to the limit to show that this threshold policy is also optimal in the original model and we characterize the optimal policy. A particular care is given to the completeness of the proof. We also illustrate the difficulties involved in the proof with numerical examples

Contrôle optimal de l’admission en service dans une file d’attente avec impatience et coûts de mise en route

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.Nous considérons un modèle d’une file d’attente à un serveur en temps continu, danslaquelle les clients doivent être servis avant une durée de séjour finie aléatoire, de distribution expo-nentielle. Un client qui n’est pas servi avant cette limite quitte le système: il est impatient. Le fait deservir les clients et le fait de perdre des clients par impatience induisent des coûts. Le fait de les garderdans la file induit également un coût constant par client et par unité de temps. Il s’agit de décider defaçon optimale quand servir les clients. Nous utilisons un processus de décision Markovien à horizoninfini et à coûts actualisés. La méthode standard d’uniformisation ne s’appliquant pas à cette situation,nous introduisons une famille de modèles approchés uniformisables pour lesquels nous établissons lespropriétés structurelles de l’opérateur de programmation dynamique stochastique, et nous déduisonsque la politique optimale est à seuil. Le seuil est calculé explicitement. Nous passons ensuite à lalimite pour montrer que cette politique à seuil est également optimale dans le modèle initial. Une at-tention particulière est apportée à la complétude de la preuve. Nous illustrons également les difficultésrencontrées à l’aide d’exemples numériques

Efficient Content Delivery in the Presence of Impatient Jobs

We consider a content delivery problem in which jobs are processed in batches and may abandon before their service has been initiated. We model the problem as a Markovian single-server queue and analyze two different settings: (1) th