A new factorization of mechanical words

In this paper, we introduce a morphism on Sturmian words which is tightly related to the coefficients of a particular continued fraction the ceiled continued fraction. This morphism will be applied to factorize periodic Sturmian words called Christoffel words, as well as, to characterize and construct discrete geometric objects called cellular line

Optimal Routing in Deterministic Queues in Tandem

In this paper we address the problem of routing a stream of customers in two parallel networks of queues in tandem with deterministic service times in order to minimize the average response time of the whole system. We show that the optimal routing is a Sturmian word which density depends on the decomposition in continuous fraction of the maximum service time on each route. In order to do this we study the output process of deterministic queues when the input process is Sturmian

Optimal Routing in two parallel Queues with exponential service times

In this paper we investigate the problem of the effective computation of the optimal routing sequence in a queuing system made of two queues with exponential service time in parallel. We first show that the optimal policy (minimizing the expected waiting time) is a Sturmian sequence and we establish several qualitative properties of this policy (monotonicity, continuity, convexity) Then, we propose an algorithm to compute the optimal routing sequence. We address the issues of time complexity as well as numerical stability of this algorithm. We then run an extensive set of experiments which show several interesting features of the optimal policy with apparent discontinuities and a fractal behavior

Optimal Routing Policy in Two Deterministic Queues

We consider the problem of routing customers to one of two parallel queues where inter-arrival times and service times are deterministic. We provide an explicit formula for the average waiting time of the customers sent to one of the queues when the routing policy is an upper mechanical word. This formula is based on a special continued fraction decomposition of the service time in the queue. Using this formula we provide an algorithm computing the optimal routing policy for two queues. In general, this policy is an upper mechanical word with a rational ratio, and hence is periodic

Scheduling services in a queuing system with impatience and setup costs

International audienceWe 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 sys- tem. 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

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

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

Critical level policies in lost sales inventory systems with different demand classes

We consider a single-item lost sales inventory model with different classes of customers. Each customer class may have different lost sale penalty costs. We assume that the demands follow a Poisson process and we consider a single replenishment hypoexponential server. We give a Markov decision process associated with this optimal control problem and prove some structural properties of its dynamic programming operator. This allows us to show that the optimal policy is a critical level policy. We then discuss some possible extensions to other replenishment distributions and give some numerical results for the hyperexponential server case

Routage en boucle ouverte dans deux files {./M/1} en parallèle

National audienceDans cet article nous nous intéressons au routage en boucle ouverte dans deux files d'attente en parallèle lorsque les services et les inter arrivées avant routage sont exponentielles. Notre but est de trouver la politique de routage optimale au sens où elle va minimiser le temps d'attente moyen. Pour ce faire nous étudions une file d'attente dont le processus d'entrée est un processus de Poisson échantillonné

Ordonnancement sous contraintes (m,k)-firm et combinatoire des mots

Dans ce papier, nous montrons une nouvelle méthode d'analyse de l'ordonnançabilité des ensembles de tâches sous contraintes (m,k)-firm en utilisant des propriétés des mots mécaniques (théorie des mots). Nous nous intéressons à l'ordonnancement sous contrainte de (m,k) pattern fixe. Dans un premier temps, nous montrons que les patterns introduits dans la littérature se caractérisent bien sous la forme de mots mécaniques. Les preuves d'ordonnançabilité en sont ainsi simplifiées. En identifiant les défauts de ces patterns, nous proposons ensuite une nouvelle technique basée sur la ligne cellulaire pour déterminer les (m,k) patterns des tâches. Les résultats expérimentaux montrent que cette nouvelle technique permet une amélioration de la région ordonnançable