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40
Applications of Markov Decision Processes in Communication Networks: a Survey
 in Markov Decision Processes, Models, Methods, Directions, and Open Problems, E. Feinberg and A. Shwartz (Editors) Kluwer
, 2001
"... We present in this Chapter a survey on applications of MDPs to communication networks. We survey both the different applications areas in communication networks as well as the theoretical tools that have been developed to model and to solve the resulting control problems. 1 ..."
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Cited by 27 (2 self)
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We present in this Chapter a survey on applications of MDPs to communication networks. We survey both the different applications areas in communication networks as well as the theoretical tools that have been developed to model and to solve the resulting control problems. 1
Exact Covers of Balanced Sequences and Fraenkel's Conjecture.
, 2001
"... A striking conjecture of Fraenkel asserts that the only decomposition of Z into m 3 sequences fb i n+ i cg n2Z with i and i real, i > 1 and i 's distinct for i = 1; :::; m is given by f 1 ; :::; m g = f 2 m 1 2 k : 0 k < mg: A weaker conjecture states that then i = j is an ..."
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Cited by 23 (2 self)
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A striking conjecture of Fraenkel asserts that the only decomposition of Z into m 3 sequences fb i n+ i cg n2Z with i and i real, i > 1 and i 's distinct for i = 1; :::; m is given by f 1 ; :::; m g = f 2 m 1 2 k : 0 k < mg: A weaker conjecture states that then i = j is an integer for some i 6= j. The weaker statement has been proved by Simpson if some i is at most 2. Fraenkel's conjecture was proved by Simpson if some i is at most 3=2 and by Morikawa if m = 3 and, under some condition, if m = 4. An unconditional proof for m = 4 was given by Altman, Gaujal and Hordijk and for m = 5; 6 by the author. Proofs for the more general case of balanced sequences have been given for m = 3 by the author and for m = 4 by Altman, Gaujal and Hordijk. In the present paper we extend the other results mentioned above to balanced sequences. We start with a survey of the related literature and give selfcontained proofs. Keywords: complementary sequences, eventual coveri...
Fraenkel's Conjecture For Six Sequences
, 2001
"... A striking conjecture of Fraenkel asserts that every decomposition of Z>0 into m 3 sets fb i n + i cg n2Z>0 with i and i real, i > 1 and i 's distinct for i = 1; :::; m satis es f 1 ; :::; m g = f 2 m 1 2 k : 0 k < mg: Fraenkel's conjecture was proved by Morikawa if m = 3 and, ..."
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Cited by 21 (3 self)
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A striking conjecture of Fraenkel asserts that every decomposition of Z>0 into m 3 sets fb i n + i cg n2Z>0 with i and i real, i > 1 and i 's distinct for i = 1; :::; m satis es f 1 ; :::; m g = f 2 m 1 2 k : 0 k < mg: Fraenkel's conjecture was proved by Morikawa if m = 3 and, under some condition, if m = 4. Proofs in terms of balanced sequences have been given for m = 3 by the author and for m = 4 by Altman, Gaujal and Hordijk. In the present paper we use the latter approach to establish Fraenkel's conjecture for m = 5 and for m = 6.
Admission Control in Stochastic Event Graphs
 IEEE Transaction on Automatic Control
, 1997
"... In this paper, we show that the expected workload and the expected waiting time in (max,+) linear system under a single input sequence is multimodular. We use this result to construct the optimal deterministic admission control in the (max,+) system under rate constraints. ..."
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Cited by 18 (10 self)
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In this paper, we show that the expected workload and the expected waiting time in (max,+) linear system under a single input sequence is multimodular. We use this result to construct the optimal deterministic admission control in the (max,+) system under rate constraints.
Balance Properties of MultiDimensional Words
 Theor. Comput. Sci
, 2000
"... A word u is called 1balanced if for any two factors v and w of u of equal length, we have 1 jvj i jwj i 1 for each letter i, where jvj i denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multidimensional words. We rst characteri ..."
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Cited by 16 (1 self)
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A word u is called 1balanced if for any two factors v and w of u of equal length, we have 1 jvj i jwj i 1 for each letter i, where jvj i denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multidimensional words. We rst characterize all 1balanced words on Z n . In particular we prove they are fully periodic for n > 1. We then give a quantitative measure of balancedness for some words on Z 2 with irrational density, including twodimensional Sturmian words. 1
Optimal Routing Policy in Two Deterministic Queues
 Calculateurs Parallèles
, 2000
"... : We consider the problem of routing customers to one of two parallel queues where interarrival 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 ..."
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Cited by 12 (11 self)
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: We consider the problem of routing customers to one of two parallel queues where interarrival 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. Keywords: Mechanical Words, Continued Fractions, Deterministic Queues LORIA, 615 route du jardin botanique, BP 101, F54606 VillerslesNancy, France. Email: Bruno.Gaujal@loria.fr. y LORIA, 615 route du jardin botanique, BP 101, F54606 VillerslesNancy, France. Email: Emmanuel.Hyon@loria.fr. Politique de routage optimale de deux files dterministes Rsum : Dans cet article nous tudions le rout...
A characterization of balanced episturmian sequences
, 2008
"... It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a k ..."
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Cited by 12 (1 self)
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It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a kletter alphabet, k ≥ 3, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a kletter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel’s conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed k> 2, there is only one way to cover Z by k Beatty sequences. The problem can be translated to combinatorics on words: for a kletter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies.
Optimal Routing Policies 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 maximu ..."
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Cited by 10 (3 self)
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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 maximum service time on each route. In order to do this, we particularly study the output process of deterministic queues when the input process is Sturmian.
Optimal Routing in two parallel queues with exponential service times, in "Journal of Discrete Event Dynamic Systems
, 2005
"... ap por t de r ech er ch e THÈME 1 ..."
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Optimal admission, routing and service assignment control: the case of single buffer queues
 Proc 37th IEEE CDC, Tampa
, 1998
"... We consider the problem of optimal routing of arriving packets into N servers having no waiting room. Packets that are routed to a busy server are lost. We consider two problems where the objective is to maximize the expected throughput (or equivalently, minimize the loss rate). We assume that the c ..."
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Cited by 8 (4 self)
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We consider the problem of optimal routing of arriving packets into N servers having no waiting room. Packets that are routed to a busy server are lost. We consider two problems where the objective is to maximize the expected throughput (or equivalently, minimize the loss rate). We assume that the controller has no information on the state of the server. We establish the optimality of the so called “balanced” policies, for exponential service times and general stationary arrival processes, which include, in particular, the interrupted Poisson process, Markov modulated Poisson Process (MMPP) and Markov arrival process (MAP). Based on this solution, we solve the dual problem of optimal assignment of a single server to several single server queues to which packets arrive according to Poisson processes. This general model is then applied to solve an optimal scheduling problem for robots of Web search engines.