E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Gametheoretic distributed algorithms. to appear in J. Distributed and Parallel Comupting, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... and su#cient conditions (on the network topology, the amount of flow that users control, and on the edge latency functions) for the existence and uniqueness of (pure strategy) Nash equilibria [4, 8, 25, 59, 138] and for convergence to a Nash equilibrium under natural models of user behavior [5, 138]. Some of these results have been extended to networks in which users cannot split flow and must instead route all of their tra#c on a single path [117] Finally, the game theory community has generalized the tra#c model studied here to broad classes of games that need not take place in a ....

....In this section we extend the basic model to the case of finitely many network users, each of whom controls a strictly positive amount of tra#c. In this section we allow a network user to split flow among any number of paths; this model has been studied extensively in the networking literature [4, 5, 8, 25, 59, 138]. In the next section we will investigate the setting in which each network user must route all of its flow on a single path. We are given a network G with continuous nondecreasing latency functions # as before, and in addition k users. We assume that user i intends to send r i units of flow from ....

E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Game-theoretic distributed algorithms. Journal of Parallel and Distributed Computing, 61(9):1367--1381, 2001.

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E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Gametheoretic distributed algorithms. to appear in J. Distributed and Parallel Comupting, 2000.

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Altman E,Basar T,Jimenez T,Shimkin N. Routing into two parallel links: game-theoretic distributed algorithms. Journal of Parallel and Distributed Computing 2001;61(9):1367--81.

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E. Altman, T. Basar, T. Jimenez and N. Shimkin, "Routing into two parallel links: game-theoretic distributed algorithms", to appear in the Special Issue of Journal of Parallel and Distributed Computing on "Routing in Computer and Communication Networks ", 2001.

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E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Game-theoretic distributed algorithms. to appear in J. Distributed and Parallel Comupting, 2000. 17

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E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Game-theoretic distributed algorithms. to appear in J. Distributed and Parallel Comupting, 2000. 17

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E. Altman, T. Ba#ar, T. Jim#nez and N. Shimkin, Routing into two parallel links: game-theoretic distributed algorithms, to appear in Journal of Parallel and Distributed Computing, 2001.

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E. Altman, T. Basar, T. Jimenez, and N. Shimkin. Routing into two parallel links: Game-theoretic distributed algorithms. to appear in J. Distributed and Parallel Comupting, 2000.

....the example studied in [13] in which an addition of capacity may, in general, increase both the price and the cost of each and every user. In all cases below, we computed the equilibrium iteratively with relaxation (which has been proven for some topologies to converge to an equilibrium, see [1]) as follows: 1. Define a candidate solution (0) tE for the total link flows which is obtained by minimizing the function V defined in (4) The flow of each player i in the initial iteration is then defined as (0) t(0)ri[Y .jz rJ] 2. At iteration n 0, we first compute the best responses ....

E. Altman, T. Ba]ar, T. Jimnez and N. Shimkin, Routing into two parallel links: game-theoretic distributed algorithms, to appear in Journal of Parallel and Distributed Computing, 2001.

....of the Wardrop equilibrium which involves side constraints has been studied in [18] and references therein. When all class use a class centralized optimization approach then the optimization concept is the Nash equilibrium. There has been much recent interest in this framework in recent years [1] [2], 3] 11] 16] 17] 19] In the context of road traffic, a class, or a group user, may correspond to a transportation company, or to a bus company; in both examples we may assume that the route of each vehicle is indeed determined by the company and not by the individual driver. The concept ....

E. Altman, T. Basar, T. Jimenez and N. Shimkin, "Routing into two parallel links: game-theoretic distributed algorithms", to appear in the Special Issue of Journal of Parallel and Distributed Computing on "Routing in Computer and Communication Networks", 2001.

....the example studied in [13] in which an addition of capacity may, in general, increase both the price and the cost of each and every user. In all cases below, we computed the equilibrium iteratively with relaxation (which has been proven for some topologies to converge to an equilibrium, see [1]) as follows: 1. De ne a candidate solution f x(0)g l2L for the total link ows which is obtained by minimizing the function V de ned in (4) The ow of each player i in the initial iteration is then de ned as x l (0) x l (0)r [ j2I r ] 2. At iteration n 0, we rst ....

E. Altman, T. Basar, T. Jimenez and N. Shimkin, Routing into two parallel links: game-theoretic distributed algorithms, to appear in Journal of Parallel and Distributed Computing, 2001.

....to use the equilibrium immediately. Results on the convergence of such schemes to the equilibrium (in the context of routing games, which we also use here) have been limited to the very special case of two parallel links and with either two users [9] or with N 2 users but with linear link costs [1]. As discussed in that reference, such costs are useful as they can well approximate a wide range of other costs under a light load regime. We would like to mention that other learning approaches have been considered in the past with other types of assumptions on updating. For example, in [8] ....

....but the convergence goes twice less fast than the previous algorithm, since we make twice more computations. Convergence of a greedy algorithm with relaxation An algorithm frequently used in information technology, of which ESS is a special case is the greedy algorithm with relaxation (see e.g. [1, 4]) The users update their ow with a convex combination between the best reply and the previous action. More precisely, if y (n 1) and (n) then the nth action used by player i is (n) n) 1 )y where is a relaxation factor within (0; 1] Lemma 3.1 also holds ....

E. Altman, T. Basar, T. Jimenez and N. Shimkin, "Routing into two parallel links: game-theoretic distributed algorithms", to appear in the Special Issue of Journal of Parallel and Distributed Computing on "Routing in Computer and Communication Networks", 2001. 15

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E. Altman, T. Basar, T. Jimenez, and N. Shimkin, "Routing into two parallel links: Game-theoretic distributed algorithms," A special Issue of Journal of Parallel and Distributed Computing on "Routing in Computer and Communication Networks", vol. 61, pp. 1367--1381, September 1 2001.

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