Network Control and Optimization: First EuroFGI by Eitan Altman, Konstantin Avrachenkov, Andrey Garnaev

By Eitan Altman, Konstantin Avrachenkov, Andrey Garnaev (auth.), Tijani Chahed, Bruno Tuffin (eds.)

This quantity 4465 of the Lecture Notes in desktop technology sequence is a coll- tion of the papers of the NET-COOP 2007 convention, a ?rst-of-a-series Euro- NGI/FGI convention on community regulate and Optimization. the development happened within the appealing urban of Avignon, France, June 5–7, 2007, used to be together or- nized by way of INRIA and the college of Avignon and used to be hosted through the latter. web communications and companies areexperiencing an increasein quantity and variety either of their capability and of their call for. This comes on the rate of a rise within the complexity in their keep watch over and optimization, customarily as a result of heterogeneity in structure in addition to utilization. the necessity for brand spanking new methods of e?ectively and reasonably allocating assets belonging to a large set of now not unavoidably cooperative networks to a set of very likely competing clients is pressing and is the purpose of this convention. Speci?cally, this convention goals at constructing study on keep watch over and op- mization of the web, starting from functionality overview and optimization of basic stochastic networks to extra speci?c goals reminiscent of lower-layer fu- tionalities in cellular networks, routing for computational grids, online game theoretic methods to entry keep watch over, cooperation, festival and adversary capacities in different environments.

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Network Control and Optimization: First EuroFGI International Conference, NET-COOP 2007, Avignon, France, June 5-7, 2007. Proceedings

This quantity 4465 of the Lecture Notes in machine technological know-how sequence is a coll- tion of the papers of the NET-COOP 2007 convention, a ? rst-of-a-series Euro- NGI/FGI convention on community keep watch over and Optimization. the development happened within the attractive urban of Avignon, France, June 5–7, 2007, was once together or- nized via INRIA and the college of Avignon and used to be hosted by way of the latter.

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Additional info for Network Control and Optimization: First EuroFGI International Conference, NET-COOP 2007, Avignon, France, June 5-7, 2007. Proceedings

Example text

Since limn→∞ (1 − n1 )n−1 = e−1 , a total throughput demand which is less or equal than this quantity guarantees the existence of an equilibrium. It can be shown that the simple bound obtained above holds for non-symmetric users as well, implying that the symmetric case is worst in terms of system utilization. Our result is summarized below. Due to lack of space the (somewhat lengthy) proof is omitted. 30 I. Menache and N. Shimkin Theorem 7 (Asymmetric users). For any set of n users, an equilibrium point exists if n 1 ρi ≤ (1 − )n−1 .

Consider the non-cooperative game whose Nash equilibrium point is defined in (2). Define i Ji (pi ) as the social performance criterion. , the better equilibrium point coincides with the social optimum, and (ii) the price of anarchy is generally unbounded. Proof. (i) immediate, as a social optimum obeys the equilibrium equations (2). (ii) we establish that the price of anarchy is unbounded by means of an example. Consider a network with two identical users with a throughput requirement of ρi = → 0 (i = 1, 2), and an average power function given by J(p) = W p, where p is the user’s transmission probability (user indexes are omitted in this example, as users are identical).

N (9) Proof. In every equilibrium of the symmetric case pi = pj = p, for every i, j (immediate by Theorem 3). Thus, the equilibrium equations (2) diminish into a single (scalar) equation: h(p) = p(1 − p)n−1 = ρ. (10) We next investigate the function h(p). The derivative of h(p) is given below. h (p) = (1 − p)n−2 1 − p − (n − 1)p = (1 − p)n−2 (1 − np). (11) It can be seen that the maximum value of the function h(p) is obtained at p = 1/n. An equilibrium exists if and only if the maximizer of the function obtains a value which is greater than ρ.

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