Hierarchical-routing-2 - Atomic Hierarchical Routing Games...

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Atomic Hierarchical Routing Games in Communication Networks Vijay Kamble * , Eitan Altman , Rachid El-Azouzi and Vinod Sharma § * Dept. of Industrial Engineering and Management, IIT - Kharagpur, West Bengal, India Maestro group, INRIA, 2004 Route des Lucioles, Sophia Antipolis, France LIA, University of Avignon, 339, chemin des Meinajaries, Avignon, France § Dept. of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India Abstract —Theoretical studies on routing games in networks have so far dealt with reciprocal congestion effects between rout- ing entities. But, with the advent of technologies like Cognitive Radio, we have networks which support differentiation of flows. In a two priority level model a user can be high priority or low priority and there is a cost for such a classification. The point of departure of this model from the traditional routing scenarios is the absence of reciprocity in the congestion effects: The low priority flow faces congestion from both high priority as well as low priority flow while the high priority flow is immune to the congestion effects from the low priority flow. This hierarchy is naturally present in contexts where there are primary (licensed) users and secondary (unlicensed) users who can sense their environment because there are equipped with a cognitive radio [9]. We study such kind of routing scenarios for the cases of atomic users. We establish the existence and the uniqueness of Nash equilibrium and further we show the existence of a potential function for linear congestion costs and a certain priority classification pricing scheme. Natural applications of this model to Cognitive Radio are also pointed out. I. I NTRODUCTION Non-cooperative routing has long been studied both in the framework of road-traffic as well as in the framework of wireless networks. Such frameworks allow to model the flow configuration that results in networks in which routing deci- sions are made in a non-cooperative and distributed manner between the users. In the case of a finite (not very large) number of agents, the resulting flow configuration corresponds to the so called Nash equilibrium defined as a situation in which no agent has an incentive to deviate unilaterally. In other words, the Nash equilibrium does not state what can or cannot happen when more than one decision maker changes their strategy (route) simultaneously. When the decision makers in a Nash game are discrete and finite in number, a Nash equilibrium can be achieved without the costs of all used routes being equal, contrary to Wardrop’s equilibrium principle [11]. In some cases, Wardrop’s principle has been shown to represent a limiting case of the Nash equilibrium principle, as the number of users becomes very large [4], [5], [13]. In the context of routing in networks, the Nash equilibrium has been introduced by Orda et al [10] as well as [2] in order to predict the traffic pattern that occurs when each of several non-cooperative sources of traffic (say a service provider) has
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