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A scaling analysis of a star network with logarithmic weights
Authors:Philippe Robert  Amandine Véber
Institution:1. INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France;2. CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
Abstract:The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has L requests to transmit and is idle, it tries to access the channel at a rate proportional to log(1+L). A stochastic model of such an algorithm is investigated in the case of the star network, in which J nodes can transmit simultaneously, but interfere with a central node 0 in such a way that node 0 cannot transmit while one of the other nodes does. One studies the impact of the log policy on these J+1 interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter N being the norm of the initial state. It is shown that the asymptotic fluid behavior of the system is a consequence of the evolution of the state of the network on a specific time scale (Nt,t(0,1)). The main result is that, on this time scale and under appropriate conditions, the state of a node with index j1 is of the order of Naj(t), with 0aj(t)<1, where t?aj(t) is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.
Keywords:primary  60K25  60K30  60F05  secondary  68M20  90B22  Communication networks  Fluid scaling  Separation of timescales
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