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Fluid model for a network operating under a fair bandwidth-sharing policy. (English) Zbl 1066.60093
The paper starts from an exponential model for active transmission flows in the internet with Poisson arrivals of flows and with a fair resource sharing policy. Resource sharing is a generalization of traditional processor sharing: Flows interfere by requesting simultaneously the same transmission capacity, and the portions of capacity dedicated to the different classes of flows are computed from a constrained (by the overall capacities) optimization procedure which yields so-called wighted alpha-fair allocation. Several favourable properties of this policy are proved. It is assumed that some of the resources operate in heavy traffic, which means that their capacity is fully used, expressed via the constraints in the optimization procedure being saturated. From the stochastic network model a fluid limit is derived and used to study the behaviour of the active flows under a law of large numbers scaling. The set of invariant states of the fluid model (invariant manifold) is studied and several characterisations of invariant states are given. The authors discuss in some detail possible consequences for obtaining diffusion approximations for which their fluid limit results open the way.

MSC:
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
90B15 Stochastic network models in operations research
60K25 Queueing theory (aspects of probability theory)
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