FLUID LIMITS FOR NETWORKS WITH BANDWIDTH
SHARING AND GENERAL DOCUMENT SIZE DISTRIBUTIONS
H. C. Gromoll and R. J. Williams
We consider a model of Internet congestion control, introduced by Roberts
and Massoulie, that represents the randomly varying
number of flows in a network where bandwidth is shared fairly among
document transfers. In contrast to an earlier work by Kelly and Williams,
the present paper allows interarrival times and document
sizes to be generally distributed, rather than exponentially distributed.
Furthermore, we allow a fairly
general class of bandwidth sharing policies that includes the weighted
alpha-fair policies of Mo and Walrand, as well as certain other utility
based scheduling policies. To describe the evolution of the system,
measure valued processes are used to keep track of the residual document
sizes of all flows through the network. We propose a fluid model (or
formal functional law of large numbers approximation) associated with
the stochastic flow level model.
Under mild conditions, we show that
the appropriately rescaled measure valued processes corresponding to a
sequence of such models (with fixed network structure) are tight, and
that any weak limit point of the sequence is almost surely a fluid model
solution. For the special case of weighted alpha-fair policies, we
also characterize the invariant states of the fluid model.
Annals of Applied Probability, 19 (2009), 243-280.
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Last updated October 27, 2007.