Asymptotic Behavior of a Critical Fluid Model for Bandwidth
Sharing with General File Size Distributions
Yingjia Fu and Ruth J. Williams
Abstract
This work concerns the asymptotic behavior of solutions to a critical
fluid model for a data
communication network, where file sizes are
generally distributed and the network operates under a fair
bandwidth sharing policy, chosen from the family of (weighted)
alpha-fair policies introduced by Mo and
Walrand. Solutions of the
fluid model are measure-valued functions of time. Under law of large
numbers scaling, Gromoll and Williams
proved that these solutions approximate dynamic solutions
of a
flow level model for congestion control in data communication networks,
introduced by Massoulie
and Roberts.
In a recent work, we proved stability of the strictly subcritical version of
this
fluid model under
mild assumptions. In the current work, we study the asymptotic behavior (as time goes to infinity) of
solutions of the critical
fluid model, in which the nominal load on each network resource is less than or
equal to its capacity and at least one resource is fully loaded. For this we introduce a new Lyapunov
function, inspired by the work of Kelly and Williams, Mulvany et al. and Paganini et al.
Using this, under moderate conditions on the file size distributions, we prove that critical
fluid model
solutions converge uniformly to the set of invariant states as time goes to infinity, when started in
suitable relatively compact sets. We expect that this result will play a key role in developing a diffusion
approximation for the critically loaded
flow level model of Massoulie and Roberts. Furthermore,
the techniques developed here may be useful for studying other stochastic network models with resource
sharing.
Published in the Annals of Applied Probability 2022, Vol. 32, No. 3, 1862-1901. The journal website where the paper is published is accessible by
clicking here.
For the final published version of the paper for scientific, non-commercial use,
click here.