Stability of a Subcritical Fluid Model for Fair Bandwidth
Sharing with General File Size Distributions
Yingjia Fu and Ruth J. Williams
Abstract
This work concerns the asymptotic behavior of solutions to a (strictly) subcritical
fluid
model for a data communication network, where file sizes are generally distributed and the
network operates under a fair bandwidth sharing policy. Here we consider fair bandwidth
sharing policies that are a slight generalization of the alpha-fair policies of Mo and Walrand (2000).
It has been a standing problem to prove stability of the data communications network
model of Massoulie and Roberts (2000), operating under fair bandwidth sharing policies, when
the offered load is less than capacity (subcritical conditions). A crucial step in an approach to
this problem is to prove stability of subcritical
fluid model solutions. Paganini et al. (2012)
introduced a Lyapunov function for this purpose and gave an argument, assuming that
fluid
model solutions are sufficiently smooth in time and space that they are strong solutions of a
partial differential equation and assuming that no
fluid level on any route touches zero before
all route levels reach zero. The aim of the current paper is to prove stability of the subcritical
fluid model without these strong assumptions.
Starting with a slight generalization of the Lyapunov function proposed by Paganini et
al. (2012), assuming that each component of the initial state of a measure-valued
fluid model
solution, as well as the file size distributions, have no atoms and have finite first moments,
we prove absolutely continuity in time of the composition of the Lyapunov function with any
subcritical
fluid model solution and describe the associated density. We use this to prove that
the Lyapunov function composed with such a subcritical
fluid model solution converges to zero
as time goes to infinity. This implies that each component of the measure-valued
fluid model
solution converges vaguely on (0, infinity) to the zero measure as time goes to infinity. Under the
further assumption that the file size distributions have finite pth moments for some p > 1 and
that each component of the initial state of the
fluid model solution has finite pth moment, it
is proved that the
fluid model solution reaches the measure with all components equal to the
zero measure in finite time and that the time to reach this zero state has a uniform bound for
all
fluid model solutions having a uniform bound on the initial total mass and the pth moment
of each component of the initial state. In contrast to the analysis in Paganini et al. (2012), we
do not need their strong smoothness assumptions on
fluid model solutions and we rigorously
treat the realistic, but singular situation, where the
fluid level on some routes becomes zero
while other route levels remain positive.
Published in Stochastic Systems, 10 (2020), 251-273.
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