Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions
Yingjia Fu and Ruth J. Williams

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. For the journal copy, click here.