UC San Diego

We study the asymptotic behavior of solutions to a 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) α-fair policies introduced by Mo and Walrand (2000). Solutions of the fluid model are measure-valued functions of time. Under law of large numbers scaling, Gromoll and Williams (2009) proved that these solutions approximate dynamic solutions of a flow level model for congestion control in data communication networks, introduced by Massoulié and Roberts (2000). Our first result is the stability of the strictly subcritical version of this fluid model under mild assumptions. For this, using a slight generalization of a Lyapunov function proposed by Paganini et al. (2012), and taking into account that some fluid model solution components may reach zero while others are positive, we prove that the Lyapunov function composed with a subcritical fluid model solution converges to zero as time goes to infinity. Our second result is on 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 (2004), Mulvany et al. (2019) and Paganini et al. (2012). 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 Massoulié and Roberts (2000). Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing.