Asymptotic behavior of a critical fluid model for bandwidth sharing with general file size distributions

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) α-fair policies introduced by Mo and Walrand [18]. Solutions of the fluid model are measure-valued functions of time. Under law of large numbers scaling, Gromoll and Williams [8] proved that these solutions approximate dynamic solutions of a flow level model for congestion control in data communication networks, introduced by Massoulié and Roberts [17]. In a recent work [6], 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 [14], Mulvany et al. [19] and Paganini et al. [20]. 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 [17]. Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing. ∗Research supported in part by NSF grants DMS-1206772 and DMS-1712974, and the Charles Lee Powell Foundation. A preliminary form of some of the material in this paper was featured in the Le Cam Lecture delivered by RJW at the IMS Annual Meeting held in Vilnius, Lithuania, in July 2018. †Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: yif051@ucsd.edu. ‡Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: rjwilliams@ucsd.edu. MSC2020 Mathematics Subject Classification: Primary 60F99, 60K30, 90B10; Secondary 60J25, 60K25, 90B18.