Applied Mathematics Colloquium: Ms Yingjia Fu
University of California, San Diego
In a recent work, we proved stability of the strictly subcritical version of this fluid model under mild assumptions. In this talk, 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 Massoulié and Roberts. Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing.