Before revision:  However, except for the case where the weights of all the routes are equal, the steady-state distribution of the limiting diffusion process is unknown and thus there are no explicit-form characterizations, even when exponential file size distributions are assumed. For more general file size distributions, the diffusion approximation was derived for the equal-weights case by Vlasiou, Zhang and Zwart (2014), but an interchange-of-limits result was lacking. We take a Lyapunov-drift-based approach that is different from the diffusion approximation approach, where we directly analyze the steady state of the system. We first establish a state-space collapse result in steady state, and then obtain explicit-form bounds on the weighted sum of the expected numbers of flows on different routes , where the weights are the same as those used in the weighted proportionally fair sharing policy.
After revision:  However, except for the case where the weights of all the routes are equal, the steady-state distribution of the limiting diffusion process is unknown and thus there are no explicit-form bounds on the weighted sum of the expected numbers of flows on different routes , where the weights are the same as those used in the weighted proportionally fair sharing policy.