Before revision:  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. Our bounds hold for a class of phase-type file size distributions;
After revision:  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 in heavy traffic. The bounds are linear in the number of critically loaded links in the network, and they hold for a class of phase-type file size distributions;