  We consider a connection-level model proposed by Massouli\'{e} and Roberts
for bandwidth sharing among file transfer flows in a communication network. We
study weighted proportionally fair sharing policies and establish 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; i.e., the bounds are heavy-traffic insensitive to the
distributions in this class. Our approach is Lyapunov-drift based, which is
different from the widely used diffusion approximation approach. A key
technique we develop is to construct a novel inner product in the state space,
which then allows us to obtain a multiplicative type of state-space collapse in
steady state. Furthermore, this state-space collapse result implies the
interchange of limits as a by-product for the diffusion approximation of the
equal-weight case under phase-type file-size distributions, demonstrating the
heavy-traffic insensitivity of the stationary distribution.

Performance;Probability