Heavy-Traffic Insensitive Bounds for Weighted Proportionally Fair Bandwidth Sharing Policies

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.