Heavy-Traffic Insensitive Bounds for Weighted Proportionally Fair Bandwidth Sharing Policies
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We consider a connection-level model proposed by Massoulié and Roberts for bandwidth sharing among file transfer flows in a communication network, and we study weighted proportionally fair sharing policies where the weights represent the relative importance of flows on different routes. We are interested in characterizing performance in the heavy-traffic regime. Existing work on this problem has focused on diffusion approximations, which were first studied by Kang et al. (2009). 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. Our bounds hold for a class of phase-type file size distributions; i.e., the bounds are heavy-traffic insensitive to the distributions in this class. For the equal-weights case, the upper and lower bounds coincide, which implies the heavy-traffic insensitivity of the expectation of the total number of flows. Furthermore, our state-space collapse result implies an interchange of limits as a by-product for the diffusion approximation by Vlasiou, Zhang and Zwart (2014).
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