Large Deviations for the Stationary Measure of Networks Under Proportional Fair Allocations

We address a conjecture introduced by Massoulie [Massoulie L (2007) Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab. 17(3):809–839], concerning the large deviations of the stationary measure of bandwidth-sharing networks functioning under the proportional fair allocation. For Markovian networks, we prove that proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. For monotone networks, we give a more direct proof of the same result, relying on stochastic comparisons, that holds for general service time distribution. These results support the intuition that proportional fairness is “close” to allocations of service being insensitive to the service time distribution.

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