  We address a conjecture introduced by Massouli\'e (2007), 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
hold for general service requirement distribution. These results comfort the
intuition that Proportional fairness is 'close' to allocations of service being
insensitive to the service time requirement.

Optimization and Control;Probability