We address a conjecture introduced by Massoulié (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.
In this short article we show how the techniques presented in [9] can be extended to a variety of non continuous and multivariate processes. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein-Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among others gaussian processes.
It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, that is, Burke's theorem in this context. In this short note we prove convergence to this invariant measure: if we have an arbitrary continuous process satisfying some mild conditions as initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.
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