Structural properties of proportional fairness: stability and insensitivity

In this note we provide a novel characterization of the proportionally fair bandwidth allocation of network capacities, in terms of the Legendre-Fenchel transform of the network capacity region. We use this characterization to prove stability of network dynamics under proportionally fair sharing, by exhibiting a suitable Lyapunov function. Our stability result extends previously known results to a more general model including Markovian users routing. In particular, it implies that the stability condition previously known under exponential service time distributions remains valid under so-called phase-type service time distributions. We then exhibit a modification of proportional fairness, which coincides with it in some asymptotic sense, is reversible (and thus insensitive), and has explicit stationary distribution. Finally we show that the stationary distributions under modified proportional fairness and balanced fairness, a sharing criterion proposed because of its insensitivity properties, satisfy the same large deviations characteristics. These results give support to the choice of proportional fairness as a default bandwidth allocation criterion, combining the desirable properties of ease of implementation with performance and insensitivity.