Qualitative properties of $α$-fair policies in bandwidth-sharing networks

We consider a flow-level model of a network operating under an $α$-fair bandwidth sharing policy (with $α>0$) proposed by Roberts and Massoulié [Telecomunication Systems 15 (2000) 185-201]. This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network. We study the transient properties as well as the steady-state distribution of the model. In particular, for $α\geq1$, we obtain bounds on the maximum number of flows in the network over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property for all $α\geq1$. For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $α>0$, by relying on a norm-like Lyapunov function. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009) 1719-1780], in steady state, for the case where $α=1$ and under a local traffic condition.

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