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We consider a flow-level model of a network operating under an $\alpha$-fairbandwidth sharing policy (with $\alpha>0$) proposed by Roberts andMassouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is aprobabilistic model that captures the long-term aspects of bandwidth sharingbetween users or flows in a communication network. We study the transientproperties as well as the steady-state distribution of the model. Inparticular, for $\alpha\geq1$, we obtain bounds on the maximum number of flowsin the network over a given time horizon, by means of a maximal inequalityderived from the standard Lyapunov drift condition. As a corollary, weestablish the full state space collapse property for all $\alpha\geq1$. For thesteady-state distribution, we obtain explicit exponential tail bounds on thenumber of flows, for any $\alpha>0$, by relying on a norm-like Lyapunovfunction. As a corollary, we establish the validity of the diffusionapproximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009)1719-1780], in steady state, for the case where $\alpha=1$ and under a localtraffic condition.

Qualitative properties of $\alpha$-fair policies in bandwidth-sharing networks