Qualitative properties of .-fair policies in bandwidth-sharing networks

We consider a flow-level model of a network operating under an .-fair bandwidth sharing policy (with .>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 ..1 , 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 ..1 . For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any .>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 .=1 and under a local traffic condition.