We consider a flow-level model of a network operating under an α- fair bandwidth sharing policy (with α > 0) proposed by Roberts and Massouli´e [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 ≥ 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 α ≥ 1. 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.