  We consider a flow-level model of a network operating under an $\alpha$-fair
bandwidth sharing policy (with $\alpha>0$) 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 $\alpha\geq1$, 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 $\alpha\geq1$. For the
steady-state distribution, we obtain explicit exponential tail bounds on the
number of flows, for any $\alpha>0$, 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 $\alpha=1$ and under a local
traffic condition.
