  There has recently been considerable interest in the stability of different
fair bandwidth sharing policies for models that arise in the context of
Internet congestion control. Here, we consider a connection level model,
introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000)
185--201], that represents the randomly varying number of flows present in a
network. The weighted $\alpha$-fair and weighted max-min fair bandwidth sharing
policies are among important policies that have been studied for this model.
Stability results are known in both cases when the interarrival times and
service times are exponentially distributed. Partial results for general
service times are known for weighted $\alpha$-fair policies; no such results
are known for weighted max--min fair policies. Here, we show that weighted
max--min fair policies are stable for subcritical networks with general
interarrival and service distributions, provided the latter have $2+\delta_1$
moments for some $\delta_1>0$. Our argument employs an appropriate Lyapunov
function for the weighted max--min fair policy.
