Network stability under max--min fair bandwidth sharing

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.