We consider a multi-class single server queueing network as a model of a
packet switching network. The rates packets are sent into this network are
controlled by queues which act as congestion windows. By considering a sequence
of such congestion windows we allow the network to become congested. We show
the stationary throughput of routes on this sequence of networks converges to
an allocation that maximizes aggregate utility subject to the network's
capacity constraints. To perform this analysis we require that our utility
functions satisfy an exponential concavity condition. This family of utilities
includes weighted $\alpha$-fair utilities for $\alpha >1$.