Insensitive, maximum stable allocations converge to proportional fairness

We describe a queueing model where service is allocated as a function of queue sizes. We consider allocations policies that are insensitive to service requirements and have a maximal stability region. We take a limit where the queueing model becomes congested. We study how service is allocated under this limit. We demonstrate that the only possible limit allocation is one that maximizes a proportionally fair optimization problem.