 Stationary law of a linear flow network

For the proportionally fair allocation on a linear flow network, independent Poisson flow arrivals at rates ${\nu }_r$ and exponential document sizes of rates ${\mu }_r$ give a continuous-time Markov chain with departures ${\mu }_r{n}_r{x}_r(n)$. Put ${\rho }_r={\nu }_r/{\mu }_r$. Under ${\rho }_0+{\rho }_i<1$ for each local route, the normalizing constant is $Z=(1-{\rho }_0{)}^{I-1}/{\prod }_i(1-{\rho }_0-{\rho }_i)$. The binomial coefficient weight has adjacent-state ratios equal to the aggregate service rates, proving detailed balance. For more general network topologies, proportional fairness need not give this reversible product form.