 Stationary law of a proportionally fair four-cycle

The proportionally fair allocation on a four-cycle has aggregate service $P/(P+Q)$ on each active odd route and $Q/(P+Q)$ on each active even route. With Poisson arrival rates ${\nu }_r$ and exponential document-size parameters ${\mu }_r$, the departure intensity is ${\mu }_r{n}_r{x}_r$. The adjacent ratios of the displayed binomial coefficient weight cancel those intensities in detailed balance, with ${\rho }_r={\nu }_r/{\mu }_r$.