= Stationary law of a proportionally fair four-cycle
{title2=$\pi(n)\propto\binom{\sum_rn_r}{n_1+n_3}\prod_r\rho_r^{n_r}$}
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 $\mu_rn_rx_r$. The adjacent ratios of the displayed <binomial coefficient> weight cancel those intensities in <detailed balance>, with $\rho_r=\nu_r/\mu_r$.
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