Reversible four-cycle flow model

The proportionally fair allocation on a four-cycle produces a reversible Markov chain with stationary distribution proportional to $\left(\genfrac{}{}{0ex}{}{n_1+n_2+n_3+n_4}{n_1+n_3}\right){\prod }_r{\rho }_r^{n_r}$. This is normalizable exactly when every resource's offered load is strictly below capacity, equivalently $\max ({\rho }_1,{\rho }_3)+\max ({\rho }_2,{\rho }_4)<1$.