Proportionally fair allocation on a four-cycle

For the four routes consisting of adjacent pairs of a unit-capacity four-cycle, put $P=n_1+n_3$, $Q=n_2+n_4$, and $N=P+Q$. Every active route in the first group has total service $P/N$, and every active route in the second has total service $Q/N$. Its per-flow rate is this total divided by its active flow count.

Write $z_r=n_r{x}_r$ for aggregate route service. The four resource constraints are all sums of one odd and one even route service, so they are equivalent to the displayed maximum constraint. For fixed maxima, increasing each active service to its group's maximum improves weighted logarithmic utility. Maximizing $P\log p+Q\log q$ with $p+q\le 1$ gives $p=P/(P+Q)$ and $q=Q/(P+Q)$, where $P=n_1+n_3$, $Q=n_2+n_4$. Inactive routes are omitted; the same formula covers states where one group is empty.