Proportional fairness

A rate allocation is proportionally fair when the sum of proportional changes toward any feasible competitor is nonpositive. If route $r$ has $n_r$ identical active flows, this condition is ${\sum }_r{n}_r({\tilde{x}}_r-x_r)/x_r\le 0$ and is equivalent to maximizing weighted logarithmic utility subject to capacity constraints.

When a route has zero active flows, its term is absent from weighted logarithmic utility. Its aggregate allocation is taken to be zero; a per-flow rate on it has no operational meaning and need not be uniquely determined. One must not divide by its zero population. Empty-network and empty-group cases can therefore be treated separately while preserving uniqueness of active-flow rates.

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.

The utility ${\sum }_r{n}_r\log x_r$ sums logarithmic utilities over individual flows. Its first-order condition gives proportional fairness. Its negative is a strictly convex function on active flow rates, giving uniqueness of those rates on a convex set of feasible allocations.