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.

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.

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.