Proportionally fair allocation on a two-resource linear network ($x_1=(a+b)/(aN),x_2=(a+b)/(bN),x_0=1/N$)

For local counts $a,b$, through count $c$, and $N=a+b+c>0$, proportional fairness gives aggregate through service $c/N$. Every active local route receives aggregate service $(a+b)/N$, divided equally among its active flows. Inactive routes receive zero aggregate service. This maximizes weighted logarithmic utility subject to both unit-capacity constraints.