= Proportionally fair allocation on a two-resource linear network
{title2=$x_1=(a+b)/(aN),\quad x_2=(a+b)/(bN),\quad x_0=1/N$}
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.
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