= Proportionally fair allocation on a linear flow network
{title2=$x_0=1/N,\quad x_i=M/(Nn_i)$}
On unit-capacity links $1,\ldots,I$, route zero uses every link and route $i$ uses only link $i$. Let $N=\sum_{r=0}^I n_r$ and $M=\sum_{i=1}^I n_i$. Under <proportional fairness>, every active local route has $n_0x_0+n_ix_i=1$. Maximizing $n_0\log w_0+M\log(1-w_0)$ for $w_0=n_0x_0$ gives $w_0=n_0/N$ and the displayed per-flow rates on active routes. At the empty state departures vanish, and rates of absent flows may be set to zero.
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