 Proportionally fair allocation on a linear flow network

ID: proportionally-fair-allocation-on-a-linear-flow-network

On unit-capacity links

1, \dots, I

, route zero uses every link and route

i

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_{0} x_{0} + n_{i} x_{i} = 1

. Maximizing

n_{0} lo g w_{0} + M lo g (1 - w_{0})

for

w_{0} = n_{0} x_{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.

 New to topics? Read the docs here!