Max-min is a frequently praised criterion for flow control despite its
limitations. In practice, the fast rate of changes in the connection
lay-out in modern networks makes it hard for any flow control algorithm
to converge to an optimal point. It such an environment, it might be
better to trade accuracy with speed.
We present algorithms that relax the optimality criterion of the
max-min flow fairness but achieve a fast convergence time that is
logarithmic in the link bandwidth and not a function of the number of
active connections (or sessions). The relaxation we use requires rates
to be increased or decreased by a certain factor, $1+\varepsilon$, or
in other words, assigned rates can only be a natural power of some
basic bandwidth $1+\varepsilon$. Under this criterion, the quiescent
time of our flow control algorithms is $\log_{1+\varepsilon}B$, where
$B$ is the maximum link bandwidth in minimum allocation units. This is
a great improvement over the super-linear quiescent time of known
algorithms both exact and approximated.