  We propose a self-stabilizing leader election protocol on directed rings in
the model of population protocols. Given an upper bound $N$ on the population
size $n$, the proposed protocol elects a unique leader within $O(nN)$ expected
steps starting from any configuration and uses $O(N)$ states. This convergence
time is optimal if a given upper bound $N$ is asymptotically tight, i.e.,
$N=O(n)$.
