The Secret Recipe

$N$ people (numbered $1$ through $N$) are trying to catch Chef because he has a secret recipe for the most delicious biryani ever. Chef does not have a number. The chase starts at time $t = 0$ and takes place on a circular road with length $L$. Positions on the road are described by one real-valued coordinate; for each real number $x$, position $x+L$ is the same as position $x$. Chef's initial po