Prisoners, Rooms, and Light Switches

We examine a new variant of the classic prisoners and light switches puzzle: A warden leads his $n$ in out $r$ rooms, one at time, some order, with each prisoner eventually visiting every room an arbitrarily large number times. The rooms are indistinguishable, except that has $s$ switches; win their freedom if point can correctly declare been least once. What is minimum per room, $s$, such manage this? show do not know switches' starting configuration, then they have no chance escape—but sufficient surprisingly small. analysis gives rise to puzzling open questions, as well.