@David: I do not think this falls under ASEP, but then again, I have not studied them. But you are correct in that the idea of looking for different rates in distinct regions stems from ASEP.

@JDH, a deterministic model is not stochastic in my opinion, so the counter example is not really a counter example I think. The counter examples are deterministic, and its really difficult to talk about what happens if you move from those cases. As for whether it is synchronized, yes, they all move at once. (coins are tossed prior for all people who can move, and if they are granted permission, all advance at the same time.)

NOTE: I have been very vague about the definition of "section", I am open to suggestions about that.

Erm, think about it as a finite loop that has spots added to it. We are in the limiting case when there are infinitely many spots.