  We consider the problem of stochastic flow of multiple particles traveling on
a closed loop, with a constraint that particles move without passing. We use a
Markov chain description that reduces the problem to a generalized random walk
on a hyperplane (with boundaries). By expressing positions via a moving
reference frame, the geometry of the no-passing criteria is greatly simplified,
with the resultant condition expressible as the coordinate system planes which
bound the first orthant. To determine state transition probabilities, we
decompose transitions into independent events and construct a digraph
representation in which calculating transition probability is reduced to a
shortest path determination on the digraph. The resultant decomposition digraph
is self-converse, and we exploit that property to establish the necessary
symmetries to find the stationary density for the process.
