  The asymmetric simple exclusion process (ASEP) is an important model from
statistical physics describing particles that hop randomly from one site to the
next along an ordered lattice of sites, but only if the next site is empty.
ASEP has been used to model and analyze numerous multiagent systems with local
interactions ranging from ribosome flow along the mRNA to pedestrian traffic.
In ASEP with periodic boundary conditions a particle that hops from the last
site returns to the first one. The mean field approximation of this model is
referred to as the ribosome flow model on a ring (RFMR). We analyze the RFMR
using the theory of monotone dynamical systems. We show that it admits a
continuum of equilibrium points and that every trajectory converges to an
equilibrium point. Furthermore, we show that it entrains to periodic transition
rates between the sites. When all the transition rates are equal all the state
variables converge to the same value. Thus, the RFMR with homogeneous
transition rates is a nonlinear consensus algorithm. We describe an application
of this to a simple formation control problem.
