  We consider an exhaustive polling system with three nodes in its transient
regime under a switching rule of generalized greedy type. We show that, for the
system with Poisson arrivals and service times with finite second moment, the
sequence of nodes visited by the server is eventually periodic almost surely.
To do this, we construct a dynamical system, the triangle process, which we
show has eventually periodic trajectories for almost all sets of parameters and
in this case we show that the stochastic trajectories follow the deterministic
ones a.s. We also show there are infinitely many sets of parameters where the
triangle process has aperiodic trajectories and in such cases trajectories of
the stochastic model are aperiodic with positive probability.
