  We consider discrete-space continuous-time Markov models of reaction networks
and provide sufficient conditions for the following stability condition to
hold: each state in a closed, irreducible component of the state space is
positive recurrent; moreover the time required for a trajectory to enter such a
component has finite expectation. The provided analytical results depend solely
on the underlying structure of the reaction network and not on the specific
choice of model parameters. Our main results apply to binary systems and our
main analytical tool is the "tier structure" previously utilized successfully
in the study of deterministic models of reaction networks.
