  We discuss Stein's method for approximation by the stationary distribution of
a single-birth Markov chain, in conjunction with stochastic monotonicity and
similar assumptions. We use bounds on the increments of the solution of
Poisson's equation for such a process. Applications include rates of
convergence to stationarity, and bounding the total variation distance between
the stationary distributions of two Markov chains in the case where one
transition matrix dominates the other.
