  For downward skip-free continuous-time Markov chains on non-negative integers
stopped at zero, existence of a quasi-stationary distribution is studied. The
scale function for these processes is introduced and the boundary is classified
by a certain integrability condition on the scale function, which gives an
extension of Feller's classification of the boundary for birth-and-death
processes. The existence and the set of quasi-stationary distributions are
characterized by the scale function and the new classification of the boundary.
