We consider a bivariate Markov process {(
U(
t),
S(
t));
t ≥ 0 }, where
U(
t) (
t ≥ 0) takes values in [0, ∞) and
S(
t) (
t ≥ 0) takes values in a finite set. We assume that
U(
t) (
t ≥ 0) is skip-free to the left, and therefore we call it the M/G/1-type Markov process. The M/G/1-type Markov process was first introduced as a generalization of the workload process in the MAP/G/1 queue and its stationary distribution was analyzed under a strong assumption that the conditional infinitesimal generator of the underlying Markov chain
S(
t) given
U(
t) > 0 is irreducible. In this paper, we extend known results for the stationary distribution to the case that the conditional infinitesimal generator of the underlying Markov chain given
U(
t) > 0 is reducible. With this extension, those results become applicable to the analysis of a certain class of queueing models.
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