  Dating from the work of Neuts in the 1980s, the field of matrix-analytic
methods has been developed to analyse discrete or continuous-time Markov chains
with a two-dimensional state space in which the increment of a level variable
is governed by an auxiliary phase variable. More recently, matrix-analytic
techniques have been applied to general Markov additive models with a finite
phase space. The basic assumption underlying these developments is that the
process is skip-free (in the case of QBDs or fluid queues) or that it is
one-sided, that is it is jump-free in one direction.
  From the Markov additive perspective, traditional matrix-analytic models can
be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments
in the level are constrained to be lattice random variables and for fluid
queues, they have to be piecewise linear.
  In this paper we discuss one-sided lattice and non-lattice Markov additive
processes in parallel. Results that are standard in one tradition are
interpreted in the other, and new perspectives emerge. In particular, using
three fundamental matrices, we address hitting, two-sided exit, and creeping
probabilities.
