  In this paper, we derive a simple drift condition for the stability of a
class of two-dimensional Markov processes, for which one of the coordinates
(also referred to as the {\em phase} for convenience) has a well understood
behaviour dependent on the other coordinate (also referred as {\em level}). The
first (phase) component's transitions may depend on the second component and
are only assumed to be eventually independent. The second (level) component has
partially bounded jumps and it is assumed to have a negative drift given that
the first one is in its stationary distribution. The results presented in this
work can be applied to processes of the QBD (quasi-birth-and-death) type on the
quarter- and on the half-plane, where the phase and level are interdependent.
Furthermore, they provide an off-the-shelf technique to tackle stability issues
for a class of two-dimensional Markov processes. These results set the stepping
stones towards closing the existing gap in the literature of deriving easily
verifiable conditions/criteria for two-dimensional processes with unbounded
jumps and interdependence between the two components.
