  In this paper we consider (upward skip-free) discrete-time and discrete-space
Markov additive chains (MACs) and develop the theory for the so-called
$\tilde{W}$ and $\tilde{Z}$ scale matrices. which are shown to play a vital
role in the determination of a number of exit problems and related fluctuation
identities. The theory developed in this fully discrete setup follows similar
lines of reasoning as the analogous theory for Markov additive processes in
continuous-time and is exploited to obtain the probabilistic construction of
the scale matrices, identify the form of the generating function and produce a
simple recursion relation for $\tilde{W}$, as well as its connection with the
so-called occupation mass formula. In addition to the standard one and
two-sided exit problems (upwards and downwards), we also derive distributional
characteristics for a number of quantities related to the one and two-sided
`reflected' processes.
