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In this paper we consider (upward skip-free) discrete-time and discrete-spaceMarkov additive chains (MACs) and develop the theory for the so-called$\mathbf{W}$ and $\mathbf{Z}$ scale matrices. which are shown to play a vitalrole in the determination of a number of exit problems and related fluctuationidentities. The theory developed in this fully discrete setup follows similarlines of reasoning as the analogous theory for Markov additive processes incontinuous-time and is exploited to obtain the probabilistic construction ofthe scale matrices, identify the form of the generating function and produce asimple recursion relation for $\mathbf{W}$, as well as its connection with theso-called occupation mass formula. In addition to the standard one andtwo-sided exit problems (upwards and downwards), we also derive distributionalcharacteristics for a number of quantities related to the one and two-sided`reflected' processes.

Exit Times for a Discrete Markov Additive Process