In a previous paper we demonstrated that the well known matrix-geometric solution of Quasi-Birth-and-Death processes is valid also if we introduce Rational Arrival Process (RAP) components. Here we extend those results and we offer an alternative proof by using results obtained by Tweedie. We prove the matrix-geometric form for a certain kind of operators on the stationary measure for discrete time Markov chains of GI/M/1 type. We apply this result to an embedded chain with RAP components. We then discuss the straight- forward modification of the standard algorithms for calculating the matrix R in the traditional QBD framework. Finally we present examples demonstrating great reductions in dimensionality from the traditional QBD framework to the QBD - RAP framework.