The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.
