Truncation and augmentation of level-independent QBD processes

In the study of quasi-birth-and-death (QBD) processes, the first passage probabilities from states in level one to the boundary level zero are of fundamental importance. These probabilities are organized into a matrix, usually denoted by G.The matrix G is the minimal nonnegative solution of a matrix quadratic equation. If the QBD process is recurrent, then G is stochastic. Otherwise, G is sub-stochastic and the matrix equation has a second solution G_sto, which is stochastic. In this paper, we give a physical interpretation of G_sto in terms of sequences of truncated and augmented QBD processes.As part of the proof of our main result, we derive expressions for the first passage probabilities that a QBD process will hit level k before level zero and vice versa, which are of interest in their own right.The paper concludes with a discussion of the stability of a recursion naturally associated with the matrix equation which defines G and G_sto. In particular, we show that G is a stable equilibrium point of the recursion while G_sto is an unstable equilibrium point if it is different from G.