  We revisit the shift technique applied to Quasi-Birth and Death (QBD)
processes (He, Meini, Rhee, SIAM J. Matrix Anal. Appl., 2001) by bringing the
attention to the existence and properties of canonical factorizations. To this
regard, we prove new results concerning the solutions of the quadratic matrix
equations associated with the QBD. These results find applications to the
solution of the Poisson equation for QBDs.
