Shift techniques for Quasi-Birth and Death processes: canonical   factorizations and matrix equations
  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. 
Dario A. Bini, Guy Latouche, Beatrice Meini
math.NA