Convergence analysis of the Latouche--Ramaswami algorithm for null recurrent quasi-birth-death processes (Q2784378)

The minimal nonnegative solution \(G\) of the matrix equation \(G=A_0+A_1G+A_2G^2\), where the matrices \(A_i, i=0,1,2,\) are nonnegative and \(A_0+A_1+A_2\) is stochastic, plays an important role in the study of quasi-birth-death processes (QBDs). The algorithm of \textit{G. Latouche} and \textit{V. Ramaswami} [J. Appl. Probab. 30, No.~3, 650-674 (1993; Zbl 0789.60055)] is a highly efficient algorithm for finding the matrix \(G\). The convergence of the algorithm has been shown to be quadratic for positive recurrent QBDs and for transient QBDs.NEWLINENEWLINENEWLINEIn this paper the author shows that the convergence is linear with a rate \(1/2\) for null recurrent QBDs under mild assumptions. This result explains up to some extend the experimental observation that the convergence of the algorithm is still quite fast for nearly null recurrent QBDs.