Convergence analysis of the Latouche-Ramaswami algorithm or null recurrent quasi-birth-death processes

The minimal nonnegative solution G of the matrix equation G = A(0) + A(1)G + A(2)G(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 Latouche Ramaswami algorithm is a highly e cient algorithm for nding the matrix G. The convergence of the algorithm has been shown to be quadratic for positive recurrent QBDs and for transient QBDs. In this paper, we show that the convergence of the algorithm is linear with rate 1/2 for null recurrent QBDs under mild assumptions. This new result explains the experimental observation that the convergence of the algorithm is still quite fast for nearly null recurrent QBDs.