A fast Newton’s iteration for M/G/1-type and GI/M/1-type Markov chains

In this paper we revisit Newton’s iteration as a method to find the G or R matrix in M/G/1type and GI/M/1-type Markov chains. We start by reconsidering the method proposed in [14] which required O(m +Nm) time per iteration, and show that it can be reduced to O(Nm), where m is the block size and N the number of blocks. Moreover, we show how this method is able to further reduce this time complexity to O(Nr + Nmr + mr) when A0 has rank r < m. In addition, we consider the case where [A1A2 . . . AN ] is of rank r < m and propose a new Newton’s iteration method which is proven to converge quadratically and that has a time complexity of O(Nm +Nmr +mr) per iteration. The computational gains in all the cases are illustrated through numerical examples.