A fast newton's iteration for M/G/1-type and GI/M/1-type Markov Chains

In this article we revisit Newton's iteration as a method to find the G or R matrix in M/G/1-type and GI/M/1-type Markov chains. We start by reconsidering the method proposed in Ref.^[15], which required O(m ⁶+Nm ⁴) time per iteration, and show that it can be reduced to O(Nm ⁴), 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 ³+Nm ²r²+m ³r) when A 0 has rank r<m. In addition, we consider the case where [A 1^{A 2}A N] 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 ³+Nm ² r ²+mr ³) per iteration. The computational gains in all the cases are illustrated through numerical examples.