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 2 r 2 + m 3 r) 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.

• Fecha de publicación: 2012
• Institución: Universidad del Rosario
• Repositorio: Repositorio EdocUR - U. Rosario
• Idioma: eng
• OAI Identifier: oai:repository.urosario.edu.co:10336/27431
• Acceso en línea: https://doi.org/10.1080/15326349.2012.726038; https://repository.urosario.edu.co/handle/10336/27431
• Palabra clave: Markov chains; Matrix-anaytic methods; Newton iteration; Numerical methods for Markov chains
• License: Restringido (Acceso a grupos específicos)