A family of fast fixed point iterations for M/G/1-type Markov chains

TL;DR

This paper introduces a new family of fixed point iterations for efficiently computing the minimal nonnegative solution of a key matrix equation in M/G/1-type Markov chains, demonstrating faster convergence than classical methods.

Contribution

A novel family of fixed point iterations is proposed, extending classical methods with proven faster convergence for solving the matrix equation in Markov chain analysis.

Findings

New fixed point iteration family converges faster than classical methods.

Numerical experiments confirm improved efficiency.

Theoretical analysis supports convergence speed improvements.

Abstract

We consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix $G$ provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of $G$, that includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.