# Solving quadratic matrix equations arising in random walks in the quarter plane

## TL;DR

This paper develops efficient algorithms leveraging Toeplitz structure to solve quadratic matrix equations in queueing models, providing convergence analysis and numerical validation for computing minimal nonnegative solutions.

## Contribution

It introduces specialized fixed point iteration algorithms with acceleration strategies for quadratic matrix equations with Toeplitz structure, including a structured perturbation analysis.

## Key findings

- Algorithms effectively compute the minimal nonnegative solution G.
- Numerical experiments demonstrate the efficiency and accuracy of the proposed methods.
- Structured perturbation analysis provides insights into solution stability.

## Abstract

Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models, described by random walks in the quarter plane, the coefficients $A_1,A_0,A_{-1}$ are infinite tridiagonal matrices with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton's iteration, for the computation of $G$ and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution $G$. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported.