A computational framework for two-dimensional random walks with restarts

09/25/2019
by   Dario A. Bini, et al.
0

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. Finding the steady state probability distribution of the process requires to perform operations involving these structured matrices. We propose an extension of the framework of [5] which allows to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. Algebraically, this gives rise to corrections with infinite support that cannot be treated using the tools currently available in the literature. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional Quasi-Birth-Death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. This methodology reduces the problem to solving a quadratic matrix equation with coefficients of infinite size. We provide conditions on the transition probabilities which ensure that the solution of interest of the matrix equation belongs to the enriched algebra. The reliability of our approach is confirmed by extensive numerical experimentation on some case studies.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/23/2019

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

Quadratic matrix equations of the kind A_1X^2+A_0X+A_-1=X are encountere...
research
07/05/2019

Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

We consider a class of linear matrix equations involving semi-infinite m...
research
12/19/2022

A defect-correction algorithm for quadratic matrix equations, with applications to quasi-Toeplitz matrices

A defect correction formula for quadratic matrix equations of the kind A...
research
07/24/2021

Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

Markov-modulated Lévy processes lead to matrix integral equations of the...
research
02/08/2021

Geometric means of quasi-Toeplitz matrices

We study means of geometric type of quasi-Toeplitz matrices, that are se...
research
09/16/2019

Hipster random walks

We introduce and study a family of random processes on trees we call hip...

Please sign up or login with your details

Forgot password? Click here to reset