On the Solution to a Countable System of Equations Arising in Stochastic Processes

TL;DR

This paper introduces a fast, exact method for solving countable systems of equations common in stochastic processes, improving computational efficiency over existing techniques, and also provides a way to determine the matrix eigenvalues.

Contribution

The paper presents a novel method for efficiently computing the inverse of infinite or finite matrices in stochastic systems, along with a procedure to find their eigenvalues.

Findings

Provides a fast, exact inverse computation method for countable matrices

Applicable to Markov, birth-death, and inventory process models

Enhances computational efficiency over existing inverse techniques

Abstract

In this paper we develop a method to compute the solution to a countable (finite or infinite) set of equations that occurs in many different fields including Markov processes that model queueing systems, birth-and-death processes and inventory systems. The method provides a fast and exact computation of the inverse of the matrix of the coefficients of the system. In contrast, alternative inverse techniques perform much slower and work only for finite size matrices. Furthermore, we provide a procedure to construct the eigenvalues of the matrix under consideration.