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

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.