A method for calculating successive approximate solutions for a class of block banded M/G/1 type Markovian models

Abstract

This paper presents an efficient equilibrium solution algorithm for a class of infinite block banded M/G/1 type Markov chains. By re-blocking the states, these are a class of the so-called quasi-birth-and-death (QBD) type chains. The proposed algorithm is not based on an iterative approach, so that the exact solution can be computed in a known finite number of steps. The key point on which the algorithm is based is the identification of a linear dependence among variables. This dependence is expressed in terms of a companion matrix. The equilibrium solution of the Markov chain is obtained operating on this matrix. An attractive feature of the algorithm is that it allows the computation of a succession of approximate solutions with growing accuracy, until the exact solution is obtained in a finite number of steps. The class of block-banded M/G/1 type Markov chains we consider requires that the lower diagonal block is invertible and that the chain is ergodic. However, many models arising from telecommunication systems satisfy this restriction. Results for a case study show that the proposed algorithm is efficient and quite accurate, even when providing approximate solutions.