Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish

Abstract

In this paper, we discuss the use of spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth–death processes for the case where some eigenvalues are zero. Since this leads to multiple eigenvalues at zero, a difficult problem to analyze, we suggest to eliminate such eigenvalues. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The method is demonstrated by two examples, one dealing with a tandem queue, the other one with a shorter queue problem.