β-Invariant Measures for Transition Matrices of GI/M/1 Type

Abstract

In this paper, we study transition matrices of GI/M/1 type by using the approach proposed in Li and Zhao.[13] We obtain conditions on the α-classification of states for the transition matrix of GI/M/1 type. Unlike for matrices of M/G/1 type where association of the matrix multiplication can be easily justified, for matrices of GI/M/ type, we first construct formal expressions for the β-invariant measure based on a representation of factorization of the transition matrix, and then show that it is a β-invariant measure directly. We also prove some spectral properties for the matrix of GI/M/1 type, which are not only used in constructing a formal expression for the β-invariant measure, but also of their own interest. We point out that the spectral analysis required for studying matrices of GI/M/1 type is much more sophisticated than that for matrices of M/G/1 type. Finally, we discuss connections of expressions for the β-invariant measure provided in this paper and in the literature.

Keywords:

• β-Invariant measures
• Quasi-stationary distributions
• Markov chains of GI/M/1 type
• Radius of convergence
• The RG-factorizations
• Spectral analysis

Acknowledgments

The authors thank the two referees for their valuable comments and remarks, and Dr. K. Hardy for his help in preparing the graphs in this paper, and acknowledge that this work was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). Dr. Li also acknowledges the support provided by Carleton University.