The classification of matrix GI/M/1-type Markov chains with a tree structure and its applications to queueing

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron-Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.