The quasi-birth–death type Markov chain with a tree structure

In this paper, we extend the notion of the quasi-birth–death (QBD) Markov chain to the QBD-type Markov chain with a d-ary tree structure, and investigate various structural properties of such Markov chains. We first obtain an explicit solution for the matrices Gk, k = 1, 2, ..., d. The result is then applied to the preemptive MAP/PH/1 queue with last come first served discipline. We obtain a duality result when the QBD-type Markov chain satisfies a certain symmetrical condition. Finally, we establish a relationship between Gk and Rk, k = 1, 2, ..., d, which generalizes the previous result by Latouche and Ramaswami.