The quasi-birth-death type markov chain with a tree structure

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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 . 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 , which generalizes the previous result by Latouche and Ramaswami