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

Raymond W. Yeung · Attahiru Sule Alfa

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Advanced Queuing Theory Analysis Probability and Statistical Research Petri Nets in System Modeling

Abstract

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

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References

1993A logarithmic reduction algorithm for quasi-birth-death processes 1990A duality theorem for the matrix paradigms in queueing theory 1994Matrix product-form solutions for Markov chains with a tree structure 1995A generalization of the matrix M/G/l paradigm for Markov chains with a tree structure

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