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On the link between Markovian trees and tree-structured Markov chains

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  • Hautphenne, Sophie
  • Houdt, Benny Van

Abstract

In this paper, we describe a link between Markovian binary trees (MBT) and tree-like quasi-birth-and-death processes (TLQBD) by associating a specific TLQBD to each MBT. The algorithms to compute the matrices Gk in the TLQBD then correspond to the algorithms calculating the extinction probability vector of the MBT. This parallelism leads to a new quadratic algorithm, based on the Newton iteration method, which converges to the extinction probability of an MBT. We also present a one-to-one correspondence between a general Markovian tree (GMT) and a scalar tree-structured M/G/1-type Markov chain. This allows us to prove the equivalence between the main result on the positive recurrence, null recurrence or transience of a scalar tree-structured M/G/1-type Markov chain and the criticality of a GMT.

Suggested Citation

  • Hautphenne, Sophie & Houdt, Benny Van, 2010. "On the link between Markovian trees and tree-structured Markov chains," European Journal of Operational Research, Elsevier, vol. 201(3), pages 791-798, March.
    • Handle: RePEc:eee:ejores:v:201:y:2010:i:3:p:791-798
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    References listed on IDEAS

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    1. Nigel Bean & Nectarios Kontoleon & Peter Taylor, 2008. "Markovian trees: properties and algorithms," Annals of Operations Research, Springer, vol. 160(1), pages 31-50, April.
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