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Matrix Analytic Methods in Branching processes

Authors: Sophie Hautphenne, Guy Latouche, and Marie-Ange Remiche

Published in: Dagstuhl Seminar Proceedings, Volume 7461, Numerical Methods for Structured Markov Chains (2008)

Abstract

We examine the question of solving the extinction probability of a particular class of continuous-time multi-type branching processes, named Markovian binary trees (MBT). The extinction probability is the minimal nonnegative solution of a fixed point equation that turns out to be quadratic, which makes its resolution particularly clear. We analyze first two linear algorithms to compute the extinction probability of an MBT, of which one is new, and, we propose a quadratic algorithm arising from Newton's iteration method for fixed-point equations. Finally, we add a catastrophe process to the initial MBT, and we analyze the resulting system. The extinction probability turns out to be much more difficult to compute; we use a $G/M/1$-type Markovian process approach to approximate this probability.

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Sophie Hautphenne, Guy Latouche, and Marie-Ange Remiche. Matrix Analytic Methods in Branching processes. In Numerical Methods for Structured Markov Chains. Dagstuhl Seminar Proceedings, Volume 7461, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{hautphenne_et_al:DagSemProc.07461.9,
  author =	{Hautphenne, Sophie and Latouche, Guy and Remiche, Marie-Ange},
  title =	{{Matrix Analytic Methods in Branching processes}},
  booktitle =	{Numerical Methods for Structured Markov Chains},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7461},
  editor =	{Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07461.9},
  URN =		{urn:nbn:de:0030-drops-13935},
  doi =		{10.4230/DagSemProc.07461.9},
  annote =	{Keywords: Branching Processes, Matrix Analytic Methods, Extinction Probability, Catastrophe Process}
}

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Matrix Analytic Methods in Branching processes

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