Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Hautphenne, Sophie; Latouche, Guy; Remiche, Marie-Ange License
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URN: urn:nbn:de:0030-drops-13935
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Matrix Analytic Methods in Branching processes

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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.


BibTeX - Entry

@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/opus/volltexte/2008/1393},
  URN =		{urn:nbn:de:0030-drops-13935},
  doi =		{10.4230/DagSemProc.07461.9},
  annote =	{Keywords: Branching Processes, Matrix Analytic Methods, Extinction Probability, Catastrophe Process}
}

Keywords: Branching Processes, Matrix Analytic Methods, Extinction Probability, Catastrophe Process
Seminar: 07461 - Numerical Methods for Structured Markov Chains
Issue date: 2008
Date of publication: 07.04.2008


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