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Keywords
(7)
Branching Process
Complex System
Fixed Point Equation
Iteration Method
Matrix Analytic Method
Rate of Convergence
Relative Efficiency
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Publications
Algorithmic Approach to the Extinction Probability of Branching Processes
Algorithmic Approach to the Extinction Probability of Branching Processes,10.1007/s1100900991417,Methodology and Computing in Applied Probability,S
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Algorithmic Approach to the Extinction Probability of Branching Processes
(
Citations: 5
)
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Sophie Hautphenne
,
Guy Latouche
,
MarieAnge Remiche
The extinction probability of a
branching process
is characterized as the solution of a fixedpoint equation which, for a fairly general class of Markovian branching processes, is vector quadratic. We address the question of solving that equation, using a mixture of algorithmic and probabilistic arguments. We compare the
relative efficiency
of three iterative methods based on functional iteration, on the basis of the probabilistic interpretation of the successive iterations as well as on the basis of traditional
rate of convergence
analysis. We illustrate our findings through a few numerical examples and conclude by showing how they extend to more complex systems.
Journal:
Methodology and Computing in Applied Probability  METHODOL COMPUT APPL PROBAB
, vol. 13, no. 1, pp. 171192, 2011
DOI:
10.1007/s1100900991417
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Citation Context
(3)
...This equation relies on the assumption of independence between individuals and may be solved numerically: linear algorithms have been developed in Bean, Kontoleon, and Taylor
^{[}
^{3}
^{]}
and in Hautphenne, Latouche, and Remiche
^{[}
^{9}
^{]}
, and quadratic algorithms have been proposed in Hautphenne, Latouche, and Remiche
^{[}
^{8}
^{]}
and in Hautphenne and van Houdt
^{[}
^{12}
^{]}
...
Sophie Hautphenne
,
et al.
Markovian Trees Subject to Catastrophes: Transient Features and Extinc...
...This family of random processes has been extensively analytically characterized by Hautphenne et al. [
8
] among others...
...The proofs of these relations are given in [
8
]...
...It has been established in [
8
] that the matrix M respects the following identity:...
...In [
8
], three different algorithms are proposed to numerically evaluate q. This vector gives us the probability that the dissemination process may eventually stop at a certain time...
Kenji Leibnitz
,
et al.
Modeling the Extinction in the Information Diffusion Process in Wirele...
...Bean et al. (2008) and in
Hautphenne et al. (2010)
, or with the quadratic Newton algorithm in Hautphenne et al. (2008)...
Sophie Hautphenne
,
et al.
The Markovian binary tree applied to demography
References
(9)
Transient Markov arrival processes
(
Citations: 11
)
Guy Latouche
,
MarieAnge Remiche
,
Peter Taylor
Journal:
Annals of Applied Probability  ANN APPL PROBAB
, vol. 13, no. 2003, pp. 628640, 2003
Markovian trees: properties and algorithms
(
Citations: 9
)
Nigel G. Bean
,
Nectarios Kontoleon
,
Peter G. Taylor
Journal:
Annals of Operations Research  Annals OR
, vol. 160, no. 1, pp. 3150, 2008
Newton’s iteration for the extinction probability of a Markovian binary tree
(
Citations: 9
)
Sophie Hautphenne
,
Guy Latouche
,
MarieAnge Remiche
Journal:
Linear Algebra and Its Applications  LINEAR ALGEBRA APPL
, vol. 428, no. 11, pp. 27912804, 2008
Branching processes
(
Citations: 794
)
K. B. Athreya
,
A. N. Vidyashankar
Journal:
Handbook of Statistics
, vol. 19, pp. 3553, 2001
Nonnegative matrices in the mathematical sciences
(
Citations: 2060
)
A. Berman
,
R. J. Plemmons
Published in 1994.
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Citations
(5)
Markovian Trees Subject to Catastrophes: Transient Features and Extinction Probability
Sophie Hautphenne
,
Guy Latouche
Journal:
Stochastic Models  STOCH MODELS
, vol. 27, no. 4, pp. 569590, 2011
On the solution of a quadratic vector equation arising in Markovian Binary Trees
Dario A. Bini
,
Beatrice Meini
,
Federico Poloni
Published in 2010.
Modeling the Extinction in the Information Diffusion Process in Wireless Sensor Networks
Kenji Leibnitz
,
MarieAnge Remiche
Conference:
International Conference on Computer Communications and Networks  ICCCN
, pp. 15, 2010
A Perron iteration for the solution of a quadratic vector equation arising in Markovian Binary Trees
Beatrice Meini
,
Federico Poloni
Published in 2010.
The Markovian binary tree applied to demography
Sophie Hautphenne
,
Guy Latouche
Journal:
Journal of Mathematical Biology  J MATH BIOL
, pp. 127