Limiting Distributions and Regeneration Times for Multitype Branching Processes with Immigration in a Random Environment

Key, Eric S.

doi:10.1214/aop/1176992273

Cited by 56 publications

(78 citation statements)

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“…To obtain these results, we use some general statements on Markov processes (Section 3.2), classical arguments for Galton-Watson processes with immigration (see [17], which was inspired by [1]), and the tail of the time when a branching process in a random environment with immigration returns to 0 in the subcritical case, which was proved in [13].…”

Section: Resultsmentioning

confidence: 99%

“…when the process returns to 0 in the subcritical case, which was proved in [13], and we use the results of Section 3.2 in the critical case.…”

Section: Branching Processes In a Random Environment With Immigrationmentioning

confidence: 99%

“…The subcritical case with the assumption that E(log + (Y )) < ∞ is handled in [13]: the first part of (i) is [13,Theorem 3.3] and (ii) is a consequence of [13,Theorem 4.2].…”

Section: Proposition 1 (I)mentioning

confidence: 99%

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Cell contamination and branching processes in a random environment with immigration

Bansaye

2009

Adv. Appl. Probab.

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We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the cell population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law governing the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We first determine the asymptotic behavior of branching processes in a random environment with state-dependent immigration, which gives the convergence in distribution of the number of parasites in a cell line. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in a random environment and laws of large numbers for a Markov tree.Keywords: Branching processes in a random environment with immigration; Markov chain indexed by a tree; empirical measure; renewal theorem 2000 Mathematics Subject Classification: Primary 60J80; 60J85; 60K37; 92C37; 92D25; 92D30

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Section: Resultsmentioning

confidence: 99%

“…when the process returns to 0 in the subcritical case, which was proved in [13], and we use the results of Section 3.2 in the critical case.…”

Section: Branching Processes In a Random Environment With Immigrationmentioning

confidence: 99%

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Cell contamination and branching processes in a random environment with immigration

Bansaye

2009

Adv. Appl. Probab.

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“…MBPRE were analyzed in [6] and [40] and MBPIRE were considered by [32] and [37]. MBPFPRE were introduced in [16] to study the tail distribution of busy periods of the branching type polling systems with zero switch-over times.…”

Section: Introductionmentioning

confidence: 99%

Multitype branching processes with immigration in random environment, and polling systems

Vatutin

2011

Sib. Adv. Math.

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For multitype branching processes with immigration evolving in a random environment and producing a final product we find the tail distribution of the size of the final product accumulated in the system for a life period. Using this result we investigate the tail distribution of the busy periods of the branching type polling systems with random service disciplines and random positive switch-over times.

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“…For example, conditions are presented for the extinction when the random environment is stationary ergodic. The stability, strong law of large numbers and central limit theorems for multitype branching processes with immigration in a random environment have been studied in [18,24]. These processes find applications in very diverse fields, including biological systems and queueing theory.…”

Section: Introductionmentioning

confidence: 99%

Applying Branching Processes to Delay-Tolerant Networks

Fiems

Altman

2010

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

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Abstract. Mobility models that have been used in the past to study delay tolerant networks (DTNs) have been either too complex to allow for deriving analytical expressions for performance measures, or have been too simplistic. In this paper we identify several classes of DTNs where the dynamics of the number of nodes that have a copy of some packet can be modelled as branching process with migration. Using recent results on such processes in a random environment, we obtain explicit formulae for the first two moments of the number of copies of a file that is propagated in the DTN, for quite general mobility models. Numerical examples illustrate our approach.

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Limiting Distributions and Regeneration Times for Multitype Branching Processes with Immigration in a Random Environment

Cited by 56 publications

References 0 publications

Cell contamination and branching processes in a random environment with immigration

Cell contamination and branching processes in a random environment with immigration

Multitype branching processes with immigration in random environment, and polling systems

Applying Branching Processes to Delay-Tolerant Networks

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