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Multitype branching process with non-homogeneous Poisson and contagious Poisson immigration

Part of: Operations research and management science Markov processes Special processes

Published online by Cambridge University Press:  22 November 2021

Landy Rabehasaina Open the ORCID record for Landy Rabehasaina [Opens in a new window]  and
Jae-Kyung Woo
Landy Rabehasaina*
Affiliation:
University Bourgogne Franche-Comté
Jae-Kyung Woo*
Affiliation:
University of New South Wales
*
*Postal address: Laboratory of Mathematics of Besançon, University Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon CEDEX, France. Email address: lrabehas@univ-fcomte.fr
**Postal address: School of Risk and Actuarial Studies, University of New South Wales, Sydney, Australia. Email address: j.k.woo@unsw.edu.au
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Abstract

In a multitype branching process, it is assumed that immigrants arrive according to a non-homogeneous Poisson or a contagious Poisson process (both processes are formulated as a non-homogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time t for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, we provide some transient expectation results when there are only two types of particles.

Keywords

Multitype branching process with immigrationnon-homogeneous Poisson processcontagious Poisson processconvergence in distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K25: Queueing theory 90B15: Network models, stochastic
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1007 - 1042
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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