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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

Information

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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References

Allison, P. D. (1980). Estimation and testing for a Markov model of reinforcement. Sociol. Methods Res. 8, 434–453.10.1177/004912418000800405CrossRefGoogle Scholar
Altman, E. (2005). On stochastic recursive equations and infinite server queues. In Proceedings of IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies.10.1109/INFCOM.2005.1498355CrossRefGoogle Scholar
Asmussen, S. and Bladt, M. (1996). Renewal theory and queueing algorithms for matrix exponential distributions. In Matrix Analytic Methods in Stochastic Models, eds A. S. Alsfa and S. Chakravarty, pp. 313–341. Marcel Dekker, New York.Google Scholar
Athreya, A. B. (1969). Limit theorems for multitype continuous time Markov branching processes, II: The case of an arbitrary linear functional. Z. Wahrscheinlichkeitsth. 13, 204–214.10.1007/BF00539201CrossRefGoogle Scholar
Athreya, A. B. and Ney, P. E. (1972). Branching Processes. Springer.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Badía, F. G., Sangüesa, C. and Cha, J. H. (2018). Univariate and multivariate stochastic comparisons and ageing properties of the generalized Pólya process. J. Appl. Prob. 55, 233–253.10.1017/jpr.2018.15CrossRefGoogle Scholar
Bates, G. E. (1955). Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya scheme. Ann. Math. Statist. 26, 705–720.10.1214/aoms/1177728429CrossRefGoogle Scholar
Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Springer.Google Scholar
Cha, J. H. (2014). Characterization of the generalized Pólya process and its applications. Adv. Appl. Prob. 46, 1148–1171.10.1239/aap/1418396247CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2016). New shock models based on the generalized Polya process. Europ. J. Operat. Res. 251, 135–141.10.1016/j.ejor.2015.11.032CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2016). Justifying the Gompertz curve of mortality via the generalized Polya process of shocks. Theoret. Pop. Biol. 109, 54–62.10.1016/j.tpb.2016.03.001CrossRefGoogle ScholarPubMed
Durham, S. D. (1971). A problem concerning generalized age-dependent branching processes with immigration. Ann. Math. Statist. 42, 1121–1123.10.1214/aoms/1177693344CrossRefGoogle Scholar
Feller, W. (1943). On a general class of ‘contagious’ distributions. Ann. Math. Statist. 14, 389–400.10.1214/aoms/1177731359CrossRefGoogle Scholar
Hyrien, O., Mitov, K. V. and Yanev, N. M. (2016). Supercritical Sevastyanov branching processes with non-homogeneous Poisson immigration. In Branching Processes and their Applications (Lecture Notes Statist. 219), eds I. del Puerto et al. pp. 151–166. Springer, Cham.10.1007/978-3-319-31641-3_9CrossRefGoogle Scholar
Hyrien, O., Mitov, K. V. and Yanev, N. M. (2017). Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration. J. Appl. Prob. 54, 569–587.10.1017/jpr.2017.18CrossRefGoogle ScholarPubMed
Hyrien, O., Peslak, S. A., Yanev, N. M. and Palis, J. (2015). Stochastic modeling of stress erythropoiesis using a two-type age-dependent branching process with immigration. J. Math. Biol. 70, 1485–1521.10.1007/s00285-014-0803-xCrossRefGoogle ScholarPubMed
Greenwood, M. and Yule, G. U. (1920). An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. J. R. Statist. Soc. A 83, 255–279.10.2307/2341080CrossRefGoogle Scholar
Jeanblanc, M., Yor, M. and Chesney, M. (2009). Poisson processes and ruin theory. Chapter 8 of Mathematical Methods for Financial Markets. Springer Finance, London.10.1007/978-1-84628-737-4_8CrossRefGoogle Scholar
Konno, H. (2010). On the exact solution of a generalized Polya process. Adv. Math. Phys. 2010, 504267.10.1155/2010/504267CrossRefGoogle Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer.Google Scholar
Landriault, D., Willmot, G. E. and Xu, D. (2014). On the analysis of time dependent claims in a class of birth process claim count models. Insurance Math. Econom. 58, 168–173.10.1016/j.insmatheco.2014.07.001CrossRefGoogle Scholar
Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213.10.1016/S1007-5704(03)00037-6CrossRefGoogle Scholar
Le Gat, Y. (2014). Extending the Yule process to model recurrent pipe failures in water supply networks. Urban Water J. 11, 617–630.10.1080/1573062X.2013.783088CrossRefGoogle Scholar
Mitov, K. V., Yanev, N. M. and Hyrien, O. (2018). Multitype branching processes with inhomogeneous Poisson immigration. Adv. Appl. Prob. 50, 211–228.10.1017/apr.2018.81CrossRefGoogle Scholar
Pólya, G. (1930). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré Prob. Statist. 1, 117–161.Google Scholar
Qi, J., Ju, W. and Sun, K. (2017). Estimating the propagation of interdependent cascading outages with multi-type branching processes. IEEE Trans. Power Systems 32, 1212–1223.Google Scholar
Resing, J. A. C. (1993). Polling systems and multitype branching processes. Queueing Systems 13, 409–426.10.1007/BF01149263CrossRefGoogle Scholar
Rolski, D. T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1998). Stochastic Processes for Insurance and Finance. John Wiley.Google Scholar
Sevast’yanov, B. A. (1971). Branching Processes (in Russian). Nauka, Moscow.Google Scholar
van der Mei, R. D. (2007). Towards a unifying theory on branching-type polling systems in heavy traffic. Queueing Systems 57, 29–46.10.1007/s11134-007-9044-7CrossRefGoogle Scholar
Vatutin, V. A. (1977). A critical Bellman–Harris branching process with immigration and several types of particles. Theory Prob. Appl. 21, 435–442.10.1137/1121055CrossRefGoogle Scholar
Wasserman, S. (1983). Distinguishing between stochastic models of heterogeneity and contagion. J. Math. Psychol. 27, 201–215.10.1016/0022-2496(83)90043-3CrossRefGoogle Scholar
Weiner, H. J. (1970). On a multi-type critical age-dependent branching process. J. Appl. Prob. 7, 523–543.10.2307/3211936CrossRefGoogle Scholar
Weiner, H. J. (1972). A multi-type critical age-dependent branching process with immigration. J. Appl. Prob. 9, 697–706.10.2307/3212609CrossRefGoogle Scholar
Willmot, G. E. (2010). Distributional analysis of a generalization of the Polya process. Insurance Math. Econom. 47, 423–427.10.1016/j.insmatheco.2010.09.002CrossRefGoogle Scholar