Science China Mathematics

, Volume 61, Issue 3, pp 545–562 | Cite as

Generalized Markov interacting branching processes



We consider a very general interacting branching process which includes most of the important interacting branching models considered so far. After obtaining some key preliminary results, we first obtain some elegant conditions regarding regularity and uniqueness. Then the extinction vector is obtained which is very easy to be calculated. The mean extinction time and the conditional mean extinction time are revealed. The mean explosion time and the total mean life time of the processes are also investigated and resolved.


generalized Markov interacting branching process regularity extinction probability mean extinction time mean explosive time total mean life time 


60J27 60J35 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 11371374 and 11571372) and Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110162110060).


  1. 1.
    Anderson W. Continuous-Time Markov Chains: An Applications-Oriented Approach. New York: Springer-Verlag, 1991CrossRefMATHGoogle Scholar
  2. 2.
    Asmussen S, Hering H. Branching Processes. Boston: Birkhäuser, 1983CrossRefMATHGoogle Scholar
  3. 3.
    Athreya K, Jagers P. Classical and Modern Branching Processes. Berlin: Springer, 1997CrossRefMATHGoogle Scholar
  4. 4.
    Athreya K, Ney P. Branching Processes. Berlin: Springer-Verlag, 1972CrossRefMATHGoogle Scholar
  5. 5.
    Chen A Y. Uniqueness and extinction properties of generalised Markov branching processes. J Math Anal Appl, 2002, 274: 482–494MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen A Y, Li J P. General collision branching processes with two parameters. Sci China Ser A, 2009, 52: 1546–1568MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen A Y, Li J P, Chen Y Q, et al. Extinction probability of interacting branching collision processes. Adv Appl Prob, 2012, 44: 226–259MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen A Y, Li J P, Hou Z T, et al. Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues. Queueing Syst, 2010, 66: 275–311MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen A Y, Pollett P, Li J P, et al. Uniqueness, extinction and explosivity of generalised Markov branching processes with pairwise interaction. Methdol Comput Appl Probab, 2010, 12: 511–531MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen A Y, Pollett P, Li J P, et al. Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst, 2010, 64: 267–304MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen A Y, Pollett P K, Zhang H J, et al. The collision branching process. J Appl Probab, 2004, 41: 1033–1048MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chen A Y, Renshaw E. Markov bulk-arriving queues with state-dependent control at idle time. Adv in Appl Probab, 2004, 36: 499–524MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen R R. An extended class of time-continuous branching processes. J Appl Probab, 1997, 34: 14–23MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ezhov I I. Branching processes with group death. Theory Probab Appl, 1980, 25: 202–203Google Scholar
  15. 15.
    Harris T. The Theory of Branching Processes. Berlin: Springer-Verlag, 1963CrossRefMATHGoogle Scholar
  16. 16.
    Hunter J J. Mathematical Techniques of Applied Probability. New York: Academic Press, 1983MATHGoogle Scholar
  17. 17.
    Jagers P, Klebaner F C. Population-size-dependent and age-dependent branching processes. Stochastic Process Appl, 2000, 87: 235–254MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kalinkin A V. Extinction probability of a branching process with interaction of particles. Theory Probab Appl, 1982, 27: 201–205MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kalinkin A V. Markov branching processes with interaction. Russian Math Surveys, 2002, 57: 241–304MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kalinkin A V. On the extinction probability of a branching process with two kinds of interaction of particles. Theory Probab Appl, 2003, 46: 347–352MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kingman J F C. Regenerative Phenomena. London: Wiley & Sons, 1972MATHGoogle Scholar
  22. 22.
    Li J P, Chen A Y. Markov branching processes with immigration and instantaneous resurrection. Sci China Ser A, 2008, 51: 1266–1286MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Olofsson P. The x log x condition for general branching processes. J Appl Probab, 1998, 35: 537–544MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pakes A G. Extinction and explosion of nonlinear Markov branching processes. J Aust Math Soc, 2007, 82: 403–428MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sevast’yanov B A. The theory of branching random processes (in Russian). Uspekhi Mat Nauk, 1951, 6: 47–99Google Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  3. 3.Department of Mathematical SciencesThe University of LiverpoolLiverpoolUK

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