Generalized Markov interacting branching processes. (English) Zbl 1390.60283
Summary: 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.
MSC:
| 60J27 | Continuous-time Markov processes on discrete state spaces |
| 60J35 | Transition functions, generators and resolvents |
Keywords:
generalized Markov interacting branching process; regularity; extinction probability; mean extinction time; mean explosive time; total mean life timeReferences:
| [1] | Anderson W. Continuous-Time Markov Chains: An Applications-Oriented Approach. New York: Springer-Verlag, 1991 · Zbl 0731.60067 · doi:10.1007/978-1-4612-3038-0 |
| [2] | Asmussen S, Hering H. Branching Processes. Boston: Birkhäuser, 1983 · Zbl 0516.60095 · doi:10.1007/978-1-4615-8155-0 |
| [3] | Athreya K, Jagers P. Classical and Modern Branching Processes. Berlin: Springer, 1997 · Zbl 0855.00027 · doi:10.1007/978-1-4612-1862-3 |
| [4] | Athreya K, Ney P. Branching Processes. Berlin: Springer-Verlag, 1972 · Zbl 0259.60002 · doi:10.1007/978-3-642-65371-1 |
| [5] | Chen A Y. Uniqueness and extinction properties of generalised Markov branching processes. J Math Anal Appl, 2002, 274: 482-494 · Zbl 1010.60076 · doi:10.1016/S0022-247X(02)00251-2 |
| [6] | Chen A Y, Li J P. General collision branching processes with two parameters. Sci China Ser A, 2009, 52: 1546-1568 · Zbl 1186.60085 · doi:10.1007/s11425-009-0129-0 |
| [7] | Chen A Y, Li J P, Chen Y Q, et al. Extinction probability of interacting branching collision processes. Adv Appl Prob, 2012, 44: 226-259 · Zbl 1260.60179 · doi:10.1017/S0001867800005516 |
| [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: 275-311 · Zbl 1255.90039 · doi:10.1007/s11134-010-9194-x |
| [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: 511-531 · Zbl 1218.60070 · doi:10.1007/s11009-009-9121-y |
| [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: 267-304 · Zbl 1186.60094 · doi:10.1007/s11134-009-9162-5 |
| [11] | Chen A Y, Pollett P K, Zhang H J, et al. The collision branching process. J Appl Probab, 2004, 41: 1033-1048 · Zbl 1069.60063 · doi:10.1017/S0021900200020805 |
| [12] | Chen A Y, Renshaw E. Markov bulk-arriving queues with state-dependent control at idle time. Adv in Appl Probab, 2004, 36: 499-524 · Zbl 1046.60080 · doi:10.1017/S0001867800013586 |
| [13] | Chen R R. An extended class of time-continuous branching processes. J Appl Probab, 1997, 34: 14-23 · Zbl 0874.60078 · doi:10.1017/S0021900200100658 |
| [14] | Ezhov I I. Branching processes with group death. Theory Probab Appl, 1980, 25: 202-203 |
| [15] | Harris T. The Theory of Branching Processes. Berlin: Springer-Verlag, 1963 · Zbl 0117.13002 · doi:10.1007/978-3-642-51866-9 |
| [16] | Hunter J J. Mathematical Techniques of Applied Probability. New York: Academic Press, 1983 · Zbl 0539.60065 |
| [17] | Jagers P, Klebaner F C. Population-size-dependent and age-dependent branching processes. Stochastic Process Appl, 2000, 87: 235-254 · Zbl 1045.60090 · doi:10.1016/S0304-4149(99)00111-8 |
| [18] | Kalinkin A V. Extinction probability of a branching process with interaction of particles. Theory Probab Appl, 1982, 27: 201-205 · Zbl 0501.60087 · doi:10.1137/1127022 |
| [19] | Kalinkin A V. Markov branching processes with interaction. Russian Math Surveys, 2002, 57: 241-304 · Zbl 1042.60056 · doi:10.1070/RM2002v057n02ABEH000496 |
| [20] | Kalinkin A V. On the extinction probability of a branching process with two kinds of interaction of particles. Theory Probab Appl, 2003, 46: 347-352 · Zbl 1020.60072 · doi:10.1137/S0040585X97978981 |
| [21] | Kingman J F C. Regenerative Phenomena. London: Wiley & Sons, 1972 · Zbl 0236.60040 |
| [22] | Li J P, Chen A Y. Markov branching processes with immigration and instantaneous resurrection. Sci China Ser A, 2008, 51: 1266-1286 · Zbl 1146.60070 · doi:10.1007/s11425-008-0027-x |
| [23] | Olofsson P. The x log x condition for general branching processes. J Appl Probab, 1998, 35: 537-544 · Zbl 0926.60063 |
| [24] | Pakes A G. Extinction and explosion of nonlinear Markov branching processes. J Aust Math Soc, 2007, 82: 403-428 · Zbl 1126.60069 · doi:10.1017/S1446788700036193 |
| [25] | Sevast’yanov B A. The theory of branching random processes (in Russian). Uspekhi Mat Nauk, 1951, 6: 47-99 · Zbl 0044.33801 |
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