Informatics and Applications
2018, Volume 12, Issue 2, pp 35-43
MAXIMAL BRANDING PROCESSES IN RANDOM ENVIRONMENT
Abstract
The work continues the author's long research in the theory of maximal branching processes that are obtained from classical branching processes by replacing the sum of offsping numbers by the maximum. One can say that the next generation is formed by the offspring of the most productive particle. Earlier, the author generalized processes with integer values up to processes with arbitrary nonnegative values, investigated their properties, and proved the limit theorems. Further, maximal branching processes with several types ofparticles were introduced and studied. In this paper, the author introduces the concept of maximal branching processes in random environment (with one type of particles) and an important case of the "power" random environment. In the latter case, the basic properties of maximal branching processes are studied and the ergodic theorem is proved. As an application, the author considers gated infinite-server queues.
[+] References (21)
- Harris, T. 1963. The theory of branching processes. Berlin: Springer-Verlag. 230 p.
- Lamperti, J. 1970. Maximalbranching processes and long- range percolation. J. Appl. Probab. 7(1): 89-96.
- Lamperti, J. 1972. Remarks on maximal branching pro-cesses. Theor. Probab. Appl. 17(1): 44-53.
- Vatutin, V.A., and A. M. Zubkov. 1993. Branching pro-cesses. II. J. Sov. Math. 67(6):3407-3485.
- Lebedev, A. V. 2006. Maximal branching processes with nonnegative values. Theor. Probab. Appl. 50(3):482-488.
- Lebedev, A. V. 2009. Maksimal'nye vetvyashchiesya prot- sessy [Maximal branching processes]. Sovremennyeprob- lemy matematiki i mekhaniki [Modern Problems of Math-ematics and Mechanics] 4(1):93-106.
- Lebedev, A. V. 2012. Maximal branching processes with several types ofparticles. Mosc. Univ. Math. Bull. 67(3):97- 101.
- Lebedev, A. V. 2016. Neklassicheskie zadachi stokhasti- cheskoy teorii ekstremumov [Nonclassical problems of extreme value theory]. Moscow. DSc Diss. 257 p.
- Aydogmus, O., A. P. Ghosh, S. Ghosh, and A. Roiter- shtein. 2015. Colored maximal branching process. Theor. Probab. Appl. 59(4): 663-672.
- Browne, S., E.G. Coffman, E.N. Gilbert, and P. E. Wright. 1992. Gated, exhaustive, parallel service. Probab. Eng. Inform. Sc. 2(2): 217-239.
- Browne, S., E. G. Coffman, E. N. Gilbert, and P. E. Wright. 1992. The gated infinite-server queue: Uniform service times. SIAM J. Appl. Math. 52(6):1751-1762.
- Tan, X., and Ch. Knessl. 1994. Heavy traffic asymptotics for a gated, infinite-server queue with uniform service times. SIAMJ. Appl. Math. 54(6):1768-1779.
- Pinotsi, D., and M.A. Zazanis. 2004. Stability conditions for gated M|G|to queues. Probab. Eng. Inform. Sc. 18(1):103-110.
- Lebedev, A. V. 2003. The gated infinite-server queue with unbounded service times and heavy traffic. Probl. Inf. Transm. 39(3):309-316.
- Lebedev, A. V. 2004. Gated infinite-server queue with heavy traffic andpowertail. Probl. Inf. Transm. 40(3):237- 242.
- Gorbunova, A. V., I. S. Zaryadov, S. I. Matyushenko, K. E. Samouylov, and S.Ya. Shorgin. 2015. Approksi- matsiya vremeni otklika sistemy oblachnykh vychisleniy [The approximation of responce time of a cloud computing system]. Informatika i ee Primeneniya - Inform. Appl. 9(4):32-38.
- Jirina, M. 1958. Stochastic branching processes with con-tinuous state space. Chech. Math. J. 8(2):292-313.
- Borovkov, A. A. 1998. Ergodicity and stability of stochastic processes. Wiley. 618 p.
- Esary, J., F. Prochan, and D. Walkup. 1967. Association of random variables with applications. Ann. Math. Stat. 38(5):1466-1474.
- Bulinski, A. V., and A. P. Shashkin. 2007. Limit theorems for associated random fields and related systems. World Scientific Publishing. 448 p.
- Bhattacharya, R. N., and C. Lee. 1995. Ergodicity of non-linear first order autoregressive models. J. Theor. Probab. 8(1):207-219.
[+] About this article
Title
MAXIMAL BRANDING PROCESSES IN RANDOM ENVIRONMENT
Journal
Informatics and Applications
2018, Volume 12, Issue 2, pp 35-43
Cover Date
2018-05-30
DOI
10.14357/19922264180206
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
maximal branching processes; random environment; ergodic theorem; stable distributions; extreme value theory
Authors
A. V. Lebedev 
Author Affiliations
Department of Probability Theory, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Main Building, 1 Leninskiye Gory, Moscow 119991, Russian Federation
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