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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Catalytic branching random walk and queueing systems with random number of independent servers

Authors: V. A. Vatutin, V. A. Topchii and E. B. Yarovaya
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 69 (2003).
Journal: Theor. Probability and Math. Statist. 69 (2004), 1-15
MSC (2000): Primary 60J80, 60J15; Secondary 60K25, 60K05
Published electronically: February 7, 2005
MathSciNet review: 2110900
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Abstract: A continuous time branching random walk on the lattice $\mathbf{Z}$ in which particles may produce children only at the origin is considered. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical, we find the asymptotic behavior of the survival probability of the process at time $t$ as $t\to\infty $ and the probability that the number of particles at the origin at time $t$ is positive. We also prove a Yaglom type conditional limit theorem for the total number of particles existing at time $t$. A relation between the model considered and a queueing system with a random number of independently operating servers is discussed.

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Additional Information

V. A. Vatutin
Affiliation: Steklov Mathematical Institute, Gubkina Street 8, 117966, GSP-1 Moscow, Russia

V. A. Topchii
Affiliation: Omsk Branch of Sobolev Institute of Mathematics, Pevtsova Street 13, 644099 Omsk, Russia

E. B. Yarovaya
Affiliation: Faculty of Mathematics and Mechanics, Moscow State University, 119992, GSP-2 Moscow, Russia

Received by editor(s): February 24, 2003
Published electronically: February 7, 2005
Additional Notes: Supported by the RFBR grants 02-01-00266, 00-15-96136 and the INTAS grants 99-01317, 00-0265.
Article copyright: © Copyright 2005 American Mathematical Society