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

DOI:
https://doi.org/10.1090/S0094-9000-05-00609-5

Published electronically:
February 7, 2005

MathSciNet review:
2110900

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A continuous time branching random walk on the lattice 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 as and the probability that the number of particles at the origin at time is positive. We also prove a Yaglom type conditional limit theorem for the total number of particles existing at time . 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

Email:
vatutin@mi.ras.ru

**V. A. Topchii**

Affiliation:
Omsk Branch of Sobolev Institute of Mathematics, Pevtsova Street 13, 644099 Omsk, Russia

Email:
topchij@iitam.omsk.net.ru

**E. B. Yarovaya**

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

Email:
yarov@cardpl.msk.ru

DOI:
https://doi.org/10.1090/S0094-9000-05-00609-5

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