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On the subcritical Bellman-Harris process with immigration

Published online by Cambridge University Press:  14 July 2016

A. G. Pakes  and
Norman Kaplan
A. G. Pakes
Affiliation:
Monash University
Norman Kaplan
Affiliation:
University of California, Berkeley
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Abstract

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

Keywords

BRANCHING PROCESSESIMMIGRATIONAGE-DEPENDENCELIMITING DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 652 - 668
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1]Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2]Cheong, C. K. (1970) Note on the key renewal theorem. Nanta Math. IV, 69–71.Google Scholar
[3]De Bruijn, N.G. (1959) Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform. Nieuw. Arch. Wiskunde (3) 7, 20–26.Google Scholar
[4]Erickson, K. B. (1970) Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.Google Scholar
[5]Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[6]Heathcote, C. R. (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. R. Statist. Soc. B 28, 213–217.Google Scholar
[7]Jagers, P. (1968) Age-dependent branching processes allowing immigration. Theor. Probability Appl. 13, 225–236.Google Scholar
[8]Karlin, S. and Mcgregor, J. (1967) The number of mutant forms maintained in a population. Proc. 5th. Berkeley Symp. Math. Statist. Prob. IV, 415–438.Google Scholar
[9]Katz, M. (1963) The probability in the tail of a distribution. Ann. Math. Statist. 34, 312–318.Google Scholar
[10]Pakes, A. G. (1971) Some results for the supercritical branching process with immigration. Math. Biosci. 11, 355–363.Google Scholar
[11]Pakes, A. G. (1972) Limit theorems for an age-dependent branching process with immigration. Math. Biosci. 14, 221–234.Google Scholar
[12]Seneta, E. (1967) The Galton-Watson process with mean one. J. Appl. Prob. 4, 489–495.Google Scholar
[13]Seneta, E. (1968) The stationary distribution of a branching process allowing immigration: A remark on the critical case. J. R. Statist. Soc. B 30, 176–179.Google Scholar
[14]Yang, Y. S. (1972) On branching processes allowing immigration. J. Appl. Prob. 9, 24–31.Google Scholar
[15]Pakes, A. G. (1974) Limit theorems for the integrals of some branching processes Stoch. Proc. and Appl. To appear.Google Scholar