Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T08:54:56.057Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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