Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T08:11:13.529Z Has data issue: false hasContentIssue false
  • English
  • Français

Modes of growth of counting processes with increasing arrival rates

Published online by Cambridge University Press:  14 July 2016

W. A. O'N. Waugh
W. A. O'N. Waugh*
Affiliation:
The University of Toronto
Get access Rights & Permissions [Opens in a new window]

Abstract

Jumping processes which grow by unit jumps, with decreasing sojourn times between successive jumps, are studied. Markovian birth processes, non-Markovian branching processes, and some generalizations of these are special cases. Three classes are described, in one of which growth is explosive, in the second asymptotically continuous, and in the third oscillatory. A theorem is proved which gives an explicit functional expression in the asymptotically continuous case, and borderline cases between the classes are investigated.

Keywords

COUNTING PROCESSESBIRTH PROCESSESBRANCHING PROCESSESSOJOURN TIMESFIRST ENTRY TIMESNON-MARKOVIAN GROWTHASYMPTOTIC GROWTH
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 2 , June 1974 , pp. 237 - 247
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

Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. (2nd ed.) Springer, Berlin.Google Scholar
Feller, W. (1943) The general form of the so-called law of the iterated logarithm. Trans. Amer. Math. Soc. 54, 373–402.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
Hirschman, I. I. and Widder, D. V. (1965) The Convolution Transform. Princeton University Press.Google Scholar
John, P. W. M. (1961) A note on the quadratic birth process. J. London Math. Soc. 36, 159–160.CrossRefGoogle Scholar
Karlin, S. (1968) Total Positivity. (Vol. 1) Stanford University Press.Google Scholar
Kendall, D. G. (1948) On the role of a variable generation time in the development of a stochastic birth process. Biometrika 35, 316–330.CrossRefGoogle Scholar
Waugh, W. A. O'N. (1970a) Transformation of a birth process into a Poisson process. J. Roy. Statist. Soc. B 32, 418–431.Google Scholar
Waugh, W. A. O'N. (1970b) Uses of the sojourn time series for the Markovian birth process. Proc. Sixth Berkeley Symposium on Math. Statist. and Prob. Google Scholar
Waugh, W. A. O'N. (1972) Taboo extinction, sojourn times, and asymptotic growth for the Markovian birth and death process. J. Appl. Prob. 9, 486–506.CrossRefGoogle Scholar
Waugh, W. A. O'N. (1974) Asymptotic growth of a class of size-and-age-dependent birth processes. J. Appl. Prob. 11, 248–254.Google Scholar