Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T21:09:33.083Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic behaviour of a divergent linear birth and death process

Published online by Cambridge University Press:  14 July 2016

John Haigh
John Haigh*
Affiliation:
University of Sussex
Get access Rights & Permissions [Opens in a new window]

Abstract

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

Keywords

BIRTH AND DEATH PROCESSESASYMPTOTIC GROWTHFIRST ENTRY TIMES
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 1 , March 1978 , pp. 187 - 191
Copyright
Copyright © Applied Probability Trust 1978 

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

Keilson, J. (1964) A review of transient behaviour in regular diffusion and birth-death processes. Part I. J. Appl. Prob. 1, 247–266.CrossRefGoogle Scholar
Keilson, J. (1965) A review of transient behaviour in regular diffusion and birth-death processes. Part II. J. Appl. Prob. 2, 405–428.Google Scholar
Keilson, J. (1971) Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.Google Scholar
Siegel, M. J. (1976) The asymptotic behaviour of a divergent birth and death process. Adv. Appl. Prob. 8, 315–338.Google Scholar
Soloviev, A. D. (1972) Asymptotic distribution of the moment of first crossing of a high level by a birth and death process. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 71–86.Google Scholar
Takács, L. (1966) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.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
Williams, T. (1965a) The distribution of response times in a birth and death process. Biometrika 52, 581–585.CrossRefGoogle Scholar
Williams, T. (1965b) The basic birth-death model for microbial infections. J. R. Statist. Soc. B 27, 338–360.Google Scholar