Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T14:45:42.119Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic behavior of a divergent linear birth and death process

Published online by Cambridge University Press:  01 July 2016

Martha J. Siegel
Martha J. Siegel*
Affiliation:
Towson State College, Baltimore, Maryland
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

Keywords

BIRTH AND DEATH PROCESSESASYMPTOTIC GROWTHFIRST-ENTRY TIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 315 - 338
Copyright
Copyright © Applied Probability Trust 1976 

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] Copson, E. T. (1965) Asymptotic Expansions. Cambridge University Press.CrossRefGoogle Scholar
[2] Erdelyi, A. (1953) Higher Transcendental Functions. McGraw-Hill, New York.Google Scholar
[3] Karlin, S. and McGregor, J. L. (1958) Linear growth birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
[4] Karlin, S. and McGregor, J. L. (1957) The differential equations of the birth and death process and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
[5] Kemperman, J. H. B. (1962) An analytical approach to the differential equations of the birth and death process. Michigan J. Math. 9, 321–361.CrossRefGoogle Scholar
[6] Milch, P. R. (1968) A multidimensional linear growth birth and death process. Ann. Math. Statist. 39, 727–754.Google Scholar
[7] Reuter, G. E. H. (1957) Denumerable Markov processes and associated contraction semigroups. Acta. Math. 97, 1–46.Google Scholar
[8] Whittaker, E. T. and Watson, G. N. (1965) A Course of Modern Analysis. Cambridge University Press.Google Scholar
[9] Williams, T. (1965) The distribution of response times in a birth and death process. Biometrika 52, 581–585.CrossRefGoogle Scholar
[10] Williams, T. (1965) The basic birth-death model for microbial infections. J. R. Statist. Soc. B 27, 338–360.Google Scholar