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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
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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 

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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