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Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

Published online by Cambridge University Press:  01 July 2016

J. Gani and
D. R. Mcneil
J. Gani
Affiliation:
University of Sheffield
D. R. Mcneil*
Affiliation:
The Johns Hopkins University, Baltimore, Md.
*
∗ Present address: Department of Statistics, Princeton University.
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Extract

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0 t X(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional W x = ∫0 Tx g{X(τ)}dτ, where T x is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form W x arise naturally in traffic and storage theory; for example W x may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 2 , Autumn 1971, pp. 339 - 352
Copyright
Copyright © Applied Probability Trust 1971 

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Footnotes

Research supported by u.s. Department of Transportation, Bureau of Public Roads, Research Contracts FH-II-6884 (McNeil) and FH-11-6885 (Gani).

References

Abramovttz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. U. S. National Bureau of Standards, Washington D. C.Google Scholar
Bailey, N. T. J. (1954) A continuous time treatment of a simple queue using generating functions. J. R. Statist. Soc. B 16, 288–291.Google Scholar
Daley, D. J. (1969) The total waiting time in a busy period of a stable single-server queue, I. J. Appl. Prob. 6, 550–564.CrossRefGoogle Scholar
Daley, D. J. and Jacobs, D. R. (1969) The total waiting time in a busy period of a stable single-server queue, II. J. Appl. Prob. 6, 565–592.CrossRefGoogle Scholar
Erdélyi, A. et al. (1954) Tables of Integral Transforms, Volume 1. McGraw Hill, New York.Google Scholar
Gaver, D. P. (1969) Highway delays resulting from flow stopping incidents. J. Appl. Prob. 6, 137–153.Google Scholar
Gnedenko, B. V. (1962) The Theory of Probability. Chelsea, New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications. Volume 2. J. Wiley, New York.Google Scholar
Iglehart, D. (1965) Limiting diffusion approximations for the many server queue and the repairman problem. J. Appl. Prob. 2, 429–441.CrossRefGoogle Scholar
Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes. Part II. J. Appl. Prob. 2, 405–428.CrossRefGoogle Scholar
Kendall, D. G. (1948) On the generalized birth-and-death process. Ann. Math. Statist. 19, 1–15.CrossRefGoogle Scholar
McNeil, D. R. (1970) Integral functionals of birth and death processes and related limiting distributions. Ann. Math. Statist. 41, 480–485.CrossRefGoogle Scholar
Puri, Prem S. (1966) On the homogeneous birth-and-death process and its integral. Biometrika 53, 61–71.CrossRefGoogle ScholarPubMed
Puri, Prem S. (1968) Some further results on the birth-and-death process and its integral. Proc. Camb. Phil. Soc. 64, 141–154.CrossRefGoogle Scholar
Stone, C. (1961) Limit Theorems for Birth and Death Processes and Diffusion Processes. Ph.D. Thesis, Stanford University.Google Scholar
Stone, C. (1963) Limit theorems for random walks, birth and death processes and diffusion processes. Ill. J. Math. 4, 638–660.Google Scholar
Watson, G. N. (1944) A Treatise on the Theory of Bessel Functions. Cambridge U.P. (Second Edition).Google Scholar