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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) = ∫0tX(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 Wx = ∫0Txg{X(τ)}dτ, where Tx 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 Wx arise naturally in traffic and storage theory; for example Wx 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).

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).

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