Joint distributions of random variables and their integrals for certain birth-death and diffusion processes
- 1 January 1971
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 3 (2) , 339-352
- https://doi.org/10.2307/1426175
Abstract
For the linear growth birth-death process with parameters λn=nλ, μn=nμ, Puri ((1966), (1968)) has investigated the joint distribution of the numberX(t) of survivors in the process and the associated integralY(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results ast→ ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functionalWx= ∫0Txg{X(τ)}dτ, whereTxis the first passage time to the origin in a general birth-death process withX(0) =xandg(·) is an arbitrary function. Functionals of the formWxarise naturally in traffic and storage theory; for exampleWxmay 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 ofM/G/1 andGI/M/1 queues by Gaver (1969) and Daley (1969).Keywords
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