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

For the linear growth birth-death process with parameters λ _n = nλ, μ _n = 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 ^tX(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 ^Txg{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).