On integrals of birth-death processes at random time

In this paper, we consider a time-changed path integral of the homogeneous birth-death process. Here, the time changes according to an inverse stable subordinator. It is shown that its joint distribution with the time-changed birth-death process is governed by a fractional partial differential equation. In a linear case, the explicit expressions for the Laplace transform of their joint generating function, means, variances and covariance are obtained. The limiting behavior of this integral process has been studied. Later, we consider the fractional integrals of linear birth-death processes and their time-changed versions. The mean values of these fractional integrals are obtained and analyzed. In a particular case, it is observed that the time-changed path integral of the linear birth-death process and the fractional integral of time-changed linear birth-death process have equal mean growth.