  In this paper, we consider a generalized birth-death process (GBDP) and
examined its linear versions. Using its transition probabilities, we obtain the
system of differential equations that governs its state probabilities. The
distribution function of its waiting-time in state $s$ given that it starts in
state $s$ is obtained. For a linear version of it, namely, the generalized
linear birth-death process (GLBDP), we obtain the probability generating
function, mean, variance and the probability of ultimate extinction of
population. Also, we obtain the maximum likelihood estimate of one of its
parameter. The differential equations that govern the joint cumulant generating
functions of the population size with cumulative births and cumulative deaths
are derived. In the case of constant birth and death rates in GBDP, the
explicit forms of the state probabilities, joint probability mass functions of
population size with cumulative births and cumulative deaths, and their
marginal probability mass functions are obtained. It is shown that the Laplace
transform of a stochastic integral of GBDP satisfies its Kolmogorov backward
equation with certain scaled parameters. Also, the first two moments of the
stochastic path integral of GLBDP are obtained. Later, we consider the
immigration effect in GLBDP for two different cases. An application of a linear
version of GBDP and its stochastic path integral to vehicles parking management
system is discussed.
