Analytic expression of the probability density function for the first-passage time in birth-death processes

Birth-death processes occur throughout the universe, and the rates of birth and death generally depend on the system size (the number of products or customers), varying with each event. Moreover, deriving an analytic expression for the probability density function for birth-death processes remains challenging despite the interest in the first-passage time (when the system size first reaches a specific threshold). This work derives the probability density function of the first-passage time without approximations based on considering all cases where the number of products reaches a threshold at each birth instance, providing an analytic expression for the statistics of the first-passage time. This work offers a promising new approach to investigating the first-passage time in birth-death processes.