I am working on a biological stochastic modeling problem which involves the simple continuous time birth death markov process of one species: $$*\rightarrow X\rightarrow *$$ Where $X$ can either increase by 1 (a random event with constant propensity) or decrease by one (a random event with propensity directly proportional to the value of $X$).
Possible values of $X$ are all non-negative integers. I need a method to calculate the probability distribution of the first passage time where $X\rightarrow 0$, for initial conditions drawn from the steady state distribution of $X$ (which I know is a Poisson distribution). Alternatively, the initial condition could be fixed at a given value if this would make the calculations simpler. Alternatively, some arbitrary upper bound for $X$ could be chosen if this makes calculations simpler.