Consider a birth-death Markov chain, whose state space is $X=0,1,\ldots$ with the transition probabilities $P_{i,i+1}=p, P_{i+1,i}=q>p, \forall i\geq 0$. $P_{0,0}=1-p$ and $P_{i,i}=1-p-q, i>0$. Can I determine the (closed-form) probability mass function (PMF) of the first-passage time from state $1$ to $0$?
It seems that this first-passage time has close relationship to the busy period time of a discrete-time M/M/1 queue. It is known that the probability density function (PDF) of busy period of an (continuous-time) M/M/1 queue with arrival rate $\lambda$ and mean service time $1/\mu$ is given by $$ f(t)=\frac{1}{t\sqrt{\rho}}e^{-(\lambda+\mu)t}I_1(2t\sqrt{\lambda\mu}), $$ where $\rho=\lambda/\mu$. The mean and variance of the busy period time are $1/(\mu-\lambda)$ and $\frac{1+\rho}{\mu^2(1-\rho)^3}$, respectively.
To what extent can the classical M/M/1 results be applied in the discrete-time scenario? I found that the (empirical) mean first-passage time from state $1$ to $0$ is indeed $1/(q-p)$. Can someone please point me to some references for a detail treatment of such a discrete-time case?