Reference Request: Hitting Times in Birth-and-Death Chains I am looking for a general formula for hitting times in a standard birth-and-death chain.
I'm absolutely convinced that I've seen a paper with such a formula in it in the past, but I cannot for the life of me find it now.
The formula looks something like this:
$$
E_{i-1}(\tau_i) =
\prod_{j < i} \frac{q_{j,j+1} ... }{...}.
$$
I've looked through any papers on birth–death process which seem at all relevant.
These include the following.

*

*Ding, Lubetzky, Peres; Total Variation Cutoff in Birth-and-Death Chains

*Fill; The Passage Time Distribution for a Birth-and-Death Chain

*Smith; The Cutoff Phenomenon for Random Birth and Death Chains

*Zhang; Moments of First Hitting Times for Birth–Death Processes on Trees

I've even tried to use the formula for my own research in the past.
Try as I might, I can't find it now.
If anyone knows the reference, that would be much appreciated!
 A: Let use compute $E_0(\tau_1)$, the general expression follows by shift.
For simplicity I assume that $|X_{i+1}-X_i|=0$, ie that the probability to stay put vanishes (one can also treat the general case by essentially the method described below). Let $\omega_i=P(X_1=i+1|X_0=i)$.
Write, starting at $X_0=0$,
$$\tau_1=1_{X_1=1}+1_{X_1=-1}(1+ \tau_0'+\tau_1')$$
where $\tau_0',\tau_1'$ are of the same law as $\tau_0$ starting at $-1$ and $\tau_1$ starting at $0$.
Let $\omega_i$ be the probability to jump right at $i$.
Taking expectations and rearranging you get
$$ E_0(\tau_1)=1/\omega_0+\rho_0 E_{-1}(\tau_0),$$
where $\rho_i=(1-\omega_i)/\omega_i$.
Now you can iterate:
$$E_0(\tau_1)=1/\omega_0+\rho_0/\omega_{-1}+\rho_0\rho_{-1} E_{-2}(\tau_{-1})$$
So
$$ E_0(\tau_1)=\sum_{i=0}^\infty \frac{1}{\omega_{-i}} \prod_{j=0}^{i-1} \rho_{-j}$$
(If the right side diverges, then the expectation is indeed infinite).
These formulae appear in the study of (one dimensional) random walk in random environment. Look up my lecture notes (Springer LNM) for an introduction. This is equation (2.1.14) there.
A: There is a discussion of birth an death chains in [1]; see page 27 for the hitting time formulas. See also [2], [3], [4].
[1] Markov Chains and Mixing Times: Second Edition by Levin and Peres, with contributions by Wilmer, https://bookstore.ams.org/mbk-107
https://darkwing.uoregon.edu/~dlevin/MARKOV/mcmt2e.pdf
[2] Chen, Mu-Fa. "Speed of stability for birth-death processes." Frontiers of Mathematics in China 5, no. 3 (2010): 379-515.
[3] Fill, James Allen. "On hitting times and fastest strong stationary times for skip-free and more general chains." Journal of Theoretical Probability 22, no. 3 (2009): 587.
[4] Palacios, J. and Tetali, P., 1996. A note on expected hitting times for birth and death chains. Statistics & Probability Letters, 30(2), pp.119-125.
