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!



Reference Request: Hitting Times in Birth-and-Death Chains
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](https://darkwing.uoregon.edu/%7Edlevin/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.


