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!