The time to absorption in a Markov chain is Phase-type (PH) distributed. There are discrete PH distributions if you have a discrete time Markov chain or continuous PH distributions otherwise. PH distributions are very easy to work with, the density functions, moments, etc. have nice and explicit formulas.

If you need a nice book with many introductory examples, here is one:

Latouche, Guy, and Vaidyanathan Ramaswami. Introduction to matrix analytic methods in stochastic modeling. Vol. 5. Siam, 1999.

Look at Kemeny and Snell. Absorbing Markov chains