Earlier work by Diaconis and Saloff-Coste gives a spectral criterion for a maximum separation cutoff to occur for birth and death chains. Ding, Lubetzky and Peres gave a related criterion for a maximum total variation cutoff to occur in the same setting. Here, we provide complementary results which allow us to compute the cutoff times and windows in a variety of examples.
"Computing cutoff times of birth and death chains." Electron. J. Probab. 20 1 - 47, 2015. https://doi.org/10.1214/EJP.v20-4077