It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passge times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.
