The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt
and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt
classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.