Comparison of level-crossing times for Markov and semi-Markov processes

We derive sufficient conditions for the level-crossing ordering of continuous-time Markov chains (CTMCs) and semi-Markov processes (SMPs). The former ones constitute a relaxation of Kirstein's conditions for stochastic ordering of CTMCs in the usual sense, whereas the latter ones are an extension of conditions established by Di Crescenzo and Ricciardi and Irle for skip-free-to-the-right SMPs.