The crossing barrier of a non-homogeneous semi-Markov chain

In this paper, we derive the probability distribution function of the first entry time in a time varying subset of the phase space of a non-homogeneous semi-Markov chain. Deterministic and stochastic time varying subsets are considered and results with a ready algorithmic implementation are provided. A possible application is debated by defining and analysing deterministic and stochastic time variable level crossing order for semi-Markov chains.