Geodesic
Distribution of traversal time relative to sequences of states in a~semi-Markov process
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 167-173

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One considers a semi-Markov process with trajectories from the space $\mathscr D$. One derives a formula for the conditional generating function of the traversal time relative to a sequence of states, similar to Levy's formula for a monotone process with independent increments. 
@article{ZNSL_1985_142_a19,
     author = {B. P. Harlamov},
     title = {Distribution of traversal time relative to sequences of states in {a~semi-Markov} process},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--173},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a19/}
}
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B. P. Harlamov. Distribution of traversal time relative to sequences of states in a~semi-Markov process. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 167-173. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a19/