Some Transient Results on the M/SM/1 Special Semi-Markov Model in Risk and Queueing Theories

We consider a usual situation in risk theory for which the arrival process is a Poisson process and the claim process a positive (J - x ) process inducing a semi-Markov process. The equivalent in queuing theory is the IX'[/SM/I model introduced for the first time by Ncuts (1966) For both models, we give an explicit expression of the probability of non-ruin on [o, t] starting with u as initial reserve and of the waiting time distribution of the last customer arrived before t. "Explicit expression" means in terms of the matrix of the aggregate claims distributions.