On the limiting distributions in Markov renewal processes with finitely many states

We are interested in some limiting distributions of M.R.P.N(t)={Nj(t), 1≤j≤r} or the total sojourn times in each state for a special M.R.P., in which transition distribution functions have the formQij(t)=pijHi(t). To obtain the same results for the general M.R.P. and the limiting joint distribution of {Sj(t), 1≤j≤r}, we have to use a different method, which will be stated in a paper, to appear in the near future, with examples in practical applications.