Stationary semi-Markov models in risk and queueing theories

For the semi-Markov model in risk and queueing theories, we introduce the natural extension of the concept of stationarity in classical models (i.e. the passage of a renewal process to a general stationary renewal process for the arrivals). It is shown that some interesting relations can be obtained between positive capital risk models using a new concept of duality called *-duality.