Weak convergence for generalized semi-Markov processes

Generalized semi-Markov schemes were introduced by Matthes in 1962 under the designation ‘Bedienungsschemata’ (service schemes). They include a large variety of familiar stochastic models. It is shown in this paper that under appropriate regularity conditions the associated stochastic process describing the state at timet,t≥0, and the stationary distribution are continuous functions of the life-times of the active components. The supplementary-variable Markov process is shown to be the limit process of a sequence of discrete-state-process obtained through approximating the life-time distributions by mixtures of Erlang distributions and measuring ages and residual life-times in phases. This approach supplements the phase method.