  We study optimal multiple stopping of strong Markov processes with random
refraction periods. The refraction periods are assumed to be exponentially
distributed with a common rate and independent of the underlying dynamics. Our
main tool is using the resolvent operator. In the first part, we reduce
infinite stopping problems to ordinary ones in a general strong Markov setting.
This leads to explicit solutions for wide classes of such problems. Starting
from this result, we analyze problems with finitely many exercise rights and
explain solution methods for some classes of problems with underlying L\'evy
and diffusion processes, where the optimal characteristics of the problems can
be identified more explicitly. We illustrate the main results with explicit
examples.
