  We consider an additive functional driven by a time-inhomogeneous Markov
chain with a finite state space. Our study focuses on the joint distribution of
the two-sided exit time and the state of the driving Markov chain at the time
of exit, given in terms of expectation operators. These operators can be
expressed as compositions of other operators related to some relevant one-sided
exit (or first passage) problems. In addition, we study the law of the driving
Markov chain at times prior to the exit time.
