We illustrate in this short chapter an argument which has been used in different contexts, to analyse the behaviour of a Markov-modulated process until it leaves a finite interval for the first time. Among others, the same approach has been used for QBDs and for Fluid flows. In general, the stationary drift is assumed to be different from zero, an assumption which simplifies considerably the analysis. Things become more involved, indeed, when \(\boldsymbol \pi ^{\mathrm {T}} \boldsymbol \mu = 0\) , and we show how to overcome the technical difficulty. In the last sections, we assume that the process is regulated at one boundary, so that there is only one way to escape.