namely piece-wise constant. This allows to use an appropriately tailored randomization technique. Besides the mathematical importance of the WHf, there is an important computational aspect: this methodology allows for very efficient computation of important functionals of Markov chains.

The Wiener-Hopf factorization for finite-state Markov chains was originally derived in [@Barlow1980] in the time-homogeneous case; see also [@London1982] and [@Williams1991]. For the WHf in case of time-homogeneous Feller Markov processes we refer to [@Williams2008]. For some related applied work we refer to [@Avram2003], which deals with the ruin problem, and to [@Asmussen1995; @rogers1994fluid; @rogers_shi_1994] that study fluid models. In addition, [@Kennedy1990] studies the so called “noisy” Wiener-Hopf factorizations; for applications see [@Asmussen1995; @rogers1994fluid; @rogers_shi_1994; @Jobert2006; @Jiang2008; @Mijatovic2011; @Jiang2012; @Hieber2014; @Hainaut2016].

It needs to be stressed that even though the classical WHf of [@Barlow1980] can be applied to the generator matrix, say $\mG_t$, of a time-inhomogeneous Markov chain $X$ at every time $t$, these factorizations do not have any probabilistic meaning with regard to the process $X$. In particular, they are of no use for computing functionals such as - below. So, a relevant WHf for a time-inhomogeneous Markov chain requires a different approach than the one that would just directly apply 