numerical inverse Laplace transform methods, not necessary the Gaver-Stehfest formula from Section \[subsubsec:specialcase\]. In particular, one can use Talbot approximation formula presented in Section \[sec:inverLaplaceTransform\], which is more efficient than the Gaver-Stehfest under fairly general assumptions (cf. [@abate2006unified]).

Application in Fluid flow problems
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The Wiener-Hopf factorization for a time-homogeneous finite Markov chain was applied in [@rogers1994fluid] in the context of fluid models of queues. In this section, we will apply our results to a time-inhomogeneous Markov chain fluid flow problem.

First, we briefly review the classical fluid flow problem (cf [@mitra1988] and [@rogers1994fluid] for detailed discussion). Suppose we have a large water tank with capacity $a\in (0,\infty]$. On the top of the tank, there are $I_{t}\in \mathcal{I}$ pipes open at time $t$, with each pipe pouring water into the tank at rate $r^{+}$. At the bottom of the tank, there are $O_{t}\in \mathcal{O}$ taps open at time $t$, with each tap allowing water to flow out at rate $r^{-}$. We assume that $\mathcal{I}$ and $\mathcal{O}$ are finite sets.

Then, the volume $\xi_{t}$ of water in the tank at time $t$ satisfies $$\begin{aligned}
\frac{\operatorname{d \!}\xi_{t}}{\operatorname{d \!}t}=r^{+}I_{t}-r^{-}O_{t}, \quad\text{if }0<\xi_{t}<a.\end{aligned}$$ Moreover, if $\xi_{t}=0$, i.e. if the tank is empty, then the outflow 