be computed using Wiener-Hopf factorization theory of [@Barlow1980].

Define $\bN_{n}:={\{0,\ldots,n\}}$, ${\widetilde}{\mathbf{E}}:=\bN_{n}\times\mathbf{E}$ and let $({\widetilde}{\Omega},{\widetilde}{\cF},{\widetilde}{\bP})$ be a complete probability space. Next, let us consider a [*time-homogeneous*]{} Markov chain, say $Z=(N,Y):=(N_{t},Y_{t})_{t\geq 0}$, defined on $({\widetilde}{\Omega},{\widetilde}{\cF},{\widetilde}{\bP})$, taking values in ${\widetilde}{\mathbf{E}}$ and with generator matrix ${\widetilde}{\mG}\left((n_{1},j_{1}),(n_{2},j_{2}) \right)_{(n_{1},j_{1}),(n_{2},j_{2})\in {\widetilde}{\mathbf{E}}}$ given as $$\begin{aligned}
{\widetilde}{\mG}=\kbordermatrix{ & \{0\}\times\mathbf{E} & \{1\}\times\mathbf{E} & \cdots & \{n-1\}\times\mathbf{E} & \{n\}\times\mathbf{E} \\ \{0\}\times\mathbf{E} & \mG_{1}-q_{1}\mI & q_{1}\mI & \cdots & 0 & 0 \\ \{1\}\times\mathbf{E} & 0 & \mG_{2}-q_{2}\mI & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \{n-1\}\times\mathbf{E} & 0 & 0 & \cdots & \mG_{n}-q_{n}\mI & q_{n}\mI \\ \{n\}\times\mathbf{E} & 0 & 0 & \cdots & 0 & \mG_{n+1}},\end{aligned}$$ where $q_{1},\ldots,q_{n}$ are positive constants and $\mI$ is the identity matrix. For each $i\in\mathbf{E}$, we define the probability measure ${\widetilde}{\bP}^{i}$ on $({\widetilde}{\Omega},{\widetilde}{\cF})$ by $$\begin{aligned}
\label{eq:TildePi}
{\widetilde}{\bP}^{i}(A):={\widetilde}{\bP}\left(\left.A\,\right|Z_{0}=(0,i)\right),\quad A\in{\widetilde}{\cF}.\end{aligned}$$ The next result regards the Markov property of process $N$.

\[prop:NMC\] For any $i\in\mathbf{E}$, the process $N$ is a time-homogeneous Markov chain under ${\widetilde}{\bP}^{i}$, with generator matrix given by $$\begin{aligned}
{\widetilde}{\mG}_{N}=\kbordermatrix{ & 0 & 1 & \cdots & n-1 & n \\ 0 & -q_{1} & q_{1} & \cdots & 0 & 0 \\ 1 