model. Note that if we change the labels of the states from $\{e_+,e_-\}$ to $\{e_-,e_+\}$ and modify the inputs accordingly, we can compute $\Pi_{c}^{+}(e_-,e_+)$ using the same algorithm that computes $\Pi_{c}^{-}(e_+,e_-)$.

Appendix: Approximation of Multivariate Inverse Laplace Transform {#sec:inverLaplaceTransform}
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For the convenience of the reader, we will briefly recall the basics of Laplace transform and its inverse. Then, we will proceed with an important result regarding the approximation of the multivariate inverse Laplace transform.

Let $f:[0,\infty)^{n}\rightarrow [0,\infty)$ be a Borel-measurable function such that $$\begin{aligned}
\int_{0}^{\infty}\cdots\int_{0}^{\infty}f(t_{1},\ldots,t_{n})\prod_{k=1}^{n}e^{-q_{k}t_{k}}\operatorname{d \!}t_{1}\cdots\operatorname{d \!}t_{n}\end{aligned}$$ exists for any $q_{1},\ldots,q_{n}>0$. Then, the multivariate Laplace transform $\widehat{f}$ of $f$, defined by $$\begin{aligned}
{\widehat}{f}(q_{1},\ldots,q_{n})=\cL(f)(q_{1},\ldots,q_{n}):=\int_{0}^{\infty}\cdots\int_{0}^{\infty}f(t_{1},\ldots,t_{n})\prod_{k=1}^{n}e^{-q_{k}t_{k}}\operatorname{d \!}t_{1}\cdots\operatorname{d \!}t_{n},\end{aligned}$$ is well-defined for any $q_{k} \in \bC^+$, $k=1,\ldots,n$, where[^2] $\bC^+:= {\{z\in\bC {\;|\;}\Re(z)>0\}}$ with $\Re(z)$ denoting the real part of $z\in\bC$. The inverse multivariate Laplace transform of function $g:(\bC^+)^n \to\bC$, is the function $\check{g}$, such that $\cL(\check{g}) = g$. We will also write $\check{g} = \cL^{-1}(g)$. The existence and uniqueness of the inverse Laplace transform is a well understood subject (cf. [@Widder1941]). Although there are explicit formulas of the inverse Laplace transform for many functions, generally speaking, in many practical situations the inverse Laplace transform of a function is computed by numerical approximation technics. We 