Let us consider a phase-type renewal process, where renewal intervals have $PH(\boldsymbol{\beta},S)$ distribution,
http://en.wikipedia.org/wiki/Phase-type_distribution

We set $S=\begin{pmatrix}-\lambda& \lambda\\\ 0 & -\mu\end{pmatrix}$ and $\boldsymbol{\beta}=(1,0)$

Denote by $P_{ij}(k,t)$ the probability that there occurs $k$ renewals in the interval [0,t] and the phase at $t$ is $j$ on condition that the phase at 0 is $i$. 

Then it is well-known that the matrix $P(k,t)=(P_{ij}(k,t))$ can be defined by its generating function
$$P(z,t)=\sum_{k=0}^{\infty}P(k,t)z^k=e^{G(z)t}$$
where $G(z) = S-S\boldsymbol{1}\boldsymbol{\beta}z= \begin{pmatrix}-\lambda& \lambda\\\ \mu z & -\mu\end{pmatrix} $.

Clearly, this probability is equivalent to the above-mentioned problem.