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 $R_{ij}(k,t)$ 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 $R(k,t)=(R_{ij}(k,t))$ can be defined by its generating function $$R(z,t)=\sum_{k=0}^{\infty}R(k,t)z^k=e^{G(z)t}$$ where $G(z) = S-S\boldsymbol{1}\boldsymbol{\beta}z, \ \boldsymbol{1}=(1,1)^T$.

Finally, we see that $P(k,z)= \begin{pmatrix}R_{1,0}(z,t) & R_{1,1}(z,t) \\\ R_{1,0}(z,t) & R_{1,1}(z,t)\end{pmatrix}$

How do you think about this?

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.