Let us consider the homogeneous continuous time Markov chain $(X_t)_{t\ge 0}$ with two states {0,1} and the intensity matrix

$Q=\begin{pmatrix}-\lambda& \lambda\\\mu& -\mu\end{pmatrix}$

Let $N_t$ be the number of $1 \to 0$ transitions of $X_t$ in the interval [0, t].

The main interesting question is to find the probability  $p_{ij}(k,t)=P(N_t=k, X_t= j| X_0=i)$.

May be we can use its generating functions $P_{ij}(z,t)=\sum_{k=0}^{\infty}z^kp_{i,j}(k,t)$ and recursion to obtain the result. 

Have any ideas for a calculation?