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

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

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

The main interesting question is to find the the probability $p_k(i,j)=P(N_t=k, X_t= j| X_0=i)$.

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

Have any ideas for a calculation?