First entrance time distribution CTMC Let $Z$ be a continuous time markov chain on $S=\{A,D,V\}$ with $Q$-matrix
\begin{pmatrix}-(1+d)& d& 1\\a & -a & 0\\0 & 0 & 0 \end{pmatrix} 
and $Z(0)=A$. How to find a closed expression for the distribution of the time
$$T=\inf\{t\geq 0: Z(t)=V\}$$?
 A: EDIT 20.12.2019 (12/20/19)
I tried the following. For simplicity, let's assume $a=d=1$.
Just consider the markov chain $Y$ on $\{A,D\}$ with Generator
$$Q=\begin{pmatrix} -d & d\\ a & -a\end{pmatrix}$$
We consider the number of jumps $N(t)$, where $N(0)=0$. We know then that $N(t)$ is poisson distributed with rate $1$. Distinguish two cases:
$$P(T>t)=\sum_{n=0}^{\infty}\left(P(T>t\mid N(t)=2n)P(N(t)=2n)+P(T>t\mid N(t)=2n+1)P(N(t)=2n+1)\right)$$
so we compute $P(T>t\mid N(t)=2n), P(T>t\mid N(t)=2n+1)$.
If $N(t)=2n+1$, then $Y(t)=D$. By the construction of the orgininal markov chain $Z$, it is not possible to jump into $V$. Hence $P(T>t\mid N(t)=2n+1)=1$.
If $N(t)=2n$, then $Y(t)=A$. Then $T$ has not occured in the past if in after every time point when $Y$ just has jumped in $A$ there was an Exp($1$)-time $T_V^i$ (for jumping in $V$) larger than an Exp($1$)-time $T_D^i$ for jumping in $D$. Since $N(t)=2n$, this has happend $n$ times independently. 
Further, before the last jump into $A$ there occured $n$ phases in $A$ and also $n$ phases in $D$ which durations are $2n$ independent Exp($1$) random variables. This sum is now denoted by $S_n$ and is Erlang(2n)-distributed. Now, since the last jumping point into $A$ there is again an Exp($1$)-time $T_V$ for the possibility in jumping into $V$. So $T$ has not occured if $S_n+T_V\leq T$.
Then we get
$$P(T>t\mid N(t)=2n)=P(t<S_n+T_V,\bigcap_{i=1}^n\{T_D^i\leq T_V^i\})$$
Now $T_D^i\leq T_V^i=1/2$ by property of Exp-Distribution and $S_n$ is independent from $T_V$ so $S_n+T_V$ is Erlang(2n+1)-distributed, denoted by the random variable $E$. By independency,
$$P(T>t\mid N(t)=2n)=P(t<E,\bigcap_{i=1}^n\{T_D^i\leq T_V^i\})=P(t<E)\frac{1}{2}^n$$
We can finally proceed by plugging in the poisson probabilties. What do you think about this?
