Consider a $3$ state Continuous time Process $\{X_t\}_{t \geq0}$ with state space $\mathcal{S} = \{0,1,2\}$ where state $0$ denotes the absorbing state. Let the generator of this process be: $$Q = \left(\begin{matrix} 0 & 0 & 0 \\ 0 & -\mu & \mu \\ \lambda & \phi & -(\lambda+\phi) \end{matrix} \right)$$
Letting $N_t$ denote the total number of transitions (jumps between states) made within the time interval $[0,t)$, I am interested in: $$\mathbb{P}(\Gamma^2_{t}| N_t = 1, X_0 =1),$$ where $\Gamma^2_{t}$ is a random variable denoting the holding time in state $2$ in the censored interval $[0,t)$.
The issue is because the process is not time reversible, $\Gamma^2_{t}$ cannot really be exponential. Does anyone have any ideas?
Thanks!