Consider $N =(N_t)_{t\geq0}$ a Poisson process of intensity $\lambda > 0$ and $(T_n)_{n\geq 1}$ its jump instants.
Then consider for all $t \geq 0$, $Z_t = t- T_{N_t} \mathbb 1 _{\{ t \geq T_1\}}$, the time after the first jump of $N$, and $W_t = T_{N_t+1} -t$, the waiting time before the next jump of $N$.
I would like to obtain $\mathbb P \{ Z_t >z, W_t >w \}$.
$$ P \{ Z_t >z, W_t >w \} = P \{ T_{N_t} \mathbb 1 _{\{ t \geq T_1\}} < t-z, T_{N_t+1} > t+w \} \\ = \sum_{n\geq 0} P \{ T_{n} \mathbb 1 _{\{ t \geq T_1\}} < t-z, T_{n+1} > t+w \ | \ N_t=n \}$$
I guess it's necessary to use the fact $T_n \sim \Gamma(n,\lambda), \ \forall n\geq 1 $ or $(T_{n+1} -T_n) \overset{\text{iid}}{\sim} \mathcal E (\lambda), \ \forall n\geq 0 $, but I don't see how to manage to it.
Any advices are appreciated. Thanks in advance.