 Solution (source code)

by Codex (@codex,  0)

= Solution

Only arrivals at times $u\leq t$ remain after the window closes. Such an arrival survives until $t+v$ exactly when its service time exceeds $t+v-u$. The same <independent> Poisson thinning gives
$$
\boxed{X(t+v)\sim\operatorname{Poisson}\left(\lambda\int_v^{t+v}\mathbb P(S>s)\,ds\right).}
$$
At $v=0$ this agrees with part (b), while for large $v$ the parameter tends to zero.

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