This question is inspired by this question.
Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ be the number of jobs in the system at time $t$ with $X(0) = 0$. Let $T$ be a fixed time.
I wish to determine
\begin{equation} \mathbb{P}(\max_{t \in (0,T)} X(t) > n), \quad n = 0,1,\ldots \tag{1} \end{equation}
For $d > T$ we have a pure birth process and it is obvious how to determine $(1)$. For $n = 0$ we need at least one arrival in $(0,T)$ so we can determine $(1)$ in that case as well. How would one tackle the case $d < T$ and $n > 1$?