Probability that a birth--death process crosses level $n$ in $(0,T)$ 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 $n > 1$?
 A: I would suggest the following answer. The queueing system you are talking about in the research community is referred to as the $M/D/\infty$ queue with arrival rate $\lambda$ and fixed service time $d$. I think there is no need to repeat here the analytic solution for the number of jobs $X(t)$ in this system (you can kindly refer, for example, to Section 6.1 in P.P. Bocharov, C. D'Apice, A.V. Pechinkin and S. Salerno: Queueing Theory, Modern Probability and Statistics, VSP, The Netherlands, 2004.).
Coming back to your question, denote $p_i(t)=\mathbb{P}(X(t)=i$, $i\ge 0$. In the cited book (and, of course, in many other sources) you can find that $p_i(t)$ have very simple form:
$$
p_i(t)={[\Lambda(t)]^i \over i!} e^{-\Lambda(t)}, i \ge 0,
$$
where
$$
\Lambda(t)=\lambda \int_0^t (1-B(y))dy, \ \ 
B(y)=
\begin{cases}
0, y\le d\\
1, y > d.
\end{cases}
$$
Thus for any $T>0$
$$
\mathbb{P} \left (\max_{t\in (0,T)} X(t)>0 \right )
=1-\mathbb{P} \left (\max_{t\in (0,T)} X(t)=0 \right )
=1- \int_0^T p_0(t) dt.
$$
Now i think it is straghtforward to find the solution for $n\ge 1$.
