Poisson counting process subinterval distribution

Question

Suppose $N(\omega,t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda,\,\forall\omega \in\Omega$ where $\Omega$ is the sample space. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(\omega,t)$ counts exactly $n$ points within at least one subinterval $[t,t+\tau]$ of $[0,T]$, or Prob$\big(\bigcup_t\big\{\omega\,\big|\, [t,t+\tau]\subseteq [0,T] \wedge N(\omega,t+\tau)-N(\omega,t)=n\big\}\big)$?

Answer 1

I think it is unlikely that this probability can be expressed in closed form. However, we can express your union of uncountably many events under the probability sign as the countable union $$A:=\bigcup_{k=1}^\infty A_k,$$ where $$A_k:=\{S_{k+n-1}-S_k\le\tau<S_{k+n}-S_{k-1},\ S_{k-1}\le T-\tau\}$$ and $S_1,S_2,\dots$ are the times of successive jumps of the Poisson process, with $S_0:=0$. In principle, the probability $P(A)$ of the union $A$ of the $A_k$'s can be expressed by the inclusion–exclusion principle, which reduces the calculation of the probability of $A$ to the calculation of the probabilities of the finite intersections of the $A_k$'s. In turn, the latter probabilities can be expressed as iterated integrals, taking into account that the increments $X_j:=S_j-S_{j-1}$ for natural $j$ are iid exponential random variables with rate $\lambda$.

Answer 2

Too long for a comment and maybe a partial answer. First I'm not sure what you really mean, so my comment may be completely wrong. May be you are considering an $M/D/n-1/0$ queuing system with an arrival process, which is $PP(\lambda)$, $n-1$ counters and deterministic service time (of length $\tau$). There is no waiting room. Customers are lost if all counters are busy. Then an interesting characteristic is the mean fraction of customers which are rejected.

Of course also the probability that an arriving customer in $[0,T]$ is rejected may be of interest. This probability corresponds to the probability that there is an interval $[t,t+\tau]$ with not less than $n$ arriving customers. If you know these probabilities it's easy to compute the probability you want. The actual computations are tedious. Maybe you find a solution in the queuing theory literature (google for $M/D$-queue). Unfortunately I have no access to the relevant literature.