Consider the following two random variables,
In first case you record the number of people arriving at a queue, for a random amount of time. Note, here the arrival of people in the queue is random (governed by Poisson distribution) and the amount of time the queue is observed is also random. Denote the number of people observed here by $N(T)$
In second case you record the number of people arriving in same queue, for fixed amount of time $t_1$. Denote this by $N(t_1)$.
Is $N(T)$ independent of $N(t_1)$ ? In general, if we observe a set queues (not just a single queue) will the two random variables recording the total number of people arrived across all the queues, be independent?