When a Queue Looks the Same to an Arriving Customer as to an Observer - Strauch, Ralph E. (Management Science, 1970)
In a queue with a Poisson arrival distribution, the probability distribution state of the queue at any time is independent of whether or not an arrival occurs at that time. In particular, in a stationary queue with Poisson arrivals, the stationary distribution of the number of waiting customers in the queue at an arbitrary time is equal to the stationary distribution of the number of customers waiting in the queue when a new customer arrives.