Queueing systems with pre-scheduled random arrivals

We consider a point process i + xi(i) , where i is an element of Z and the xi(i)'s are i.i.d. random variables with compact support and variance sigma(2). This process, with a suitable rescaling of the distribution of xi(i)'s, is well known to converge weakly, for large sigma, to the Poisson process. We then study a simple queueing system with this process as arrival process. If the variance sigma(2) of the random translations xi(i) is large but finite, the resulting queue is very different from the Poisson case. We provide the complete description of the system for traffic intensity rho = 1, where the average length of the queue is proved to be finite, and for rho < 1 we propose a very effective approximated description of the system as a superposition of a fast process and a slow, birth and death, one. We found interesting connections of this model with the statistical mechanics of Fermi particles. This model is motivated by air traffic systems.