We consider a point process i + ξi, where
i∈ Z and the ξi’s are i.i.d. random variables with compact support and variance σ2. This process, with a suitable rescaling of the distribution of ξi’s, is well known to converge weakly, for large σ, to the Poisson process. We then study a simple queueing system with this process as arrival process. If the variance σ2 of the random translations ξi 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 ρ = 1, where the average length of the queue is proved to be finite, and for ρ < 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.
