We consider a point process i + ξ i , where i Î \mathbbZiZ and the ξ i ’s are i.i.d. random variables with compact support and variance σ 2. This process, with a suitable rescaling of the distribution of ξ i ’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 σ 2 of the random translations ξ 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 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.