Queueing systems with pre-scheduled random arrivals

We consider a point process i + ξ <Subscript> i </Subscript>, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${i\in \mathbb{Z}}$$</EquationSource> </InlineEquation> and the ξ <Subscript> i </Subscript>’s are i.i.d. random variables with compact support and variance σ <Superscript>2</Superscript>. This process, with a suitable rescaling of the distribution of ξ <Subscript> i </Subscript>’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 <Superscript>2</Superscript> of the random translations ξ <Subscript> i </Subscript> 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. Copyright Springer-Verlag 2011