Queueing systems with pre-scheduled random arrivals

TL;DR

This paper analyzes a queueing system with pre-scheduled random arrivals, showing convergence to Poisson processes under certain conditions, and explores its unique properties and connections to Fermi particle statistics, motivated by air traffic applications.

Contribution

It introduces a novel queueing model with pre-scheduled arrivals, providing a complete analytical description and highlighting its differences from Poisson-based queues.

Findings

Process converges to Poisson in total variation for large variance

Queue exhibits negative autocorrelation, unlike Poisson queues

Connections to Fermi particle statistics in statistical mechanics

Abstract

We consider a point process $i+\xi_i$, where $i\in \bZ$ and the $\xi_{i}$'s are i.i.d. random variables with variance $\sigma^{2}$. This process, with a suitable rescaling of the distribution of $\xi_i$'s, converges to the Poisson process in total variation for large $\sigma$. We then study a simple queueing system with our process as arrival process, and we provide a complete analytical description of the system. Although the arrival process is very similar to the Poisson process, due to negative autocorrelation the resulting queue is very different from the Poisson case. We found interesting connections of this model with the statistical mechanics of Fermi particles. This model is motivated by air traffic systems.