Out-of-equilibrium effects for queueing system with prescheduled random arrivals

In this paper we present a model to describe inbound congestion in air tra c systems. The model is based on an arrival process which adds a random delay to each arrival time of an homogeneous schedule. The deviation of the delay is much larger than the expected time between two consecutive arrivals 1 . The fact that groups of aircraft may request to land with very small interarrival times, due to the randomness of the system, causes congestion, with consequent additional queueing delay of aircraft to be spent in holding paths. This model has been proposed numerically by M. Ball and collaborators. From an experimental point of view the arrival process de ned by the model above causes a congestion that is quite similar to the observed one. This study represents an improvement of the state-of-art in this eld, mainly for the following reasons. First, we have been able to provide a complete theoretical control of the system and to do this it has been natural to split the description of the system on two di erent time scales, and this gives an additional insight to the problem. In particular the evolution of congestion, even in non-equilibrium conditions, can be viewed as the superposition of two processes. A relatively simple process that evolves on time scales of the order of the standard deviation of the delay of the aircraft (slow varying process) and a more complicate process, evolving on a time scale of the order of the landing time, and hence much faster than the rst one. The slow process can be identi ed in terms of easily measurable quantities (the number of aircraft that are arriving with a positive delay and the number of aircraft that are anticipating their arrival) and it is the relevant quantity in order to compute the average congestion in the system. The fast process describes the deviation from the average length of the queue, and it is important to evaluate the safety problems due to rare but observable peaks of tra c. All this description is absolutely non standard in the classical approach to this problem, that is based on Poisson arrivals hypothesis and/or on purely numerical approach. The second reason for considering the proposed model as state-of-art is that it bene ts of concepts and ideas coming from totally di erent elds, namely equilibrium and non- equilibrium statistical mechanics. In particular the similarity of our model of arrivals with the Poisson model on short time scales is related to the evolution of the many particle systems starting from non equilibrium conditions, while the complete control of the fast process mentioned above, which is quite complicated, has been possible due to its similarity with the computations related with a particular class of systems of particles obeying an exclusion principle. In the paper the comparison between the exact results of our model and the observed data is discussed, and a detailed control of the non-equilibrium features of the system is also 1 presented. The latter is important when one wants to control the transient times, i.e., the time needed in order to return to a stationary distribution once the system is perturbed by a shock (e.g., a sudden reduction of arrival capacity at airports and/or unforeseen peak of demand)