Understanding interarrival and interdeparture time statistics from interactions in queuing systems

We discuss the statistics of long queues, in which the interdeparture time statistics is dominated by spatial interactions among the elements in a queue rather than the arrival or exit processes. Based on a Fokker–Planck approach, it is possible to calculate the stationary distance distribution among the elements in a queue as a function of their interaction potential. The results relate to the ones known from Random Matrix Theory. Together with the velocity distribution, one can determine the time-gap distribution as well. This yields an analytical approach to the interdeparture and interarrival time distributions of queuing systems with spatially interacting elements. While these distributions are usually determined from empirical data or from theoretical assumptions about the arrival or exit process, we offer here an alternative interpretation of interdeparture time distributions as an effect of interactions in a queue. This is relevant for the understanding of traffic and production systems and for the optimization of the statistical behavior of some queuing systems.