Air Traffic, Boarding and Scaling Exponents

Abstract

The air traffic is a very important part of the global transportation network. In distinction from vehicular traffic, the boarding of an airplane is a significant part of the whole transportation process. Here we study an airplane boarding model, introduced in 2012 by Frette and Hemmer, with the aim to determine precisely the asymptotic power–law scaling behavior of the mean boarding time 〈t _b 〉 and other related quantities for large number of passengers N. Our analysis is based on an exact enumeration for small system sizes N ≤ 14 and Monte Carlo simulation data for very large system sizes up to $N = 2^{16} = 65,536$. It shows that the asymptotic power–law scaling 〈t _b 〉 ∝ N ^α holds with the exponent $\alpha = 1/2$ (α = 0. 5001 ± 0. 0001). We have estimated also other exponents: $\nu = 1/2$ for the mean number of passengers taking seats simultaneously in one time step, β = 1 for the second moment of 〈t _b 〉 and γ ≈ 1∕3 for its variance. We have found also the correction–to–scaling exponent θ ≈ 1∕3 and have verified that a scaling relation $\gamma = 1 - 2\theta$, following from some analytical arguments, holds with a high numerical accuracy.