Abstract
We study airplane boarding in the limit of a large number of passengers using geometric optics in a Lorentzian metric. The airplane boarding problem is naturally embedded in a (1+1)-dimensional space-time with a flat Lorentzian metric. The duration of the boarding process can be calculated based on a representation of the one-dimensional queue of passengers attempting to reach their seats in a two-dimensional space-time diagram. The ability of a passenger to delay other passengers depends on their queue positions and row designations. This is equivalent to the causal relationship between two events in space-time, whereas two passengers are timelike separated if one is blocking the other and spacelike if both can be seated simultaneously. Geodesics in this geometry can be utilized to compute the asymptotic boarding time, since space-time geometry is the many-particle (passengers) limit of airplane boarding. Our approach naturally leads to the introduction of an effective refractive index that enables an analytical calculation of the average boarding time for groups of passengers with different aisle-clearing time distribution. In the past, airline companies attempted to shorten the boarding times by trying boarding policies that allow either slow or fast passengers to board first. Our analytical calculations, backed by discrete-event simulations, support the counterintuitive result that the total boarding time is shorter with the slow passengers boarding before the fast passengers. This is a universal result, valid for any combination of the parameters that characterize the problem: the percentage of slow passengers, the ratio between aisle-clearing times of the fast and the slow group, and the density of passengers along the aisle. We find an improvement of up to 28% compared with the fast-first boarding policy. Our approach opens up the possibility to unify numerous simulation-based case studies under one framework.
Highlights
A main theme in statistical physics is the connection between the microscopic dynamics of an ensemble of interacting particles or units and the macroscopic observables
The microscopic level is the structure of the passenger queue and the main macroscopic observable is the required time for all N passengers to get settled in their assigned seats
The third parameter C is the ratio of the aisle-clearing time of the fast passengers to the aisle-clearing time of the slow passengers
Summary
A main theme in statistical physics is the connection between the microscopic dynamics of an ensemble of interacting particles or units and the macroscopic observables. In this paper we investigate two simple group-based policies that can bring us closer to a near-optimal solution, which is practical, namely, passengers who tend to use more time to complete the seating are separated from the others and can be requested to enter the airplane either before or after the remaining passengers. One such type of passengers are those with overhead bin luggage. We show by simulations that the large-N limit results hold for a realistic number of passengers
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