Analysis of aeroplane boarding via spacetime geometry and random matrix theory

Abstract

We show that airplane boarding can be asymptotically modeled by 2-dimensional Lorentzian geometry. Boarding time is given by the maximal proper time among curves in the model. Discrepancies between the model and simulation results are closely related to random matrix theory. We then show how such models can be used to explain why some commonly practiced airline boarding policies are ineffective and even detrimental.Airplane boarding is a process experienced daily by millions of passengers worldwide. Airlines have developed various strategies in the hope of shortening boarding time, typically leading to announcements of the form "passengers from rows 40 and above are now welcome to board the plane", often heard around airport terminals. We will show how the airplane boarding process can be asymptotically modeled by spacetime geometry. The discrepancies between the asymptotic analysis and finite population results will be shown to be closely related to random matrix theory (RMT). Previously, airplane boarding has only been analyzed via discrete event simulations [1,2,3].We model the boarding process as follows: Passengers 1, ..., N are represented by coordinates X i = (q i , r i ), where q i is the index of the passenger along the boarding queue (1st, 2nd, 3rd and so on), and r is his/her assigned row number. We rescale (q, r) to [0, 1] × [0, 1]. It is assumed that the main cause of delay in airplane boarding is the time it takes passengers to organize their luggage and seat themselves once they have arrived at their assigned row. The input parameters for our model are:u -Average amount of aisle length occupied by a passenger.w -Distance between successive rows. b -Number of passengers per row. D -Amount of time (delay) it takes a passenger to clear the aisle, once he has arrived at his designated row. p(q, r) -The joint distribution of a passenger's row and queue joining time. p(q, r) is directly affected by the airline policy and the way passengers react to the policy.For the purposes of presentation, we shall assume that u, w, b, D are all fixed. The airplane boarding process produces a natural partial order relation of blocking among passengers. Passenger X blocks passenger Y if it is impossible for passenger Y to reach his assigned row before passenger X (and others blocked by X) has sat down and cleared the aisle. Airplane boarding functions as a peeling process for the partial order defined by the blocking relation. At first, passengers who are not blocked by others sit down; these passengers are the minimal elements under the blocking relation. In the second round, passengers who are not blocked by passengers other than those of the first round are seated, and so forth. Boarding time thus coincides with the size of the longest chain in the partial order.We assign to the boarding process with parameters u, b, w, D, p(q, r) a Lorentz metric defined on the (q, r) unit square bywhere k = bu/w and α(q, r) = 1 r p(q, z)dz. There are two properties of the metric which relate it to the boarding process:• The volume...

Analysis of Airplane Boarding Times

“…The analysis carried out in Bachmat et al (2005Bachmat et al ( , 2006 suggests that, as the number of passengers n becomes large, t ≈ T √ n. More precisely, for any ε > 0, with probability approaching 1 as n −→ ∞, 1 − ε < t T √ n < 1 + ε.…”

“…We will recall the basic facts about this case from Bachmat et al (2005Bachmat et al ( , 2006 and state a few more which will serve us later. When there is no boarding policy, the joint row/queue position distribution is uniform, namely, p(q, r) = 1.…”

“…An analytical model which takes airplane design parameters into account was recently introduces in Bachmat et al (2005Bachmat et al ( , 2006. The authors express the expected boarding time T , as the number of passengers is large, in terms of the solution to a variational problem.…”

“…The variational problem turns out to have a beautiful geometric interpretation in terms of spacetime (Lorentzian) geometry. A major issue that is not addressed in Bachmat et al (2005Bachmat et al ( , 2006 is the validation of the model against detailed simulations of the boarding process. This is particularly important since the model makes many simplifying assumptions.…”

“…In Section 3 we present, following Bachmat et al (2005Bachmat et al ( , 2006, a constrained variational problem whose solution provides the expected critical path and boarding time in the limit as the number of passengers tends to infinity. The input to the variational problem is a probability measure p, determined by the airline boarding policy and a congestion parameter k which is related to the interior design of the airplane.…”