Lorentzian-geometry-based analysis of airplane boarding policies highlights "slow passengers first" as better.

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.

Simulation of fingering behavior in smoldering combustion using a cellular automaton.

Smoldering is the slow, low-temperature, flameless burning of porous fuels and the most persistent type of combustion phenomena. It is a complex physical process that is not yet completely understood, but it is known that it is driven by heat transfer, mass transfer, and fuel chemistry. A specific case of high interest and complexity is fingering behavior. Fingering is an instability that occurs when a thin fuel layer burns against an oxygen current. These instabilities appear when conduction rather than convection is the dominant mode of heat transfer to the fuel ahead and the availability of oxygen is limited during the combustion of a thin fuel, such as paper. The pattern of the fingers can be characterized through the distance between them and their width, and can be classified into three different regimes: isolated fingers, tip-splitting fingers, or no fingers forming and a smooth continuous front. In this paper, a multilayer cellular automaton based on three governing principles (heat, oxygen, and fuel) is shown to reproduce all the regimes and the details of finger structures observed in previous experiments. It is shown how when oxygen is not limited, a smooth smoldering front is formed. If the oxygen speed decreases beyond a critical value, fingers appear first as tip-splitting fingers and later as isolated fingers, increasing the distance between them and decreasing their thickness. The oxygen consumed during oxidation influences these critical values with a positive correlation. This cellular automaton provides an alternative approach to simulate smoldering combustion in large systems over long times. That the model is able to reproduce the complex pattern formation seen in a fingering experiment validates the model. In the future, we could apply the model in various other geometries to make predictions on the outcome of smoldering combustion processes.

Time needed to board an airplane: a power law and the structure behind it.

A simple model for the boarding of an airplane is studied. Passengers have reserved seats but enter the airplane in arbitrary order. Queues are formed along the aisle, as some passengers have to wait to reach the seats for which they have reservation. We label a passenger by the number of his or her reserved seat. In most cases the boarding process is much slower than for the optimal situation, where passenger and seat orders are identical. We study this dynamical system by calculating the average boarding time when all permutations of N passengers are given equal weight. To first order, the boarding time for a given permutation (ordering) of the passengers is given by the number s of sequences of monotonically increasing values in the permutation. We show that the distribution of s is symmetric on [1,N], which leads to an average boarding time (N+1)/2. We have found an exact expression for s and have shown that the full distribution of s approaches a normal distribution as N increases. However, there are significant corrections to the first-order results, due to certain correlations between passenger ordering and the substrate (seat ordering). This occurs for some cases in which the sequence of the seats is partially mirrored in the passenger ordering. These cases with correlations have a boarding time that is lower than predicted by the first-order results. The large number of cases with reduced boarding times have been classified. We also give some indicative results on the geometry of the correlations, with sorting into geometry groups. With increasing N, both the number of correlation types and the number of cases belonging to each type increase rapidly. Using enumeration we find that as a result of these correlations the average boarding time behaves like N(α), with α≃0.69, as compared with α=1.0 for the first-order approximation.

Velocity and cluster distributions in a bottleneck system.

Velocity and cluster distributions for particles with unidirectional motion in one dimension are studied. The particles never pass each other, like cars on a narrow road that does not allow overtaking. As a result, particles cluster behind slow particles (queues are formed behind slow cars). Thus, the actual velocity of each particle is to a large extent determined by slow particles further ahead. Considering all possible permutations of N particles with initial velocities {vi}, the average number of particles with actual velocity vi is (N+1)/[i(i+1)] (in the sequence {vi}, the initial velocities are listed with monotonically increasing values). For i large and vi proportional, variant i the average number of actual velocities is thus a power law in vi, even though the average cluster density is found to be independent of cluster size, L. On the other hand, the cluster density varies significantly with cluster velocity; we obtain [(N-i)!(N-L)!]/[NN!(N-L-i+1)!]. The average velocity at a given position in the sequence of N particles, and the average global velocity are determined. Explicit results for several distributions of the initial velocities show that the global velocity depends sensitively on the form of this distribution.