Lenses in Minkowski space optimize queues by hiding bottleneck units

Abstract

We mathematically construct focal lenses for positive-mass particles in Minkowski space. Particles emanating from a point source, refocus with synchronized proper time regardless of initial direction and velocity. Minkowski space offers a useful representation of the passenger queue formed during airplane boarding. The blocking relations between passengers coincide with the past-future relation in spacetime, and the boarding time is determined by the longest blocking chain. All blocking chains can be kept equally long if slow passengers are placed according to a lens in Minkowski space. The two media of the lens correspond to a grouping of fast and slow passengers, and the lens specifies the optimal placement of fast and slow passengers in the queue at the gate. Not only the boarding time, but also its variance is minimized.