Airplane boarding meets express line queues

Airlines have been implementing several policies which partition passengers into groups of fast and slow passengers and placing them in different queue/row combinations. For example, the policy which provides boarding precedence to passengers needing assistance and families traveling with small children, or a policy which provides precedence to passengers with no overhead bin luggage. The idea of separating customers into groups with different service times also appears in the very different setting of express line queues, for example, in the supermarket or in server farms. Such queues have been extensively explored in the last 20 years in the queueing theoretic literature. We show that the two systems, airplane boarding with slow and fast passenger groups and express line queues are intimately related in the asymptotic regime where the ratio between the slowest and fastest customer becomes large. We produce good algorithms for placing different groups of passengers in the airplane boarding setting with competitive guarantees compared to optimal placements. We then show how we can analyze the airplane boarding setting using results from the express line setting. This relation provides a novel bridge between the theory of project management and the critical path method to which airplane boarding belongs and queueing theory to which express line queues belong. The analysis uses very basic notions and arguments from geometry, but in the setting of space-time (Lorentzian) geometry, a geometry which was invented to study relativity theory.