Waves in a Spatial Queue: Stop-and-Go at Airport Security

TL;DR

This paper models stop-and-go waves in a human queue at airport security using a continuous-space model, revealing how movement waves propagate and decay over distance.

Contribution

It introduces a novel continuous-space queue model capturing wave phenomena and provides asymptotic analysis relating wave size to Brownian motion.

Findings

Probability of large waves decreases as order k^{-1/2}

On average, the k'th customer moves once every order k^{1/2} service times

Wave behavior is linked to coalescing Brownian motion

Abstract

We model a long queue of humans by a continuous-space model in which, when a customer moves forward, they stop a random distance behind the previous customer, but do not move at all if their distance behind the previous customer is below a threshold. The latter assumption leads to ``waves" of motion in which only some random number $W$ of customers move. We prove that $\Pr(W > k)$ decreases as order $k^{-1/2}$; in other words, for large $k$ the $k$'th customer moves on average only once every order $k^{1/2}$ service times. A more refined analysis relies on a non-obvious asymptotic relation to the coalescing Brownian motion process; we give a careful outline of such an analysis without attending to all the technical details.