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

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.