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We model a long queue of humans by a continuous-space model in which, when acustomer moves forward, they stop a random distance behind the previouscustomer, but do not move at all if their distance behind the previous customeris below a threshold. The latter assumption leads to ``waves" of motion inwhich 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 $k$'thcustomer moves on average only once every order $k^{1/2}$ service times. A morerefined analysis relies on a non-obvious asymptotic relation to the coalescingBrownian motion process; we give a careful outline of such an analysis withoutattending to all the technical details.

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