Waves in a Spatial Queue

Envisaging a physical queue of humans, we model a long queue 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 ℙ(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.