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 Stochastic Systems > Vol. 7 (2017) open journal systems 


Waves in a spatial queue

David Aldous, UC Berkeley


Abstract
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 \(\mathbb{P} (W > k)\) 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.

AMS 2000 subject classifications: Primary 60K25; secondary 60J05, 60J70.

Keywords: Coalescing Brownian motion, scaling limit, spatial queue.

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Aldous, David, Waves in a spatial queue, Stochastic Systems, 7, (2017), 197-236 (electronic). DOI: 10.1214/15-SSY208.

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