Open Access
2017 Waves in a spatial queue
David Aldous
Stoch. Syst. 7(1): 197-236 (2017). DOI: 10.1214/15-SSY208


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.


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David Aldous. "Waves in a spatial queue." Stoch. Syst. 7 (1) 197 - 236, 2017.


Received: 1 November 2015; Published: 2017
First available in Project Euclid: 26 May 2017

zbMATH: 1364.60108
MathSciNet: MR3663341
Digital Object Identifier: 10.1214/15-SSY208

Primary: 60K25
Secondary: 60J05 , 60J70

Keywords: coalescing Brownian motion , Scaling limit , spatial queue

Vol.7 • No. 1 • 2017
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