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

Waves in a spatial queue

David Aldous, UC Berkeley

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.


[1]    R. A. Arratia. Coalescing Brownian motions on the line. PhD thesis, University of Wisconsin–Madison, 1979. MR2630231

[2]    P. Billingsley. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR0233396

[3]    M. Blank. Metric properties of discrete time exclusion type processes in continuum. J. Stat. Phys., 140(1):170–197, 2010. MR2651444

[4]    A. Borodin, P. L. Ferrari, M. Prähofer, and Tomohiro Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys., 129(5–6):1055–1080, 2007. MR2363389

[5]    D. Bullock, R. Haseman, J. Wasson, and R. Spitler. Automated measurement of wait times at airport security. Transportation Research Record: Journal of the Transportation Research Board, 2177:60–68, 2010.

[6]    D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. II. Probability and its Applications (New York). Springer, New York, second edition, 2008. MR2371524

[7]    L. Gray and D. Griffeath. The ergodic theory of traffic jams. J. Statist. Phys., 105(3–4):413–452, 2001. MR1871652

[8]    L. Kleinrock. Queueing Systems. Volume 1: Theory. Wiley-Interscience, 1975. MR1383820

[9]    E. Schertzer, R. Sun, and J. M. Swart. The Brownian web, the Brownian net, and their universality. ArXiv 1506.00724, June 2015. MR2712322

[10]    B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields, 111(3):375–452, 1998. MR1640799

[11]    R. Tribe, S. K. Yip, and O. Zaboronski. One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes. Electron. Commun. Probab., 17(40):7, 2012. MR2981896

[12]    R. Tribe and O. Zaboronski. Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab., 16(76):2080–2103, 2011. MR2851057

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