## Stochastic Systems

- Stoch. Syst.
- Volume 7, Number 1 (2017), 197-236.

### Waves in a spatial queue

#### 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.

#### Article information

**Source**

Stoch. Syst. Volume 7, Number 1 (2017), 197-236.

**Dates**

Received: November 2015

First available in Project Euclid: 26 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ssy/1495785620

**Digital Object Identifier**

doi:10.1214/15-SSY208

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Keywords**

Coalescing Brownian motion scaling limit spatial queue

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Aldous, David. Waves in a spatial queue. Stoch. Syst. 7 (2017), no. 1, 197--236. doi:10.1214/15-SSY208. https://projecteuclid.org/euclid.ssy/1495785620.