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Bounded-hop percolation and wireless communication

Part of: Geometric probability and stochastic geometry Special processes

Published online by Cambridge University Press:  24 October 2016

Christian Hirsch
Christian Hirsch*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany. Email address: christian.hirsch@wias-berlin.de
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Abstract

Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

Keywords

Ad hoc networkchemical distanceconnection probabilitycontinuum percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 833 - 845
Copyright
Copyright © Applied Probability Trust 2016 

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