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Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

Part of: Geometric probability and stochastic geometry Stochastic processes Graph theory Markov processes

Published online by Cambridge University Press:  01 July 2016

F. Baccelli ,
K. Tchoumatchenko  and
S. Zuyev
F. Baccelli*
Affiliation:
École Normale Supérieure
K. Tchoumatchenko*
Affiliation:
École Normale Supérieure
S. Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: École Normale Supérieure, Dept. of Mathematics and Computer Science, 45 rue d'Ulm, 75230 Paris cedex 05, France. Email address: [email protected]
∗∗ Postal address: France Telecom CNET/DAC/OAT, 38-40 rue du General Leclerc, 92 794 Essy-les-Moulineaux, Cedex 9, France. Email address: [email protected]
∗∗∗ Postal address: Statistics and Modelling Science Dept., University of Strathclyde, Livingston Tower, 26 Richmond St., Glasgow, G1 1XH, UK. Email address: [email protected]
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Abstract

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

Keywords

Poisson processDelaunay triangulationVoronoi tessellationshortest pathfirst-passage percolationrouteingmobile networks

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60G55: Point processes 05C12: Distance in graphs 05C38: Paths and cycles
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 1 - 18
Copyright
Copyright © Applied Probability Trust 2000 

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