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Scaling limits for shortest path lengths along the edges of stationary tessellations

Part of: Limit theorems Stochastic processes Geometric probability and stochastic geometry Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Florian Voss ,
Catherine Gloaguen  and
Volker Schmidt
Florian Voss*
Affiliation:
Ulm University
Catherine Gloaguen*
Affiliation:
Orange Labs
Volker Schmidt*
Affiliation:
Ulm University
*
∗ Current address: Medical Data Services, Boehringer Ingelheim Pharma GmbH & Co. KG, Binger Str. 173, 55216 Ingelheim, Germany. Email address: florian.voss@boehringer-ingelheim.com
∗∗ Postal address: Orange Labs, 38-40, rue du Général Leclerc, 92794 Issy-les-Moulineaux, France.
∗∗∗ Postal address: Insitute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, Germany.
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Abstract

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We consider spatial stochastic models, which can be applied to, e.g. telecommunication networks with two hierarchy levels. In particular, we consider Cox processes XL and XH concentrated on the edge set T(1) of a random tessellation T, where the points XL,n and XH,n of XL and XH can describe the locations of low-level and high-level network components, respectively, and T(1) the underlying infrastructure of the network, such as road systems, railways, etc. Furthermore, each point XL,n of XL is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of XH. We investigate the typical shortest path length C* of the resulting marked point process, which is an important characteristic in, e.g. performance analysis and planning of telecommunication networks. In particular, we show that the distribution of C* converges to simple parametric limit distributions if a scaling factor κ converges to 0 or ∞. This can be used to approximate the density of C* by analytical formulae for a wide range of κ.

Keywords

Stochastic geometryrandom geometric graphCox processPalm distributionPoisson approximationuniform integrabilitysubadditive ergodic theoremBlaschke-Petkantschin formulatelecommunication network

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60F99: None of the above, but in this section 90B15: Network models, stochastic
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 936 - 952
Copyright
Copyright © Applied Probability Trust 2010 

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