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

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

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 

References

Aldous, D. J. and Kendall, W. S. (2008). Short-length routes in low-cost networks via Poisson line patterns. Adv. Appl. Prob. 40, 1–21.Google Scholar
Baccelli, F. and Zuyev, S. (1996). Poisson–Voronoi spanning trees with applications to the optimization of communication networks. Operat. Res. 47, 619–631.Google Scholar
Baccelli, F., Gloaguen, C. and Zuyev, S. (2000). Superposition of planar Voronoi tessellations. Commun. Statist. Stoch. Models 16, 69–98.CrossRefGoogle Scholar
Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). Markov paths on the Poisson–Delaunay graph with applications to routing in mobile networks. Adv. Appl. Prob. 32, 1–18.CrossRefGoogle Scholar
Baccelli, F., Klein, M., Lebourges, M. and Zuyev, S. (1997). Stochastic geometry and architecture of communication networks. Telecommun. Systems 7, 209–227.Google Scholar
Bauer, H. (1981). Probability Theory and Elements of Measure Theory, 2nd edn. Academic Press, London.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Fleischer, F., Gloaguen, C., Schmidt, V. and Voss, F. (2009). Simulation of the typical Poisson–Voronoi–Cox–Voronoi cell. J. Statist. Comput. Simul. 79, 939–957.Google Scholar
Gloaguen, C., Voss, F. and Schmidt, V. (2009). Parametric distance distributions for fixed access network analysis and planning. In Proc. 21st Internat. Teletraffic Congress (Paris, September 2009), pp. 1–8.Google Scholar
Gloaguen, C., Coupé, P., Maier, R. and Schmidt, V. (2002). Stochastic modelling of urban access networks. In Proc. 10th Internat. Telecommun. Network Strategy Planning Symp. (Munich, June 2002), VDE, Berlin, pp. 99–104.Google Scholar
Gloaguen, C., Fleischer, F., Schmidt, H. and Schmidt, V. (2005). Simulation of typical Cox–Voronoi cells with a special regard to implementation tests. Math. Meth. Operat. Res. 62, 357–373.Google Scholar
Gloaguen, C., Fleischer, F., Schmidt, H. and Schmidt, V. (2006). Fitting of stochastic telecommunication network models via distance measures and Monte–Carlo tests. Telecommun. Systems 31, 353–377.Google Scholar
Gloaguen, C., Fleischer, F., Schmidt, H. and Schmidt, V. (2010). Analysis of shortest paths and subscriber line lengths in telecommunication access networks. Networks Spatial Econom. 10, 15–47.Google Scholar
Haenggi, M. (2005). On distances in uniformly random networks. IEEE Trans. Inf. Theory 51, 3584–3586.CrossRefGoogle Scholar
Haenggi, M. et al. (2009). Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE J. Sel. Areas Commun. 27, 1029–1046.Google Scholar
Jensen, E. B. V. (1998). Local Stereology. World Scientific, River Edge, NJ.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Prob. 1, 883–909.CrossRefGoogle Scholar
Lachièze-Rey, R. (2009). Strong mixing property for STIT tessellations. Preprint. Available at http://arxiv.org/abs/0905.1145v3.Google Scholar
Lautensack, C. and Zuyev, S. (2008). Random Laguerre tessellations. Adv. Appl. Prob. 40, 630–650.Google Scholar
Maier, R. and Schmidt, V. (2003). Stationary iterated tessellations. Adv. Appl. Prob. 35, 337–353.CrossRefGoogle Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.Google Scholar
Nagel, W. and Weiss, V. (2005). Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob. 37, 859–883.Google Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Tchoumatchenko, K. and Zuyev, S. (2001). Aggregate and fractal tessellations. Prob. Theory Relat. Fields 121, 198–218.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.Google Scholar
Voss, F., Gloaguen, C. and Schmidt, V. (2009). Capacity distributions in spatial stochastic models for telecommunication networks. Image Anal. Stereology 28, 155–163.CrossRefGoogle Scholar
Voss, F., Gloaguen, C. and Schmidt, V. (2009). Palm calculus for stationary Cox processes on iterated random tessellations. In Proc. 7th Internat. Symp. Modeling and Optimization of Mobile, Ad Hoc and Wireless Networks, IEEE Press, Piscataway, NJ, pp. 534–539.Google Scholar
Voss, F., Gloaguen, C. and Schmidt, V. (2009). Scaling limits for shortest path lengths along the edges of stationary tessellations – Supplementary material. Preprint. Available at http://arxiv.org/abs/0912.4516v1.Google Scholar
Voss, F., Gloaguen, C., Fleischer, F. and Schmidt, V. (2010). Densities of shortest path lengths in spatial stochastic networks. To appear in Stoch. Models.Google Scholar
Voss, F., Gloaguen, C., Fleischer, F. and Schmidt, V. (2009). Distributional properties of Euclidean distances in wireless networks involving road systems. IEEE J. Sel. Areas Commun. 27, 1047–1055.Google Scholar
Weiss, V. and Nagel, W. (1999). Interdependences of directional quantities of planar tessellations. Adv. Appl. Prob. 31, 664–678.Google Scholar
Zuyev, S. (1999). Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar
Zuyev, S. (2009). Stochastic geometry and telecommunications networks. In New Perspectives in Stochastic Geometry, eds Kendall, W. S. and Molchanov, I., Oxford University Press, pp. 520–554.Google Scholar