Abstract
We consider modulated Poisson-Voronoi tessellations, intended as models for telecommunication networks on a nationwide scale. By introducing an algorithm for the simulation of the typical cell of the latter tessellation, we lay the mathematical foundation for such a global analysis. A modulated Poisson-Voronoi tessellation has an intensity which is spatially variable and, hence, is able to provide a broad spectrum of model scenarios. Nevertheless, the considered tessellation model is stationary and we consider the case where the modulation is generated by a Boolean germ-grain model with circular grains. These circular grains may either have a deterministic or random but bounded radius. Furthermore, based on the introduced simulation algorithm for the typical cell and on Neveu’s exchange formula for Palm probability measures, we show how to estimate the mean distance from a randomly chosen location to its nearest Voronoi cell nucleus. The latter distance is interpreted as an important basic cost characteristic in telecommunication networks, especially for the computation of more sophisticated functionals later on. Said location is chosen at random among the points of another modulated Poisson process where the modulation is generated by the same Boolean model as for the nuclei. The case of a completely random placement for the considered location is thereby included as a special case. The estimation of the cost functional is performed in a way such that a simulation of the location placement is not necessary. Test methods for the correctness of the algorithm based on tests for random software are briefly discussed. Numerical examples are provided for characteristics of the typical cell as well as for the cost functional. We conclude with some remarks about extensions and modifications of the model regarded in this paper, like modulated Poisson—Delaunay tessellations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Baccelli and B. Błaszczyszyn, On a coverage process ranging from the Boolean model to the Poisson-Voronoi tessellation. Advances in Applied Probability,33 (2001), 293–323.
F. Baccelli, C. Gloaguen and S. Zuyev, Superposition of Planar Voronoi Tessellations. Communications in Statistics, Series Stochastic Models,16 (2000), 69–98.
F. Baccelli, M. Klein, M. Lebourges and S. Zuyev, Geéométrie aléatoire et architecture de réseaux. Annales des Télécommunication,51 (1996), 158–179.
F. Baccelli, M. Klein, M. Lebourges and S. Zuyev, Stochastic geometry and architecture of communication networks. Telecommunications Systems,7 (1997), 209–227.
F. Baccelli, D. Kofman and J.L. Rougier, Self organizing hierarchical multicast trees and their optimization. Proceedings of IEEE Infocom ’99, New York, 1999, 1081–1089.
F. Baccelli and S. Zuyev, Poisson-Voronoi spanning trees with applications to the optimization of communication networks. Operations Research,47 (1996), 619–631.
B. Błaszczyszyn and R. Schott, Approximate decomposition of some modulated Poisson-Voronoi tessellations. Advances in Applied Probability,35 (2003), 847–862.
B. Błaszczyszyn and R. Schott, Approximations of functionals of some modulated Poisson-Voronoi tessellations with applications to modeling of communication networks. Japan Journal of Industrial and Applied Mathematics,22 (2004), 179–204.
D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Springer, New York, 1988.
C. Gloaguen, P. Coupé, R. Maier and V. Schmidt, Stochastic modelling of urban access networks. Proc. 10th Internat. Telecommun. Network Strategy Planning Symp. (Munich, June 2002), VDE, Berlin, 2002, 99–104.
C. Gloaguen, F. Fleischer, H. Schmidt and V. Schmidt, Simulation of typical Cox-Voronoi cells with a special regard to implementation tests. Mathematical Methods of Operations Research,62 (2005), 357–373.
C. Gloaguen, F. Fleischer, H. Schmidt and V. Schmidt, Analysis of shortest paths and subscriber line lengths in telecommunication access networks. Networks and Spatial Economics,9 (2008), in print.
R. Maier, J. Mayer and V. Schmidt, Distributional properties of the typical cell of stationary iterated tessellations. Mathematical Methods of Operations Research,59 (2004), 287–302.
J. Mayer and R. Guderlei, Test oracles and randomness. Lecture Notes in Informatics,P-58, Köllen Druck+Verlag GmbH, Bonn, 2004, 179–189.
J. Mayer, V. Schmidt and F. Schweiggert, A unified simulation framework for spatial stochastic models. Simulation Modelling Practice and Theory,12 (2004), 307–326.
M.P. Quine and D.F. Watson, Radial generation ofn-dimensional Poisson processes. Journal of Applied Probability,21 (1984), 548–557.
R. Schneider and W. Weil, Stochastische Geometrie. Teubner, Stuttgart, 2000.
D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and Its Applications, 2nd edition. J. Wiley & Sons, Chichester, 1995.
K. Tchoumatchenko and S. Zuyev, Aggregate and fractal tessellations. Probability Theory Related Fields,121 (2001), 198–218.
J.G. Wendel, A problem in geometric probability. Math. Scand.,11 (1962), 109–111.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Fleischer, F., Gloaguen, C., Schmidt, H. et al. Simulation algorithm of typical modulated Poisson-Voronoi cells and application to telecommunication network modelling. Japan J. Indust. Appl. Math. 25, 305 (2008). https://doi.org/10.1007/BF03168553
Received:
Revised:
DOI: https://doi.org/10.1007/BF03168553