Abstract
We consider stationary Poisson line processes in the Euclidean plane and analyze properties of Voronoi tessellations induced by Poisson point processes on these lines. In particular, we describe and test an algorithm for the simulation of typical cells of this class of Cox–Voronoi tessellations. Using random testing, we validate our algorithm by comparing theoretical values of functionals of the zero cell to simulated values obtained by our algorithm. Finally, we analyze geometric properties of the typical Cox–Voronoi cell and compare them to properties of the typical cell of other well-known classes of tessellations, especially Poisson–Voronoi tessellations. Our results can be applied to stochastic–geometric modelling of networks in telecommunication and life sciences, for example. The lines can then represent roads in urban road systems, blood arteries or filament structures in biological tissues or cells, while the points can be locations of telecommunication equipment or vesicles, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Binder RV (2000) Testing object-oriented systems: models, patterns and tools. Addison-Wesley, Boston
Błaszczyszyn B, Schott R (2005) Approximations of functionals of some modulated–Poisson Voronoi tessellations with applications to modeling of communication networks. Jp J Ind Appl Math (to appear) 22(2) (INRIA Rap port RR-5323)
Casella C, Berger RL (2002) Statistical inference. Wadsworth, Duxbury
Gloaguen C, Coupé P, Maier R, Schmidt V (2002) Stochastic modelling of urban access networks. In: Proceedings of IEEE Infocom 2002, München
Gloaguen C, Fleischer F, Schmidt H, Schmidt V (2004) Fitting of stochastic telecommunication network models, via distance measures and Monte-Carlo tests. Preprint
Levene H (1960) Robust tests for the equality of variances. In: Olkin I (eds). Contributions to probability and statistics. Stanford University Press, Palo Alto, pp. 278–292
Maier R, Mayer J, Schmidt V (2004) Distributional properties of the typical cell of stationary iterated tessellations. Math Methods Oper Res 59:287–302
Mayer J, Guderlei R (2004) Test oracles and randomness. Lecture Notes Informatics P-58:179–189
Mayer J, Schmidt V, Schweiggert F (2004) A unified simulation framework for spatial stochastic models. Simul Model Pract Theory 12:307–326
Møller J (1989) Random tessellations in \(\mathbb{R}^d\). Adv Appl Probab 21:37–73
Okabe A, Boots B, Sugihara K, Chiu SN (2000) Spatial tessellations. 2nd ed. Wiley, Chichester
Schneider R, Weil W (2000) Stochastische geometrie. Teubner, Stuttgart
Schütz GJ, Axmann M, Freudenthaler S, Schindler H, Kandror K, Roder JC, Jeromin A (2004) Visualization of vesicle transport along and between distinct pathways in neurites of living cells. Microsc Res Tech 63:159–167
Sneed HM, Winter M (2002) Testen objektorientierter software, 1st edn. Hanser, München
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. 2nd edn. Wiley, Chichester
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gloaguen, C., Fleischer, F., Schmidt, H. et al. Simulation of typical Cox–Voronoi cells with a special regard to implementation tests. Math Meth Oper Res 62, 357–373 (2005). https://doi.org/10.1007/s00186-005-0036-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-005-0036-2