Abstract
We consider random geometric models for telecommunication access networks and analyse their serving zones which can be given, for example, by a class of so-called Cox–Voronoi tessellations (CVTs). Such CVTs are constructed with respect to locations of network components, the nucleii of their induced cells, which are scattered randomly along lines induced by a Poisson line process. In particular, we consider two levels of network components and investigate these hierarchical models with respect to mean shortest path length and mean subscriber line length, respectively. We explain point-process techniques which allow for these characteristics to be computed without simulating the locations of lower-level components. We sustain our results by numerical examples which were obtained through Monte Carlo simulations, where we used simulation algorithms for typical Cox–Voronoi cells derived in a previous paper. Also, briefly, we discuss tests of correctness of the implemented algorithms. Finally, we present a short outlook to possible extensions concerning multi-level models and iterated random tessellations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baccelli F, Blaszczyszyn B (2001) On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation. Adv Appl Probab 33:293–323
Baccelli F, Gloaguen C, Zuyev S (2000) Superposition of planar Voronoi tessellations. Commun Stat Series Stoch Models 16:69–98
Baccelli F, Klein M, Lebourges M, Zuyev S (1996) Géométrie aléatoire et architecture de réseaux. Ann Télécommun 51:158–179
Baccelli F, Kofman, D, Rougier JL (1999) Self organizing hierarchical multicast trees and their optimization. In: Proceedings of IEEE infocom ’99, New York, pp 1081–1089
Baccelli F, Zuyev S (1996) Poisson–Voronoi spanning trees with applications to the optimization of communication networks. Oper Res 47:619–631
Błaszczyszyn B, Schott R (2003) Approximate decomposition of some modulated Poisson–Voronoi tessellations. Adv Appl Probab 35:847–862
Błaszczyszyn B, Schott R (2005) Approximations of functionals of some modulated Poisson–Voronoi tessellations with applications to modeling of communication networks. Japan J Ind Appl Math 22(2):179–204
Daley DJ, Vere-Jones D (1988) An introduction to the theory of point processes. Springer, New York
Diestel R (1997) Graph theory. Springer, New York
Gloaguen C, Coupé P, Maier R, Schmidt V (2002) Stochastic modelling of urban access networks. In: Proc. Networks 2002, June 23–27, Munich, Germany, pp 99–104
Gloaguen C, Fleischer F, Schmidt H, Schmidt V (2005) Simulation of typical Cox–Voronoi cells with a special regard to implementation tests. Math Methods Oper Res 62(3):357–373
Gloaguen C, Fleischer F, Schmidt H, Schmidt V (2006) Fitting of stochastic telecommunication network models, via distance measures and Monte-Carlo tests. Telecommun Syst 31:353–377
Jungnickel D (1997) Graphs, networks and algorithms. Springer, Berlin
Maier R (2003) Iterated random tessellations with applications in spatial modelling of telecommunication networks. Doctoral Dissertation, University of Ulm
Maier R, Schmidt V (2003) Stationary iterated tessellations. Adv Appl Probab 35:337–353
Mayer J, Guderlei R (2004) Test oracles and randomness. Lecture notes in informatics, P-58. Köllen Druck+Verlag GmbH, Bonn, pp 179–189
Mayer J, Schmidt V, Schweiggert F (2004) A unified simulation framework for spatial stochastic models. Simulation Modelling Practice and Theory 12:307–326
Miyazawa M (1995) Note on generalizations of Mecke’s formula and extensions of H=λ G. J Appl Probab 32:105–122
Neveu J (1976) Processus ponctuels. In: Dold A, Eckman B (eds) Lecture notes in mathematics, vol. 598. Springer, Berlin, pp 249–445
Schneider R, Weil W (2000) Stochastische Geometrie. Teubner, Stuttgart
Serfozo R (1999) Introduction to stochastic networks. Springer, Berlin
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd edn. Wiley, Chichester
Tchoumatchenko K, Zuyev S (2001) Aggregate and fractal tessellations. Probab Theory Relat Fields 121:198–218
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gloaguen, C., Fleischer, F., Schmidt, H. et al. Analysis of Shortest Paths and Subscriber Line Lengths in Telecommunication Access Networks. Netw Spat Econ 10, 15–47 (2010). https://doi.org/10.1007/s11067-007-9021-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-007-9021-z