Distributional properties of the typical cell of stationary iterated tessellations

Maier, Roland; Mayer, Johannes; Schmidt, Volker

doi:10.1007/s001860300326

Cited by 19 publications

(12 citation statements)

References 16 publications

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“…We emphasize that even for the simple street models considered in the present paper, the empirical distance densities, which have been computed in [6] from real network data, could be approximated very well by parametric densities. We expect even better results using more sophisticated street models like iterated tessellations ( [18]), which is a subject of ongoing research.…”

Section: Discussionmentioning

confidence: 99%

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Distributional properties of euclidean distances in wireless networks involving road systems

Voß

Gloaguen²,

Fleischer³

et al. 2009

IEEE J. Select. Areas Commun.

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Abstract-Stochastic models for hierarchical telecommunication networks are considered, which can be applied to the analysis and planning of large wireless networks. The network geometry is modelled by random geometric graphs, and the locations of network nodes by point processes on the edges of these graphs. In particular, the locations of high-level components ( Representation formulas are derived for the distribution function and density of the typical Euclidean connection distance between LLC and HLC. They lead to suitable estimators of these characteristics, which can be computed by Monte Carlo simulation of the typical serving zone and the typical segment system in it, respectively.

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Section: Discussionmentioning

confidence: 99%

“…Using the notation ψ = (ψ (1) , ψ (2) ) for the elements of N M1,M2 , Neveu's exchange formula takes the following form, see e.g. [18].…”

Section: Appendixmentioning

confidence: 99%

Distributional properties of euclidean distances in wireless networks involving road systems

Voß

Gloaguen²,

Fleischer³

et al. 2009

IEEE J. Select. Areas Commun.

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“…With this algorithm it is possible to simulate cells from which all distributional properties of the typical cell of Cox-Voronoi tessellations T X c can be obtained for Cox processes [4]) whose random driving measures depend on Boolean models ( [11]). In [9] a related technique is used in order to obtain distributional properties of the typical cell of iterated tessellations. There the typical cell is also not simulated directly, but cells are simulated from which the distribution of characteristics of the typical cell can be obtained.…”

Section: Neveu's Exchange Formula and An Indirect Simulation Algorithmmentioning

confidence: 99%

“…Using the notation introduced above, Neveu's exchange formula takes the following form (see e.g. [9]). …”

Section: Lemma 33mentioning

confidence: 99%

Simulation of the typical Poisson–Voronoi–Cox–Voronoi cell

Fleischer

Gloaguen

Schmidt³

et al. 2009

Journal of Statistical Computation and Simulation

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We consider stationary Poisson-Voronoi tessellations (PVT) in the Euclidean plane and study properties of Voronoi tessellations induced by linear Poisson processes on the edges of the PVT. We are especially interested in simulation algorithms for the typical cell. Two different simulation algorithms are introduced. The first algorithm directly simulates the typical cell whereas the second algorithm simulates cells from which distributional properties of the typical cell can be obtained. This second algorithm can also be used for simulating the typical cell of other Cox-Voronoi tessellations. The implementation of both algorithms is tested for their correctness using random software tests. Then different cell characteristics are studied by simulation and compared to the typical cell of Poisson-Voronoi tessellations and Cox-Voronoi tessellations based on linear Poisson processes on the lines of Poisson line processes. Our results can be applied for example in the analysis of telecommunication networks and vesicle paths on cytoskeletal networks.

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“…We give a formula for the distribution of Z with this center function, extending some related but less explicit results found in the literature (Miles [17], Ambartzumian [1, Chap. 9], Favis and Weiß [25], Mecke [14], Maier et al [11]). As did Calka [2], we use the SlivnyakMecke formula (following the terminology of Møller [21], see also [20]; a general version appears in Mecke [13]).…”

Section: The Typical Cell Of a Poisson Hyperplane Tessellationmentioning

confidence: 99%