Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (1) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process inwith density ρ, and add the origin as an extra point. Now connect any two pointsxandyof the process with probabilityg(x − y), independently of the point process and all other pairs, wheregis a function which depends only on the Euclidean distance betweenxandy, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if, then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that ifgis of the form, for somer > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (01) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (2) ◽  
pp. 455-470 ◽  
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (02) ◽  
pp. 455-470
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

Anisotropic Growth of Voronoi Cells

1994 ◽  
Vol 26 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Poisson Process ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Simple Extension ◽  
Anisotropic Growth ◽  
Homogeneous Poisson Process ◽  
Translation Invariant ◽  
Voronoi Cells ◽  
Invariant Distance ◽  
The Mean

This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.

scholarly journals Percolation in the signal to interference ratio graph

2006 ◽  
Vol 43 (2) ◽  
pp. 552-562 ◽  
Author(s):  
Olivier Dousse ◽  
Massimo Franceschetti ◽  
Nicolas Macris ◽  
Ronald Meester ◽  
Patrick Thiran
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Euclidean Distance ◽  
Critical Density ◽  
Continuum Percolation ◽  
Two Dimensional ◽  
Radio Communications ◽  
Communications Networks ◽  
Infinite Range ◽  
Independent Model

Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communications networks. Our main result shows that, despite the infinite-range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).

scholarly journals Percolation in the signal to interference ratio graph

2006 ◽  
Vol 43 (02) ◽  
pp. 552-562 ◽  
Author(s):  
Olivier Dousse ◽  
Massimo Franceschetti ◽  
Nicolas Macris ◽  
Ronald Meester ◽  
Patrick Thiran
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Euclidean Distance ◽  
Critical Density ◽  
Continuum Percolation ◽  
Two Dimensional ◽  
Radio Communications ◽  
Communications Networks ◽  
Infinite Range ◽  
Independent Model

Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communications networks. Our main result shows that, despite the infinite-range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).

Anisotropic Growth of Voronoi Cells

1994 ◽  
Vol 26 (01) ◽  
pp. 43-53 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Poisson Process ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Simple Extension ◽  
Anisotropic Growth ◽  
Homogeneous Poisson Process ◽  
Translation Invariant ◽  
Voronoi Cells ◽  
Invariant Distance ◽  
The Mean

This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.

SEGMENTED POINT PROCESS MODELS FOR MAINTAINED SYSTEMS

2007 ◽  
Vol 14 (05) ◽  
pp. 431-458 ◽  
Author(s):  
A. SYAMSUNDAR ◽  
V. N. A. NAIKAN
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Single Point ◽  
Process Models ◽  
Parameter Estimates ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stable Conditions ◽  
Point Process Models

A maintained system is generally modeled using point processes. The most common processes used are the renewal process and the non homogeneous Poisson process corresponding to maximal and minimal repair situations with homogeneous Poisson process being a special case of both. A general repair formulation with a factor indicating the degree of repair is introduced into the minimal repair model to form an Arithmetic Reduction of Intensity model. These processes are generally able to model maintained systems with a fair degree of accuracy when the system is operating under stable conditions. However whenever there is a change in the environment these models which are monotonic in nature are not able to accommodate this change. Such systems operating under different environments need to be modelled by segmented models with the system domain divided into segments at the points of changes in the environment. The individual segments can then be modeled by any of the above point process models and these can be combined to form a composite model. This paper proposes a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such phenomena can be obtained and such models are shown to more accurately describe the maintained system in a changing environment than the single point process models usually used.

Applying the Anderson-Darling Test to Suicide Clusters

Crisis ◽  
2013 ◽  
Vol 34 (6) ◽  
pp. 434-437 ◽  
Author(s):  
Donald W. MacKenzie
Keyword(s):  
Poisson Process ◽  
Goodness Of Fit ◽  
Small Samples ◽  
Massachusetts Institute Of Technology ◽  
Homogeneous Poisson Process ◽  
Cornell University ◽  
The Difference ◽  
Suicide Clusters ◽  
Institute Of Technology ◽  
Anderson Darling

Background: Suicide clusters at Cornell University and the Massachusetts Institute of Technology (MIT) prompted popular and expert speculation of suicide contagion. However, some clustering is to be expected in any random process. Aim: This work tested whether suicide clusters at these two universities differed significantly from those expected under a homogeneous Poisson process, in which suicides occur randomly and independently of one another. Method: Suicide dates were collected for MIT and Cornell for 1990–2012. The Anderson-Darling statistic was used to test the goodness-of-fit of the intervals between suicides to distribution expected under the Poisson process. Results: Suicides at MIT were consistent with the homogeneous Poisson process, while those at Cornell showed clustering inconsistent with such a process (p = .05). Conclusions: The Anderson-Darling test provides a statistically powerful means to identify suicide clustering in small samples. Practitioners can use this method to test for clustering in relevant communities. The difference in clustering behavior between the two institutions suggests that more institutions should be studied to determine the prevalence of suicide clustering in universities and its causes.

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