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Queueing Systems

, Volume 71, Issue 1–2, pp 199–220 | Cite as

On the distribution of typical shortest-path lengths in connected random geometric graphs

  • D. Neuhäuser
  • C. Hirsch
  • C. Gloaguen
  • V. SchmidtEmail author
Article

Abstract

Stationary point processes in ℝ2 with two different types of points, say H and L, are considered where the points are located on the edge set G of a random geometric graph, which is assumed to be stationary and connected. Examples include the classical Poisson–Voronoi tessellation with bounded and convex cells, aggregate Voronoi tessellations induced by two (or more) independent Poisson processes whose cells can be nonconvex, and so-called β-skeletons being subgraphs of Poisson–Delaunay triangulations. The length of the shortest path along G from a point of type H to its closest neighbor of type L is investigated. Two different meanings of “closeness” are considered: either with respect to the Euclidean distance (e-closeness) or in a graph-theoretic sense, i.e., along the edges of G (g-closeness). For both scenarios, comparability and monotonicity properties of the corresponding typical shortest-path lengths C e and C g are analyzed. Furthermore, extending the results which have recently been derived for C e, we show that the distribution of C g converges to simple parametric limit distributions if the edge set G becomes unboundedly sparse or dense, i.e., a scaling factor κ converges to zero and infinity, respectively.

Keywords

Point process Aggregate tessellation β-skeleton Shortest path Palm mark distribution Stochastic monotonicity Scaling limit 

Mathematics Subject Classification (2000)

60D05 60G55 60F99 90B15 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • D. Neuhäuser
    • 1
  • C. Hirsch
    • 1
  • C. Gloaguen
    • 2
  • V. Schmidt
    • 1
    Email author
  1. 1.Institute of StochasticsUlm UniversityUlmGermany
  2. 2.Orange LabsIssy-les-MoulineauxFrance

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