Skip to main content
SpringerLink
Log in

Joint distributions for total lengths of shortest-path trees in telecommunication networks

  • Published:
annals of telecommunications - annales des télécommunications Aims and scope Submit manuscript

Abstract

Shortest-path trees play an important role in the field of optimising fixed-access telecommunication networks with respect to costs and capacities. Distributional properties of the corresponding two half-trees originating from the root of such a tree are of special interest for engineers. In the present paper, we derive parametric approximation formulas for the marginal density functions of the total lengths of both half-trees. Besides, a parametric copula is used in order to combine the marginal distributions of these functionals to a bivariate joint distribution as, naturally, the total lengths of the half-trees are not independent random variables. Asymptotic results for infinitely sparse and infinitely dense networks are discussed as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Daley DJ, Vere-Jones D (2005) An introduction to the theory of point processes, vol I/II. Springer, New York

    Google Scholar 

  2. Fleischer F, Gloaguen C, Schmidt V, Voss F (2009) Simulation of the typical Poisson-Voronoi-Cox-Voronoi cell. J Stat Comput Simul 79:939–957

    Article  MATH  MathSciNet  Google Scholar 

  3. Friedman JH, Rafsky LC (1979) Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Ann Stat 7: 967–717

    Article  MathSciNet  Google Scholar 

  4. Gloaguen C, Voss F, Schmidt V (2011) Parametric distributions of connection lengths for the efficient analysis of fixed access network. Ann Telecommun 66:103–118

    Article  Google Scholar 

  5. Hirsch C, Neuhäuser D, Schmidt V Asymptotic properties of Euclidean shortest-path trees and applications to telecommunication networks. Working paper (under preparation)

  6. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, Chichester

    MATH  Google Scholar 

  7. Mai J-F, Scherer M (2012) Simulating copulas: stochastic models, sampling algorithms, and applications. Imperial College Press, London

    Book  Google Scholar 

  8. Nelsen RB (2006) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  9. Neuhäuser D, Hirsch C, Gloaguen C, Schmidt V (2013) A parametric copula approach for modelling shortest-path trees in telecommunication networks. In: Dudin A, Turck K (eds) Analytical and stochastic modeling techniques and applications. Lecture notes in computer science, vol 7984. Springer, Berlin, pp 324–336

    Google Scholar 

  10. Neuhäuser D, Hirsch C, Gloaguen C, Schmidt V (2012) On the distribution of typical shortest-path lengths in connected random geometric graphs. Queueing Syst 71:199–220

    Article  MATH  MathSciNet  Google Scholar 

  11. Ruppert D (2010) Statistics and data analysis for financial engineering. Springer texts in statistics, New York

  12. Schneider R, Weil W (2008) Stochastic and integral geometry. Springer, Berlin

    Book  MATH  Google Scholar 

  13. Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley, Chichester

    MATH  Google Scholar 

  14. Voss F, Gloaguen C, Fleischer F, Schmidt V (2011) Densities of shortest path lengths in spatial stochastic networks. Stoch Model 27:141–167

    Article  MATH  MathSciNet  Google Scholar 

  15. Voss F, Gloaguen C, Schmidt V (2010) Scaling limits for shortest path lengths along the edges of stationary tessellations. Adv Appl Probab 42:936–952

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ulm University, Helmholtzstraße 18, 89069, Ulm, Germany

    David Neuhäuser, Christian Hirsch & Volker Schmidt

  2. Orange Labs, 38-40 rue du Général Leclerc, 92794 Cedex 9, Issy-Moulineaux, France

    Catherine Gloaguen

Authors
  1. David Neuhäuser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Hirsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Catherine Gloaguen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Volker Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Neuhäuser.

Additional information

This work was supported by Orange Labs through Research grant No. 46146063-9241. Christian Hirsch was supported by a research grant from DFG Research Training Group 1100 at Ulm University. Parts of the numerical results were obtained by the help of Jan Sommer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Neuhäuser, D., Hirsch, C., Gloaguen, C. et al. Joint distributions for total lengths of shortest-path trees in telecommunication networks. Ann. Telecommun. 70, 221–232 (2015). https://doi.org/10.1007/s12243-014-0440-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12243-014-0440-9

Keywords

Search

Navigation