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Coverage of Random Discs Driven by a Poisson Point Process

  • Guo-Lie LanEmail author
  • Zhi-Ming Ma
  • Su-Yong Sun
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 205)

Abstract

Motivated by the study of large-scale wireless sensor networks, in this paper we discuss the coverage problem that a pre-assigned region is completely covered by the random discs driven by a homogeneous Poisson point process. We first derive upper and lower bounds for the coverage probability. We then obtain necessary and sufficient conditions, in terms of the relation between the radius r of the discs and the intensity \(\lambda\) of the Poisson process, in order that the coverage probability converges to 1 or 0 when \(\lambda\) tends to infinity. A variation of Stein-Chen method for compound Poisson approximation is well used in the proof.

Notes

Acknowledgments

The authors would like to thank the referees for their encouragement and valuable suggestions. Zhi-Ming Ma would like to thank the organizers for inviting him to participate in the stimulating conference in honor of Louis Chen on his 70th birthday.

References

  1. 1.
    Aldous D (1989) Probability approximations via the poisson clumping heuristic. Springer, New YorkCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldous D (1989) Stein’s method in a two-dimensional coverage problem. Statist Probab Lett 8:307–314CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Akyildiz IF, Wei LS, Sankarasubramaniam Y, Cayirci E (2002) A survey on sensor networks. IEEE Commun Mag 40:102–114CrossRefGoogle Scholar
  4. 4.
    Barbour AD, Chen LHY, Loh WL (1992) Compound poisson approximation for nonnegative random variables via Stein’s method. Ann Prob 20:1843–1866CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Baccelli F, Blaszczyszyn B (2001) On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications. Adv Appl Prob 33:293–323zbMATHMathSciNetGoogle Scholar
  6. 6.
    Hall P (1988) Introduction to the theory of coverage processes. John Wiley and Sons Inc., New YorkzbMATHGoogle Scholar
  7. 7.
    Karr Alan F (1991) Point processes and their statistical inference. Marcel Dekker, Inc., New YorkzbMATHGoogle Scholar
  8. 8.
    Lan GL, Ma ZM, Sun SY (2007) Coverage problem of wireless sensor networks. Proc CJCDGCGT 2005 LNCS 4381:88–100MathSciNetGoogle Scholar
  9. 9.
    Peköz EA (2006) A compound Poisson approximation inequality. J Appl Prob 43:282–288CrossRefzbMATHGoogle Scholar
  10. 10.
    Philips TK, Panwar SS, Tantawi AN (1989) Connectivity properties of a packet radio network model. IEEE Trans Inform Theor 35:1044–1047CrossRefGoogle Scholar
  11. 11.
    Shakkottai S, Srikant R, Shroff N (2003) Unreliable sensor grids: coverage, connectivity and diameter. Proc IEEE INFOCOM 2:1073–1083Google Scholar
  12. 12.
    Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. John Wiley and Sons, Inc., New YorkzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Guangzhou UniversityGuangzhouChina
  2. 2.CASAcademy of Math and Systems ScienceBeijingChina

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