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.
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References
Aldous D (1989) Probability approximations via the poisson clumping heuristic. Springer, New York
Aldous D (1989) Stein’s method in a two-dimensional coverage problem. Statist Probab Lett 8:307–314
Akyildiz IF, Wei LS, Sankarasubramaniam Y, Cayirci E (2002) A survey on sensor networks. IEEE Commun Mag 40:102–114
Barbour AD, Chen LHY, Loh WL (1992) Compound poisson approximation for nonnegative random variables via Stein’s method. Ann Prob 20:1843–1866
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 33:293–323
Hall P (1988) Introduction to the theory of coverage processes. John Wiley and Sons Inc., New York
Karr Alan F (1991) Point processes and their statistical inference. Marcel Dekker, Inc., New York
Lan GL, Ma ZM, Sun SY (2007) Coverage problem of wireless sensor networks. Proc CJCDGCGT 2005 LNCS 4381:88–100
Peköz EA (2006) A compound Poisson approximation inequality. J Appl Prob 43:282–288
Philips TK, Panwar SS, Tantawi AN (1989) Connectivity properties of a packet radio network model. IEEE Trans Inform Theor 35:1044–1047
Shakkottai S, Srikant R, Shroff N (2003) Unreliable sensor grids: coverage, connectivity and diameter. Proc IEEE INFOCOM 2:1073–1083
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. John Wiley and Sons, Inc., New York
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.
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Lan, GL., Ma, ZM., Sun, SY. (2012). Coverage of Random Discs Driven by a Poisson Point Process. In: Barbour, A., Chan, H., Siegmund, D. (eds) Probability Approximations and Beyond. Lecture Notes in Statistics(), vol 205. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1966-2_4
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DOI: https://doi.org/10.1007/978-1-4614-1966-2_4
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