Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T21:25:10.325Z Has data issue: false hasContentIssue false
  • English
  • Français

On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications

Part of: Special processes Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

François Baccelli  and
Bartłomiej Błaszczyszyn
François Baccelli*
Affiliation:
INRIA and ENS, Paris
Bartłomiej Błaszczyszyn*
Affiliation:
University of Wrocław
*
∗ Postal address: ENS/INRIA, 45 rue d'Ulm, 75005 Paris, France. Email address: francois.baccelli@ens.fr
∗∗ Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
Get access Rights & Permissions [Opens in a new window]

Abstract

We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.

Keywords

shot noiseBoolean modelVoronoi tessellationcoverage processpoint processPoisson processrandom closed setperturbation analysiswireless communicationsignal to interference ratioCDMA protocol

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 293 - 323
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baccelli, F. and Błaszczyszyn, B. (2000). On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation, with applications to wireless communications. INRIA Rept 4019. Available at http://www.inria.fr/rrrt/.Google Scholar
[2] Baccelli, F., Klein, M., Lebourges, M. and Zuyev, S. (1997). Stochastic geometry and architecture of communication networks. Telecommun. Systems 7, 209–227.CrossRefGoogle Scholar
[3] Błaszczyszyn, B., Merzbach, E. and Schmidt, V. (1997). A note on expansion for functionals of spatial marked point processes. Statist. Prob. Lett. 36, 299–306.CrossRefGoogle Scholar
[4] Ehrenberger, U. and Leibnitz, K. (1999). Impact of clustered traffic distributions in CDMA radio network planning. In Teletraffic Engineering in a Competitive World, eds Key, P. and Smith, D. Elsevier, Amsterdam, pp. 129–138.Google Scholar
[5] Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York.Google Scholar
[6] Gakhov, F. D. (1990). Boundary Value Problems. Dover, New York.Google Scholar
[7] Gubner, J. (1996). Computation of shot-noise probability distributions and densities. SIAM J. Sci. Comput. 17, 750–761.CrossRefGoogle Scholar
[8] Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inf. Theory 46, 388–404.CrossRefGoogle Scholar
[9] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
[10] Heinrich, L. and Molchanov, I. S. (1994). Some limit theorems for extremal and union shot-noise processes. Math. Nacht. 168, 139–159.CrossRefGoogle Scholar
[11] Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot-noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.CrossRefGoogle Scholar
[12] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, London.Google Scholar
[13] Orsingher, E. and Battaglia, F. (1982). Probability distributions of level crossings of shot-noise models. Stochastics 8, 45–61.CrossRefGoogle Scholar
[14] Rice, J. (1977). On a generalized shot-noise. Adv. Appl. Prob. 17, 709–730.Google Scholar
[15] Schmidt, V. (1985). On finiteness and continuity of shot-noise process. Optimization 16, 921–933.CrossRefGoogle Scholar
[16] Stoyan, D., Kendall, W. and Mecke, J. (1995). Stochastic Geometry and its Applications. John Wiley, Chichester.Google Scholar
[17] Westcott, M. (1976). On the existence of a generalized shot-noise process. In Studies in Probability and Statistics. Papers in Honour of Edwin J. G. Pitman ed. Williams, E. J. North-Holland, Amsterdam, pp. 73–88.Google Scholar
[18] Veeravalli, V. and Sendonaris, A. (1999). The coverage-capacity tradeoff in cellular CDMA systems. IEEE Trans. Vehicular Technol. 48, 1443–1451.CrossRefGoogle Scholar