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Large deviation principles for connectable receivers in wireless networks

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  11 January 2017

Christian Hirsch ,
Benedikt Jahnel ,
Paul Keeler  and
Robert I. A. Patterson
Christian Hirsch*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
Benedikt Jahnel*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
Paul Keeler*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
Robert I. A. Patterson*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
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Abstract

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We study large deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large deviation principle for the rescaled process of these receivers as the connection threshold tends to 0. Finally, we show how these results can be used to develop importance sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

Keywords

Wireless networksignal-to-interference-and-noise ratiolarge deviation principleimportance sampling

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1061 - 1094
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Adams, S., Collevecchio, A. and König, W. (2011). A variational formula for the free energy of an interacting many-particle system. Ann. Prob. 39, 683–728.Google Scholar
[2] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
[3] Baccelli, F. and Błaszczyszyn, B. (2009). Stochastic geometry and wireless networks, Vol. 1, Theory. Foundations Trends Networking 3, 249–449.Google Scholar
[4] Baccelli, F. and Błaszczyszyn, B. (2009). Stochastic geometry and wireless networks, Vol. 2, Applications. Foundations Trends Networking 4, 1–312.Google Scholar
[5] Baccelli, F. and Zhang, X. (2015). A correlated shadowing model for urban wireless networks. In 2015 IEEE Conference of Computer Communications (INFOCOM), IEEE, New York, pp. 801–809.CrossRefGoogle Scholar
[6] Baldi, P., Frigessi, A. and Piccioni, M. (1993). Importance sampling for Gibbs random fields. Ann. Appl. Prob. 3, 914–933.Google Scholar
[7] Bangerter, B., Talwar, S., Arefi, R. and Stewart, K. (2014). Networks and devices for the 5G era. IEEE Commun. Magazine 52, 90–96.Google Scholar
[8] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities, Oxford University Press.Google Scholar
[9] Daly, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
[10] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
[11] Eichelsbacher, P. and Schmock, U. (1998). Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem. Stoch. Process. Appl. 77, 233–251.Google Scholar
[12] Ganesh, A. J. and Torrisi, G. L. (2008). Large deviations of the interference in a wireless communication model. IEEE Trans. Inf. Theory 54, 3505–3517.Google Scholar
[13] Georgii, H.-O. (1993). Large deviations and maximum entropy principle for interacting random fields on Z d . Ann. Prob. 21, 1845–1875.Google Scholar
[14] Georgii, H.-O. (1994). Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Prob. Theory Relat. Fields 99, 171–195.Google Scholar
[15] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd edn. De~Gruyter, Berlin.Google Scholar
[16] Georgii, H.-O. and Zessin, H. (1993). Large deviations and the maximum entropy principle for marked point random fields. Prob. Theory Relat. Fields 96, 177–204.Google Scholar
[17] Giesecke, K. and Shkolnik, A. (2014). Optimal importance sampling of default losses. Preprint. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract\_id=1924135.Google Scholar
[18] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[19] Kroese, D. P., Taimre, T. and Botev, Z. I. (2011). Handbook of Monte Carlo Methods. John Wiley, Hoboken, NJ.Google Scholar
[20] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9 36–58.Google Scholar
[21] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
[22] Schreiber, T. and Yukich, J. E. (2005). Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs. Stoch. Process. Appl. 115, 1332–1356.Google Scholar
[23] Seppäläinen, T. and Yukich, J. E. (2001). Large deviation principles for Euclidean functionals and other nearly additive processes. Prob. Theory Relat. Fields 120, 309–345.Google Scholar
[24] Torrisi, G. L. and Leonardi, E. (2013). Simulating the tail of the interference in a Poisson network model. IEEE Trans. Inf. Theory 59, 1773–1787.Google Scholar
[25] Torrisi, G. L. and Leonardi, E. (2014). Large deviations of the interference in the Ginibre network model. Stoch. Systems 4, 173–205.CrossRefGoogle Scholar