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Optimal paths on the space-time SINR random graph

Part of: Graph theory Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

François Baccelli ,
Bartłomiej Błaszczyszyn  and
Mir-Omid Haji-Mirsadeghi
François Baccelli*
Affiliation:
INRIA/ENS
Bartłomiej Błaszczyszyn*
Affiliation:
INRIA/ENS and University of Wrocław
Mir-Omid Haji-Mirsadeghi*
Affiliation:
Sharif University of Technology and INRIA/ENS
*
∗ Postal address: INRIA, 23 Avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
∗ Postal address: INRIA, 23 Avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
∗∗∗∗ Postal address: Department of Mathematical Sciences, Sharif University of Technology, Azadi Av., Tehran, Iran. Email address: mirsadeghi@mehr.sharif.edu
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Abstract

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We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. In this paper we study optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both ‘positive’ and ‘negative’ results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to ∞. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that, when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

Keywords

Poisson point processrandom graphfirst passage percolationshot noise processSINR

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05C80: Random graphs
Secondary: 90C27: Combinatorial optimization 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 131 - 150
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

The work of this author was part of a joint PhD programe, co-advised by Prof. Amir Daneshgar of Sharif University of Technology.

References

Asmussen, S. et al. (2008). Asymptotic behavior of total times for Jobs that must start over if a failure occurs. Math. Operat. Res. 33, 932–944.Google Scholar
Baccelli, F. and Bordenave, C. (2007). The radial spanning tree of a Poisson point process. Ann. Appl. Prob. 17, 305–359.Google Scholar
Baccelli, F., Błaszczyszyn, B. and Mühlethaler, P. (2006). An aloha protocol for multihop mobile wireless networks. IEEE Trans. Inf. Theory 52, 421–436.CrossRefGoogle Scholar
Baccelli, F., Błaszczyszyn, B. and Mühlethaler, P. (2009). Stochastic analysis of spatial and opportunistic aloha. IEEE J. Sel. Areas Commun. 27, 1109–1119.Google Scholar
Baccelli, F., Błaszczyszyn, B. and Mühlethaler, P. (2010). Time–space opportunistic routing in wireless ad hoc networks: algorithms and performance optimization by stochastic geometry. Comput. J. 53, 592–609.CrossRefGoogle Scholar
Bordenave, C. (2008). Navigation on a Poisson point process. Ann. Appl. Prob. 18, 708–746.Google Scholar
Broadbent, S. R. and Hammersley, J. M. (1957). Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629–641.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II. Springer, New York.CrossRefGoogle Scholar
Dousse, O., Baccelli, F. and Thiran, P. (2005). Impact of interferences on connectivity in ad hoc networks. IEEE/ACM Trans. Networking 13, 425–436.CrossRefGoogle Scholar
Dousse, O. et al. (2006). Percolation in the signal to interference ratio graph. J. Appl. Prob. 43, 552–562.CrossRefGoogle Scholar
Ganti, R. K. and Haenggi, M. (2009). Bounds on information propagation delay in interference-limited ALOHA networks. In Proc. Workshop Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, IEEE Press, pp. 513–519.Google Scholar
Gilbert, E. N. (1961). Random plane networks. J. SIAM 9, 533–543.Google Scholar
Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170.CrossRefGoogle Scholar
Jacquet, P., Mans, B., Mühlethaler, P. and Rodolakis, G. (2009). Opportunistic routing in wireless ad hoc networks: upper bounds for the packet propagation speed. IEEE J. Sel. Areas Commun. 27, 1192–1202.Google Scholar
Jelenkovic, P. R. and Tan, J. (2009). Stability of finite population ALOHA with variable packets. Preprint. Available at http://arxiv.org/abs/0902.4481v2.Google Scholar
Jelenković, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. INFOCOM 2007 (Anchorage, AL, May 2007), pp. 892–900.Google Scholar
Jelenković, P. R. and Tan, J. (2007). Is ALOHA causing power law delays? In Managing Traffic Performance in Converged Networks (Lecture Notes Comput. Sci. 4516), eds Mason, L., Drwiega, T. and Yan, J., Springer, Berlin, pp. 1149–1160.Google Scholar
Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Prob. 1, 883–909.CrossRefGoogle Scholar
Kleinberg, J. (2000). The small-world phenomenon: an algorithmic perspective. In Proc. 32nd Ann. ACM Symp. Theory of Computing, ACM, New York, pp. 163–170.Google Scholar
Pimentel, L. P. R. (2006). The time constant and critical probabilities in percolation models. Electron Commun. Prob. 11, 160–167.Google Scholar
Pugh, C. and Shub, M. (1971). Ergodic elements of ergodic actions. Compositio Math. 23, 115–122.Google Scholar
Tse, D. and Viswanath, P. (2005). Foundamentals of Wireless Communication. Cambridge University Press.Google Scholar
Vahidi-Asl, M. Q. and Wierman, J. C. (1990). First-passage percolation on the Voronoi tessellation and Delaunay trangulation. In Random Graphs '87 (Poznań, 1987), eds Karośki, M., Jaworski, J. and Ruciński, A., John Wiley, Chichester, pp. 341–359.Google Scholar