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Optimal paths on the space-time SINR random graph

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

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: [email protected]
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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.

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