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# Asymptotic properties of wireless multi-hop networks

In this dissertation, we consider wireless multi-hop networks, where the nodes are randomly placed. We are particularly interested in their asymptotic properties when the number of nodes tends to infinity. We use percolation theory as our main tool of analysis. As a first model, we assume that nodes have a fixed connectivity range, and can establish wireless links to all nodes within this range, but no other (Boolean model). We compute for one-dimensional networks the probability that two nodes are connected, given the distance between them. We show that this probability tends exponentially to zero when the distance increases, proving that pure multi-hopping does not work in large networks. In two dimensions however, an unbounded cluster of connected nodes forms if the node density is above a critical threshold (super-critical phase). This is known as the percolation phenomenon. This cluster contains a positive fraction of the nodes that depends on the node density, and remains constant as the network size increases. Furthermore, the fraction of connected nodes tends rapidly to one when the node density is above the threshold. We compare this partial connectivity to full connectivity, and show that the requirement for full connectivity leads to vanishing throughput when the network size increases. In contrast, partial connectivity is perfectly scalable, at the cost of a tiny fraction of the nodes being disconnected. We consider two other connectivity models. The first one is a signal-to-interference- plus-noise-ratio based connectivity graph (STIRG). In this model, we assume deterministic attenuation of the signals as a function of distance. We prove that percolation occurs in this model in a similar way as in the previous model, and study in detail the domain of parameters where it occurs. We show in particular that the assumptions on the attenuation function dramatically impact the results: the commonly used power-law attenuation leads to particular symmetry properties. However, physics imposes that the received signal cannot be stronger than the emitted signal, implying a bounded attenuation function. We observe that percolation is harder to achieve in most cases with such an attenuation function. The second model is an information theoretic view on connectivity, where two arbitrary nodes are considered connected if it is possible to transmit data from one to the other at a given rate. We show that in this model the same partial connectivity can be achieved in a scalable way as in the Boolean model. This result is however a pure connectivity result in the sense that there is no competition and interferences between data flows. We also look at the other extreme, the Gupta and Kumar scenario, where all nodes want to transmit data simultaneously. We show first that under point-to-point communication and bounded attenuation function the total transport capacity of a fixed area network is bounded from above by a constant, whatever the number of nodes may be. However, if the network area increases linearly with the number of nodes (constant density), or if we assume power-law attenuation function, a throughput per node of order 1/√n can be achieved. This latter result improves the existing results about random networks by a factor (log n)1/2. In the last part of this dissertation, we address two problems related to latency. The first one is an intruder detection scenario, where a static sensor network has to detect an intruder that moves with constant speed along a straight line. We compute an upper bound to the time needed to detect the intruder, under the assumption that detection by disconnected sensors does not count. In the second scenario, sensors switch off their radio device for random periods, in order to save energy. This affects the delivery of alert messages, since they may have to wait for relays to turn on their radio to move further. We show that asymptotically, alert messages propagate with constant, deterministic speed in such networks.

Section des systèmes de communication

Faculté informatique et communications

Institut de systèmes de communication

Laboratoire pour les communications informatiques et leurs applications 3

Jury: François Baccelli, Panganamala R. Kumar, Emre Telatar, Martin Vetterli

Public defense: 2005-8-26

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Record created on 2005-07-12, modified on 2017-05-12