Random Geometric Graphs and the Initialization Problem for Wireless Networks

32 pages. Tutorial invitéInternational audienceThe initialization problem, also known as naming, assigns one unique identifier (ranging from 1 to $n$) to a set of n indistinguishable nodes (stations or processors) in a given wireless network $N$. $N$ is composed of $n$ nodes randomly deployed within a square (or a cube) $X$. We assume the time to be slotted and $N$ to be synchronous; two nodes are able to communicate if they are within a distance at most $r$ of each other ($r$ is the transmitting/receiving range). Moreover, if two or more neighbors of a processor $u$ transmit concurrently at the same round, $u$ does not receive either messages. After the analysis of various critical transmitting/sensing ranges for connectivity and coverage of randomly deployed sensor networks, we design sub-linear randomized initialization and gossip algorithms achieving $O(n^1/2 \log(n)^1/2)$ and $O(n^1/3 \log(n)^2/3) rounds in the two-dimensional and the three-dimensional cases, respectively. Next, we propose energy-efficient initialization and gossip algorithms running in$O(n^3/4 \log (n)^1/4)$rounds, with no station being awake for more than O(n^1/4 \log (n)^3/4)$ rounds

Optimal Initialization and Gossiping Algorithms for Random Radio Networks

The initialization problem, also known as naming, consists to give a unique identifier ranging from $1$ to $n$ to a set of $n$ indistinguishable nodes in a given network. We consider a network where $n$ nodes (processors) are randomly deployed in a square (resp. cube) $X$. We assume that the time is slotted and the network is synchronous, two nodes are able to communicate if they are within distance at most of $r$ of each other ($r$ is the transmitting/receiving range). Moreover, if two or more neighbors of a processor $u$ transmit concurrently at the same time slot, then $u$ would not receive either messages. We suppose also that the anonymous nodes know neither the topology of the network nor the number of nodes in the network. Under this extremal scenario, we first show how the transmitting range of the deployed processors affects the typical characteristics of the network. Then, by allowing the nodes to transmit at various ranges we design sub-linear randomized initialization protocols~: In the two, resp. three, dimensional case, our randomized initialization algorithms run in $O(n^{1/2} \log{n}^{1/2})$, resp. $O(n^{1/3} \log{n}^{2/3})$, time slots. These latter protocols are based upon an optimal gossiping algorithm which is of independent interest

Extremal Properties of Three Dimensional Sensor Networks with Applications

In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems: Assume that $n$ nodes, each capable of sensing events within a radius of $r$, are randomly and uniformly distributed in a 3-dimensional region $\mathcal{R}$ of volume $V$, how large must the sensing range be to ensure a given degree of coverage of the region to monitor? For a given transmission range, what is the minimum (resp. maximum) degree of the network? What is then the typical hop-diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks

Outage and Local Throughput and Capacity of Random Wireless Networks

Outage probabilities and single-hop throughput are two important performance metrics that have been evaluated for certain specific types of wireless networks. However, there is a lack of comprehensive results for larger classes of networks, and there is no systematic approach that permits the convenient comparison of the performance of networks with different geometries and levels of randomness. The uncertainty cube is introduced to categorize the uncertainty present in a network. The three axes of the cube represent the three main potential sources of uncertainty in interference-limited networks: the node distribution, the channel gains (fading), and the channel access (set of transmitting nodes). For the performance analysis, a new parameter, the so-called {\em spatial contention}, is defined. It measures the slope of the outage probability in an ALOHA network as a function of the transmit probability $p$ at $p=0$. Outage is defined as the event that the signal-to-interference ratio (SIR) is below a certain threshold in a given time slot. It is shown that the spatial contention is sufficient to characterize outage and throughput in large classes of wireless networks, corresponding to different positions on the uncertainty cube. Existing results are placed in this framework, and new ones are derived. Further, interpreting the outage probability as the SIR distribution, the ergodic capacity of unit-distance links is determined and compared to the throughput achievable for fixed (yet optimized) transmission rates.Comment: 22 pages, 6 figures. Submitted to IEEE Trans. Wireles

Optimal Power Allocation over Multiple Identical Gilbert-Elliott Channels

We study the fundamental problem of power allocation over multiple Gilbert-Elliott communication channels. In a communication system with time varying channel qualities, it is important to allocate the limited transmission power to channels that will be in good state. However, it is very challenging to do so because channel states are usually unknown when the power allocation decision is made. In this paper, we derive an optimal power allocation policy that can maximize the expected discounted number of bits transmitted over an infinite time span by allocating the transmission power only to those channels that are believed to be good in the coming time slot. We use the concept belief to represent the probability that a channel will be good and derive an optimal power allocation policy that establishes a mapping from the channel belief to an allocation decision. Specifically, we first model this problem as a partially observable Markov decision processes (POMDP), and analytically investigate the structure of the optimal policy. Then a simple threshold-based policy is derived for a three-channel communication system. By formulating and solving a linear programming formulation of this power allocation problem, we further verified the derived structure of the optimal policy.Comment: 10 pages, 7 figure

High-resolution thermal expansion measurements under Helium-gas pressure

We report on the realization of a capacitive dilatometer, designed for high-resolution measurements of length changes of a material for temperatures 1.4 K $\leq T \leq$ 300 K and hydrostatic pressure $P \leq$ 250 MPa. Helium ($^4$He) is used as a pressure-transmitting medium, ensuring hydrostatic-pressure conditions. Special emphasis has been given to guarantee, to a good approximation, constant-pressure conditions during temperature sweeps. The performance of the dilatometer is demonstrated by measurements of the coefficient of thermal expansion at pressures $P \simeq$ 0.1 MPa (ambient pressure) and 104 MPa on a single crystal of azurite, Cu$_3$(CO$_3$)$_2$(OH)$_2$, a quasi-one-dimensional spin S = 1/2 Heisenberg antiferromagnet. The results indicate a strong effect of pressure on the magnetic interactions in this system.Comment: 8 pages, 7 figures, published in Rev. Sci. Instrum with minor change

Pseudo-backscatter laser Doppler velocimeter employing antiparallel-reflector in the forward direction

A laser Doppler velocimeter for measuring the velocity of a flowing fluid was discussed. It comprises laser means for providing first and second beams of collimated coherent monochromatic electromagnetic radiation which are focused to intersect at a predetermined location in a flowing fluid. The movement of the particles of the fluid serve to scatter radiation in substantially all directions. The scattered radiation is shifted in frequency from that of the first beam by an amount corresponding to the velocity of the fluid at the predetermined location. A corner cube is disposed generally forward of the predetermined location, the corner cube responding to a portion of the radiation scattered in the forward direction and being operative to reflect the portion in a direction substantially antiparallel to the forward direction