468 research outputs found

    Extremal Properties of Three Dimensional Sensor Networks with Applications

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    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 nn nodes, each capable of sensing events within a radius of rr, are randomly and uniformly distributed in a 3-dimensional region R\mathcal{R} of volume VV, 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

    On the Catalyzing Effect of Randomness on the Per-Flow Throughput in Wireless Networks

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    This paper investigates the throughput capacity of a flow crossing a multi-hop wireless network, whose geometry is characterized by general randomness laws including Uniform, Poisson, Heavy-Tailed distributions for both the nodes' densities and the number of hops. The key contribution is to demonstrate \textit{how} the \textit{per-flow throughput} depends on the distribution of 1) the number of nodes NjN_j inside hops' interference sets, 2) the number of hops KK, and 3) the degree of spatial correlations. The randomness in both NjN_j's and KK is advantageous, i.e., it can yield larger scalings (as large as Θ(n)\Theta(n)) than in non-random settings. An interesting consequence is that the per-flow capacity can exhibit the opposite behavior to the network capacity, which was shown to suffer from a logarithmic decrease in the presence of randomness. In turn, spatial correlations along the end-to-end path are detrimental by a logarithmic term

    Randomized Initialization of a Wireless Multihop Network

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    Address autoconfiguration is an important mechanism required to set the IP address of a node automatically in a wireless network. The address autoconfiguration, also known as initialization or naming, consists to give a unique identifier ranging from 1 to nn for a set of nn indistinguishable nodes. We consider a wireless network where nn nodes (processors) are randomly thrown in a square XX, uniformly and independently. We assume that the network is synchronous and two nodes are able to communicate if they are within distance at most of rr of each other (rr is the transmitting/receiving range). The model of this paper concerns nodes without the collision detection ability: if two or more neighbors of a processor uu transmit concurrently at the same time, then uu would not receive either messages. We suppose also that nodes know neither the topology of the network nor the number of nodes in the network. Moreover, they start indistinguishable, anonymous and unnamed. Under this extremal scenario, we design and analyze a fully distributed protocol to achieve the initialization task for a wireless multihop network of nn nodes uniformly scattered in a square XX. We show how the transmitting range of the deployed stations can affect the typical characteristics such as the degrees and the diameter of the network. By allowing the nodes to transmit at a range r= \sqrt{\frac{(1+\ell) \ln{n} \SIZE}{\pi n}} (slightly greater than the one required to have a connected network), we show how to design a randomized protocol running in expected time O(n3/2log2n)O(n^{3/2} \log^2{n}) in order to assign a unique number ranging from 1 to nn to each of the nn participating nodes

    Optimal Paths on the Space-Time SINR Random Graph

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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 the 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. This paper studies 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 infinity. 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

    Random graph models for wireless communication networks

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    PhDThis thesis concerns mathematical models of wireless communication networks, in particular ad-hoc networks and 802:11 WLANs. In ad-hoc mode each of these devices may function as a sender, a relay or a receiver. Each device may only communicate with other devices within its transmission range. We use graph models for the relationship between any two devices: a node stands for a device, and an edge for a communication link, or sometimes an interference relationship. The number of edges incident on a node is the degree of this node. When considering geometric graphs, the coordinates of a node give the geographical position of a node. One of the important properties of a communication graph is its connectedness | whether all nodes can reach all other nodes. We use the term connectivity, the probability of graphs being connected given the number of nodes and the transmission range to measure the connectedness of a wireless network. Connectedness is an important prerequisite for all communication networks which communication between nodes. This is especially true for wireless ad-hoc networks, where communication relies on the contact among nodes and their neighbours. Another important property of an interference graph is its chromatic number | the minimum number of colours needed so that no adjacent nodes are assigned the same colour. Here adjacent nodes share an edge; adjacent edges share at least one node; and colours are used to identify di erent frequencies. This gives the minimum number of frequencies a network needs in order to attain zero interference. This problem can be solved as an optimization problem deterministically, but is algorithmically NP-hard. Hence, nding good asymptotic approximations for this value becomes important. Random geometric graphs describe an ensemble of graphs which share common features. In this thesis, node positions follow a Poisson point process or a binomial point process. We use probability theory to study the connectedness of random graphs and random geometric graphs, which is the fraction of connected graphs among many graph samples. This probability is closely related to the property of minimum node degree being at least unity. The chromatic number is closely related to the maximum degree as n ! 1; the chromatic number converges to maximum degree when graph is sparse. We test existing theorems and improve the existing ones when possible. These motivated me to study the degree of random (geometric) graph models. We study using deterministic methods some degree-related problems for Erda}os-R enyi random graphs G(n; p) and random geometric graphs G(n; r). I provide both theoretical analysis and accurate simulation results. The results lead to a study of dependence or non-dependence in the joint distribution of the degrees of neighbouring nodes. We study the probability of no node being isolated in G(n; p), that is, minimum node degree being at least unity. By making the assumption of non-dependence of node degree, we derive two asymptotics for this probability. The probability of no node being isolated is an approximation to the probability of the graph being connected. By making an analogy to G(n; p), we study this problem for G(n; r), which is a more realistic model for wireless networks. Experiment shows that this asymptotic result also works well for small graphs. We wish to nd the relationship between these basic features the above two important problems of wireless networks: the probability of a network being connected and the minimum number of channels a network needs in order to minimize interference. Inspired by the problem of maximum degree in random graphs, we study the problem of the maximum of a set of Poisson random variables and binomial random variables, which leads to two accurate formulae for the mode of the maximum for general random geometric graphs and for sparse random graphs. To our knowledge, these are the best results for sparse random geometric graphs in the literature so far. By approximating the node degrees as independent Poisson or binomial variables, we apply the result to the problem of maximum degree in general and sparse G(n; r), and derived much more accurate results than in the existing literature. Combining the limit theorem from Penrose and our work, we provide good approximations for the mode of the clique number and chromatic number in sparse G(n; r). Again these results are much more accurate than existing ones. This has implications for the interference minimization of WLANs. Finally, we apply our asymptotic result based on Poisson distribution for the chromatic number of random geometric graph to the interference minimization problem in IEEE 802:11b/g WLAN. Experiments based on the real planned position of the APs in WLANs show that our asymptotic results estimate the minimum number of channels needed accurately. This also means that sparse random geometric graphs are good models for interference minimization problem of WLANs. We discuss the interference minimization problem in single radio and multi-radio wireless networking scenarios. We study branchand- bound algorithms for these scenarios by selecting di erent constraint functions and objective functions
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