468 research outputs found
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 nodes, each
capable of sensing events within a radius of , are randomly and uniformly
distributed in a 3-dimensional region of volume , 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
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 inside hops' interference sets, 2)
the number of hops , and 3) the degree of spatial correlations. The
randomness in both 's and is advantageous, i.e., it can yield larger
scalings (as large as ) 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
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 for a set of indistinguishable
nodes. We consider a wireless network where nodes (processors) are randomly
thrown in a square , 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 of each other ( 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 transmit concurrently at
the same time, then 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 nodes
uniformly scattered in a square . 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 in order to assign a
unique number ranging from 1 to to each of the participating nodes
Optimal Paths on the Space-Time SINR Random Graph
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
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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