In this work we introduce a new mathematical tool for optimization
of routes, topology design, and energy efficiency in wireless
sensor networks. We introduce a vector field formulation that
models communication in the network, and routing is performed in
the direction of this vector field at every location of the
network. The magnitude of the vector field at every location
represents the density of amount of data that is being transited
through that location. We define the total communication cost in
the network as the integral of a quadratic form of the vector
field over the network area.
With the above formulation, we introduce a mathematical machinery
based on partial differential equations very similar to the
Maxwell's equations in electrostatic theory. We show that in order
to minimize the cost, the routes should be found based on the
solution of these partial differential equations. In our
formulation, the sensors are sources of information, and they are
similar to the positive charges in electrostatics, the
destinations are sinks of information and they are similar to
negative charges, and the network is similar to a non-homogeneous
dielectric media with variable dielectric constant (or
permittivity coefficient).
In one of the applications of our mathematical model based on the
vector fields, we offer a scheme for energy efficient routing. Our
routing scheme is based on changing the permittivity coefficient
to a higher value in the places of the network where nodes have
high residual energy, and setting it to a low value in the places
of the network where the nodes do not have much energy left. Our
simulations show that our method gives a significant increase in
the network life compared to the shortest
path and weighted shortest path schemes.
Our initial focus is on the case where there is only one
destination in the network, and later we extend our approach to
the case where there are multiple destinations in the network. In
the case of having multiple destinations, we need to partition the
network into several areas known as regions of attraction of the
destinations. Each destination is responsible for collecting all
messages being generated in its region of attraction. The
complexity of the optimization problem in this case is how to
define regions of attraction for the destinations and how much
communication load to assign to each destination to optimize the
performance of the network. We use our vector field model to solve
the optimization problem for this case. We define a vector field,
which is conservative, and hence it can be written as the gradient
of a scalar field (also known as a potential field). Then we show
that in the optimal assignment of the communication load of the
network to the destinations, the value of that potential field
should be equal
at the locations of all the destinations.
Another application of our vector field model is to find the
optimal locations of the destinations in the network. We show that
the vector field gives the gradient of the cost function with
respect to the locations of the destinations. Based on this fact,
we suggest an algorithm to be applied during the design phase of a
network to relocate the destinations for reducing the
communication cost function. The performance of our proposed
schemes is confirmed by several examples and simulation
experiments.
In another part of this work we focus on the notions of
responsiveness and conformance of TCP traffic in communication
networks. We introduce the notion of responsiveness for TCP
aggregates and define it as the degree to which a TCP aggregate
reduces its sending rate to the network as a response to packet
drops. We define metrics that describe the responsiveness of TCP
aggregates, and suggest two methods for determining the values of
these quantities. The first method is based on a test in which we
drop a few packets from the aggregate intentionally and measure
the resulting rate decrease of that aggregate. This kind of test
is not robust to multiple simultaneous tests performed at
different routers. We make the test robust to multiple
simultaneous tests by using ideas from the CDMA approach to
multiple access channels in communication theory. Based on this
approach, we introduce tests of responsiveness for aggregates, and
call it CDMA based Aggregate Perturbation Method (CAPM). We use
CAPM to perform congestion control. A distinguishing feature of
our congestion control scheme is that it maintains a
degree of fairness among different aggregates.
In the next step we modify CAPM to offer methods for estimating
the proportion of an aggregate of TCP traffic that does not
conform to protocol specifications, and hence may belong to a DDoS
attack. Our methods work by intentionally perturbing the aggregate
by dropping a very small number of packets from it and observing
the response of the aggregate. We offer two methods for
conformance testing. In the first method, we apply the
perturbation tests to SYN packets being sent at the start of the
TCP 3-way handshake, and we use the fact that the rate of ACK
packets being exchanged in the handshake should follow the rate of
perturbations. In the second method, we apply the perturbation
tests to the TCP data packets and use the fact that the rate of
retransmitted data packets should follow the rate of
perturbations. In both methods, we use signature based
perturbations, which means packet drops are performed with a rate
given by a function of time. We use analogy of our problem with
multiple access communication to find signatures. Specifically, we
assign orthogonal CDMA based signatures to different routers in a
distributed implementation of our methods. As a result of
orthogonality, the performance does not degrade because of cross
interference made by simultaneously testing routers. We have shown
efficacy of our methods through mathematical analysis and
extensive simulation
experiments
