Download Electric-field based

Routing in WSNs through
analogies with electrostatics
December 2005
L. Tzevelekas
I. Stavrakakis
Recent developments in routing for
WSNs
•
Papers on this topic:
1.
2.
3.
4.
M. Kalantari and M. Shayman, “Routing in wireless ad hoc
networks by analogy to electrostatic theory,” in Proc. IEEE ICC,
Paris, France, June 2004.
M. Kalantari and M. Shayman, “Energy efficient routing in
wireless sensor networks,” in Proc. Conference on Information
Sciences and Systems, Princeton University, NJ, Mar. 2004.
N. T. Nguyen, A.-I A. Wang, P. Reiher, and G. Kuenning,
“Electric-field based routing: A reliable framework for routing in
MANETs,” ACM Mobile Computing and Communications
Review, vol. 8, no. 1, pp. 35– 49, Apr. 2004.
S. Toumpis and L. Tassiulas, “Packetostatics: Deployment of
Massively Dense Sensor Networks as an Electrostatics Problem,”
Proc. INFOCOM 2005.
Back to the basics
• Sensors are sources of information => positive charges
in electromagnetism
• Central node is the sink of information => negative
charges in electromagnetism
• Network is like a non-homogeneous dielectric media
• Routing scheme based on changing the permittivity
factor of media:
high values in places with high residual energy, i.e. more traffic
through that region
low values for regions with low residual energy, i.e fewer traffic
through that region
Mathematical formulation
• N sensors on a plane region A
• Sensors communicate through wireless multi-hop links to a
central node entity called the sink
• On happening of events, the nearest sensor chooses to report
it to the sink
• Sensors have limited amounts of residual energy (known at
each time unit)
• Events happening with a known spatial density rate:
r(z) ≥ 0, z denotes a point (x,y) on the plane
Rate of events inside α:
Mathematical formulation
• Sensors are not mobile and their locations are known
• For routing purposes in the network:
• We keep a direction for every point z of the network
• Sensor placed at point z uses direction defined at that point and
forwards traffic to the closest sensors to this direction
• Assume the direction to be a continuous function of z
• Each sensor i is associated with a weight wi>0 denoting
the average rate of messages generated by sensor i
• Assign weight to the central node as follows:
N
w0=   wi
i 1
Mathematical formulation
• Define a path as a directed curved line starting at sensor i and
ending at the central node
• Each sensor i is associated with a path pi (‘abstract’ paths, not
constrained by locations of intermediate nodes)
• The amount of load on that path is wi (same as the rate of
messages generated at the starting node)
• Define a vector field on A to be the load density vector field at
point z, denoted by D(z):
Mathematical formulation
• Based on the definition of D(z), one can prove that
• C is a closed contour in the region A
• dn is a differential vector normal to the contour at each
point of its boundary
• w is the algebraic sum of the weights of the nodes
inside the closed contour
• the above equation is analogous to Gauss’s law in the
electrostatics theory
Mathematical formulation
• Analytical explanation of the
formula:
the rate at which messages
exit a contour is the
algebraic sum of the
weights of the nodes that are
inside that contour (net
amount of sources inside that
contour)
Defining the net amount of sources in the network as:
one can express the equation in differential form as follows:
where
Energy efficient routing
• Define the concept of load flow lines
–
–
–
–
–
–
–
Similar to the electric flux lines
Family of 2-D curved lines
Always tangent to the direction of D(z)
Orientation same as orientation of D(z)
Cannot intersect except at a source or a destination
Start at sources and end at the destination
To find pi we start at the place of sensor i and follow the
direction of the load flow lines
Energy efficient routing
• To associate the paths with the residual energy at each sensor, we
define the scalar field of the residual energy of the network as a
function of z:
• Make use of the fact that |D(z)| represents the communication
activity at point z (= the energy dissipation at point z)
• To achieve uniform energy dissipation overall in the region A we
form the following quadratic minimization problem:
Energy efficient routing
• Solving the previous minimization problem leads to
the introduction of the potential function U
it can be shown that the optimal solution for D*(z) satisfies
and this is the necessary and sufficient condition for having a conservative
vector field D*(z)
Such a vector field can equivalently be described through the gradient of a
scalar function called the potential function
And the differential equations then become:
(poisson equation)
Energy efficient routing
• Finally taking into account the scalar field ω(z, t) of the residual
energy, we obtain the following quadratic minimization problem
where
K(z) is a scalar weight function representing where in the network we
prefer to have a higher load and where we want to have less communication
activity in order to save the energy of the nodes
Results
• sensors of the network are distributed in a
1000m x 1000m square
• the number of sensors is N = 441
• the central node has been placed in the
center of the network area
• a Poisson process is responsible for
generating events
• distributed the initial energy of 20000 units
among the sensors.
•the initial energy of the sensors is inversely
proportional to their distance from the
central node.
• the transmission range of each sensor is
85m
Results
Numerical computation of the
values of the potential function U
Direction of D*(z) is calculated as
the gradient of U
Paths from sensors to the destination
are found by following these segment
lines
Results
Comparison of
1. the performance of
routes found based on
D (z)
2. the performance of
routes found with the
shortest path method
3. for several
experiments