1. Routing in Wireless Ad Hoc Networks by Analogy to Electrostatic Theory
Mehdi Kalantari
Mark Shayman
University of Maryland, College Park

2. Electrostatic Routing
Motivation: Natural phenomena with similarity in structure
Fluid dynamics: How the flow forms among the sources and
the sinks
Heat exchange: How heat is exchanged between hot and
cold environments
Electrostatic: How the electric field is propagated in a dielectric
with a density of positive and negative charges
Common features: Transfer something from sources to sinks,
Gauss’ Law

3. Assumptions Basic assumptions:
A network of N wireless nodes that can communicate through radio frequency wireless links. Nodes are densely distributed in the network.
Nodes have no mobility.
There are M source-destination pairs in the network, the
communication demand for each of them is known.

4. Definition of Load Vector Field
Given a set of source-destination pairs and a set of paths between each pair, the load vector field represents the actual load or demand at every place of network. Its direction shows the desired direction of relaying packets.
Example: Line of sight routing
: The weight of the demand of source-destination pair i
: The unit vector aiming from the source to the destination
: The line of sight of the source and destination

5. One important property of
Gauss' law: The amount of flow that crosses the boundary of a closed
contour is the amount of net sources inside that contour:
In the other words: C
: A differential element
normal to the boundary of C
: The net sum of sources--positive for sources, negative for destinations.

6. PDE form of the Gauss Law
: The density of sources
Divergence of load vector field is equal to density of sources.

7. One major problem of definition of
Cancellation: The value of may not represent the actual load density
at a place of the network.
See example below.
However, there is one special case in which represents the actual
demand and load of the network: Many to one communication: There is a
single destination in the network.
In the example here, we assume line of sight routing. The resulting D involves cancellation so the routes cannot be recovered from the flow lines of the vector field.

8. Many to One Communication
Many sources in the network want to communicate to a single node.
Nodes can use multi-path and multi-hop routes to communicate.
In this case routes uniquely specify , and uniquely specifies routes.
The value of at each place of the network specifies the actual value of the communication demand at that point.

9. Abstract Definition of
In general is any vector field that satisfies the following equations:
: The normal component of on the boundary of the network
Routes are generated from by following the direction of this vector field by relaying a packet in the direction of We assume the nodes are distributed densely in the network.
Routes
Observation: The above equation
does NOT specify uniquely.

10. Spreading the Load As Evenly As Possible
Intuition: spreading the load as evenly as possible.
-The number of collisions in the network decreases.
-Network resources are utilized better.
Mathematical formulation of even load distribution:
The curl of the load vector field is 0.
s.t.
and
Solution:

11. The Analogy to the Maxwell’s Equations
The above equations with boundary condition
give uniquely.

12. A Useful Observation
This means is a conservative vector field, so it can
be written as the gradient of a scalar field.
is known as a potential function. The routing is
done in direction of the gradient of This potential
function satisfies:
This equation is analogous to the Poisson equation in
electrostatic theory.

13. Simulation Experiment
Consider a wireless network of nodes randomly
distributed in a 1000m x 1000m square.
The transmission range of the nodes is 85m.
40 source nodes want to communicate to a node in the
center of the network.
Each source node uses multi-path routing with 8 paths
to the destination.

14. A Simulation Experiment (cont’d)
Legend:
X Sources
Destination
Other Nodes
The placement of the nodes

15. A Simulation Experiment (cont’d)
The direction of optimal

16. A Simulation Experiment (cont’d)
The value of

17. A Simulation Experiment (cont’d)
Electrostatic Routes
Shortest Path Routes

18. Performance Comparison
Throughput (bits/s)
Shortest path:
Electrostatic:
70% improvement
Other experiments with random seeds:
Average improvement: 34%

19. Conclusion
Electrostatic routing is inspired by many phenomena in nature, and it can improve network performance.
Possible Future Work:
Making the approach decentralized
Trying to extend the results to the case in which we have partial location information
Extending the results to many to many communication case
Adding mobility to the model