Wireless Continuum
Networks
Stavros Toumpis
Department of Informatics
Athens University of Economics and Business
SNOW 2012
1
Scope
•
Nodes communicating exclusively over a
shared
wireless
medium
•
Nodes are so many as to form a
continuum
:
1.
There is a node at practically every location
2.
Distance between neighbors is much smaller
than network dimensions
•
Other terms:
massive
,
massively dense
,
dense
, etc.
2
Emergence of Two Spatial Scales
•
Macroscopic scale
:
–
Distances on the order of typical network
dimensions
–
Protocols of interest: Routing, Load Balancing
•
Microscopic scale
:
–
Distances on the order of nearest neighbor
separation
–
Protocols of Interest:
PHY, MAC
,
power control
,
next
hop
selection.
Constitutive
Relations
3
Physics Example: Electromagnetism
•
Macroscopic scale:
–
Distances on the order of 10

5
meters and
larger
–
Electric field
E
, Magnetic field intensity
H
, etc.
•
Microscopic scale:
•
Distances on the order of atomic scale
•
Singular charges interacting with each other
according to material properties (conducting,
semiconducting, insulating, etc.)
Constitutive Relations (Ohm’s
law,
D
=
ε
E
, etc.)
4
The Standard Roadmap
1.
Define a microscopic setting
2.
Define a macroscopic setting in terms of
macroscopic quantities
3.
Find the constitutive relations between the
macroscopic quantities describing the
macroscopic
setting using the
microscopic
setting
4.
Solve an optimization problem on the
macroscopic setting
Excellent
modeling tradeoff
between keeping
complexity down and results useful/interesting
!
5
Applications of the Roadmap
1.
‘
Packetostatics
’: Node placement
optimization and analogies with
Electrostatics
2.
‘
Packetoptics
’
: Route optimization and
analogies with Optics
3.
Minimax Traffic Load Balancing
4.
Cooperative Broadcast
6
1. ‘Packetostatics’
[KS04, TT05, TG05, TT06, SA10, SA10a]
7
Example
Setting
•
Wireless Sensor Network
:
1.
Sense the data at the source
2.
Transport the data from the sources to the sinks
3.
Deliver the data to the sinks
•
Problem: Minimize number of nodes needed
•
What is the best placement for the wireless nodes?
What is the traffic flow it induces?
8
Macroscopic Quantities
1.
Node Density Function
d(
x,y
),
measured in
nodes/
m
2
–
In area of size
dA
centered at
(
x,y
)
there are
d(
x,y
)
dA
nodes
2.
Information Density Function
ρ
(
x,y
),
measured in
bps/
m
2
–
If
ρ
(
x,y
)>0
(
<0
),
information is created (absorbed) with rate
ρ
dA
over an area of size
dA
,
centered at
(
x,y
)
3.
Traffic flow function
T
(
x,y
)
,
measured in
bps/m
–
Traffic through incremental
line segment is
T
(
x,y
)

dl
9
Gauss’s Law
•
The net amount of information leaving a
surface
A
0
through its boundary
B(A
0
),
must be equal to the net amount of
information created in that surface:
•
Taking
A
0
→0
, we get the requirement:
)
(
0
0
)
,
(
)
(
ˆ
A
B
A
dS
y
x
ds
s
n
T
y
x
y
x
T
T
T
10
The Constitutive Relation
1.
Nodes only need to transfer data from
sources to sinks
–
They do not need to sense them at the
sources
–
They do not need to deliver them to the sinks
once their location is reached
2.
The traffic flow function and the node
density function are related by:
)
,
(

)
,
(

max
y
x
d
c
y
x
T
11
Traffic Must Be Irrotational
•
Remember that we must minimize the number
of nodes
•
If the constitutive relation is satisfied, then the
traffic must be irrotational:
dA
y
x
c
dA
y
x
d
N
2
2
)
,
(
1
)
,
(
T
0
y
x
x
y
T
T
T
12
Motivation
of Constitutive Relation
•
n
=
ε
2
d
(
x
,
y
)
nodes are placed
randomly in square of side
ε
•
Power decays according to
power law
•
Transmissions (with rate
W
),
successful only if SINR
exceeds threshold
•
A ‘highway system’ on the order of
Θ(
n
½
)
=
Θ
[
ε
(
d
(
x
,
y
))
½
]
highways going from left to right
can be created
[GK01, FD04]
13
‘Packetostatics’
•
The traffic flow
T
and information density
ρ
must
satisfy:
•
In free space, the electric field
E
and the charge
density
ρ
are uniquely determined by:
•
Therefore, the optimal traffic distribution is the
same with the electric field when we substitute
the sources and sinks with positive and negative
charges!
0
,
T
T
0
,
E
E
14
15
Analogy is Uncanny!
Electrostatics
Networks
Potential differences
Number of hops
Non

homogeneous
dielectrics
Non

homogeneous
propagation environments
Conductors
Mobile sources and sinks
Thomson’s theorem
Source/Sink placement
optimization
Intersection of electric field
lines and equipotential
lines
Node locations
16
General Setting
•
Let
be the density of nodes (or, more generally, the cost)
needed to support the sensing/transport/delivery
•
Optimization Problem:
•
Minimization over all possible traffic flows
T
(
x
,
y
) that
satisfy the constraint
•
Standard tool for such problems: Calculus of Variations
)
)
,
(

,
,
(
)
,
(
y
x
y
x
G
y
x
d
T
).
,
(
)
,
(
:
subject to
)
)
,
(

,
,
(
:
minimize
y
x
y
x
dS
y
x
y
x
G
N
T
T
17
Result
•
The traffic flow is given by:
•
where the potential function
φ
is given by the scalar
non

linear partial differential equation:
•
together with appropriate boundary conditions, and
G’
,
H
, properly defined functions
,
)
,
,
(
,
,
(
2
1
)
,
(
y
x
H
y
x
G
y
x
T
)
,
,
(
,
,
(
2
y
x
H
y
x
G
18
Example: Gupta/Kumar
2
1
1
max
)
,
(
)
,
(
y
x
d
c
y
x
T
19
Example: Super Gupta/Kumar
3
2
1
max
)
,
(
)
,
(
y
x
d
c
y
x
T
20
Example: Sub Gupta/Kumar
8
3
1
max
)
,
(
)
,
(
y
x
d
c
y
x
T
21
Example:
Mixed
case
below
)
,
(
above
)
,
(
)
,
(
8
3
1
3
2
1
max
y
x
d
c
y
x
d
c
y
x
T
22
Alternative Microscopic Layers
1.
UWB Physical Layer [NR04]
2.
When nodes are mobile, optimization must take
place across space and time [SA10]
3.
When nodes use directional antennas, network
is anisotropic, and things become complicated
[SA10b]
–
Analogies with macroscopic road traffic engineering
[B52]
•
And the spoilsports:
–
The hierarchical cooperation scheme of [OL07] is
incompatible to our formulation
23
2. Packetoptics
[JA04, CT07, CT09, ST12]
24
Motivation
•
Problem: find route
between (0,0) and
(0,200) with minimum
cost
•
Nodes distributed
according to spatial
Poisson process
•
Cost per hop increases
quadratically with hop
distance
05
.
0
10
30
1
)
,
(
2
4
x
y
x
2
)
(
ad
d
c
25
05
.
0
10
10
1
)
,
(
2
4
x
y
x
26
05
.
0
10
5
1
)
,
(
2
4
x
y
x
27
05
.
0
10
2
1
)
,
(
2
4
x
y
x
Question: what
happens in the limit?
28
Single Macroscopic Quantity
•
Cost Function
:
•
Cost of route
C
that starts at
A
and ends at
B
:
•
Macroscopic Problem: Find route from
A
to
B
that minimizes cost
)
,
(
lim
)
(
0
r
r
dc
c
B
A
C
d
c
AB
r
r
)
(
]
[
29
Relation to Optics
•
Fermat’s Principle: To travel from
A
to
B
, light will
take the route that
locally
minimizes the integral:
•
Therefore we have the following analogy:
–
Index of refraction
n(
r
)
becomes the cost function
c(
r
)
–
Rays of light become minimum

cost routes
B
A
B
A
B
A
ds
n
c
ds
u
dt
)
(
1
1
r
30
Microscopic Model [CT09]
1.
Node placement: spatial Poisson
process with density
λ
(
r
)
2.
Cost per hop:
–
Proportional to distance covered:
C
H
(
d
)=
d
–
Conserving bandwidth:
C
H
(
d
)=
d
2
–
Conserving energy:
C
H
(
d
)=
ad
b
+
f
3.
Routing rule:
–
Greedy routing
–
Forward packet to node for which cost to
progress ratio is minimized
31
Computing the Cost Function
•
Tools: spatial Poisson processes, law of
large numbers, some approximations
•
Results:
(Minimizes distance covered)
(Conserves bandwidth)
(Conserves energy)
const
c
d
d
C
H
)
(
)
(
r
)
(
1
)
(
)
(
2
r
r
c
d
d
C
H
x
const
x
f
x
x
f
f
c
f
ad
d
C
b
H
,
)
(
,
0
,
)
(
)),
(
(
)
(
)
(
r
r
32
Choice of Cost Function Important!
R1:
c
(
r
)
=
[
λ
(
r
)
]
½
[JA04]
R2:
c
(
r
)
=const
R3:
c
(
r
)
= 1/
[
λ
(
r
)
]
½
R4:
c
(
r
)
=f
(
λ
(
r
)
)
33
Optics Analogy
→
The mathematics of Optimal Routing
•
Unicast routing
: Optimal routing paths
r
satisfy the equation
•
Broadcast routing
: Wave fronts of
optimally broadcast packets satisfy the
eikonal
equation:
c
S


c
ds
d
c
ds
d
)
(
r
34
Broadcast Routing
35
Any Practical Gain by Knowing Limit?
•
With finite but many nodes, the optimum
route is hard to find
•
So let us find the optimum route in the
macroscopic limit, and use it to create a near
optimum route
36
The Elephant in the Room
•
We did not
prove
that optimal routes
converge to the optical limit
•
We
assumed
that they converge to a limit,
and showed it is the optical one
(400 nodes)
(2000 nodes)
(10000 nodes)
37
Related Work
•
Using trajectories other than a straight line
has already been proposed (TBF) [NN03].
Now we know optimum!
•
Cost created by static external
interference investigated in [BB06]
•
The
eikonal
equation can be used to
predict how a packet propagates
throughout the network
–
Of particular interest in Delay Tolerant
Networks [JM10], [ST12]
38
3. Minimax Macroscopic
Traffic Load Balancing
[PP03, PR07, HV07, DK08, HV09]
39
Setting
•
Until now, we supposed only one type of
traffic, or at most a few
•
In general case, if there are
n
nodes, there
will be
n
(
n

1)
distinct traffics (and that
ignoring multicasting!)
•
We assume that everyone is interested in
sending traffic to everyone else, and we
would like to minimize the maximum of the
traffic flows experienced at all locations
40
Macroscopic Formulation (1/2)
•
Location
r
1
creates traffic for location
r
2
with
traffic generating rate
λ
(
r
1
,
r
2
),
measured in
bps
/
m
4
•
Set of all paths is
P
•
Distance between
r
1
and
r
2
using
P
is
s
(
P
,
r
1
,
r
2
)
•
Total packet generation rate is
Λ
and mean
packet length is
l
, where
)
,
,
(
)
,
(
1
,
)
,
(
2
1
2
1
2
2
1
2
2
1
2
2
1
2
r
r
r
r
r
r
r
r
r
r
P
s
d
d
l
d
d
A
A
A
A
41
Macroscopic Formulation (
2
/2)
•
Traffic through location
r
with direction
θ
has
angular
flux
Φ
(
P
,
r
,
θ
), measured in bps/m/
rad
•
Total volume that passes through location
r
is given
by
scalar flux
Φ
(
P
,
r
):
•
Problem: Find optimal distribution of paths, so that
maximum traffic load is minimized:
2
0
)
,
)
,
d
r
r
P
P
Φ(
Φ(
)
,
max
),
,
max
argmin
opt
opt
opt
r
r
r
r
P
P
P
Φ(
Φ
Φ(
P
42
Results
(1/2)
•
Lower bounds:
where
A
1
,
A
2
are subdivisions of
A
created
by some curve of length
L
•
Simplified formulas for the scalar flux in
special topologies and routing classes
)
,
(
)
,
(
1
1
2
2
1
2
2
1
2
opt
path
shortest
opt
2
1
r
r
r
r
r
r
A
A
d
d
L
A
l
Φ
Φ
43
Results (2/2)
•
Structure of the optimal traffic flow
–
Optimal routes are uniquely defined in a
bottleneck area
–
‘Field

line’ routing suffices to achieve
optimality
–
Optimization is possible in terms of single
scalar function
•
Load Balancing in the Unit Disk:
44
Related Results
•
One

turn Rectilinear Routing Optimization
in [DK08]
•
Multipath routing on a disk explored in
[PP03]
•
Optics analogy and ‘Curveball Routing’
shown in [PR07]
The ‘Grand’ Open Problem: given an arbitrary
2D connected shape and a traffic generating
rate
λ
(
r
1
,
r
2
)
,
find
Φ
opt
and
P
opt
45
4. Cooperative Broadcast
in the continuum limit
[SS05, SS06, SS07, KS09, SC10]
46
Toy Setting [SS05]
•
Topology:
source placed on left side of strip,
destination placed on right side of strip, relays
are placed in strip, Poisson distributed
•
Reception model:
nodes susceptible to thermal
noise, power decays with distance as
p
r
(
d
)=
kd

2
,
reception successful if
SINR
>
γ
•
Protocol:
We slot time. In first slot, source
transmits. In
i

th slot, everyone transmits if he
received for first time in previous slot.
Transmission powers add up at potential
receivers
47
48
What the Simulations Say
•
For sufficiently low threshold, a wave is
formed that propagates along the strip
After a while, wave achieves fixed width
and goes on for ever
•
For high threshold, wave eventually dies
out, irrespective of how many nodes
initially had the packet
•
Position of initial relays critical
49
The Continuum Assumption
•
Analysis very hard because of random
placement of nodes
•
Assumption: We have so many nodes,
that there is a node practically everywhere
•
Not interested in which node
receives
the packet in
i

th slot
•
Interested in which region of space
receives the packet in
i

th slot
50
•
Result: evolution of the
strip widths can be
predicted in straightforward manner
•
Extensions:
–
Various channel models (fading, etc.)
–
Multiple sources of data traffic
51
Conclusions
•
New framework for studying problems,
based on macroscopic approach
•
Many optimization problems with a
pronounced spatial aspect can be handled
•
Some detail is sacrificed, but solutions are
insightful
•
Math often borrowed from Physics
•
An important open problem: we do not
have convergence rates!
52
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[SA10b] A. Silva, E. Altman, P. Bernhard, M. Debbah, “Continuum equilibria and global
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[SC10] Anna Scaglione, "Cooperative Multiplexing of Broadcast Information in Dense
Wireless Networks", Communication Theory Workshop (CTW), March 2010,
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[ST12] A. Sidera and S. Toumpis, “Delay Tolerant Firework Routing: A Geographic
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57
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