Basic results and recent advances in physics-inspired wireless ...

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21 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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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54

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55

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[SA10b] A. Silva, E. Altman, P. Bernhard, M. Debbah, “Continuum equilibria and global
optimization in dense static ad hoc networks,” in Computer Networks, Vol. 54, No.
6, Apr. 2010, pp. 1005
-
1018.

[SA10] A. Silva, E. Altman, M. Debbah, G.
Alfano
, “
Magnetworks
: how mobility impacts
the design of mobile ad hoc networks,” in Infocom 2010, Mar. 2010.



56

[
JM10] P. Jacquet, B. Mans and G. Rodolakis, Information Propagation Speed in Mobile
and Delay Tolerant Networks, IEEE Transactions on Information Theory, 56(1), pp.
5001
-
5015, Oct. 2010 (ISSN
-
0018
-
9448).

[SC10] Anna Scaglione, "Cooperative Multiplexing of Broadcast Information in Dense
Wireless Networks", Communication Theory Workshop (CTW), March 2010,
Invited Paper.

[KS09] S.
Kirti

and A. Scaglione, "Cooperative Broadcast in Dense Networks with
Multiple Sources," 10th IEEE Intl. Workshop on Signal Processing Advances for
Wireless Communications (SPAWC), Perugia, Italy, June 21
-
24 2009.

[HV11] E. Hyytia, J. Virtamo, P.
Lassila
, J.
Kangasharju
, J.
Ott
, “When Does Content
Float? Characterizing Availability of Anchored Information in Opportunistic Content
Sharing,” in Infocom 2011, Apr. 2011.

[ST12] A. Sidera and S. Toumpis, “Delay Tolerant Firework Routing: A Geographic
Routing Protocol for Wireless Delay Tolerant Networks,” submitted work.


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