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
References
[SK83] J. Sylvester and L. Kleinrock, “On the Capacity of multihop slotted ALOHA
networks with regular structures,” IEEE Trans. Commun. Vol. 31, No. 8, pp. 974

982, Aug. 1983.
[B52] M. Beckman, ‘A continuum model of transportation,’ Econometrica 20, 1952, pp.
643

660.
[GK01] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. on
Information Theory, Vol. 46, No. 2, pp. 388

404, Mar. 2000.
[NN03] D. Niculescu and B. Nath, ‘Trajectory Based Forwarding and Its Applications, in
Proc. MOBICOM 2003, San Diego, CA, 2003, pp. 260

272.
[PP03] P. P. Pham and S. Perreau, ‘Performance Analysis of Reactive Shortest Path and
Multi

path Routing Mechanism With Load Balance,’ in IEEE Infocom 2003, San
Francisco, CA, Apr.

May 2003.
[KS04] M. Kalantari and M. Shayman, “Energy efficient routing in wireless sensor
networks,” in Proc. CISS, Princeton University, NJ, Mar. 2004.
[J04] P. Jacquet, “Geometry of information propagation in massively dense ad hoc
networks, in Proc. ACM MobiHOC,
Roppongi
Hills, Japan, 2004
[KS04b]
M. Kalantari and M. Shayman, “
Routing
in wireless
ad hoc networks
by
analogy
to
electrostatic
theory
,” in
Proc
. IEEE ICC
,
2004.
[J04]
P. Jacquet, “
Space

time
information
propagation in
mobile
ad hoc wireless
networks,” in
Proc
. Information
Theory Workshop, San Antonio, TX, Oct. 2004.
[NR04] R. Negi and A.
Rajeswaran
, “Capacity of power constrained ad

hoc networks, in
Proc. Infocom 2004.
53
[FD04] M. Franceschetti, O. Dousse, D.N.C. Tse, P.
Thiran
, ‘Closing the gap in the
capacity of random wireless networks,’ in Proc. ISIT, Chicago, IL, June

July 2004.
[TT05] S. Toumpis and L. Tassiulas, “Packetostatics: Deployment of massively dense
sensor networks as an electrostatics problem,” in Proc. IEEE Infocom 2005
[SS05] B.
Sirkeci

Mergen
and A. Scaglione, “A continuum approach to dense wireless
networks with cooperation,” in Proc. Infocom 2005
[TG05] S. Toumpis and G. A. Gupta, “Optimal placement of nodes in large sensor
networks under a general physical layer model,”, in Proc. IEEE SECON 2005.
[CM06] R. Catanuto, G. Morabito, and S. Toumpis, “Optical Routing in massively dense
networks: Practical issues and dynamic programming interpretation,” in Proc.
International Symposium on Wireless Communication Systems, Valencia, Spain,
2006.
[SS06] B.
Sirkeci

Mergen
, A. Scaglione, G.
Mergen
, ``Asymptotic Analysis of Multi

Stage
Cooperative Broadcast in Wireless Networks”, Joint special issue of the IEEE
Transactions on Information Theory and IEEE/ACM Trans. On Networking, Volume
52, Issue 6, June 2006 Page(s):2531

2550.
[KS06] M. Kalantari and M. Shayman, “Routing in multi

commodity sensor networks
based on partial differential equation,” in Proc. Conference on Information
Sciences and Systems, Princeton University, NJ, Mar 2006
[KS06b] M. Kalantari and M. Shayman, “Design optimization of multi

sink sensor
networks by analogy to electrostatic theory,” in Proc. WCNC, Las Vegas, NV, Apr.
2006
54
[HV06] E.
Hyyti
ä
and J. Virtamo, “On load balancing in a dense wireless multihop network,”
in Proc. 2
nd
EuroNGI
Conference on Next Generation Internet Design and Engineering
(NGI 2006), Valencia, Spain, Apr. 2006.
[BB06] F. Baccelli, N. Bambos, and C. Chan, “Optimal power, throughput, and routing for
wireless link arrays,” in Proc. IEEE Infocom, 2006.
[CM06] R. Catanuto and G. Morabito, “Optimal routing in dense wireless multihop networks
as a geometrical optics solution to the problem of variations,” in Proc. IEEE ICC,
Istanbul, Turkey, June 2006.
[SS06] B.
Sirkeci

Mergen
, A. Scaglione, and G.
Mergen
, “Asymptotic analysis of multistage
cooperative broadcast in wireless networks,” IEEE Trans. Inform. Theory, vol. 52, no.
6, pp. 2531

2550.
[TT06] S. Toumpis and L. Tassiulas, “Optimal deployment of large wireless sensor networks,
“ IEEE Trans. Inform. Theory, July 2006.
[HV07] E. Hyytia, J. Virtamo, On Traffic Load Distribution and Load Balancing in Dense
Wireless Multihop Networks, in EURASIP Journal on Wireless Communications and
Networking, special issue on ‘Novel Techniques for Analysis and Design of Cross

Layer Optimized Wireless Sensor Networks,’ May 2007.
[PR07] L.
Popa
and A.
Rostami
and R. M. Karp and C. Papadimitriou, and I.
Stoica
,
‘Balancing the traffic load in wireless networks with curveball routing,’ in Proc. Mobihoc
2007, Montreal, Canada, 2007.
[SS07] B.
Sirkeci

Mergen
and A. Scaglione, "On the Power Efficiency of Cooperative
Broadcast in Dense Wireless Networks," IEEE Journal on Selected Areas in
Communications, vol. 25, no. 2, pp. 497

507, February 2007.
[OL07] Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks, IEEE
Trans. on Information Theory, vol. 53, No. 10, pp. 3549

3572, Oct. 2007.
55
[HV08] E.
Hyytiä
and J. Virtamo, Near

Optimal Load Balancing in Dense Wireless Multi

Hop Networks, in NGI 2008, 4th Conference on Next Generation Internet
Networks, pp. 181

188, 2008,
Kraków
, Poland.
[DK08] S. Durocher, E.
Kranakis
, D.
Krizane
, L. Narayanan, ‘Balancing Traffic Load
Using One

Turn Rectilinear Routing,’ TAMC 2008, 467

478
.
[HK09] M.
Haghpanani
, M. Kalantari, M. Shayman, “Implementing Information Paths in a
Dense Wireless Sensor Networks,” in Globecom 2009.
[HV09] E. Hyytia and J. Virtamo, “On the Optimality of field

line routing in massively
dense wireless multi

hop networks,” in Performance Evaluation, Vo. 66, pp. 158

172, 2009.
[JK09] S. Jung, M
Kserawi
, D. Lee, June

Koo Kevin
Ree
, “Distributed Potential Field
Based Routing and Autonomous Load Balancing for Wireless Mesh Networks,” in
IEEE Communications Letters, Vol. 13, No. 6, June 2009.
[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.
[JB09] Philippe Jacquet, Bernard Mans, Paul
Mȕhlethaler
and Georgios Rodolakis
Opportunistic Routing in Wireless Ad Hoc Networks: Upper Bounds for the Packet
Propagation Speed.
[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.
57
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