Capacity region of large wireless networks
Devavrat Shah
MIT
Urs Niesen
Piyush Gupta
MIT
Bell-Labs
The Problem
o
Given a wireless network of n nodes
2
Determine its n dimensional capacity regions
 That is, determine



n

n


{

[
]

R
feasible
ij
:is
Purpose
o
Determining the exact capacity region

o
Has remained unresolved even for three node network !
For large networks

Capacity region serves as guideline
 to evaluate performance of a given architecture

Or, as an ‘oracle’ to determine
 feasibility of desired performance
o
Reasonable approximate characterization of capacity region

Will serve the above stated purposes
 Likely to bring out key characteristics of a good network architecture
The Approximation Problem
o
Given a wireless network of n nodes
2
Determine its n dimensional capacity region up to “scaling”
 That is, determine  G that can be “nicely” characterized




G
G
G

 G
The Approximation Problem
o
Given a wireless network of n nodes
2
Determine its n dimensional capacity region up to “scaling”
 That is, determine  G that can be “nicely” characterized




G
G

*
Equivalently, determine ( ) approximately for any  R 
where
*
nn

(

)

max
{


0
:

is
feasible
G
 G

Background
o
The approximation problem

o
Does not lend itself to easy solutions
Basic problem “parameters”:

Node placement:

Nodes are placed in a geographic area

In general can be arbitrarily placed

But, a “nicer” situation is when it is random or regular
Arbitrary
Random/Regular
Background
o
The approximation problem

o
Does not lend itself to easy solutions
Basic problem “parameters”:

Channel model:

Information theoretic Gaussian Fading with power attenuation parameter   2

This allows for possibility of network-wide co-operation

Protocol or interference model: transmission do not interfere

This implies only inter-neighbor (multihop) transmissions are possible
Background
o
The approximation problem

o
Does not lend itself to easy solutions
Basic problem “parameters”:

Traffic demand:

Arbitrary: each node can transmit to all n nodes at varying rates

This corresponds to n dimensional region (or degree of freedom)

Random:

o
2

each node has only one randomly chosen destination

and all nodes wish to transmit at the same rate
*
This corresponds to one-dimensional slice of cap. region, i.e.  (1)
In summary, we want characterization

Ideally, for arbitrary placement, Info. Th. and arbitrary demand
Background
o
Gupta and Kumar (2000) took the key first steps towards this goal

Their clever assumptions made it possible to get started
 Specifically, they considered


Random placement (not arbitrary)

Protocol model (not info. theory)

Random source-destination pairing (not arbitrary traffic)
Answer: maximal per node achievable rate scales as 1

Using multi-hop and geographic routing

Yields a one-dimensional slice of the capacity region
n
Background
o
Gupta and Kumar (2000)

Random placement (not arbitrary)
 Protocol model (not info. theory)
 Random source-destination pairing (not arbitrary traffic)
o
Ozgur, Leveque and Tse (2007) (after a long evolution) considered

Random placement (not arbitrary)
 Information theoretic channel model
 Random source-destination pairing (not arbitrary traffic)
 Obtained complete scaling using “hierarchical” co-operation
multi-hop
hierarchy
Background
o
o
o
Gupta and Kumar (2000)

Random placement (not arbitrary)

Protocol model (not info. theory)

Random source-destination pairing (not arbitrary traffic)
Ozgur, Leveque and Tse (2007) (after a long evolution) considered

Random placement (not arbitrary)

Information theoretic

Random source-destination pairing (not arbitrary traffic)

Obtained complete scaling using “Hierarchical” co-operation
Niesen, Gupta and Shah (2007) obtained scaling for

Arbitrary node placement

Information theoretic channel model

Random source-destination pairing (not arbitrary traffic)

Using our novel interpolation of “multi-hop” and “hierarchical” cooperation
multi-hop
Interpolation
hierarchy
Progress
o
o
o
o
Gupta and Kumar (2000)

Random placement (not arbitrary)

Protocol model (not info. theory)

Random source-destination pairing (not arbitrary traffic)
Ozgur, Leveque and Tse (2007) (after a long evolution) considered

Random placement (not arbitrary)

Information theoretic

Random source-destination pairing (not arbitrary traffic)

Obtained complete scaling using “Hierarchical” co-operation
Niesen, Gupta and Shah (2007) obtained scaling for

Arbitrary placement

Information theoretic

Random source-destination pairing (not arbitrary traffic)
All the above results yield a one-dimensional slice of

Random placement (not arbitrary)

Information theoretic

Arbitrary traffic demand (ie.
n 2dimensional region)
. Here we consider
Progress
o
o
Our setup

Random placement (not arbitrary)

Information theoretic

Arbitrary traffic demand (ie. n dimensional region)
2
Key challenges

Random node placement provides “some regularity”

But, arbitrary traffic demand requires
•
“co-operative” schemes that depend on traffic demand

In most of the previous results, random traffic did not present this challenge

Specifically, our “interpolation” scheme did utilize regularity of traffic
Progress
o
o
Our setup

Random placement (not arbitrary)

Information theoretic

Arbitrary traffic demand (ie. n dimensional region)
2
Our solution: somewhat surprisingly, we find that

Wireless network capacity region is equal to that of a “wireline” tree networks

Tree-construction:
•
“clustering” and use of “multi-hop” or “hierarchical” cooperation
Wireless network
Equivalent tree
Progress
o
o
Our setup

Random placement (not arbitrary)

Information theoretic

Arbitrary traffic demand (ie. n dimensional region)
2
Our solution: somewhat surprisingly, we find that

Wireless network capacity region is equal to that of a “wireline” tree network

Tree utilization: given any traffic demand, route it over tree
•
as if it were a capacitated wireline tree with capacity assigned during our construction
Equivalent tree
Routing
Progress
o
o
Our setup

Random placement (not arbitrary)

Information theoretic

Arbitrary traffic demand (ie. n dimensional region)
2
Our solution: some what surprisingly, we find that

Wireless network capacity region is equal
to that of a “wireline” tree networks

Therefore, the capacity region

approx. characterized by
•
G
is
2n “weighted cuts” , each corresponding to an

“edge” in the tree we created

2
n
Thus, effectively the
dim. capacity region
is characterized by
n
• 2n out of
2
possible cuts !
 G
Overall Progress
INNOVATION
Multi-hop and Straight line
routing
Equivalent to wire-line +
Clever routing
Random cut evaluation
Hierarchical co-op +
Random cut evaluation
Interpolation Multi-hop,
Hierarchical + Geometry
aware scheme + Random
cut evaluation
Equivalent with “wireline”
TREE + Routing over TREE
+ Separation of PHY and
NET layer
Get As Close As Possible !
NODES
CHANNEL
TRAFFIC
RANDOM
PROTOCOL
RANDOM
ARBITRARY PROTOCOL
REF
GK00
ARBITRARY
MSL05,08
+ SSG07,08
RANDOM
INFO. TH.(large )
RANDOM
LT01+
RANDOM
INFO. TH.(small )
RANDOM
OLT07
INFO. TH.
RANDOM
NGS07
ARBITRARY
NGS08
ARBITRARY
IDEAL
ARBITRARY
RANDOM
ARBITRARY
INFO. TH.
INFO. TH.
Broad implications
o
We have identified capacity region scaling

With random placement
• Extends to “regular enough” placement as well
o
Optimal architecture and separation principle

A “physical layer” or capacitated tree is realized through


Combination of multi-hop and hierarchical co-operative schemes
A “network layer” is realized by routing demand on this tree

Treating it as a wireline network

An architecture oblivious to the demands!

Lots of exciting details in the poster by Urs Niesen
End of Phase Goals
o
We have made major progress towards

o
Clearly, the next step is to complete the characterization

o
Characterizing capacity region of large networks
For arbitrary node placement
And, go beyond
That is, understand the scaling of the “multicast” region
n
 This is a n  2 dimensional space and much more complicated
 We strongly believe that we will be able to resolve it building
upon the insights from the unicast case
o