1. Capacity region of large wireless networks
Devavrat Shah
MIT
Urs Niesen Piyush Gupta
MIT Bell-Labs

2. The Problem Given a wireless network of n nodes
Determine its dimensional capacity regions
That is, determine

3. Purpose Determining the exact capacity region For large networks
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
Reasonable approximate characterization of capacity region
Will serve the above stated purposes
Likely to bring out key characteristics of a good network architecture

4. The Approximation Problem
Given a wireless network of n nodes
Determine its dimensional capacity region up to “scaling”
That is, determine that can be “nicely” characterized

5. The Approximation Problem
Given a wireless network of n nodes
Determine its dimensional capacity region up to “scaling”
That is, determine that can be “nicely” characterized
Equivalently, determine approximately for any
where

6. Background The approximation problem Basic problem “parameters”:
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

7. Background The approximation problem Basic problem “parameters”:
Does not lend itself to easy solutions
Basic problem “parameters”:
Channel model:
Information theoretic Gaussian Fading with power attenuation parameter
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

8. Background The approximation problem Basic problem “parameters”:
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 dimensional region (or degree of freedom)
Random:
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.
In summary, we want characterization
Ideally, for arbitrary placement, Info. Th. and arbitrary demand

9. Background
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
Using multi-hop and geographic routing
Yields a one-dimensional slice of the capacity region

10. Background 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
Information theoretic channel model
Obtained complete scaling using “hierarchical” co-operation
multi-hop
hierarchy

11. Background Niesen, Gupta and Shah (2007) obtained scaling for
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
Information theoretic
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

12. Progress Niesen, Gupta and Shah (2007) obtained scaling for
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
Information theoretic
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 Here we consider
Random placement (not arbitrary)
Arbitrary traffic demand (ie. dimensional region)

13. Progress Our setup Key challenges Random placement (not arbitrary)
Information theoretic
Arbitrary traffic demand (ie. dimensional region)
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

14. Progress Our setup Our solution: somewhat surprisingly, we find that
Random placement (not arbitrary)
Information theoretic
Arbitrary traffic demand (ie. dimensional region)
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
Equivalent tree
Wireless network

15. Progress Our setup Our solution: somewhat surprisingly, we find that
Random placement (not arbitrary)
Information theoretic
Arbitrary traffic demand (ie. dimensional region)
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

16. Progress Our setup Our solution: some what surprisingly, we find that
Random placement (not arbitrary)
Information theoretic
Arbitrary traffic demand (ie. dimensional region)
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 is
approx. characterized by
2n “weighted cuts” , each corresponding to an
“edge” in the tree we created
Thus, effectively the dim. capacity region
is characterized by
2n out of possible cuts !

17. Overall Progress REF NODES CHANNEL TRAFFIC GK00 LT01+ OLT07 NGS07
RANDOM PROTOCOL RANDOM
ARBITRARY PROTOCOL ARBITRARY
RANDOM INFO. TH.(large ) RANDOM
RANDOM INFO. TH.(small ) RANDOM
ARBITRARY INFO. TH RANDOM
RANDOM INFO. TH ARBITRARY
ARBITRARY INFO. TH ARBITRARY
REF
GK00
MSL05,08
+ SSG07,08
LT01+
OLT07
NGS07
NGS08
IDEAL
INNOVATION
Multi-hop and Straight line routing
Equivalent to wire-line + Clever routing
Random cut evaluation
Hierarchical co-op +
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 !

18. Broad implications We have identified capacity region scaling
With random placement
Extends to “regular enough” placement as well
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

19. End of Phase Goals We have made major progress towards
Characterizing capacity region of large networks
Clearly, the next step is to complete the characterization
For arbitrary node placement
And, go beyond
That is, understand the scaling of the “multicast” region
This is a 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