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 Determine its n dimensional.
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Transcript 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 Determine its n dimensional.
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
*
nn
(
)
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