Capacity region of large wireless networks

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
• 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
• 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
• 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
• 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
• 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
11
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
• 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
hierarchy
Interpolation
12
Progress

• 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 . Here we consider
• Random placement (not arbitrary)
• Information theoretic
• Arbitrary traffic demand (ie. dimensional
  region)

13
Progress

• Our setup
• 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
• 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
• 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

Routing
Equivalent tree
16
Progress

• Our setup
• 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 GK00 MSL05,08 SSG07,08 LT01 OLT07 NGS
07 NGS08 IDEAL
NODES CHANNEL TRAFFIC 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

• 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

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

20
Background

• GK00
• RANDOM
• node placement
• PROTOCOL
• Not Info. Th.
• RANDOM
• traffic demand
• Equivalent to finding
• in above setup
• One-dimensional characterization !

• IDEALLY
• ARBITRARY
• INFO. TH.
• Gaussian Fading
• ARBITRARY
• Traffic
• Unlike, this is dimensional !

21
Background
WHO GK00 IDEAL

• INNOVATION
• Multi-hop and Straight line routing

NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D ARBITRARY INFO.
TH. ARBITRARY
22
Background
WHO GK00 MSL05,08 SSG07,08 IDEAL

• INNOVATION
• Multi-hop and Straight line routing
• Equivalent to wire-line Clever routing

NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D ARBITRARY PROTOCOL
ARBITRARY ARBITRARY INFO. TH.
ARBITRARY
23
Background
WHO GK00 MSL05,08 SSG07,08 LT01 IDEAL
NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D ARBITRARY PROTOCOL
ARBITRARY RANDOM INFO. TH.(large )
RANDOM ARBITRARY INFO. TH.
ARBITRARY

• INNOVATION
• Multi-hop and Straight line routing
• Equivalent to wire-line Clever routing
• Random cut evaluation

24
Background
WHO GK00 MSL05,08 SSG07,08 LT01 OLT07
IDEAL
NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D ARBITRARY PROTOCOL
ARBITRARY RANDOM INFO. TH.(large )
RANDOM RANDOM INFO. TH.(small )
RANDOM ARBITRARY INFO. TH. ARBITRARY

• INNOVATION
• Multi-hop and Straight line routing
• Equivalent to wire-line Clever routing
• Random cut evaluation
• Hierarchical co-op
• Random cut evaluation

25
Our Progress
WHO GK00 MSL05,08 SSG07,08 LT01 OLT07 NGS
07 IDEAL
NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D ARBITRARY PROTOCOL
ARBITRARY RANDOM INFO. TH.(large )
RANDOM RANDOM INFO. TH.(small )
RANDOM ARBITRARY INFO. TH.
RANDOM ARBITRARY INFO. TH. ARBITRARY

• 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

26
Our Progress
WHO GK00 MSL05,08 SSG07,08 LT01 OLT07 NGS
07 NGS08 IDEAL
NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D 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

• 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

27
Lesson Learnt

• Cuts play an important role
• In all the characterizations obtained thus far
• RANDOM traffic and RANDOM placement
• One appropriate (type of) cut is bottleneck,
  and
• Essentially, schemes are designed to achieve such
  cut(s)?
• For ARBITRARY traffic
• Different cuts become bottleneck depending upon
  traffic
• Therefore, scheme needs to be flexible enough
• Interestingly enough, a Tree structure is
  sufficient to achieve this flexibility

28
The Plan
WHO GK00 MSL05,08 SSG07,08 LT01 OLT07 NGS
07 NGS08 IDEAL
NODE CHANNEL TRAFFIC RANDOM
PROTOCOL RANDOM/1D 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

• 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