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Resource Allocation in OFDM Systems Jianwei Huang Princeton University Joint work with R. Berry, M. Honig, M. Chiang V. Subramanian, R. Agrawal, R. Cendrillon, M. Moonen Sponsors: NSF, Motorola, Alcatel WINLAB Seminar, April 2006 J. Huang

slide-1
SLIDE 1

Resource Allocation in OFDM Systems

Jianwei Huang

Princeton University

Joint work with R. Berry, M. Honig, M. Chiang

  • V. Subramanian, R. Agrawal, R. Cendrillon, M. Moonen

Sponsors: NSF, Motorola, Alcatel

WINLAB Seminar, April 2006

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 1 / 46

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SLIDE 2

OFDM Systems

Frequency band divided into several parallel orthogonal carriers/tones. High spectrum efficiency. Eliminate inter-symbol-interference (ICI) due to multi-path fading. Applications: WiMAX (802.16), Wi-Fi (802.11a/g), DSL, etc.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 2 / 46

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SLIDE 3

Resource Allocation in OFDM Systems

OFDMA (Orthogonal Frequency Division Multiple Access) systems

◮ Each user is assigned a subset of carriers. ◮ No interference among users. ◮ Need to determine user-carrier association and power allocation.

Interference limited OFDM systems

◮ Each user can transmit over all carriers. ◮ Interference among active users in the same carrier. ◮ Need to determine power allocation to mitigate interference.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 3 / 46

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SLIDE 4

Resource Allocation in OFDM Systems

Power Control in Wireless Ad Hoc Networks

4

R2 R4 R3

1

T1 T3 T2 T R

Spectrum Management in DSL Networks

C P CP C P CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT 2 RT2 RT 2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP C P CP C P CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

Scheduling & Resource Allocation in WiMax Networks

tim e

base station

fa ding

users

OFDM

(OFDMA) (Interf-limited)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 4 / 46

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SLIDE 5

Model Summary

Part I II III Motivation WiMAX Ad Hoc DSL Infrastructure Ad Hoc Infrastructure Network Wireless Wireless Wired No Interference Interference Interference Objective Scheduling & Power Allocation Power Allocation Power Allocation

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 5 / 46

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SLIDE 6

Main References

WiMAX: J. Huang, V. Subramanian, R. Agrawal, and R. Berry, “Downlink Scheduling and Resource Allocation for OFDM Systems,” CISS 2006 Ad Hoc: J. Huang, R. Berry and M. Honig, “Distributed Interference Compensation for Wireless Networks,” to appear in IEEE Journal on Selected Areas in Communications, May 2006 DSL: J. Huang, R. Cendrillon, M. Chiang, and M. Moonen, “Autonomous spectrum balancing (ASB) for digital subscriber lines,” submitted to ISIT 2006

More related publications can be found at www.princeton.edu/∼jianweih

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 6 / 46

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SLIDE 7

Part I: WiMAX Network

Power Control in Wireless Ad Hoc Networks

4

R2 R4 R3

1

T1 T3 T2 T R

Spectrum Management in DSL Networks

C P CP C P CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT 2 RT2 RT 2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP C P CP C P CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

Scheduling & Resource Allocation in WiMax Networks

tim e

base station

fa ding

users

OFDM

(OFDMA) (Interf-limited)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 7 / 46

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SLIDE 8

WiMAX Network

time

base station

fading

users

Based on 802.16 and provide MAN broadband connectivity.

◮ Cover a distance of up to 5Kms and shared data speed up to 70Mbps.

Defines a scheduling based MAC protocol.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 8 / 46

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SLIDE 9

Gradient-based Scheduling

We consider downlink channel aware scheduling. Many approaches accomplish this via gradient-based scheduling.

◮ Assign each user a utility, Ui(·), depending on delay, throughput, etc. ◮ Scheduler maximizes choose rate r = (r1, . . . , rN)T from the

rate region R(e) to solve: max

r∈R(e) ∇U(X(t)) · r = max r∈R(e)

  • i

wiri,

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 9 / 46

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SLIDE 10

Gradient-based Scheduling

We consider downlink channel aware scheduling. Many approaches accomplish this via gradient-based scheduling.

◮ Assign each user a utility, Ui(·), depending on delay, throughput, etc. ◮ Scheduler maximizes choose rate r = (r1, . . . , rN)T from the

rate region R(e) to solve: max

r∈R(e) ∇U(X(t)) · r = max r∈R(e)

  • i

wiri,

◮ Myopic policy, requires no knowledge of channel or arrival statistics. ◮ Depends on the utility functions, could lead to various allocation rules ⋆ Proportional fair, maximum rate, stabilizing policies, etc.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 9 / 46

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SLIDE 11

OFDMA Rate Region

R(e) =

  • r : ri =
  • j

xij log

  • 1 + pijeij

xij

  • , (x, p) ∈ X
  • ,

where

◮ xij = time fraction of tone j allocated to user i. ◮ pij = power allocated to user i on tone j. ◮ eij = received SNR/unit power (i.e., channel condition).

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 10 / 46

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SLIDE 12

OFDMA Rate Region

R(e) =

  • r : ri =
  • j

xij log

  • 1 + pijeij

xij

  • , (x, p) ∈ X
  • ,

where

◮ xij = time fraction of tone j allocated to user i. ◮ pij = power allocated to user i on tone j. ◮ eij = received SNR/unit power (i.e., channel condition). ◮ Feasible region

X :=

  • (x, p) ≥ 0 : xij ∈ {0, 1}, ∀i, j,
  • i

xij ≤ 1, ∀ j,

  • ij

pij ≤ P

  • .
  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 10 / 46

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SLIDE 13

Joint Scheduling and Resource Allocation Problem

Solve at every scheduling interval: Problem: 1A-INT max

(x,p)∈X

V (x, p) :=

  • i

wi

  • j

xij log

  • 1 + pijeij

xij

  • Technical Challenges:

◮ Integer constraints on xij. ◮ Want to obtain low-complexity fast algorithms.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 11 / 46

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SLIDE 14

Problem with Relaxed Integer Constraints

We consider the following problem with relaxed integer constraints. Problem: 1B-RELAX max

(x,p)∈ ˜ X

V (x, p) :=

  • i

wi

  • j

xij log

  • 1 + pijeij

xij

  • by replacing xij ∈ {0, 1} in X with 0 ≤ xij ≤ 1 in ˜

X.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 12 / 46

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SLIDE 15

Problem with Relaxed Integer Constraints

We consider the following problem with relaxed integer constraints. Problem: 1B-RELAX max

(x,p)∈ ˜ X

V (x, p) :=

  • i

wi

  • j

xij log

  • 1 + pijeij

xij

  • by replacing xij ∈ {0, 1} in X with 0 ≤ xij ≤ 1 in ˜

X. We will show

◮ Typically, optimal solution of Problem 1B-RELAX is also optimal for

Problem 1A-INT.

◮ If not, a near optimal solution of Problem 1A-INT can be found with

almost no additional complexity.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 12 / 46

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SLIDE 16

Solve Problem 1B-RELAX

Convex problem that satisfies Slater’s condition.

◮ No duality gap.

Consider Lagrangian: L(x, p, λ, µ) :=

  • i

wi

  • j

xij log

  • 1 + pijeij

xij

  • + λ
  • P −
  • i,j

pij

  • +
  • j

µj

  • 1 −
  • i

xij

  • .

Optimizing over x, p and µ, find close form solution of L(λ). Dual function L(λ) is convex.

◮ Find the optimal λ∗ by 1-D iterative search.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 13 / 46

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SLIDE 17

Optimal Primal Variables

Given λ∗, µ∗, let (x∗, p∗) = arg max

(x,p)∈X L(x, p, λ∗, µ∗).

which lead to x∗

ij =

1, µij (λ∗) = maxi µij (λ∗) 0, µij (λ∗) < maxi µij (λ∗)

◮ Requires a simple sort of users per tone j.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 14 / 46

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SLIDE 18

Optimal Primal Variables

Given λ∗, µ∗, let (x∗, p∗) = arg max

(x,p)∈X L(x, p, λ∗, µ∗).

which lead to x∗

ij =

1, µij (λ∗) = maxi µij (λ∗) 0, µij (λ∗) < maxi µij (λ∗)

◮ Requires a simple sort of users per tone j.

In most cases, no tie occurs on any tone j.

◮ Only one user per tone ⇒ optimal solution for Problem 1A-INT.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 14 / 46

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SLIDE 19

Optimal Primal Variables

Given λ∗, µ∗, let (x∗, p∗) = arg max

(x,p)∈X L(x, p, λ∗, µ∗).

which lead to x∗

ij =

1, µij (λ∗) = maxi µij (λ∗) 0, µij (λ∗) < maxi µij (λ∗)

◮ Requires a simple sort of users per tone j.

In most cases, no tie occurs on any tone j.

◮ Only one user per tone ⇒ optimal solution for Problem 1A-INT.

Occasionally, ties occur on some tones.

◮ Results in multiple users per tone ⇒ not primal feasible. ◮ Need to break the ties to find a feasible primal solution for Problem

1A-INT.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 14 / 46

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SLIDE 20

Break the Ties for Problem 1A-INT

Break the ties: choose one user per tone.

◮ Each choice corresponds to one power allocation p∗.

Utilize subgradient of dual function L(λ): P −

ij p∗ ij.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 15 / 46

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SLIDE 21

Break the Ties for Problem 1A-INT

Break the ties: choose one user per tone.

◮ Each choice corresponds to one power allocation p∗.

Utilize subgradient of dual function L(λ): P −

ij p∗ ij.

We choose the smallest nonnegative subgradient.

◮ This determines a primal feasible tone allocation x. ◮ Resulting power constraint might not be tight. ◮ Re-optimize the power allocation p. ◮ Solve a single-user water-filling solution in finite number of steps

(linear in number of tones and users).

All these add little complexity.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 15 / 46

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SLIDE 22

SRT Algorithm: Scheduling and Resource allocation with Tie breaking

1

Consider Problem 1B-RELAX (i.e., relax the integer constraints).

2

Solve dual function L(λ) in close form by optimizing over x, p and µ.

3

1-D iterative search for the optimal λ∗.

4

Solve for the primal variables x, p.

5

If no ties occur, found optimal solution for Problem 1A-INT. Stop.

6

Break the ties using subgradient information.

7

Re-optimize the power allocation in finite iterations.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 16 / 46

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SLIDE 23

User throughput CDFs

1000 2000 3000 4000 5000 6000 7000 8000 9000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Throughput in Kbps User Throughput CDF α=0.5 FULL GOLDEN−1 WEIGHTED−1 MO−wR

40 users, 5MHz, α fair utility (α = 0.5), a total throughput around 20Mbps.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 17 / 46

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SLIDE 24

Summary

Consider mixed integer and convex optimization for joint scheduling and resource allocation in WiMAX network. Design SRT algorithm that achieves optimal solution (when no ties

  • ccur) with low complexity.

Breaking the ties adds little complexity. Propose simple heuristics with performances close to SRT algorithm. Not covered here: consider max. SINR constraints and different channelization methods.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 18 / 46

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SLIDE 25

Part II: Ad Hoc Network

Power Control in Wireless Ad Hoc Networks

4

R2 R4 R3

1

T1 T3 T2 T R

Spectrum Management in DSL Networks

C P CP C P CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT 2 RT2 RT 2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP C P CP C P CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

Scheduling & Resource Allocation in WiMax Networks

tim e

base station

fa ding

users

OFDM

(OFDMA) (Interf-limited)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 19 / 46

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SLIDE 26

Multi-Channel Wireless Ad Hoc Network

T1 T2 T3 R1 R2 R3

h1

11

h2

11

h1

12

h2

12

Ad hoc network: no fixed infrastructure or centralized controller. M users (transmitter-receiver pairs). K parallel channels/carriers. Interference among users in each channel. Goal: distributed power control algorithm to maximize network utility.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 20 / 46

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SLIDE 27

Special Case: Single Channel Network

Problem: 2A-SC max

{Pmin

n

≤pn≤Pmax

n

,∀n}

  • n

Un(γn). Utility Un(γn) is increasing and strictly concave in SINR. Signal-to-interference plus noise ratio (SINR) of user n γn = hn,npn σn +

m=n hn,mpm

.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 21 / 46

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SLIDE 28

Special Case: Single Channel Network

Problem: 2A-SC max

{Pmin

n

≤pn≤Pmax

n

,∀n}

  • n

Un(γn). Utility Un(γn) is increasing and strictly concave in SINR. Signal-to-interference plus noise ratio (SINR) of user n γn = hn,npn σn +

m=n hn,mpm

. Technical Challenges:

◮ Coupled across users due to interferences. ◮ Could be non-convex in power. ◮ Want distributed, low complexity algorithm with fast convergence and

  • ptimal performance.
  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 21 / 46

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SLIDE 29

ADP Algorithm: Asynchronous Distributed Pricing

Price Announcing: user n announces “price” (per unit interference): πn =

  • ∂Un(γn)

∂In

  • = ∂Un(γn)

∂γn γ2

n

pnhn,n . Power Updating: user n updates power pn to maximize surplus: Sn = Un(γn) − pn

  • m=n

πmhm,n. Repeat two phases asynchronously across users. Scalable and distributed: only need to announce single price, and know adjacent channel gains (hm,n).

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 22 / 46

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SLIDE 30

ADP Algorithm

Interpretation of prices: Pigovian taxation

◮ Tax for users generating negative externality (interferences) to society. ◮ Lead to social optimal solution.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 23 / 46

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SLIDE 31

ADP Algorithm

Interpretation of prices: Pigovian taxation

◮ Tax for users generating negative externality (interferences) to society. ◮ Lead to social optimal solution.

ADP algorithm: distributed discovery of Pigovian taxes:

◮ Will it converge? ◮ What does it converge to? ◮ Will it maximize total network utility? ◮ How fast does it converge?

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 23 / 46

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SLIDE 32

Convergence

Coefficient of relative Risk Aversion (CRA) of U(γ): CRA(γ) = −γU′′(γ) U′(γ) .

◮ larger CRA ⇒ “more concave” U.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 24 / 46

slide-33
SLIDE 33

Convergence

Coefficient of relative Risk Aversion (CRA) of U(γ): CRA(γ) = −γU′′(γ) U′(γ) .

◮ larger CRA ⇒ “more concave” U.

Theorem: If for all user n:

(a) Pmin

n

> 0, and (b) CRA(γn) ∈ [1, 2] for all feasible γn;

then there is a unique optimal optimal solution of Problem 2A-SC, and the ADP algorithm globally converges to it.

◮ E.g. condition (b) is always satisfied with log utilities. ◮ Proof based on relating this algorithm to a fictitious supermodular

game.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 24 / 46

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SLIDE 34

Relationship Summary

Problem 2A-SC ADP Algorithm Fictitious Game (Supermodular)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 25 / 46

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SLIDE 35

Relationship Summary

KKT Points Fixed Points Nash Equilibria Problem 2A-SC ADP Algorithm Fictitious Game (Supermodular)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 25 / 46

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SLIDE 36

Relationship Summary

Price/Power Updates Best Response Updates KKT Points Fixed Points Nash Equilibria Problem 2A-SC ADP Algorithm Fictitious Game (Supermodular)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 25 / 46

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SLIDE 37

Relationship Summary

Multiple KKT Points Multiple Nash Equilibria / Conditional Convergence Multiple Fixed Points / Conditional Convergence Price/Power Updates Best Response Updates KKT Points Fixed Points Nash Equilibria Problem 2A-SC ADP Algorithm Fictitious Game (Supermodular)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 25 / 46

slide-38
SLIDE 38

Relationship Summary

Unique Nash Equilibrium / Global Convergence Unique Fixed Point / Global Convergence Unique Optimal Solution Multiple KKT Points Multiple Nash Equilibria / Conditional Convergence Multiple Fixed Points / Conditional Convergence Price/Power Updates Best Response Updates KKT Points Fixed Points Nash Equilibria Problem 2A-SC ADP Algorithm Fictitious Game (Supermodular)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 25 / 46

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SLIDE 39

Supermodular Games

A class of games with “strategic complementaries”

◮ Strategy sets are compact subsets of R; and each player’s pay-off Sn

has “increasing differences”: ∂2Sn ∂xn∂xm > 0.

Key properties:

(1) An N.E. exists. (2) If the N.E. is unique, then the asynchronous best response updates will globally converge to it.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 26 / 46

slide-40
SLIDE 40

Convergence Results

Construction of the fictitious game

◮ Split each user in network into two fictitious players in the game. ◮ Choose players’ payoffs such that best response updates correspond to

the power/price updates in ADP.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 27 / 46

slide-41
SLIDE 41

Convergence Results

Construction of the fictitious game

◮ Split each user in network into two fictitious players in the game. ◮ Choose players’ payoffs such that best response updates correspond to

the power/price updates in ADP.

Proof:

◮ Condition CRA(γn) ∈ [1, 2] guarantees ⋆ The fictitious game is supermodular. ⋆ The Problem 2A-MC has strictly concave objective function (under log

change of variable [Chiang’05]).

◮ Thus there is a unique global optimal solution/fixed point/N.E.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 27 / 46

slide-42
SLIDE 42

Convergence Speed

10 20 0.5 1 Power ADP Algorithm 200 400 600 0.5 1 Power Gradient−based Algorithm 5 10 15 20 20 40 60 80 Iterations Price 200 400 600 20 40 60 80 Iterations Price

log utilities, 10 users, Gradient-based Algorithm is based on [Chiang’05]

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 28 / 46

slide-43
SLIDE 43

Multi-channel Model

Problem: 2B-MC max

{pn∈Pn,∀n}

  • n
  • k

Uk

n (γk n).

Assume each user can transmit over K orthogonal channels. Received SINR in channel k for user n γk

n =

hk

n,npk n

σk

n + m=n hk n,mpk m

Can allocate power across channels subject to total power constraint: Pn :=

  • pk

n ≥ Pmin n

,

  • k

pk

n ≤ Pmax n

  • .
  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 29 / 46

slide-44
SLIDE 44

Dual ADP (DADP) Algorithm

Two classes of prices:

◮ User still announces an interference price πk

n on each channel k.

◮ User also keeps a local resource price, µn, to reflect power constraint.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 30 / 46

slide-45
SLIDE 45

Dual ADP (DADP) Algorithm

Two classes of prices:

◮ User still announces an interference price πk

n on each channel k.

◮ User also keeps a local resource price, µn, to reflect power constraint.

Primal Updates: separable across channels

◮ User chooses power pk

n to maximize:

Sk

n = Uk n (γk n) − pk n

 

m=n

hm,nπk

m + µn

  .

◮ Interference price πk

n updated as in single channel ADP algorithm.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 30 / 46

slide-46
SLIDE 46

Dual ADP (DADP) Algorithm

Two classes of prices:

◮ User still announces an interference price πk

n on each channel k.

◮ User also keeps a local resource price, µn, to reflect power constraint.

Primal Updates: separable across channels

◮ User chooses power pk

n to maximize:

Sk

n = Uk n (γk n) − pk n

 

m=n

hm,nπk

m + µn

  .

◮ Interference price πk

n updated as in single channel ADP algorithm.

Dual Iteration: resource price follows subgradient updates: µn(t) =

  • µn(t−) + κ
  • k

pk

n − Pmax n

+ .

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 30 / 46

slide-47
SLIDE 47

Convergence

Theorem: The DADP algorithm globally and geometrically converges to the unique optimal solution of Problem 2B-MC.

◮ Under similar restrictions on the utility functions in single channel case. ◮ With small constant stepsize κ.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 31 / 46

slide-48
SLIDE 48

Convergence

Theorem: The DADP algorithm globally and geometrically converges to the unique optimal solution of Problem 2B-MC.

◮ Under similar restrictions on the utility functions in single channel case. ◮ With small constant stepsize κ.

Proof:

◮ Need to show the Lipschitz condition and strong convexity of the

gradient of dual function.

◮ Separation of time-scales assumption: Primal Updates converges

between any two adjacent Dual Iterations

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 31 / 46

slide-49
SLIDE 49

Simulation Results

5 10 15 20 10

−4

10

−3

10

−2

10

−1

Primal Updates Max 1 Primal Update / Dual Iteration Max 3 Primal Updates / Dual Iteration Max 5 Primal Updates / Dual Iteration Max 7 Primal Updates / Dual Iteration (Utot

* −Utot)/Utot *

log utilities, 16 channels, 50 users

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 32 / 46

slide-50
SLIDE 50

Summary

Consider power control in multi-channel wireless ad hoc networks. Propose dual-based ADP algorithm that achieves optimal solution with fast convergence (under proper conditions of the utility functions). Not covered here:

◮ Primal-based ADP algorithm to solve both convex and non-convex

multi-channel power control problem.

◮ Convergence of ADP algorithm when each user is limited to choose one

  • ut of many channels.
  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 33 / 46

slide-51
SLIDE 51

Part III: DSL Network

Power Control in Wireless Ad Hoc Networks

4

R2 R4 R3

1

T1 T3 T2 T R

Spectrum Management in DSL Networks

C P CP C P CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT 2 RT2 RT 2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP C P CP C P CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

Scheduling & Resource Allocation in WiMax Networks

tim e

base station

fa ding

users

OFDM

(OFDMA) (Interf-limited)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 34 / 46

slide-52
SLIDE 52

Digital Subscriber Line

Discrete Multi-Tone introduced in early 90’s by John Cioffi. Utilize the spectrum unused by voice transmissions and divide into large number of subchannels. Convert traditional telephone twisted-pair copper wires to broadband communication media. Frequency (KHz) 3.4 Power Voice DSL

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 35 / 46

slide-53
SLIDE 53

Current Efforts

Performance bottleneck: crosstalks (interferences) among lines.

◮ Current practice: static spectrum management. ◮ A new wave of dynamic spectrum management since 2002.

FAST Copper project:

◮ Joint NSF project among Princeton, Stanford and Fraser Research. ◮ Collabration with AT&T. ◮ Aim at providing DSL broadband service up to 100Mbps by joint

  • ptimization over Frequency, Amplitude, Space and Time.

◮ Today we will focus on the Frequency aspect.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 36 / 46

slide-54
SLIDE 54

Multiple-line Channel Model

Downlink Transmission crosstalk

  • Customer

CO

  • Customer
  • RT

Mathematically similar as the multi-channel wireless ad hoc network.

◮ Difference: channel can be considered as time-invariant.

Crosstalks are highly distance and frequency dependent:

◮ Decrease with distance. ◮ Increase with frequency.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 37 / 46

slide-55
SLIDE 55

Multiple-line Channel Model

Downlink Transmission crosstalk

  • Customer

CO

  • Customer
  • RT

Mathematically similar as the multi-channel wireless ad hoc network.

◮ Difference: channel can be considered as time-invariant.

Crosstalks are highly distance and frequency dependent:

◮ Decrease with distance. ◮ Increase with frequency.

In the CO/RT mixed case

◮ RT generates excessive interference to the CO line. ◮ CO generates little interference to the RT line. ◮ Also called near-far problem: performance bottleneck in the US.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 37 / 46

slide-56
SLIDE 56

Spectrum Management Problem

Problem: 3 maximize {pk

n}k,n

  • n

wnRn subject to Rn =

  • k

log

  • 1 +

pk

n

  • m=n αk

n,mpk m + σk n

  • , ∀n
  • k

pk

n ≤ Pmax n

, ∀n pk

n ≥ 0, ∀k, n.

Technical Difficulty:

◮ Highly non-convex and tightly coupled problem. ◮ No explicit message passing among users is desired. ◮ Want find distributed and low complexity algorithm with near optimal

performance.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 38 / 46

slide-57
SLIDE 57

Dynamic Spectrum Management (DSM)

State-of-art of dynamic spectrum management

◮ Iterative Water-filling (IW) [Yu, Ginis, Cioffi’02] ◮ Optimal Spectrum Balancing (OSB) [Cendrillon et al.’04] ◮ Iterative Spectrum Balancing (ISB) [Cendrillon, Moonen’05]

[Liu, Yu’05]

◮ Autonomous Spectrum Balancing (ASB) [Huang et al.’06]

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 39 / 46

slide-58
SLIDE 58

Dynamic Spectrum Management (DSM)

State-of-art of dynamic spectrum management

◮ Iterative Water-filling (IW) [Yu, Ginis, Cioffi’02] ◮ Optimal Spectrum Balancing (OSB) [Cendrillon et al.’04] ◮ Iterative Spectrum Balancing (ISB) [Cendrillon, Moonen’05]

[Liu, Yu’05]

◮ Autonomous Spectrum Balancing (ASB) [Huang et al.’06]

Algorithm Operation Complexity Performance IW Autonomous O (KN) Suboptimal OSB Centralized O

  • KeN

Asymptotic Optimal ISB Centralized O

  • KN2

Near optimal ASB Autonomous O (KN) Near optimal

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 39 / 46

slide-59
SLIDE 59

Autonomous Spectrum Balancing

Finding the optimal solution requires global crosstalk information. In mixed CO/RT case, need to protect the service on the CO line.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 40 / 46

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SLIDE 60

Autonomous Spectrum Balancing

Finding the optimal solution requires global crosstalk information. In mixed CO/RT case, need to protect the service on the CO line. Reference Line:

◮ A fictitious line representative of typical CO line. ◮ All parameters are fixed and known to all users.

Each user tries to maximize a weighted sum of its own rate and reference line’s rate.

Reference Line (5km) Actual Line

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 40 / 46

slide-61
SLIDE 61

ASB Algorithm

Each user n solves the following problem

maximize

{pk

n}k≥0 wn

  • k
  • 1 +

pk

n

  • m=n αk

n,mpk m + σk n

  • +
  • k

log

  • 1 +

˜ pk ˜ αk

npk n + ˜

σk

  • subject to
  • k

pk

n ≤ Pmax n

,

Iterate through users until convergence.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 41 / 46

slide-62
SLIDE 62

ASB Algorithm

Each user n solves the following problem

maximize

{pk

n}k≥0 wn

  • k
  • 1 +

pk

n

  • m=n αk

n,mpk m + σk n

  • +
  • k

log

  • 1 +

˜ pk ˜ αk

npk n + ˜

σk

  • subject to
  • k

pk

n ≤ Pmax n

,

Iterate through users until convergence. Only need local information. Still non-convex, but can be solved by dual decomposition.

◮ Duality gap is asymptotically zero when the number of tones is high

[Cendrillon et al.’04].

◮ Per tone subproblem: find optimal solution by solving a cubic equation.

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 41 / 46

slide-63
SLIDE 63

Simulation Setup

4 ADSL lines Mixed CO/RT deployment Target rate set to 2Mbps on RT2 and RT3

CP CP CP CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT2 RT2 RT2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP CP CP CP CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 42 / 46

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SLIDE 64

Performance

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 RT1 (Mbps) CO (Mbps)

OSB IW ASB

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 43 / 46

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SLIDE 65

Summary

Consider dynamic spectrum management in DSL networks. Propose ASB algorithm, which is fully autonomous, with low complexity, and achieves near optimal performance. Not covered here:

◮ High SINR approximation on the reference line to with even lower

complexity and faster convergence.

◮ Consider asynchronous transmissions, where the channels are not

  • rthogonal to each other.
  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 44 / 46

slide-66
SLIDE 66

Conclusions

Power Control in Wireless Ad Hoc Networks

4

R2 R4 R3

1

T1 T3 T2 T R

Spectrum Management in DSL Networks

C P CP C P CP CO CO CO CO RT1 RT1 RT1 RT1 RT2 RT 2 RT2 RT 2 RT3 RT3 RT3 RT3 CP CP CP CP CP CP CP CP C P CP C P CP 5 km 4 km 3.5 km 3 km 2 km 3 km 4 km

Scheduling & Resource Allocation in WiMax Networks

tim e

base station

fa ding

users

OFDM

(OFDMA) (Interf-limited)

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 45 / 46

slide-67
SLIDE 67

Results Summary

Part I II III Motivation WiMAX Ad Hoc DSL Problem Integer/Convex Nonconvex/Convex Nonconvex Methodology Integer Relaxation Supermodular Game Reference Line Tie Breaking Theory Approximation Algorithm SRT ADP ASB Centralized Distributed Distributed Properties (Near) Optimal Optimal Near Optimal Low Complexity Low Complexity Low Complexity Converges Fast Convergence Converges in pratice

  • J. Huang (Princeton Univ.)

OFDM Resource Allocation April 2006 46 / 46