resource allocation ppt

Resource Allocation

in WiMAX

Laura [email protected]

I. Outlines 2

Outlines

1. Single-Cell OFDMA Networks

2. Multicell OFDMA Networks

3. OFDMA Networks with Relays

Cottatellucci: Resource Allocation in WiMAX c© Eurecom 11 February 2008

II. WiMAX Physical Layer 3

Variants of WiMAX PHY

Designation Function LOS/NLOS Frequency Duplexing

WiMAN-SC Point-to-Point LOS 10-66 GHz TDD, FDD

WiMAN-SCa Point-to-Point NLOS 2-10 GHz TDD, FDD

WiMAN-OFDM Point-to-Multipoint NLOS 2-10 GHz TDD, FDD

WiMAN-OFDMA Point-to-Multipoint NLOS 2-10 GHz TDD, FDD

WiMAN-HUMAN Point-to-Multipoint NLOS 2-10 GHz TDD

We focus on WiMAN-OFDMA


III. Single-Cell OFDMA Networks 4

System Model

.

base station

user 1

user 2

user K

h2N

h11

hK1 hKN

hK2

h1N

h21

h22

• K : number of users

• N : number of subcarriers

• hkn : channel gain of user k in subcarrier

n

• pkn : power of user k in subcarrier n

• γkn = |hkn|2σ2 : normalized SNR of user k

in subcarrier n

• rkn = log(1+pknγkn) : maximum achiev-

able rate by user k on tone n

In single-hop OFDMA resource allocation consists in jointly

assigning subcarriers and allocating powers.



Resource Allocation in Downlink: Problem Definition

Sum Rate Maximization Problem

Object: maximize the sumrate under a constraint on themaximum transmitted power.

• Sk : set of subcarriers allocated to user k

SUM RATE

maximize

K∑

k=1

∑

n∈Sk

log2(1 + pknγkn)

subject to

K∑

k=1

∑

n∈Sk

pkn ≤ Ptot,

Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}

pkn ≥ 0 ∀k and ∀n







MAXIMUM POWER CONSTRAINT

maximize

K∑

k=1

∑

n∈Sk

log2(1 + pknγkn)

subject to

K∑

k=1

∑

n∈Sk

pkn ≤ Ptot,

Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}








ORTHOGONALITY CONSTRAINT

maximize

K∑

k=1

∑

n∈Sk

log2(1 + pknγkn)

subject to

K∑

k=1

∑

n∈Sk

pkn ≤ Ptot,

Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}


Nonconvex optimization problem

with exponential complexity in the number of users and subcarriers.



Resource Allocation in Downlink: Problem Definition (2)

Power Minimization Problem

Object: minimize the trans-mitted power under con-straints on the minimumtransmitting rates.

• Rk : Target rate for user k

SUM POWER

minimize

K∑

k=1

∑

n∈Sk

pkn

subject to∑

n∈Sk

log2(1 + pknγkn) ≥ Rk∀k

Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}








MINIMUM RATE CONSTRAINTS

minimize

K∑

k=1

∑

n∈Sk

pkn

subject to∑

n∈Sk


Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}








ORTHOGONALITY CONSTRAINT

minimize

K∑

k=1

∑

n∈Sk

pkn

subject to∑

n∈Sk


Sj ∩ Sk = ∅ ∀j 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}



with exponential complexity in the number of users and subcarriers.



Resource Allocation in Uplink: Problem Definition


Object: maximize the sumrate under independent con-straints on the maximumtransmitted power for eachuser.

maximize

K∑

k=1

∑

n∈Sk

log2(1 + pknγkn)

subject to∑

n∈Sk

pkn ≤ P k, ∀k

Sj ∩ Sk = ∅ ∀ 6= k

∪Kk=1Sk ⊆ {1, 2, . . . , N}




IV. Fundamental Results in Optimization 8

Lagrange Duality

Primal Problem

maximize

L∑

`=1

f`(x)

subject to

M∑m=1

hm(x) ≤ P

• f`(x) : not necessarily concave

• hm(x) : not necessarily convex

Define:

• Lagrange multipliers: λ = (λ1, . . . , λM)T

• Lagrangian:L(x, λ) =

∑` f`(x) + λT (P−∑

m hm(x))

• Dual objective: g(λ) = maxx L(x, λ)

Dual Problem

minimize g(λ)

subject to λ ≥ 0

x∗ : solution primal problem (λ4,x4) : solution dual problem

For convex problems∑L

`=1(f`(x∗)− f`(x

4)), the duality gap is zero!...

....We can equivalently solve the dual unconstraint problem!Cottatellucci: Resource Allocation in WiMAX c© Eurecom 11 February 2008


Fundamental Results in Nonconvex Optimization∗

Definition of Time Sharing Condition : Consider the maximum value of the primal problem

as a function of the constraint P. If such a function is concave in P then the primal

problem satisfies the time sharing condition.

If the primal optimization problem satisfies the time sharing property,then it has zero duality gap,

i.e. the primal problem and the dual problem have the same optimal value.

λ∗∑

n fn(x∗)

P∑

hm(x∗)

g∗

f∗ 6= g∗

λ∗∑

n fn(x∗)

P

∑n fn(x̂∗)

λT (P−∑hm(x̂∗)

slope=λ

∑hm(x̂∗)

∑hm(x∗)

f∗ = g∗

g(λ)

(*) W. Yu and R. Lui, Dual methods for nonconvex spectrum optimization of multicarrier systems, 2006.



Application to Multicarrier Systems

As the number of subcarriers tends to infinity the sum rate maximization and the sum

power minimization problems in uplink and down link satisfy asymptotically the time sharing

conditions (Yu et al. 2006, Seong et al. 2006).

This property holds also when additional constraints (e.g. integer bit loading) are enforced

We can solve the unconstrained dual problemand obtain an almost optimum solution!

When applied to OFDMA systems the joint search over usersand subcarriers decouples and the search has linear complexity

in the number of users and tones, i.e. O(KN).

Joint subcarrier and power allocation is feasible in real-time systems!!!



An Example:Sum Rate Maximization in Downlink via Duality

Dual function

g(λ) = max{pkn}

N∑n=1

K∑

k=1

rkn + λ(P −∑

n

∑

k

pkn)

=

N∑n=1

max{pkn}

(K∑

k=1

rkn − λpkn

)

︸︷︷︸+λP

gn(λ)

=∑

n

gn(λ) + λP

The maximization of the dual function re-

duces to N independent concave maximiza-

tion problems!

maxλ

g(λ) with λ ≥ 0

Algorithm

Initialization: set λ = λ0 > 0, ε > 0

repeat

for n = 1 . . . N

select k∗ maximizing gn(λ)

determine pk∗n maximizing gn(λ)

endfor

update λ according to

gradient/elipsoid criterion

until |P −∑

n

pk∗n| < ε




Dual function

g(λ) = max{pkn}

N∑n=1

K∑

k=1

rkn + λ(P −∑

n

∑

k

pkn)

=

N∑n=1

max{pkn}

(K∑

k=1

rkn − λpkn

)

︸︷︷︸+λP

gn(λ)

=∑

n

gn(λ) + λP



tion problems!

maxλ

g(λ) with λ ≥ 0

LINEAR COMPLEXITY IN K!

Algorithm


repeat

for n = 1 . . . N



endfor



until |P −∑

n

pk∗n| < ε




Dual function

g(λ) = max{pkn}

N∑n=1

K∑

k=1

rkn + λ(P −∑

n

∑

k

pkn)

=

N∑n=1

max{pkn}

(K∑

k=1

rkn − λpkn

)

︸︷︷︸+λP

gn(λ)

=∑

n

gn(λ) + λP



tion problems!

maxλ

g(λ) with λ ≥ 0

LINEAR COMPLEXITY IN KN!

Algorithm


repeat

for n = 1 . . . N



endfor



until |P −∑

n

pk∗n| < ε


V. Single Cell: State of Art 12

Resource Allocation in Downlink: State of Art

• Max-min problem: it maximizes the worst user rate.

• Proportional fairness: like the sum rate maximization problem but with theadditional constraints of given ratios among the user rates, i.e. Rk is a givenfraction α of the total rate.Shen, Andrews, and Evans, Adaptive resource allocation in multiuser OFDM systemswith proportional fairness, December 2005

• Hard fairness, which coincides with the sum power minimization problem.

• Unified framework for a large class of utility functions.Song and Li, Crosslayer optimization for OFDM wireless networks. Part I: Theoreticalframework, March 2005.

These approaches do not exploit Lagrange duality for joint optimizationIf the optimization problem with fairness satisfied the time-sharing condition,

the dual approach would improve performance and complexity!!


V. Single Cell: State of Art 13

Resource Allocation in Uplink: State of Art

• Game theory tools to enforce fairness in centralized approaches.Han, Ji,Ray Liu, Fair multiuser channel allocation for OFDMA networks using NAshBargaining and coalitions, August 2005.

• Bayesian games to develop distributed algorithms with partial and/or sta-tistical knowledge of the channel state information at the transmitters.He, Gault, Debbah, and Altman, Correlated and non-correlated equilibria for multiuserOFDM systems, January 2008

Advantages: No feedback channel for information on resource allocation.Resource allocation possible also in critical situation when complete CSIis not available or not reliable.

Disadvantages: Possible performance degradation.Possible collisions on the same subcarrier with consequent nonzero out-age probability for slow fading channels.


VI. Multicell OFDMA Networks 14

Frequency Reuse

The system band is divided in different rf sub-bands. Each cell communicates only ona sub-band. Adjacent cells transmit on disjoint sub-bands. 1

rfis the reuse factor.

Frequency reuse factor 13

The inter-cell interference can be ne-glected. Single-cell resource allocation al-gorithms are applicable.

Drawbacks• Loss in spectral efficiency

• Complex coloring problem (even more complex

with relays)

• Cell planning



Dynamic Frequency Reuse

All cells can use the whole available bandwidth but the resourceallocation takes into account the global interference in the network.

• Scalability problems

• Scalability faced with distributed resource allocation algorithms

• Distributed approach available for statistical knowledge of the interference

Kiani, Øier, Gesbert, Maximizing multicell capacity using distributed power allocation and

scheduling, March 2007

Applicable to dense networks under the assumption of two power levels



Centralized Power Allocation for Two-Cell NetworkPonukumati et al. 2008

user 2user 1

base station 1 base station 2

cell 1 cell 2

h(1)2k

h(1)1k

h(2)1k

K1 users in cell 1K2 users in cell 2

h(2)2k

• Cooperative resource allocation (maximization of the sum rate in both cells).

• Centralized approach with perfect CSI.

• Single user detector at the receiver.

• Joint subchannel and power allocation via duality.

• Complexity order linear in the number of subcarrier and linear in K1K2, i.e.

O(NK1K2)



Optimum Resource Allocation vs Frequency Reuse 1/2

System Setting

• Rayleigh fading channels

• Subcarriers N = 16

• Users K1 = K2 = 2

• Variance of the channel gains

h(1)1k and h

(2)2k , σ2

d = 1


h(2)1k and h

(1)2k , σ2

i = 0.1

10 20 30 40 50 60 7040

50

60

70

80

90

100

110

120

130

Total power available in two cells in watts

Sum

cap

acity

in b

its/c

hann

el u

se

sum rate with optimal allocationsum rate with frequency reuse 0.5

Huge gain with optimum resource allocation!



Duality Gap for Finite N

16 18 20 22 24 26 28 30 3214

16

18

20

22

24

26

28

30

32

Total power available in both cells in watts

Tot

al p

ower

allo

cate

d in

wat

ts

total power allocated with optimal allocationtotal power allocated with reuse 0.5

Subcarriers N = 8

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

Total power available in both cells in watts

Tot

al p

ower

allo

cate

d in

bot

h ce

lls in

wat

ts

total power allocated with reuse 0.5total power allocated with optimal power allocation

Subcarriers N = 16

The duality gap decreases rapidly when the number of subcarriers increases



Optimum Resource Allocation vs Frequency Reuse 1/2

System Setting

• Rayleigh fading channels

• Subcarriers N = 8

• Users K1 = K2 = 2


depending on the distance

from the base station

16 18 20 22 24 26 28 30 3235

40

45

50

55

60

65

70

Total power available in two cells in watts

Sum

cap

acity

in b

its/c

hann

el u

se

Huge gain with optimum resource allocation!


VII. OFDMA Networks with Relays 20

Relay Channel and...

Sink

Relay

Source

Relay ChannelsThe source transmits a signal to the relay and des-tination, the relay forwards the received signal tothe destination• Orthogonal/Nonorthogonal

• Decode-Forward (DF), Compress-Forward (CF),

Amplify-Forward (AF)

The best strategy depends on the channel conditions.

Roughly, DF is optimum if the source-relay channel

is good compared to the relay-destination channel.

Viceversa, CF is preferable.

• Relay deployed or end users acting as relays.

• Increase capacity and diversity and decrease outage

probability.



...Cooperative Diversity

T2

T1 T3

T4

data

Tx

T2T1 T1

T1 T1T2 T2

T2

The nodes cooperate to create a virtual multiele-ment antenna system used for half of time by T1

and half time by T2.



Resource Allocation in Relay Assisted OFDMA Networks

The optimal allocation implies joint optimization of:

1. Selection/Deployment of relay nodes 3. Subcarrier assignment (OFDMA)2. Selection of a relaying strategy 4. Power allocation(decode-forward, amplify-forward)

Main Issues Arising in Relay Networks

1. High complexity of joint resource allocation

2. Signalling very costly for centralized resource allocation

(source-relay, relay-destination, source-destination CSI)

3. Scaling problems

Research Objectives

Design of resource allocation algorithms which are

• joint (relay node and strategy selection/ power and subcarrier allocation)• distributed (to reduce signaling, improve scalability)• low complexity.



Joint Resource Allocation: Problem Definition

• Uplink and downlink traffic: K + 1 nodes.

• 2K possible flows {(1, K + 1), . . . (K, K + 1), (K + 1, 1), . . . (K + 1, K)}.

• P : (K + 1)×N matrix of power allocation with at most two nonzero elements per column.

• R : 2K ×N matrix of rates with at most one nonzero elements per column.

• (P1)m : total transmitted power by node m.

• (R1)m : total achievable rate by node m.

maximizeP,R

K+1∑m=1

Um((R1)m)

subject to P1 ¹ Pmax and R ∈ C(P)



Joint Resource Allocation via DualityNg and Yu, 2007

Equivalent problem

maximizeP,R,t

K+1∑m=1

Um((t)m)

subject to P1 ¹ Pmax and R1 º t R ∈ C(P)

The dual method splits into two problems

gappl(λ) = maxt

∑(Um(t)− λmtm)

and

gphy(λ) =

maxP,R

∑λm

∑n R(m,n)

s.t.P1 ¹ Pmax,R ∈ C(P).

The physical problem can be split further.


VIII. Further Topics in Resource Allocation 25

Cross-layer Resource Allocation

Typically, power and subcarrier allocation

in uplink and downlink are performed ignoring the scheduling needs.

As a consequence, the resource allocation algorithm may allocate resourcesfor an empty queue and assign very low resources to a queue in overflow

Object: joint scheduling and resource allocation!

• Maximum weight matching scheduling (developed for OFDM downlink systems) takes

into account the instantaneous state of the queue and the channel.

• Queue proportional scheduling (developed for broadcast fading channels and OFDM

fading channels) takes into account the queuing process and channel statistics and

implies a more long term policy.


VIII. Further Topics in Resource Allocation 26

Further Topics

Bitloading

MIMO systems

Resource allocation and adaptive coding and modulation

Crosslayer: Routing and power allocation

Distributed versus centralized algorithms.

Fairness.


IX. Conclusions 27

Conclusions

Joint resource allocation in single cell is doable in practical systems.

Joint resource allocation in certain relay assisted networks can bedecomposed in sub-problems without loss in performance.

Frequency reuse 1 provides huge improvement in spectral efficiency inmulticell networks. There is still a problem of scalability forimplementation in practical systems.

The time sharing condition for optimization problems can be exploitedreally efficiently in multicarrier systems.


28

Thank You for Your Attention!

Questions?


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