Decentralized Admission Control and Resource Allocation for Power

Decentralized Admission Control and ResourceAllocation for Power-Controlled Wireless Networks

S lawomir Stanczak1,2

joint work with

Holger Boche1,2, Marcin Wiczanowski1 and Angela Feistel2

1Fraunhofer German-Sino Lab for Mobile Communications (MCI)Berlin, Germany

2Heinrich Hertz Chair for Mobile CommunicationsFaculty of EECS

Technical University of Berlin

22 September 2009

UCLA, 22.09.2009Fraunhofer German−Sino Lab

Mobile Communications

MCI1/43

Outline

1 Introduction

2 Physical-Layer Abstraction by Interference Functions

3 User-centric Approaches

4 Network-centric approachesDistributed Power Control AlgorithmsIncorporating QoS requirements



MCI2/43

Outline

1 Introduction






MCI3/43

Wireless Networks

Goal: Study resource allocation and interference management

Focus: High data rates, low or moderate channel dynamics

Energy supply is not a bottleneck.Wireless mesh networks, cellular networks



MCI4/43

Wireless Channel Characteristics

Radio propagation channel is unreliable.

channel fading, path loss, channel conditions are time-varying ...

Power and bandwidth are limited.

Wireless spectrum is a shared medium.

Link capacities are elastic.Network cannot be regarded as a collection of point-to-point links.Performance is maximized by tolerating interference in a controlled way.

Resource allocation and interference management are necessary.



MCI5/43

Wireless Network Resources and Mechanisms

Wireless resources: power, time, frequency, space, codes, routes...

Mechanisms for resource allocation and interference management

Multiple antenna techniquesMAC: power control and schedulingrouting...

Cross-layer protocols



MCI6/43

Applications

Voice transmission

Inelastic traffic: QoS requirements need to be satisfied permanently.

Data applications (WWW browsing,e-mail,ftp)

Low QoS levels are temporarily acceptable.Elastic traffic: Applications modify their data rates according to availableresources in communication networks.



MCI7/43

Quality of Service

User-centric approaches (inelastic applications):

Satisfy strict QoS requirements of applications permanently.

Network-centric approaches (elastic applications):

Maximize the aggregate utility as perceived by the network operator.Address the issue of fairness.

QoS link 1

QoS link 2max-min fairness

maxP

k QoSk (best overall efficiency)

maxP

k αkQoSk (weighted sum optimization)

minimum total power

Feasible QoS region



MCI8/43

The focus of this talk

Power control with some aspects of the physical-layer design.

Link Layer

Physical Layer

System

Constraints

Transmit/Receive

Power Control

+

Network Layer

Applications

QoS

Strategies

Single-hop communication with K > 1logical links (users)

Concurrent transmission (works withany scheduling protocol).

Each user is decoded (single-userdecoding).

Combination with routing and networkcoding strategies possible.



MCI9/43

Outline

1 Introduction






MCI10/43

Feasible QoS Region: Fγ

Given a channel, Fγ is the set of all QoS values that can be achieved by means ofpower control with all links being active concurrently.

ω2

feasible point

ω1

Fγ

infeasible point

QoS link 1

QoS

link

2

Assumption: ωk ↑ ⇔ QoS ↑Downward-comprehensiveUpper-bounded

may be non-convex.

depends on power constraints P.

Fγ depends on the physical-layer realization: Key properties of manymultiuser systems are captured by interference functions.



MCI11/43

Signal-to-Interference(+noise) Ratio (SIR)

Strictly monotonic QoS-SIR map: γ : R→ R+

For any ω ∈ Fγ , there is a power vector p ∈ P such that

γ(ωk) = SIRk(p) =pk

Ik(p)

← transmit power

← interference function

e.g. Gaussian capacity (in nats/channel use): γ(x) = ex − 1, x ≥ 0.



MCI12/43

Axiomatic Interference Function

Standard Interference Functions (SIF), Yates’95

A1 Positivity: Ik(p) > 0 for all p ≥ 0.

A2 Scalability: Ik(µp) < µIk(p) for any p ≥ 0 and for all µ > 1.

A3 Monotonicity: Ik(p(1)) ≥ Ik(p(2)) if p(1) ≥ p(2).

It models many practical interference scenarios.



MCI13/43

Interference Function: Examples

Linear interference function

Ik(p) = (Vp + z)k

Matched-filter receiverSIC receiver

Minimum interference function

Ik(p) = minuk∈Uk(V(u)p + z(u))k

MMSE receiver

QoS 3

QoS 1

2

1

2 3

3

1

interference

QoS 2

V1,1

V1,2 u(3)u(2)

u(1)

u(4)

SIR

user

2SIR user 1

G



MCI14/43

Outline

1 Introduction






MCI15/43

Problem Statement

Problem (QoS-based power control under SIFs)

p(ω) = arg minp∈P(ω)

wT pw > 0

P(ω) :={p ∈ RK

+ : ∀k SIRk(p) ≥ γ(ωk)}.

Minimum total power

QoS link 1

ω2

ω1

QoS link 2

Feasible QoS region

p2

p1

p2 = γ2I2(p)

p(ω) is the minimum point of P(ω)

p1 = γ1I1(p)

Valid power set

Zander’92, Foschini’94, Yates’95, Ulukus’98, Bambos’00, Boche&Schubert ...



MCI16/43

Fixed-Point Iteration, Yates’95

If ω is feasible, then the concurrent iterations

∀k pk(n + 1) = γk Ik(p(n))

converge to p(ω), where γk ≡ γ(ωk).

Component-wise increasing (decreasing) if p(0) = 0 (p(0) ∈ P(ω)).

Amenable to distributed implementation, scalable, works for any SIF

But how should new users join the network without disrupting theconnections of active users?



MCI17/43

Fixed-Point Iteration, Yates’95

If ω is feasible, then the concurrent iterations

∀k pk(n + 1) = γk Ik(p(n))

converge to p(ω), where γk ≡ γ(ωk).

Component-wise increasing (decreasing) if p(0) = 0 (p(0) ∈ P(ω)).

Amenable to distributed implementation, scalable, works for any SIF

But how should new users join the network without disrupting theconnections of active users?



MCI17/43

Admission Control Scheme

User k is called active at time n if SIRk(p(n)) ≥ γk .

Define An to be the set of all active users at time n and Bn := K \ An.

Power control with active link protection (δ > 1)

pk(n + 1) =

{δ γk Ik(p(n)) k ∈ An (active users)

δ pk(n) = δn+1pk(0) k ∈ Bn (inactive users)

δ > 1 can be interpreted as protection margin.

the larger δ, the faster power-up of the inactive users.δ cannot be too large for all users to be fully admissible.

Bambos’00, Chee Wei Tan’09 (only Ik (p) = (Vp + z)k )



MCI18/43

Properties of the admission control scheme

TheoremLet Ik be any standard interference function. Then,

All SIRs converge to some values.

All users are admitted in finite time if γ = (γ1, . . . , γK ) is feasible.

Transmit powers are bounded if and only if δ · γ is feasible.

Active users (k ∈ An):

Preservation of active users: An ⊆ An+1.Bounded power overshoot: pk(n + 1) < δpk(n).

Inactive users (k ∈ Bn):

SIRs of inactive users are increasing SIRk(p(n)) < SIRk(p(n + 1)).

Stanczak&Kaliszan&Bambos’09



MCI19/43

No power constraints, TX and RX beamforming

00 50 100 150 20000

10

10 20 30 40

1

2

3

4

5

6

7

8

20

30

40

50

60

70

80

90

T.4 A.6T.5A.4T.3A.3T.2A.2 A.5T.1

SIR

n

user active at n = 0user inactive at n = 0common SIR target

T.x: Transceiver optimization phasesA.x: Admission control phases

A.1

SIR

n

K = 10, nT = nR = 4, γ = 8, δγ = 9.6, An = {1, . . . , 5}The highest feasible SIR (example):

0.88 (fixed beamformers), 1.37 (RX beamforming), 8 (TX/RX beamforming)



MCI20/43

Incorporating power constraints

TheoremSuppose that δγ is feasible and

p(m) ≤ βδI(p(m))

holds for some m ∈ N0 and β ∈ [1, βmax]. Then, there exists βmax > 1 such thatAn ⊆ An+1 for all n ≥ m.

Active users send distress signals until the condition is satisfied.

Stanczak&Kaliszan&Bambos’09



MCI21/43

Outline

1 Introduction






MCI22/43

Problem Statement

Problem (Utility-based power control)

ω∗ = arg maxω∈Fγ

wTω w > 0 .

w

ω∗ is a maximal point of Fγ

Fγ

ω1

ω2

Feasible QoS regionFγ

ω2

ω1

Maximal point of Fγ

Goodman’00, Saraydar’02, Xiao’03, Neely’03, Chiang’04, Huang’05,Tassiulas’05 ...



MCI23/43

Convexity of Feasible QoS Region

q∗ = (q∗1 ,q∗2)

w

q 2

Fγ(P)

q1

Find a class of strictly increasing and concave utility functions Ψ with

γ(Ψ(x)) = x , x ≥ 0

so that Fγ is a convex set.



MCI24/43

Convexity of Feasible QoS Region: Sufficient Conditions

Theorem (Convexity under a Linear Interference Function)

If γ with γ(Ψ(x)) = x , x ≥ 0, is log-convex, then the feasible QoS region is aconvex set, regardless of the type of power constraints

Observation:γ is log-convex if and only if Ψe(x) := Ψ(ex) is concave.

Further related results:

Log-convexity of γ(x) is necessary for the region to be convex in general.Convexity of γ(x) is sufficient if V is confined to belong to some subset ofnonnegative matrices.



MCI25/43

Examples of Function Classes

Ψα(x) =

{x1−α

1−α α > 1

log(x) α = 1Ψα(x) =

log x α = 1

log x1+x α = 2

log x1+x +

α−2∑j=1

1j(1+x)j α > 2

α = 1: Throughput maximization

α→∞: Max-min fairness



MCI26/43

Arbitrarily Close Approximation of Max-Min Fairness

Let ω∗k = Ψα(SIR∗k) and let ν∗k = log(1 + SIR∗k). Then, ν∗ converges to themax-min rate allocation as α→∞.

Flow 1

Flow 2 Flow 3 Flow 4

8

8.2

8.42.4

1.8

1 6 11 16

Sou

rce

Rat

es

α

Sum of Source Rates



MCI27/43

Efficiency of the Max-Min SIR Power Allocation

Theorem

If p and q are positive right and left eigenvectors of B(k0) = V + 1Pk0

zeTk0

, then

(i) p is max-min fair power allocation if and only if p = p.

(ii) ω∗ is max-min fair ω if and only if w = w∗ = q ◦ p > 0.

α2α3 α1 = 1∞ α4

maxω∈Fγ(P) wTω

w

α1 < α2 < α3 < α4

ω1 = log(SIR1)

ω2 = log(SIR2)

max-min fairness

w∗



MCI28/43

Joint Power and Receiver Control

Alternating optimization1 Given U(t − 1) compute p(t)

(i) Compute the weight vector: w = y(B(m)) ◦ x(B(m)), m = arg maxk ρ(B(k))(ii) Compute the QoS vector: ω∗ = arg maxω∈Fγ wTω(iii) p(t) = (I− Γ(ω∗)V)−1Γ(ω∗)z,Γ(ω) = diag(γ(ω1), . . . , γ(ωK ))

2 Given p(t) compute U(t)

(i) ∀k uk(t) = arg max‖x‖2=1 SIRk(p(t),x)

monotonic convergence to max-min SIR-balancing solution

But: not amenable to distributed implementation

Theory provides a basis for novel decentralized algorithms for finding a saddlepoint of the aggregate utility function.



MCI29/43











MCI29/43











MCI29/43

Some Simulation Results

10 20 30 40 500

5

10

15

20

25

30

35

40

number of users

min

k SIR

k/γ k

Maximum power

PC

PC + RX Beamf.

PC + RX/TX Beamf.

0

0

0.2

0.1

0.4

0.2

0.6

0.3

0.8

0.4

1

0.5

1.2

0.6

1.4

0.7

0

0

10

10

20

20

30

30

40

40

50

50

ave

rag

e d

ela

ya

ve

rag

e d

ela

yarrival rate

StaticPC (Utility−Max.)PC (Queue−Weighted Utility−Max.)

PC+Beamforming (Max−Min−SINR)PC (Utility−Max.)PC (Queue−Weighted Utility−Max.)



MCI30/43

Outline

1 Introduction






MCI31/43

Utility-Based Power Control

Equivalent minimization problem: ψ(x) = −Ψ(x)

p∗ = arg minp∈P

F (p) = arg minp∈P

∑k

wkψ(SIRk(p)

).

Positivity of minimizers: p∗ > 0

Even if ψ(ex) is convex, the problem is not convex in general.



MCI32/43

Convex Statement of the Problem

Theorem

If Ik(es) is log-convex and ψ(ex) convex, the following problem is convex:

s∗ = arg mins∈S

Fe(s)

s := log p,p > 0

S := {log x : x ∈ P+}Fe(s) = F (es)

Ik(es) =∑

l vk,lesl + zk is log-convex (Hoelder inequality).



MCI33/43

Gradient-Projection Algorithm

Let τ > 0 be constant step size (small enough), and let

s(n + 1) = ΠS

[s(n)− τ∇Fe(s(n))

]s(0) ∈ S

∇Fe(s) = diag(es1 , . . . , esK )∇F (es):

∇F (p) = (I−VT Γ(p))g(p)

gk(p) = wkψ′(SIRk(p))SIRk(p)/pk (locally available)

Γ(p) = diag(SIR1(p), . . . , SIRK (p))



MCI34/43

Min-max reformulation

mins

maxu

∑k

wkψ(esk

uk

)subject to

es − p ≤ 0

u− t ≤ 0

∀k Ik(es)− tk = 0 .

Linear interference function: Ik(es) = (Ves + z)k .

The Hessian is diagonal and its diagonals are given by the gradient.



MCI35/43

Conditional Newton Algorithm

8><>:

s(n + 1)

µ(n + 1)

!=

s(n)

µ(n)

!− (∇2

(s,µ)L(z(n)))−1∇(s,µ)L(z(n))

∇(u,λu ,λ,t)L(z(n + 1)) = 0 can be solved explicitely

L(z) = L(s,u,µ,λu,λ, t): A modified Lagrangian function.

Quadratic convergence.

Global convergence if ψ(x) = − log(x) andψ(x) = 1/x .

No step size.

Distributed implementation possible!

25

K = 50

ψ(x) = − log(x)

Conditional Newton

Gradient projection

20151050-1

0

1

2



MCI36/43

Adjoint Networks: A Simple Example

Primal network:

Measure SIR at E1 and E2

E1 and E2 compute some messagesbased on localmeasurements/parameters

S1

S2

E2

E1



MCI37/43


Primal network:



S1

S2

E2

E1



MCI37/43


Primal network:



S1

S2

E2

E1



MCI37/43


Primal network:



S1

S2

E2

E1



MCI37/43


Reversed network:

The messages determine thetransmit powers of E1 and E2.

Cooperation by interference

⇒ S1 and S2 estimate the messagesfrom the received powers

S1

S2

E2

E1

Significant gains compared to schemes relying on flooding protocols.Estimation errors are dealt with stochastic approximation.



MCI37/43


Reversed network:




S1

S2

E2

E1




MCI37/43


Reversed network:




S1

S2

E2

E1




MCI37/43


Reversed network:




S1

S2

E2

E1




MCI37/43

Outline

1 Introduction






MCI38/43

Problem Statement

s∗(ω) := arg mins∈S Fe(s) s.t. ∀k fk(s) := Ik(es)/esk − 1/γk ≤ 0

��

��

��

��

FγP◦(ω)

P2

p2

ω1

ω2

P1

p1

The projection is not amenable to distributed implementation.



MCI39/43

Primal-dual algorithm based on standard Lagrangian

Primal-dual iteration under individual power constraints sk(n + 1) = min{

sk(n)− δesk (n)[hk(s(n)) + Σk

(s(n),µ(n)

)], log(Pk)

}λk(n + 1) = max

{0, λk(n) + δfk(s(n))

}Σk(s,λ) =

∑l vl,k

(λl

esl+ |SIRl(es)gl(s)|

)=∑

l vl,kml(s, µl)

Estimation of Σk using the adjoint network.



MCI40/43

Soft QoS Support

Fα(z) =∑k∈A

akψα

(SIRk(p)

γk

)︸︷︷︸

penalty term

+∑k∈B

bkψ(SIRk(p)

)︸︷︷︸

aggregate utility

.

A: QoS users need to satisfy SIRk ≥ γk , k ∈ AB: best-effort users

A \ B: pure QoS users (voice)A ∩ B: best-effort users with QoS requirements (video)B \ A: pure best-effort users (data)

Each user, say user k , determines its utility by choosing αk ≥ 1.

Slightly modified algorithms

Amenable to distributed implementation



MCI41/43

Soft QoS Support: Example

γ2

QoS

ofP

ure

QoS

Use

r2

QoS of Best Effort User 1

A = 2

B = 1

1

3

2

1 - max-min-fairness

2 - Utility-based power control with α2 = 1

3 - Utility-based power control with soft QoS support, α2 →∞

2 3 as α2 →∞→

No overshoot of user 2.

8

12

4

02 6 10 14 18

A ∩ BSIR Target

α

a = b = 1K = 4SNR=40dBψ(x) = − log(x)

SIR

A \ B

B \ A

B \ A



MCI42/43

Conclusions and Outlook

The power control problem isrelatively well-understood.

Throughput maximization in the lowSINR regime is an open problem.

Outlook

Joint optimization of

transmit powers,schedulers (time+frequency),physical-layer (single- andmulti-mode transmission).

Dynamic optimization over finiteand infinite time horizon.

Stochastic power control.

...

This book series presents monographs about fundamental topics and trends in signal processing, communications and networking in the field of information technology. The main focus of the series is to contribute on mathematical foundations and methodologies for the understanding, modeling and optimization of technical systems driven by information technology. Besides classical topics of signal processing, communications and networking the scope of this series includes many topics which are comparably related to information technology, network theory, and control. All monographs will share a rigorous mathematical approach to the addressed topics and an information technology related context.

The wireless industry is in the midst of a fundamental shift from providing voice-only services to offering customers an array of multimedia services, including a wide variety of audio, video and data communications capabilities. Future wireless networks will be integrated into every aspect of daily life, and thus could affect our life in a magnitude similar to that of the Internet and cellular phones. However, the emerging applications and directions require a fundamental understanding of how to design and control the wireless networks that lies beyond what the existing communication theory can provide. In fact, the complexity of the problems simply precludes the use of engineering common sense alone to identify good solutions, and so mathematics becomes the key avenue to cope with central technical problems in the design of wireless networks. That’s why, two fields of mathematics play a central role in this book: Perron-Frobenius theory for non-negative matrices and optimization theory. Here these theories are applied and extended to provide tools for better understanding the fundamental tradeoffs and interdependencies in wireless networks, with the goal of designing resource allocation strategies that exploit these interdependencies to achieve significant performance gains. This revised and expanded second edition consists of four largely independent parts:the mathematical framework, principles of resource allocation in wireless networks, power control algorithms and appendices. The latter contain foundational aspects to make the book more understandable to readers who are not familiar with key concepts and results from linear algebra and convex analysis.

Foundations in Signal Processing, Communications and NetworkingVol. 3W. Utschick · H. Boche · R. MatharSeries Editors

Fundamentals of Resource Allocation in Wireless NetworksTheory and AlgorithmsSecond Expanded EditionSławomir Stańczak · Marcin Wiczanowski · Holger Boche

Fundamentals of Resource Allocation in Wireless Networks

AB

StańczakW

iczanowski · Boche

Fundamentals of Resource

Allocation in Wireless Networks

1 23

Foundations in signal Processing, communications and networking

Volume 3

W. Utschick · H. Boche · R. Mathar Series Editors

Theory and Algorithms

Second Expanded Edition

Sławomir StańczakMarcin WiczanowskiHolger Boche

2nd Ed.› springer.com

isbn 978-3-540-79385-4

SPCN_Stanczak.indd 1 05.06.2008 17:12:41 Uhr



MCI43/43

Documents

Decentralized Admission Control and Resource Allocation for Power