Dynamic Spectrum

Management:

Optimization, game and

equilibriumTom Luo (Yinyu Ye)

December 18, WINE 2008

Outline Introduction of Dynamic Spectrum

Management (DSM) Social Utility Optimization Noncooperative Nash Game Competitive Spectrum Economy

Pure exchange market Budget Allocation Channel Power Production

The objective is to apply algorithmic game/equilibrium theory to solving real and challenging problems

Dynamic Spectrum Management

Dynamic Spectrum Management

Communication system DSL, cognitive radio,

cellular networks, cable TV networks,

Multiple users (each has a utility function) access multiple channels/tones

2/3 allocated spectrum is not being used at any given times

An efficient spectrum management scheme is needed

Spectrum Allocation Problem

Model Each user i has

a physical power demand

Each channel/tone j has a power supply

maximize system efficiency and utilization

. . .

user 1

user 2

power allocation

user 3

channel

D1 D2 D3

power supply

s3s2s1 sns4

Shannon Utility Function

jik

kjikjij

ijiii xa

xxxu )(1log),(

xij: the power allocation to user i on channel jx-bari: power allocations to all users other than i

бij: the crosstalk ratio to user i on channel jai

kj: the interference ratio from user k on channel jThey may time varying and stochastic

Spectrum Management Models

From the optimization perspective, the dynamic spectrum management problem can be formulated as 1. Social utility maximization

May not optimize individual utilities simultaneously

Generally hard to achieve 2. Noncooperative Nash game

May not achieve social economic efficiency 3. Competitive economy market

Price mechanism proposed to achieve social economic efficiency and individual optimality

1. Social Utility Maximization

.user toalloacated channel

on power therepresents where

,

Subject to

),( Maximizei

ij

ijx

jsx

i,i

dx

ix

ix

iu

ji

ij

jij

Social Utility Maximization- a two user and two channel example

)4

1log()2

1log(12

22

11

21

x

x

x

x

)4

1log()1

1log(22

12

21

11

x

x

x

x

u1=

u2=

1

Demand 1 1

user 1 user 2

1

Channel 1 2

Difficulty of the problem Even in the two user case, the problem

is NP-hard. No constant approximation algorithm

even for one channel and multiple users.

Problems under the Frequency Division Multiple Access (FMDA) policy can be solved efficiently

Luo and Zhang 2007

2. Noncooperative Nash Game

Model Each user

maximize its own utility under a physical power demand constraint

Ciofi and Yu 2002, etc.

. . .

user 1

user 2

power allocation

user 3

channel

D1 D2 D3

Individual rationality

;0 Subject to

),( Maximize

ix,

idx

ixe

ix

ix

iu

jij

T

The basic game assumes that there is no limit on power supply for each channel.

IWF: iterative water filling algorithm converges in certain cases

Spectrum Nash Game- the same toy example

)4

1log()2

1log(12

22

11

21

x

x

x

x

)4

1log()1

1log(22

12

21

11

x

x

x

x

u1=

u2=

1

Demand 1 1

user 1 user 2

1

Channel 1 2

Results on the problem No bound on ``price of anarchy’’ Can be solved as finding solution of a linear

complementarity problem, so that it’s PPAD hard in general

There is a FPTAS under symmetric interference condition

There is a polynomial time algorithm under symmetric and strong weak interference condition

Key to the proofs: the LCP matrix is symmetricLuo and Pang 2006, Xie, Armbruster, and Y 2008

3. Competitive Spectrum Market

The problem was first formulated by Leon Walras in 1874, and later studied by Arrow, Debreu, and Fisher, also see Brainard and Scarf.

Agents are divided into two categories: seller and buyer. Buyers have a budget to buy goods and maximize their

individual utility functions; sellers sell their goods just for money.

An equilibrium is an assignment of prices to all goods, and an allocation of goods to every buyer such that it is maximal for the buyer under the given prices and the market clears.

Market Equilibrium Condition I

budget its

is and agent by purchased good

ofamount therepresents where

;0 Subject to

),( Maximize

goods allfor pricesmarket Given

yRationalit Individual

iwij

ijx

ix,

iw

ixTp

ix

ix

iu

p

Market Equilibrium Condition II

Physical Constraint: The total purchase volume for good j should not

exceed its available supply:

; , jsx ji

ij

Market Equilibrium Condition III

Walras Law:

;0 ,

goodFor

jji

ij psx

j

user 1user 1 user 2user 2 user muser m

Power allocation in CE

equilibrium pricein CE

p1 p2 p3 p4 pnequilibrium price

in CEp1 p2 p3 p4 pn

x11 x13x11 x13

user 3user 3

x21 x24x21 x24

x31x3nx32

x31x3nx32

xm1xm3 xmn

xm1xm3 xmn

. . .

. . .

channel 1 2 3 4 nchannel 1 2 3 4 n

budget w1 w2 wmw3 wi=1budget w1 w2 wmw3 wi=1

power supply s3s2s1 sns4 sj=mpower supply s3s2s1 sns4power supply s3s2s1 sns4 sj=m

Competitive Communication Spectrum Economy

What’s the ``budget’’ in DSM?

3.1 Competitive Equilibrium in

Spectrum Economy for

Fixed Budget and Power Supply

Spectrum Management

Channel Price Adjustment ( pj )

Channel Power Allocation ( sj )

Budget Allocation ( wi )

Objectives

Fixed and given

Fixed and given

Improve channel power utilization

Competitive Spectrum Economy

Model Each user buys

channel powers under her budget constraint and maximize her own utility

Price control goal Avoid

congestion Improve

resource utilization

budget

. . .

user 1

user 2

power allocation

user 3

channel

w1 w2 w3

Price p1 p2 p3 p4 pn

Problem Formulation

m users, each has a budget wi

n channels, each with power capacity sj

Design variable xij Power allocation for i th user in jth channel pj Price for j th channel (Nash Equilibrium: pj=1

fixed) User utility (Shannon utility function )u(xi ; ¹xi ) =

nX

j =1

log

Ã

1+xi j

¾i j +P

k6=i aikj xkj

!

; i = 1;¢¢¢;m:

. . .

x11 x13

x21 x24xm4

xmn

w1 w2 wmw3

s3s2s1 sns4

x3n

x31x32

. . .. . .

x11 x13x11 x13

x21 x24x21 x24

xm4

xmn

xm4

xmn

w1 w2 wmw3

s3s2s1 sns4

x3n

x31x32

x3n

x31x32

x31x32

Competitive Equilibrium Model

Theorem A competitive equilibrium always exists for the spectrum management problem

Y 2007 based on the Lemma of AbstractEconomy developed by Debreu 1952

Equilibrium Properties Every channel has a price:

All power supply are allocated:

All budget are spent

. 1

*

m

i jij jsx

m

i

n

j jji spw1 1

*

jp j ,0*

Weak-Interference Market Weak-interference environment: the Shannon

utility function of user i is

In the weak-interference environment, An equilibrium can be computed in polynomial time. The competitive price equilibrium is unique.

Moreover, if the crosstalk ratio is strictly less than 1, then the power allocation is also unique. (Y 2007)

)(

1log),(1 ik kjijij

ijn

jiii xa

xxxu

,0ijwhere 10 ija

Two methods of solving competitive equilibrium

Centralized Solving the equilibrium conditions

Decentralized Iterative price-adjusting

user 1 user 2

s2=2s1=2

budget w1 =1 w2=1

power supply

Competitive Equilibrium Model user

1user

2

Nash Equilibrium Model

powerconstraint 5/3 7/3

Competitive Equilibrium Model- the same toy example

equilibrium

price

p1=3/5 p2=2/5

5/31/3

2

u1=0.3522 u2=0.2139

Social utility=0.5661

powerallocatio

n

5/3

u1=0.2341 u2=0.2316

Social utility=0.4657

14/3power

allocation

Computational Results Compare competitive equilibrium and

Nash equilibrium Evaluate the performance in

Individual utility and Social utility In most cases, CE results in a channel

allocation Have a higher social utility value Make more users achieve higher individual

utilities

3.2 Budget Allocation in

Competitive Spectrum Economy

Spectrum Management

Channel Price Adjustment ( pj )

Channel Power Allocation ( sj )

Budget Allocation ( wi )

Objectives

Fixed and given

Make each user meet minimum power demand or utility value threshold

Improve channel power utilization

Lin, Tasi, and Y 2008

Budget Allocation in Competitive Spectrum Economy

Budget allocation aims to satisfy a minimum physical power

demand di for each user i

or satisfy a minimum utility value ui for

each user i ; e.g., all users achieve an identical utility value

Theorem: Such a budget equilibrium always exists.

Two methods of solving competitive equilibrium

Centralized Solving entire optimal conditions which

may be nonconvex

Decentralized Iterative budget-adjusting

Budgeting for demand- computational results Number of (budget-adjusting) iterations required to

achieve individual power demands

Budgeting for demand- computational results

Number of iterations and CPU time (seconds) required to satisfy individual power demands in large scale problems, error tolerance=0.05

General cases: background noise randomly selected from (0,m], crosstalk ratio randomly selected from [0,1]

In all cases, the social utility of CE is better than that of NE.

Budgeting for demand- CE and NE comparison results

Budgeting for demand- More CE and NE comparison results

In special type of problems, the competitive equilibrium performs much better than the Nash equilibrium does.

For instance, the channels being divided into two categories: high-quality and low-quality.

(In simulations, one half of channels with background noise randomly selected from the interval (0; 0,1] and the other half of channels with background noise randomly selected from the interval [1;m].)

Two-tier channels CE with power demands v.s. NE

Budgeting for demand- More CE and NE comparison results

Budget allocation to balance utilities - Computational results

Number of iterations and CPU time (seconds) required to balance individual utilities in large scale problems, difference tolerance=0.05

Two-tier channels CE with balanced utilities v.s. NE

Budgeting to balance utilities - CE and NE comparison results

Comparison result summaries

Compare with NE, in most cases, CE with minimum power demands results in power allocation Have a higher social utility

Compare with NE, in most cases, CE with balanced utilities demands results in a power allocation Have a higher social utility Make more users have higher individual utilities Have a smaller gap between maximal individual utility

and minimal individual utility In special type of problems, for instance, two tiers

of channels, CE performs much better than NE does.

3.3 Channel Power Production in

Competitive Spectrum Economy

Spectrum Management

Channel Price Adjustment ( pj )

Channel Power Allocation ( sj )

Budget Allocation ( wi )

Objectives

To achieve higher social utility

Fixed and given

Improve channel power utilization

Produce power supply to increase social utility: the same toy example

121 ww

421 ss

)4

1log()2

1log(12

22

11

21

x

x

x

x

)4

1log()1

1log(22

12

21

11

x

x

x

x

u1=

u2=

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

the power supply of channel 1 (s1)

soci

al u

tility

social utility of CE social utility of NE (scale to used power in CE)

Future Work

How to systematically adjust channel power supply capacity to increase social utility?

The convergence of the iterative variable-adjusting method for general setting

Real-time spectrum management vs optimal policy at top levels