aris moustakas, university of athens crown kickoff nkua power control in random networks with n....

Aris Moustakas, University of Athens

CROWN KickoffNKUA

Power Control in Random Networks

with N. Bambos, P. Mertikopoulos, L. Lampiris

Aris Moustakas, University of Athens 2

• Power – Frequency allocation

• Random Networks

• min “P” subject to “SINR”– “P”: total power – power per user – power per user <Pmax

– “SINR”: SINR contraints on all (some) connections = connectivity

• Impediments in the analysis of Power Control– Randomness in

• Distance between Tx-Rx

• Fading coefficients

• Interference location/strength

– Interference (interaction – domino effect)

– Constraints (max power makes problem non-convex)

Resource Allocation

ii

ii

PSINRP

i

)(min P


• Simplify (enough) problem so that can obtain analytic solution

• Take – Minimize (total) power subject to power constraints

– Linear SINR constraints result to conical section

or

• Good news: If solution exists, can be reached using distributed algorithms (e.g. Foschini-Miljanich)

• Simplify neglect random fading,

• Problem still non-trivial due to randomness in positions and interference– Two specific examples of randomness

Power control

11

jijijiii PgPg

2/22

1

arrg

ji

ij

1iMP

ii

ii

PSINRP

i

)(min P


• Start with ordered (square) lattice of transmitter-receiver pairs.

• (a) With probability p erase transmitter– Intermittency of transmission

– Randomness of network

• (b) With probability p/(1-p) locate users at distance a1/a2

– Models randomness of location of users

Models of Randomness: The Femto-Cell Paradigm

�

� 1

� 2

�


• Both represent simple models with all important ingredients: – PC, randomness, interference

• After some algebra:

– Ei = 0,1 with probability p/(1-p) (erasures)

–

• Assume M circulant : eigenvalues

• Using Random Matrix Theory:

where

• β plays role of shift (β=0, when p=0)

Solution Approach

EuIEMEEu 1

0lim

1

1

Ttotave pN

Pp

jig

jigM

ij

iiij

1

)0(

1

1

1

ppave

j

jiiqijii eggq ||1)(

)(qdqp


• When p=0 blowup at a given γ

• p>0 moves singularity to the right.

• Pave does not diverge

• Var diverges

Hint: a finite number

(1?) of nodes diverges

Metastable state?

Analysis


• In reality system is unstable (max/ave)

• One – two dimensional systems very accurate

• Questions:– Probability of instability as a function of γ?

– Fluctuations btw samples?

Analysis


• Introduce max-power constraint

• Distributed version:

• λ=1/Pmax

• Use 3 methods to find optimum:– Foschini-Miljanic

– Best-Response

– Nash

Resource Allocation

iiP

i PSINRui

)(maxmax P


Type of problem

No Pure Nash Equilibrium

Players Best Respond

3 General Categories

Payoff:

Introduction

max

)()(P

PSINRPU i

i

γ_))-SINR(1log( rThroughput:


One Mixed Nash Equilibrium

1,1

1,

max

111

2

2max

111

iP

PP

iP

PP

ni

iii

niP

P

niP

PP

n

iii

,1

,

max

2

1

1max

21

2

niP

P

niP

PP

ni

iii

,1

,

max

2

1

1max

21

2

niP

P

niP

PP

ni

iii

,1

,

max

1

2

2max

11

1


3 mixed Nash equilibria

niP

P

niP

PP

ni

iii

,1

,

max

2

1

1max

21

2

niP

P

niP

PP

ni

iii

,1

,

max

1

2

2max

11

1

niP

PP

niP

PP

ki

ni

iii

i

,)(

1

,

2,0

max

122

1

1max

12

12

1

niP

PP

niP

PP

ki

ni

iii

i

,)(

1

,

2,0

max

11

11

2

2max

11

11

2

niP

PP

niP

PP

ki

ni

iii

i

,)(

1

,

12,0

max

122

1

1max

12

12

1

niP

P

niP

PP

ki

ni

iii

i

,1

,

2,0

max

11

2

2max

11

11

2

Aris Moustakas, University of Athens

Single & Double Pure Nash Equilibria

12

Single Equilibrium

Two Equilibria


Average Payoffs & Throughput Comparison

1 Nash:

Throughput:

)0,1( 2nP

FM: Best Response:

max

2,0P

Pk

n

i

in

i

i

P

P

P

P

1 max

2

1 max

1 13

1,1

3

1

3 Nash:FM:

BR

)0,0(

max

2,0P

Pk

n

i

in

i

i

P

P

P

P

1 max

2

1 max

1 12

1,1

2

1

Pure Nash:

Throughput:

FM:

Best Response:

)0,0( )1,0( 11

nP )0,1( 11

nP

2/

1 max

22

2/

1 max

121 1

3

1,1

3

1 n

i

in

i

i

P

P

P

P

2/

1 max

122

2/

1 max

21 1

3

1,1

3

1 n

i

in

i

i

P

P

P

P),(

max

2,0P

Pk


• Max-power constraint brings new features

• Nash – game on restricted power feasible and better than other cases

• BR not bad

• Generalisable to more users?

• Analytic estimates?

Questions


• Optimize network connectivity using collaborative methods inspired by statistical mechanics (Task 2.1)

– Power – connectivity fundamental trade-off:– Tradeoff between connectivity and number of frequency bands.– Design and validation of distributed message passing algorithms

• Develop distributed message passing methods to achieve fundamental limits of detection and localization of a network of primary sources through a network of secondary sensors (Task 2.2)

– Detection of sources using compressed sensing on random graphs and Cayley trees

– Effect of additive and multiplicative noise on detection– Application of compressed sensing on two-dimensional graphs with realistic

channel statistics

• Develop decentralized coordinated optimization approaches (Task 2.3).– design self-coordinated, fast-convergent wireless resource management techniques– convergence, stability and the impact of operation on different time scales on the

performance.

Goals


• Minimize power subject to constraints

• Interactions due to interference

• Simplifications:– Random graphs (1d-2d-inft d)

– gij = 0,1

– Power levels

• Use replica theory

Power Control – Connectivity tradeoff

1..

min

1

jijiii

ii

PgPgts

P


• Minimum number of colors needed to color network with interference constraints– E.g. no adjacent nodes in same color

• Simplifications:– Random graphs (Bethe lattice / Erdos-Renyi)

– gij = 0,1

• Use replica theory and – Graph coloring

• Message passing algorithms

Connectivity – Frequency Bandwitdth


Cooperative Sensing

Sensor Node

Transmitter Node

Signal

Sensor Communication



• Models:– H known and P discrete (on-off)

– Random graphs

– H random valued

– H in a given geometry • possible locations of sources

– With/without noise


• Compressed sensing (sparsity)

• Message passing

Collaborative Sensing and Localization

yP

zPHty ai

iaia

|Prmax

)(

aris moustakas, university of athens crown kickoff nkua power control in random networks with n....

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