Spatial Reuse in Spectrum Access:Spatial Reuse in Spectrum Access:A Matrix Spatial Congestion Games ApproachA Matrix Spatial Congestion Games Approach

Kai Zhou, Gaofei Sun, Xinbing WangDepartment of Electronic Engineering

Shanghai Jiao Tong University

Zhiyong FengKey Lab. of Universal Wireless Commun.

Beijing University of Posts and Telecommunications

OutlineOutline Introduction

Motivations Related work Objectives

System Model and Problem Formation Potential Game Approach Matrix Spatial Congestion Games Discussion Based on Numeric Results Conclusion and Future Work

MotivationsMotivationsHow to allocate spectrum in CRNs efficiently?

system throughput fairness issue … …

MotivationsMotivations Spatial reuse is a key feature in wireless networks

Interference model … …

Related worksRelated works

Model distributed spectrum competition as Congestion Games Congestion games [3] Convergence speed [8] Price of anarchy [9]

[3]R. Rosenthal, “A class of games possessing pure-strategy nash equilibria,” INTERNATIONAL JOURNAL OF GAME THEORY, vol. 2, no. 1, pp. 65–67, 1973.[8] R. Southwell and J. Huang, “Convergence dynamics of resource homogeneous congestion games,” in Proc. International Conference on Game Theory for Networks (GameNets). Shanghai, China, Apr. 2011.[9] L. Law, J. Huang, M. Liu, S. Li et al., “Price of anarchy for cognitive mac games,” in Proc. Global Telecommunications Conference. Hawaii, Dec. 2009.

Congestion !

Related worksRelated works Extend congestion games to consider spatial

reuse Virtual resource [13] Conflict graph [11] [12]

[11]C. Tekin, M. Liu, R. Southwell, J. Huang, and S. H. A. Ahmad, “Atomiccongestion games on graphs and its applications in networking,” IEEETransactions on Networking, In Press.[12] M. Liu and Y. Wu, “Spectum sharing as congestion games,” in Proc. the 46th Annual Allerton Conference on Communication, Control, and Computing. IAllerton House, Illinois, Sept. 2008.[13] S. Ahmad, C. Tekin, M. Liu, R. Southwell, and J. Huang, “Spectrumsharing as spatial congestion games,” Arxiv preprint arXiv:1011.5384,2010.

directed or undirected

weighted or unweighted

ObjectivesObjectivesMulti-radios for each SUWeighted interference level

Local

Updates

EquilibriumConverge

?

Will SUs’ local selfish updates of channel selection finally converge to an Equilibrium ?

OutlineOutline Introduction System Model and Problem Formation

System model Problem formation

Potential Game Approach Matrix Spatial Congestion Games Discussion Based on Numeric Results Conclusion and Future Work

System modelSystem model User set :Resource (channel) set :

Homogeneous Heterogeneous

Payoff functions : User-specific Non-user-specific Non-increasing with n

Utility :

{1,2,..., }U N

{1,2,..., }R R

1

( )N

ir jr ij

j

g S I

1

( ) ( )i

Ni i

r jr ijr j

p g S I

Problem formationProblem formationChannel selection matrix

Each user can access channels simultaneously ( )Interference level matrix

m m R

1 2 3 4

1 0 0.6 0.3 1

2 0.6 0 0 0.7

3 0.3 0 0 0.1

4 1 0.7 0.1 0

User number (N=4)

User

nu

mb

er

1 2 3

1 1 1 0

2 1 0 1

3 0 1 1

4 1 1 0

User n

um

ber

Channel number (R=3,m=2)

ijI

Channel selection matrix Interference level matrix

irS

Problem formationProblem formation Strategy profile: Matrix Spatial Congestion Games (MSCG)

Four types of MSCG according to payoff function:non-resource-specific and non-user-specificnon-resource-specific and user-specific resource-specific and non-user-specific resource-specific and user-specific

( , , ( ) , ( ) , , )i U r r R ir ijiU R g S I

1 2( , ,..., )N

Outline Outline Introduction System Model and Problem Formation Potential Game Approach Matrix Spatial Congestion Games Discussion Based on Numeric Results Conclusion and Future Work

Potential game approachPotential game approachPotential function [3]: Basic idea :

for each user : the value of potential function changes

correlating to the change of each user’s payoff

e.g.

potential function is boundedusers can’t update infinitely to increase

payoffs

[3] R. Rosenthal, “A class of games possessing pure-strategy nash equilibria,”INTERNATIONAL JOURNAL OF GAME THEORY, vol. 2, no. 1,pp. 65–67, 1973.

1 2( ) : N Z

'( ')i i

( ') ( ) ( ') ( ),i ip p i U

( )

Potential game approachPotential game approachFinite improvement property (FIP)

Improvement steps Improvement steps are finitePure Nash Equilibrium (PNE)

( ') ( ) 0,i ip p i U

( *) ( ), , , *i ip p i U

PNE

FIPLocal

updates

OutlineOutlineIntroduction System Model and Problem FormationPotential Game ApproachMatrix Spatial Congestion Games Discussion Based on Numeric ResultsConclusion and Future Work

MSCG –I/IVMSCG –I/IVMSCG with non-user-specific & non-

resource specific payoff functions payoff function : utility :

potential function :

proved:

1

( )N

jr ijj

g S I

1

( ) ( )i

Ni

jr ijr j

p g S I

1

( ) ( )i

ir jr iji U r R j

g S S I

( ') ( ) ( ') ( ),i ip p i U

MSCG – I/IVMSCG – I/IV Theorem 1:For matrix spatial congestion games with non-

resource-specific and non-user-specific payoff functions, every asynchronous improvement step path is finite and converges to a pure Nash Equilibrium (PNE). Furthermore, any change of the strategy profile can not result in a grater value of potential function , which indicating that this PNE is also a local optimum in potential function.• FIP PNE• Local optimum in potential function

MSCG – II/IVMSCG – II/IVMSCG with user-specific & non-

resource specific payoff functions payoff function : utility :

potential function :

, where

,

( ) ( )ij iji j U

n I

( ) ,ij i j ir jrr R

n r r r S S

1

( ) ( )i

Ni i

jr ijr j

p g S I

1

( )N

ijr ij

j

g S I

MSCG – II/IVMSCG – II/IVProved:

with increasing and non-increasing with

when increases,

decreases accordingly

Theorem 2: For matrix spatial congestion games with non-resource-specific and user-specific payoff functions, every asynchronous improvement path is finite and converges to a pure Nash Equilibrium (PNE).

1 1

( ) ( )N N

ir jr iji r R j

S S I

1

( ) ( )N

i iir jr ij

r R j

p g S S I

( )ip ( )ig x x

( )ip ( )

MSCG – III,IV/IVMSCG – III,IV/IV Proposition 1: For MSCG with resource-

specific and non-user-specific payoff functions, FIP does not hold. PNE ?

Proposition 2: For MSCG with resource-specific and user-specific payoff functions, FIP does not hold. Moreover, a PNE dose not necessarily exist.

OutlineOutlineIntroduction System Model and Problem FormationPotential Game ApproachMatrix Spatial Congestion Games Discussion Based on Numeric Results

user number channel utilization proportion updating mechanism

Conclusion and Future Work

User numberUser number using updating steps indicating convergence

speed users update in a predefined order

User numberUser number

Avera

ge im

pro

vem

en

t ste

ps

tota

l imp

rovem

en

t ste

ps

Channel utilization proportion Channel utilization proportion

Channel utilization proportion: more radios better ?

Avera

ge im

pro

vem

en

t ste

ps

Channel utilization proportion

Updating mechanismMechanism 1: updating in a predefined

orderMechanism 2: maximum improvement first

[8]

[8] R. Southwell and J. Huang, “Convergence dynamics of resource homogeneous congestion games,” in Proc.

International Conference on Game Theory for Networks (GameNets). Shanghai, China, Apr. 2011.

imp

rovem

en

t ste

ps

User number

OutlineOutlineIntroduction System Model and Problem FormationPotential Game ApproachMatrix Spatial Congestion Games Discussion Based on Numeric ResultsConclusion and Future Work

ConclusionFuture work

ConclusionConclusionMSCG model

Multi-radioSpatial reuse with weighted interference

level

Four types of MSCGGeneral payoff functionsFIPPNEConvergence speed analysis

Future workFuture workMore work on later two types of MSCG

achieve PNE?• Network topology• Charging scheme

Performance of PNE POA analysisFairness issueDistributed algorithm to achieve global

optimality

Convergence speed

Thank you !Thank you !