a game approach for multi-channel allocation in multi-hop wireless networks lin gao, xinbing wang...

A Game Approach for Multi-Channel A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless NetworksAllocation in Multi-Hop Wireless Networks

Lin Gao, Xinbing Wang

Dept. of Electronic EngineeringShanghai Jiao Tong University

Shanghai, China

A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks 2

OutlineOutline

IntroductionIntroduction MotivationsMotivations ObjectivesObjectives

System Model and Game TheorySystem Model and Game Theory

Existence of MMCPNEExistence of MMCPNE

Convergence Algorithm and SimulationConvergence Algorithm and Simulation

ConclusionsConclusions


MotivationMotivation

The appearance of Multi-hop mobile ad hoc networks (MANETs) The appearance of Multi-hop mobile ad hoc networks (MANETs) A good channel allocation scheme in multi-hop MANET can improve A good channel allocation scheme in multi-hop MANET can improve

the system performance dramatically.the system performance dramatically. Distributed algorithm shows potential ability in channel allocation Distributed algorithm shows potential ability in channel allocation

problem due to the lacking of global central node in multi-hop problem due to the lacking of global central node in multi-hop MANETs.MANETs.

MANETs


Objectives --Objectives -- How to Solve the Problem? How to Solve the Problem?

Fixed channel allocation(FCA), dynamic channel allocation(DCA) and Fixed channel allocation(FCA), dynamic channel allocation(DCA) and hybrid channel allocation(HCA), e.g., [1], [2], [3]hybrid channel allocation(HCA), e.g., [1], [2], [3]

Weighted graph coloring method, e.g., [4]Weighted graph coloring method, e.g., [4]

In this paper, We model the channel allocation problem as a static In this paper, We model the channel allocation problem as a static cooperative game, in which some players collaborate to achieve high cooperative game, in which some players collaborate to achieve high date rate.date rate.

[1] J. van den Heuvel, R. A. Leese, and M. A. Shepherd. Graph labeling and radio channel assignment. [1] J. van den Heuvel, R. A. Leese, and M. A. Shepherd. Graph labeling and radio channel assignment. Journal of Graph Theory, 29:263-283, 1998.Journal of Graph Theory, 29:263-283, 1998.

[2] I. Katzela and M. Naghshineh. Channel assignment schemes for cellular mobile telecommunication [2] I. Katzela and M. Naghshineh. Channel assignment schemes for cellular mobile telecommunication systems: a comprehensive survey. IEEE Personal Communications, 3(3):10-31, Jun 1996.systems: a comprehensive survey. IEEE Personal Communications, 3(3):10-31, Jun 1996.

[3] Hac A and Z. Chen. Hybrid channel allocation in wireless networks. In Proceedings of the IEEE [3] Hac A and Z. Chen. Hybrid channel allocation in wireless networks. In Proceedings of the IEEE Conference on Vehicular Technology Conference (VTC'99), 50(4):2329-2333, Sept. 1999.Conference on Vehicular Technology Conference (VTC'99), 50(4):2329-2333, Sept. 1999.

[4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel assignment for WLANs. [4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel assignment for WLANs. Mobile Computing and Communications Review (MC2R), 9(3), 2005.Mobile Computing and Communications Review (MC2R), 9(3), 2005.


OutlineOutline

IntroductionIntroduction

System Model and Game FormulationSystem Model and Game Formulation System Model: Multi-hop MANETSystem Model: Multi-hop MANET Game Theory: Nash EquilibriumGame Theory: Nash Equilibrium





System Model -- ISystem Model -- I

We assume that there exist several communication sessions in our We assume that there exist several communication sessions in our model and we further assume each user participates in only one model and we further assume each user participates in only one session.session.

An example of 3 communication sessions and 7 An example of 3 communication sessions and 7 users.users.


System Model -- IISystem Model -- II

We assume each user owns a device equipped with several radio We assume each user owns a device equipped with several radio transceivers, denoted by Ttransceivers, denoted by TSS and R and RSS respectively, which used to respectively, which used to

transmit the data packets respectively. transmit the data packets respectively. We assume that there is a mechanism that enables the multiple radios We assume that there is a mechanism that enables the multiple radios

in any Tin any TSS (or R (or RSS) to communicate simultaneously by using orthogonal ) to communicate simultaneously by using orthogonal

channels.channels.

RRSSTTSS

An example of 3 radios in TAn example of 3 radios in TSS (and R (and RSS).).

RRSS RRSS

TTSS TTSS


System Model -- IIISystem Model -- III

We assume that the total available bandwidth on channel We assume that the total available bandwidth on channel cc is shared is shared equally among the radios deployed on this channel. equally among the radios deployed on this channel.

Utility function:Utility function:

,

,

:

:

:

iu c i

c

c

ii

u cu c

cc C

where

k the total number of radios of u using channel c

k the total number of radios of all users using channel c

R the total available bandwidth of channel c

kR R

k


Game Formulation -- IGame Formulation -- I We formulate the channel allocation problem with a single stage We formulate the channel allocation problem with a single stage

game, which corresponds to a fixed channel allocation among the game, which corresponds to a fixed channel allocation among the players. Each player's strategy consists in defining the number of players. Each player's strategy consists in defining the number of radios on each of the channels.radios on each of the channels.

The strategy of the previous system model.The strategy of the previous system model.

s1 s2 s1

c1 c2 c3 c4

s1

s2

s3

s3r21

r21


Game Formulation -- IIGame Formulation -- II Nash Equilibrium (NE): NE expresses the resistance to the deviation Nash Equilibrium (NE): NE expresses the resistance to the deviation

of a single player in non-cooperative game. In other words, in a NE of a single player in non-cooperative game. In other words, in a NE none of the players can unilaterally change its strategy to increase its none of the players can unilaterally change its strategy to increase its utility.utility.

s1 s1 s1

c1 c2 c3 c4 c5 c6

s2 s2

s4

s3

s1 s2 s2

s3 s3

s4s4 s4

s3

An example of the Nash Equilibrium.An example of the Nash Equilibrium.


Game Formulation – IIIGame Formulation – III Non-cooperative game is not suitable for the multi-hop networks for Non-cooperative game is not suitable for the multi-hop networks for

the following two reasons:the following two reasons:

(1) In one hand, the achieved date rate of any player in multi-hop link (1) In one hand, the achieved date rate of any player in multi-hop link is not only determined by the utility itself, but also by the utilities of is not only determined by the utility itself, but also by the utilities of other players in the same link. other players in the same link.

(2) In the other hand, it is possible that the players in the same multi-(2) In the other hand, it is possible that the players in the same multi-hop link cooperatively choose their strategies for the purpose of high hop link cooperatively choose their strategies for the purpose of high achieved date rate.achieved date rate.

u2

u2

u1 u1 u1

c1 c2 c3 c4 c5 c6 channels

u2

An example of the Nash Equilibrium with poor performance An example of the Nash Equilibrium with poor performance where u1 and u2 belong to the same multi-hop link.where u1 and u2 belong to the same multi-hop link.


Game Formulation – IVGame Formulation – IV we define a novel coalition-proof Nash equilibrium in cooperative we define a novel coalition-proof Nash equilibrium in cooperative

game, named as min-max coalition-proof Nash equilibrium game, named as min-max coalition-proof Nash equilibrium (MMCPNE), in which players make their decisions so as to improve (MMCPNE), in which players make their decisions so as to improve the minimal payoff of players in the coalition.the minimal payoff of players in the coalition.

u2

u2

u1 u1 u1


u2

An example of the MMCPNE.An example of the MMCPNE.


Game Theory – VGame Theory – V By jointly searching the strategy in the strategies set of | By jointly searching the strategy in the strategies set of | lli i | players. | players.

The computation of achieving MMCPNE increases exponentially with The computation of achieving MMCPNE increases exponentially with the size of link.the size of link.

To reduce the large computation in finding MMCPNE, we introduce To reduce the large computation in finding MMCPNE, we introduce three approximate solutions, denoted by minimal coalition-proof Nash three approximate solutions, denoted by minimal coalition-proof Nash equilibrium (MCPNE), average coalition-proof Nash equilibrium equilibrium (MCPNE), average coalition-proof Nash equilibrium (ACPNE) and i coalition-proof Nash equilibrium (ICPNE). The (ACPNE) and i coalition-proof Nash equilibrium (ICPNE). The definitions of MCPNE, ACPNE and ICPNE are shown in Section IV.definitions of MCPNE, ACPNE and ICPNE are shown in Section IV.


Definition of MMCPNEDefinition of MMCPNE We define a novel coalition-proof Nash equilibrium in cooperative We define a novel coalition-proof Nash equilibrium in cooperative

game, named as min-max coalition-proof Nash equilibrium game, named as min-max coalition-proof Nash equilibrium (MMCPNE), in which players make their decisions so as to improve (MMCPNE), in which players make their decisions so as to improve the minimal payoff of players in the coalition.the minimal payoff of players in the coalition.

: (Min-Max Coalition-Proof Nash Equilibrium

- MMCPNE) :

- ,

, :

min

Definition 4

i

mm

i

u

The strategy matrix defines a novel

coalition proof Nash Equilibrium if for every coalition

co we have

X

'

'

, min ,

.

i i i i i ii i i

i

i mm mm i mmu co co u co co

co u co

co

R R

for every strategy set

X X X X

X


Definition of MCPNEDefinition of MCPNE In MCPNE, players in a coalition select their strategies to maximize In MCPNE, players in a coalition select their strategies to maximize

the minimal utilities of players in the coalition.the minimal utilities of players in the coalition.

: (Minimal Coalition-Proof Nash Equilibrium

- MCPNE) :

- ,

, :

min

Definition 5

i i

m

i

uu co

The strategy matrix defines a special

coalition proof Nash Equilibrium if for every player

u we have

R

X

'

'

, min ,

.

i i i i i ii i

i

i m m i mu u u u uu co

u

R

for every strategy

x x

x

X X


Definition of ACPNEDefinition of ACPNE In ACPNE, players in a coalition select their strategies to maximize In ACPNE, players in a coalition select their strategies to maximize

the average utility while do not decrease the minimal utility of players the average utility while do not decrease the minimal utility of players in the coalition.in the coalition.

: (Average Coalition-Proof Nash Equilibrium

- ACPNE) : -

- , , :

min

Definition 6

i i

a

i

u co

The strategy matrix defines a special coalit

ion proof Nash Equilibrium if for every player u we have

R

X

'

'

'

'

, min ,

min , min ,

, ,

.

i i i i i ii i

i i i i i ii i i i

i i i i i i

i i i i

i

i a a i au u u u u uu co

i a a i au u u u u uu co u co

i a a i au u u u u u

u co u co

u

R

or

R R

R R

for every strategy

x x

x x

x x

x

X X

X X

X X


Definition of ICPNEDefinition of ICPNE In ICPNE, however, players in a coalition select their strategies to In ICPNE, however, players in a coalition select their strategies to

maximize their own utilities while do not decrease the minimal utility maximize their own utilities while do not decrease the minimal utility of players in the same coalition.of players in the same coalition.

: (I Coalition-Proof Nash Equilibrium - ICPNE) :

-

, , :

min

Definition 7

i ii i

i

i

i iu uu co

The strategy matrix defines a special coalition proof

Nash Equilibrium if for every player u we have

R

x

X

'

'

'

'

, min ,

min , min ,

, ,

.

i i i ii i

i i i i i ii i i i

i i i i i i

i

i i iu u u uu co

i i i i iu u u u u u

u co u co

i i i i iu u u u u u

u

R

or

R R

R R

for every strategy

x

x x

x x

x

X X

X X

X X


Intuition among these NE definitionsIntuition among these NE definitions


OutlineOutline IntroductionIntroduction

System Model and Game FormulationSystem Model and Game Formulation

Existence of MMCPNEExistence of MMCPNE Theorem 1Theorem 1 Proposition 1, Lemma 2 ~ 5 Proposition 1, Lemma 2 ~ 5 Theorem 2Theorem 2




Theorem 1Theorem 1

,

,

,

:

:

(1) 1

(2) 1 ,

:

Theorem 1

i iu c u

b c

b c b c

A channel allocation is a NE iff the following

conditions hold

k and k k

for any b c C

where

k k the difference of radios number between

X

.channel b and c

The existence of Nash Equilibrium [5]:The existence of Nash Equilibrium [5]:

[5] M. Felegyhazi, M. Cagalj, S. S. Bidokhti, and J. P. Hubaux. Non-cooperative Multi-radio [5] M. Felegyhazi, M. Cagalj, S. S. Bidokhti, and J. P. Hubaux. Non-cooperative Multi-radio Channel Allocation in Wireless Networks. In Proceedings of the IEEE Conference on Channel Allocation in Wireless Networks. In Proceedings of the IEEE Conference on Computer Communications (INFOCOM '07), March 13-17 2007.Computer Communications (INFOCOM '07), March 13-17 2007.


Proposition 1Proposition 1 We divide the MMCPNE states into two sets according to theorem 1. We divide the MMCPNE states into two sets according to theorem 1.

We denote the MMCPNE states which satisfy the theorem 1 by We denote the MMCPNE states which satisfy the theorem 1 by MMCPNE-1 and denote the remainder MMCPNE states by MMCPNE-MMCPNE-1 and denote the remainder MMCPNE states by MMCPNE-2. We find that the multi-hop links in MMCPNE-1 states always occupy 2. We find that the multi-hop links in MMCPNE-1 states always occupy more bandwidth compared with those in MMCPNE-2 states.more bandwidth compared with those in MMCPNE-2 states.

The basic criterion of finding MMCPNEThe basic criterion of finding MMCPNE:

,

, . ., 1 .

Proposition 1 Assume that there exists a MMCPNE channel

allocation with high bandwidth occupied then is a Nash

equilibrium i e the conditions of theorem hold

X X

All Channel Allocations

Lemma 2 Lemma 3

Lemma 4&5

NE

MMCPNE-1 Unknow Region

The value of Proposition The value of Proposition 1 is that it provides a 1 is that it provides a method to choose the method to choose the MMCPNE with the high MMCPNE with the high bandwidth occupied, i.e., bandwidth occupied, i.e., MMCPNE-1.MMCPNE-1.


Lemma 2&3Lemma 2&3

1 2

1 2

1 2

1 2

, ,

: { , }

.

1 0, , .

Lemma 2

i i

i u u

u c u c

Assume that there exists a link l u u and R R in a NE

channel allocation If there exist two channels c C and c C such

that k and k i then is not MMCPNE

w

X

X

( ) : max (min ) .

here

C C the channel with the imum imum number of radios

Moving from NE to MMCPNE: link members relocate their radios to Moving from NE to MMCPNE: link members relocate their radios to improve the payoff of others when two members share any channels.improve the payoff of others when two members share any channels.

u2u2 u2 u2

u1 u1

u1

u1


NE

An example of a NE channel An example of a NE channel allocation corresponding to allocation corresponding to

Lemma 2.Lemma 2.


Lemma 4&5Lemma 4&5

2 1

1 1 2 1 1

1 2

1 2

, , ,

1 1: { , }

.

1 0

Lemma 4 i u u

u c u c u c

Assume that there exists a link l u u and R R in

a NE channel allocation If there exist two channels c C and c C such that

k and k whereas k

X

2 2 2, 0 1, .

( ) : ( )

u cand k then is not MMCPNE

where

the number of radios of any channel in C C

X

Moving from NE to MMCPNE: link members Moving from NE to MMCPNE: link members helping each other is that helping each other is that they mutually exchange some radios with each other.they mutually exchange some radios with each other.

u2

u2

u1 u1 u1


NE

u2 An example of a NE channel An example of a NE channel allocation corresponding to allocation corresponding to

Lemma 4.Lemma 4.


Theorem 2Theorem 2 The necessary conditions that enables a NE to be MMCPNE:The necessary conditions that enables a NE to be MMCPNE:

1 2

2 1

1 2

1 2 : { , }

. , :

1 1(1)

(2) 1:

Theorem 2 i u u

u u

u u

Assume that there exists a link l u u and R R in

a NE channel allocation If is MMCPNE the follow conditions hold

R R and

case if R R then ther

X X

1 2 1 2

1 2

1

, , , ,

1 2

1 2 ,

1 0,

2 : { , }

{ , } i

u b u b u c u c

u u

u b u

e does not exist two channels b C and

c C such that k k whereas k k

case if R R then there does not exist four channels b b C

and c c C such that k k

2 1 2, , , 1, 1, .

i j jb u c u ci whereas k k j


Conjecture 1Conjecture 1

1 2

1 2 1: { , }

.

2 , .

Conjecture i

u u

Assume that there exists a link l u u and

R R in a NE channel allocation If the conditions in

theorem hold then is MMCPNE

X

X

The sufficient conditions that enables a NE to be MMCPNE:The sufficient conditions that enables a NE to be MMCPNE:

All Channel Allocations

Lemma 2 Lemma 3

Lemma 4&5

NE

MMCPNE-1 Unknow Region

The unknown region The unknown region converges to NULL set converges to NULL set according to Conjecture 1.according to Conjecture 1.


OutlineOutline


System Model and Game FormulationSystem Model and Game Formulation


Convergence Algorithm and SimulationConvergence Algorithm and Simulation Convergence algorithmConvergence algorithm Simulation Results Simulation Results



Convergence Algorithm -- IConvergence Algorithm -- I MMCP AlgorithmMMCP Algorithm

(1) In the first stage, the coalitions move their radios to achieve high (1) In the first stage, the coalitions move their radios to achieve high utility. Thus we call this stage as inter-link competition stage. utility. Thus we call this stage as inter-link competition stage.

(2) In the second state, players in the same link mutually adjust their (2) In the second state, players in the same link mutually adjust their radios to achieve higher date rate. We call this stage as intra-link radios to achieve higher date rate. We call this stage as intra-link improvement stage.improvement stage.

link to the code


Convergence Algorithm -- IIConvergence Algorithm -- II DCP AlgorithmDCP Algorithm

By transforming the mutual operation of multiple players into multiple By transforming the mutual operation of multiple players into multiple independent operations of the players, DCP algorithm efficiently independent operations of the players, DCP algorithm efficiently reduces the computational complexity, specifically, from exponentially reduces the computational complexity, specifically, from exponentially increasing with the number of players to linear increasing with it.increasing with the number of players to linear increasing with it.

link to the code


Simulation Results -- ISimulation Results -- I Performance criterion:Performance criterion:

: the coalition utility of any coalition is

defined as the ratio of the total bandwidth it occupied to

the average bandwidth per user.

: the coalition usage fact

Coalition Utility

Coalition Usage Factor or of any

coalition is defined as the ratio of the achieved date rate

to the total bandwidth it occupied.

: the coalition efficiency of any coalition

is defined as the ability of an

Coalition Efficiency

y coalition to achieve a given

payoff (or achieved date rate).


Simulation Results -- IISimulation Results -- II Average Coalition Utility vs. TimeAverage Coalition Utility vs. Time

Channel Number: 8Channel Number: 8

User number: 5User number: 5

Link number: 4Link number: 4

Radio number: 4Radio number: 4

Coalition: {u1,u2}Coalition: {u1,u2}


Simulation Results – IIISimulation Results – III Average Coalition Efficiency vs. TimeAverage Coalition Efficiency vs. Time







Simulation Results – IVSimulation Results – IV Average Coalition Usage Factor vs. TimeAverage Coalition Usage Factor vs. Time







Simulation Results – VSimulation Results – V Average Coalition Usage Factor vs. Users NumberAverage Coalition Usage Factor vs. Users Number





OutlineOutline


System Model and Game TheorySystem Model and Game Theory



ConclusionsConclusions ConclusionsConclusions


ConclusionsConclusions In this paper, we have studied the problem of competitive channel allocation In this paper, we have studied the problem of competitive channel allocation

among devices which use multiple radios in the multi-hop system. among devices which use multiple radios in the multi-hop system. We propose a novel coalition-proof Nash equilibrium, denoted by MMCPNE, to We propose a novel coalition-proof Nash equilibrium, denoted by MMCPNE, to

ensure the multi-hop links to achieve high date rate without worsening the date ensure the multi-hop links to achieve high date rate without worsening the date rates of single-hop links. rates of single-hop links.

We investigate the existence of MMCPNE and propose the necessary We investigate the existence of MMCPNE and propose the necessary conditions for the existence of MMCPNE. conditions for the existence of MMCPNE.

Finally, we provide several algorithms to achieve the exact and approximate Finally, we provide several algorithms to achieve the exact and approximate MMCPNE states. We study their convergence properties theoretically. MMCPNE states. We study their convergence properties theoretically. Simulation results show that MMCPNE outperforms CPNE and NE schemes in Simulation results show that MMCPNE outperforms CPNE and NE schemes in terms of achieved data rates of links due to cooperation gain.terms of achieved data rates of links due to cooperation gain.

Thank you !Thank you !


ReferenceReference [1] J. van den Heuvel, R. A. Leese, and M. A. Shepherd. Graph labeling and radio [1] J. van den Heuvel, R. A. Leese, and M. A. Shepherd. Graph labeling and radio

channel assignment. Journal of Graph Theory, 29:263-283, 1998.channel assignment. Journal of Graph Theory, 29:263-283, 1998. [2] I. Katzela and M. Naghshineh. Channel assignment schemes for cellular mobile [2] I. Katzela and M. Naghshineh. Channel assignment schemes for cellular mobile

telecommunication systems: a comprehensive survey. IEEE Personal Communications, telecommunication systems: a comprehensive survey. IEEE Personal Communications, 3(3):10-31, Jun 1996.3(3):10-31, Jun 1996.

[3] Hac A and Z. Chen. Hybrid channel allocation in wireless networks. In Proceedings of [3] Hac A and Z. Chen. Hybrid channel allocation in wireless networks. In Proceedings of the IEEE Conference on Vehicular Technology Conference (VTC'99), 50(4):2329-2333, the IEEE Conference on Vehicular Technology Conference (VTC'99), 50(4):2329-2333, Sept. 1999.Sept. 1999.

[4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel [4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel assignment for WLANs. Mobile Computing and Communications Review (MC2R), 9(3), assignment for WLANs. Mobile Computing and Communications Review (MC2R), 9(3), 2005.2005.

[5] M. Felegyhazi, M. Cagalj, S. S. Bidokhti, and J. P. Hubaux. Non-cooperative Multi-[5] M. Felegyhazi, M. Cagalj, S. S. Bidokhti, and J. P. Hubaux. Non-cooperative Multi-radio Channel Allocation in Wireless Networks. In Proceedings of the IEEE Conference radio Channel Allocation in Wireless Networks. In Proceedings of the IEEE Conference on Computer Communications (INFOCOM '07), March 13-17 2007.on Computer Communications (INFOCOM '07), March 13-17 2007.


MMCP-AlgorithmMMCP-Algorithm

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DCP-AlgorithmDCP-Algorithm

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a game approach for multi-channel allocation in multi-hop wireless networks lin gao, xinbing wang...

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