Transcript Slides PPT

A Game Approach for Multi-Channel
Allocation in Multi-Hop Wireless Networks
Lin Gao, Xinbing Wang
Dept. of Electronic Engineering
Shanghai Jiao Tong University
Shanghai, China
Outline
Introduction
 Motivations
 Objectives
System Model and Game Theory
Existence of MMCPNE
Convergence Algorithm and Simulation
Conclusions
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Motivation
 The appearance of Multi-hop mobile ad hoc networks (MANETs)
 A good channel allocation scheme in multi-hop MANET can improve
the system performance dramatically.
 Distributed algorithm shows potential ability in channel allocation
problem due to the lacking of global central node in multi-hop MANETs.
MANETs
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Objectives -- How to Solve the Problem?
 Fixed channel allocation(FCA), dynamic channel allocation(DCA) and
hybrid channel allocation(HCA), e.g., [1], [2], [3]
 Weighted graph coloring method, e.g., [4]
 In this paper, We model the channel allocation problem as a static
cooperative game, in which some players collaborate to achieve high
date rate.
[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.
[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.
[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.
[4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel assignment for WLANs.
Mobile Computing and Communications Review (MC2R), 9(3), 2005.
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Outline
Introduction
System Model and Game Formulation
 System Model: Multi-hop MANET
 Game Theory: Nash Equilibrium
Existence of MMCPNE
Convergence Algorithm and Simulation
Conclusions
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System Model -- I
 We assume that there exist several communication sessions in our
model and we further assume each user participates in only one
session.
An example of 3 communication sessions and 7 users.
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System Model -- II
 We assume each user owns a device equipped with several radio
transceivers, denoted by TS and RS respectively, which used to
transmit the data packets respectively.
 We assume that there is a mechanism that enables the multiple radios
in any TS (or RS) to communicate simultaneously by using orthogonal
channels.
RS TS
RS TS
RS
TS
An example of 3 radios in TS (and RS).
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System Model -- III
 We assume that the total available bandwidth on channel c is shared
equally among the radios deployed on this channel.
 Utility function:
kui ,c
Rui  
Rc
cC kc
where
ku , c : the total number of radios of ui using channel c
i
k c : the total number of radios of all users using channel c
Rc : the total available bandwidth of channel c
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Game Formulation -- I
 We formulate the channel allocation problem with a single stage game,
which corresponds to a fixed channel allocation among the players.
Each player's strategy consists in defining the number of radios on
each of the channels.
s3
r21
s2
r21 s3
s1
s2
s1
s1
c1 c2 c3 c4
The strategy of the previous system model.
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Game Formulation -- II
 Nash Equilibrium (NE): NE expresses the resistance to the deviation
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
utility.
s4
s4
s4
s4
s2
s2
s3
s3
s3
s3
s1
s1
s1
s1
s2
s2
c1 c2 c3 c4 c5 c6
An example of the Nash Equilibrium.
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Game Formulation – III
 Non-cooperative game is not suitable for the multi-hop networks for the
following two reasons:
(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
other players in the same link.
(2) In the other hand, it is possible that the players in the same multihop link cooperatively choose their strategies for the purpose of high
achieved date rate.
u1 u1 u1 u2
u2 u2
c1
c2
c3
c4
c5
c6
channels
An example of the Nash Equilibrium with poor performance
where u1 and u2 belong to the same multi-hop link.
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Game Formulation – IV
 we define a novel coalition-proof Nash equilibrium in cooperative game,
named as min-max coalition-proof Nash equilibrium (MMCPNE), in
which players make their decisions so as to improve the minimal
payoff of players in the coalition.
u1 u1 u1 u2
u2 u2
c1
c2
c3
c4
c5
c6
channels
An example of the MMCPNE.
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Game Theory – V
 By jointly searching the strategy in the strategies set of | li | players.
The computation of achieving MMCPNE increases exponentially with
the size of link.
 To reduce the large computation in finding MMCPNE, we introduce
three approximate solutions, denoted by minimal coalition-proof Nash
equilibrium (MCPNE), average coalition-proof Nash equilibrium
(ACPNE) and i coalition-proof Nash equilibrium (ICPNE). The
definitions of MCPNE, ACPNE and ICPNE are shown in Section IV.
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Definition of MMCPNE
 We define a novel coalition-proof Nash equilibrium in cooperative
game, named as min-max coalition-proof Nash equilibrium
(MMCPNE), in which players make their decisions so as to improve
the minimal payoff of players in the coalition.
Definition 4: (Min-Max Coalition-Proof Nash Equilibrium
- MMCPNE) : The strategy matrix X mm defines a novel
coalition - proof Nash Equilibrium, if for every coalition
coi , we have :



mm
mm
i
'
mm
min Rui i X co
,
X

min
R
X
,
X
 coi
ui
coi
 coi
i
ui coi
ui coi

for every strategy set X 'coi .
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Definition of MCPNE
 In MCPNE, players in a coalition select their strategies to maximize
the minimal utilities of players in the coalition.
Definition 5: (Minimal Coalition-Proof Nash Equilibrium
- MCPNE) : The strategy matrix X m defines a special
coalition - proof Nash Equilibrium, if for every player
ui , we have :



min Rui i xumi , X mui  min Rui i xu' i , X mui
ui coi
ui coi

for every strategy xu' i .
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Definition of ACPNE
 In ACPNE, players in a coalition select their strategies to maximize
the average utility while do not decrease the minimal utility of players
in the coalition.
Definition 6: (Average Coalition-Proof Nash Equilibrium
- ACPNE) : The strategy matrix X a defines a special coalit ion - proof Nash Equilibrium, if for every player ui , we have :



min Rui i xuai , X aui  min Rui i xu' i , X aui
ui coi
ui coi

or






min Rui xua , X au  min Rui xu' , X au
i
i
i
i
i
ui coi
 ui coi i

a
'
i
a
a
i
X
,
x
R

X
,
x
R

 ui
ui
ui
  ui ui ui
co

u
co

u
i
i
i i
for every strategy xu' i .


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Definition of ICPNE
 In ICPNE, however, players in a coalition select their strategies to
maximize their own utilities while do not decrease the minimal utility
of players in the same coalition.
Definition 7: (I Coalition-Proof Nash Equilibrium - ICPNE) :
The strategy matrix X i defines a special coalition - proof
Nash Equilibrium, if for every player ui , we have :



min Rui i xui i , X i ui  min Rui i xu' i , X iui
ui coi
ui coi

or





min Rui xui , X i u  min Rui xu' , X iu
i
i
i
i
i
 ui coi i
ui coi

Rui i xui i , X i ui  Rui i xu' i , X iui

for every strategy xu' i .



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 Intuition among these NE definitions
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Outline
Introduction
System Model and Game Formulation
Existence of MMCPNE
 Theorem 1
 Proposition 1, Lemma 2 ~ 5
 Theorem 2
Convergence Algorithm and Simulation
Conclusions
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Theorem 1
 The existence of Nash Equilibrium [5]:
Theorem 1: A channel allocation X is a NE iff the following
conditions hold :
(1) kui ,c  1 and kui  k
(2) b ,c  1 for any b, c  C
where
 b ,c  kb  kc : the difference of radios number between
channel b and c.
[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
Computer Communications (INFOCOM '07), March 13-17 2007.
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Proposition 1
 We divide the MMCPNE states into two sets according to theorem 1.
We denote the MMCPNE states which satisfy the theorem 1 by
MMCPNE-1 and denote the remainder MMCPNE states by MMCPNE2. We find that the multi-hop links in MMCPNE-1 states always occupy
more bandwidth compared with those in MMCPNE-2 states.
 The basic criterion of finding MMCPNE
Proposition 1: Assume that there exists a MMCPNE channel
allocation X with high bandwidth occupied , then X is a Nash
equilibrium, i.e., the conditions of theorem 1 hold .
 The value of Proposition
1 is that it provides a
method to choose the
MMCPNE with the high
bandwidth occupied, i.e.,
MMCPNE-1.
All Channel Allocations
Lemma 2
NE
Lemma 3
Lemma 4&5
MMCPNE-1
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
Unknow
Region
21
Lemma 2&3
 Moving from NE to MMCPNE: link members relocate their radios to
improve the payoff of others when two members share any channels.
Lemma 2 : Assume that there exists a link li  {u1 , u2 } and Ru  Ru in a NE
1
2
channel allocation X. If there exist two channels c1  C  and c2  C  such
that kui ,c1  1 and kui ,c2  0, i, then X is not MMCPNE.
where
C  (C  ) : the channel with the max imum(min imum) number of radios.
NE
u1 u1
u1
u2 u2 u1 u2 u2
c1
c2
c3
c4
c5
c6
channels
An example of a NE channel
allocation corresponding to
Lemma 2.
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Lemma 4&5
 Moving from NE to MMCPNE: link members helping each other is that
they mutually exchange some radios with each other.
Lemma 4 : Assume that there exists a link li  {u1 , u2 } and Ru  Ru       in
 

1
2
1
1
a NE channel allocation X. If there exist two channels c1  C  and c2  C  such that
ku1 ,c1  1 and ku2 ,c1  0 whereas ku1 ,c2  0 and ku2 ,c2  1, then X is not MMCPNE.
where
  (  ) : the number of radios of any channel in C  (C  )
NE
u1 u1 u1 u2
An example of a NE channel
allocation corresponding to
Lemma 4.
u2 u2
c1
c2
c3
c4
c5
c6
channels
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Theorem 2
 The necessary conditions that enables a NE to be MMCPNE:
Theorem 2 : Assume that there exists a link li  {u1 , u2 } and Ru  Ru in
1
2
a NE channel allocation X. If X is MMCPNE , the follow conditions hold :
1
 1
(1) Ru2  Ru1     


(2) case 1: if Ru1  Ru2

 and

then there does not exist two channels b  C  and
c  C  such that ku1 ,b  ku2 ,b  1 whereas ku1 ,c  ku2 ,c  0,
case 2 : if Ru1  Ru2 then there does not exist four channels {b1 , b2 }  C 
and {c1 , c2 }  C  such that ku1 ,bi  ku2 ,bi  1, i whereas ku1 ,c j  ku2 ,c j  1, j.
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Conjecture 1
 The sufficient conditions that enables a NE to be MMCPNE:
Conjecture 1: Assume that there exists a link li  {u1 , u2} and
Ru1  Ru2 in a NE channel allocation X. If the conditions in
theorem 2 hold , then X is MMCPNE.
All Channel Allocations
Lemma 2
NE
Lemma 3
The unknown region
converges to NULL set
according to Conjecture 1.
Lemma 4&5
MMCPNE-1
Unknow
Region
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Outline
Introduction
System Model and Game Formulation
Existence of MMCPNE
Convergence Algorithm and Simulation
 Convergence algorithm
 Simulation Results
Conclusions
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Convergence Algorithm -- I
 MMCP Algorithm
(1) In the first stage, the coalitions move their radios to achieve high
utility. Thus we call this stage as inter-link competition stage.
(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
improvement stage.
link to the code
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Convergence Algorithm -- II
 DCP Algorithm
By transforming the mutual operation of multiple players into multiple
independent operations of the players, DCP algorithm efficiently
reduces the computational complexity, specifically, from exponentially
increasing with the number of players to linear increasing with it.
link to the code
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Simulation Results -- I
 Performance criterion:
Coalition Utility: the coalition utility of any coalition is
defined as the ratio of the total bandwidth it occupied to
the average bandwidth per user.
Coalition Usage Factor: the coalition usage factor of any
coalition is defined as the ratio of the achieved date rate
to the total bandwidth it occupied.
Coalition Efficiency: the coalition efficiency of any coalition
is defined as the ability of any coalition to achieve a given
payoff (or achieved date rate).
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Simulation Results -- II
 Average Coalition Utility vs. Time
 Channel Number: 8
User number: 5
Link number: 4
Radio number: 4
Coalition: {u1,u2}
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Simulation Results – III
 Average Coalition Efficiency vs. Time
Channel Number: 8
User number: 5
Link number: 4
Radio number: 4
Coalition: {u1,u2}
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Simulation Results – IV
 Average Coalition Usage Factor vs. Time
Channel Number: 8
User number: 5
Link number: 4
Radio number: 4
Coalition: {u1,u2}
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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Simulation Results – V
 Average Coalition Usage Factor vs. Users Number
Channel Number: 8
Radio number: 4
Coalition: {u1,u2}
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Outline
Introduction
System Model and Game Theory
Existence of MMCPNE
Convergence Algorithm and Simulation
Conclusions
 Conclusions
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Conclusions
 In this paper, we have studied the problem of competitive channel
allocation among devices which use multiple radios in the multi-hop
system.
 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 rates of single-hop links.
 We investigate the existence of MMCPNE and propose the necessary
conditions for the existence of MMCPNE.
 Finally, we provide several algorithms to achieve the exact and
approximate MMCPNE states. We study their convergence properties
theoretically. Simulation results show that MMCPNE outperforms
CPNE and NE schemes in terms of achieved data rates of links due to
cooperation gain.
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Thank you !
Reference
 [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.
 [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.
 [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.
 [4] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based channel
assignment for WLANs. Mobile Computing and Communications Review (MC2R), 9(3),
2005.
 [5] M. Felegyhazi, M. Cagalj, S. S. Bidokhti, and J. P. Hubaux. Non-cooperative Multiradio Channel Allocation in Wireless Networks. In Proceedings of the IEEE Conference
on Computer Communications (INFOCOM '07), March 13-17 2007.
A Game Approach for Multi-Channel Allocation in Multi-Hop Wireless Networks
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MMCP-Algorithm
back
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DCP-Algorithm
back
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