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Non-Cooperative Multi-Radio Channel Allocation
in Wireless Networks
Márk Félegyházi*, Mario Čagalj†, Shirin Saeedi Bidokhti*, Jean-Pierre Hubaux*
* Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland
† University of Split, Croatia
Infocom 2007
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Infocom 2007
Márk Félegyházi (EPFL) 2
Problem
► multi-radio devices► set of available channels
How to assign radios to available channels?
3d4d5d
6d
1d 2d
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Infocom 2007
Márk Félegyházi (EPFL) 3
System model (1/3)
3d4d5d
6d
1d 2d
2p
1p
3p
► – set of orthogonal channels (|| = C)
► – set of communicating pairs of devices (|| = N)
► sender controls the communication (sender and receiver are synchronized)
► single collision domain if they use the same channel
► devices have multiple radios► k radios at each device, k ≤ C
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Infocom 2007
Márk Félegyházi (EPFL) 4
System model (2/3)
► N communicating pairs of devices► C orthogonal channels► k radios at each device
,i xknumber of radios
by sender i on channel x
→
,i i xx C
k k
,x i xi N
k k
example:
3 2, 2p ck
Use multiple radios on one channel ?
, 1i xk Intuition:
23ck
34pk
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Infocom 2007
Márk Félegyházi (EPFL) 5
System model (3/3)► channels with the same properties► τ t(kx) – total throughput on any channel x
► τ(kx) – throughput per radio
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Infocom 2007
Márk Félegyházi (EPFL) 6
► selfish users (communicating pairs)► non-cooperative game GMRCA
– players → senders – strategy → channel allocation – payoff → total throughput
► strategy:
► strategy matrix:
► payoff:
Multi-radio channel allocation (MRCA) game
,1 ,,...,i i i Cs k k
1
N
s
S
s
, ( )i i i x xx C
u k k
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Infocom 2007
Márk Félegyházi (EPFL) 7
Game-Theoretic Concepts
Nash equilibrium: No player has an incentive to unilaterally deviate.* * *( , ) ( , ),i i i i i i iu s s u s s s S
Best response: Best strategy of player i given the strategies of others.
' '( ) : ( , ) ( , ),i i i i i i i i i ibr s s u s s u s s s S S
Price of anarchy: The ratio between the total payoff of players playing a socially-optimal (max. Pareto-optimal) strategy and a worst Nash equilibrium.
soi
iw NEi
i
uPOA
u
Pareto-optimality: The strategy profile spo is Pareto-optimal if:
' ': ( ) ( ),poi is u s u s i with strict inequality for at least one player i
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Infocom 2007
Márk Félegyházi (EPFL) 8
Lemma: If S* is a NE in GMRCA, then .
Use of all radios
Each player should use all of his radios.
p4 p4
,ik k i
Intuition: Player i is always better off deploying unused radios.
all channel allocations
Lem
ma
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Infocom 2007
Márk Félegyházi (EPFL) 9
Proposition: If S* is a NE in GMRCA, then dy,x ≤ 1, for any channel x and y.
Load-balancing channel allocation► Consider two arbitrary channels x and y in , where kx ≥ ky► distance: dx,y = kx – ky
all channel allocations
Lem
ma
Pro
posi
tion
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Infocom 2007
Márk Félegyházi (EPFL) 10
Nash equilibria (1/2)
Theorem 1: A channel allocation S* is a Nash equilibrium in GMRCA if for all i:
► dx,y ≤ 1 and
► ki,x ≤ 1.
p2
Nash Equilibrium: p4
Use one radio per channel.
all channel allocations
Lem
ma
Pro
posi
tion NE type 1
► Consider two arbitrary channels x and y in , where kx ≥ ky► distance: dx,y = kx – ky
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Infocom 2007
Márk Félegyházi (EPFL) 11
Nash equilibria (2/2)
Nash Equilibrium:
Theorem 2: A channel allocation S* is a Nash equilibrium in GMRCA if:
► dx,y ≤ 1,
► for any player i who has ki,x ≥ 2, x in ,
► for any player i who has ki,x ≥ 2 and x in +, ki,y ≥ ki,x – 1, for all y in –
Use multiple radios on certain channels.all channel allocations
Lem
ma
Pro
posi
tion NE type 1
NE type 2
,
( 1) ( 1)
( 1) ( )x x
i xx x
k kk
k k
► Consider two arbitrary channels x and y in , where kx ≥ ky► distance: dx,y = kx – ky
► loaded and less loaded channels: + and –
+–
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Infocom 2007
Márk Félegyházi (EPFL) 12
Efficiency (1/2)
1
1 1 1
t
t t tx x x x
POAN k
k k k kC
Corollary: If τt(kx) is constant (i.e., ideal TDMA), then any Nash equilibrium channel allocation is Pareto-optimal in GMRCA.
Theorem: In GMRCA , the price of anarchy is:
, 1x x
N k N kk k
C C
where
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Infocom 2007
Márk Félegyházi (EPFL) 13
Efficiency (2/2)
► In theory, if the total throughput function τt(kx) is constant POA = 1► In practice, there are collisions, but τt(kx) decreases slowly with kx (due to the
RTS/CTS method)
G. Bianchi, “Performance Analysis of the IEEE 802.11 Distributed Coordination Function,” in IEEE Journal on Selected Areas of Communication (JSAC), 18:3, Mar. 2000
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Infocom 2007
Márk Félegyházi (EPFL) 14
Summary► wireless networks with multi-radio devices► users of the devices are selfish players► GMRCA – multi-radio channel allocation game► results for a Nash equilibrium:
– players should use all their radios– load-balancing channel allocation– two types of Nash equilibria– NE are efficient both in theory and practice
► fairness issues► coalition-proof equilibria► algorithms to achieve efficient NE:
– centralized algorithm with perfect information– distributed algorithm with imperfect information
http://people.epfl.ch/mark.felegyhazi
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Infocom 2007
Márk Félegyházi (EPFL) 15
Future work
► general scenario – conjecture: hard► approximation algorithms► extend model to mesh networks
(multihop communication)
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Extensions
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Infocom 2007
Márk Félegyházi (EPFL) 17
Related work► Channel allocation
– in cellular networks: fixed and dynamic: [Katzela and Naghshineh 1996, Rappaport 2002]
– in WLANs [Mishra et al. 2005]– in cognitive radio networks [Zheng and Cao 2005]
► Multi-radio networks– mesh networks [Adya et al. 2004, Alicherry et al. 2005]– cognitive radio [So et al. 2005]
► Competitive medium access– Aloha [MacKenzie and Wicker 2003, Yuen and Marbach 2005]– CSMA/CA [Konorski 2002, Čagalj et al. 2005]– WLAN channel coloring [Halldórsson et al. 2004]– channel allocation in cognitive radio networks [Cao and Zheng 2005, Nie
and Comaniciu 2005]
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Infocom 2007
Márk Félegyházi (EPFL) 18
Fairness
Nash equilibria (fair) Nash equilibria (unfair)
Theorem: A NE channel allocation S* is max-min fair iff
min min
, , , ,i x j xx x
k k i j
C C
N
Intuition: This implies equality: ui = uj, i,j
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Infocom 2007
Márk Félegyházi (EPFL) 19
Centralized algorithm
Assign links to the channels sequentially.
p1 p1 p1p1 p2p2
p2p2 p3 p3 p3p3
p4 p4 p4p4
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Márk Félegyházi (EPFL) 20
Convergence to NE (1/3)
p1 p1
N = 5, C = 6, k = 3
p2 p2
p4
p1
p3 p2 p5
p4
p5
p3
p3
p4
p5
c1 c2 c3c4 c5 c6
timep5: c2→c5
c6→c4p3: c2→c5
c6→c4c1→c3
p2: c2→c5p1: c2→c5
c6→c4
p1: c4→c6c5→c2
p4: idle
channelsp5
p3
p2
p1
p1
p4
Algorithm with imperfect info:► move links from “crowded”
channels to other randomly chosen channels
► desynchronize the changes► convergence is not ensured
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Infocom 2007
Márk Félegyházi (EPFL) 21
Convergence to NE (2/3)
3UB
Algorithm with imperfect info:► move links from “crowded”
channels to other randomly chosen channels
► desynchronize the changes► convergence is not ensured
xx
N kS k
C
C
Balance:
unbalanced (UB): best balance (NE):
Efficiency: ( ) ( )
( ) ( )UB
UB NE
S SS
S S
0 1S
15UB 7S
15 7 3
15 3 4S
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Infocom 2007
Márk Félegyházi (EPFL) 22
Convergence to NE (3/3)
N (# of pairs) 10
C (# of channels) 8
k (radios per device) 3
τ(1) (max. throughput) 54 Mbps