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
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Problem
d2
d1
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multi-radio devices
set of available channels
d5
d4
d3
d6
How to assign radios to available channels?
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Márk Félegyházi (EPFL)
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System model (1/3)
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C – set of orthogonal
channels (|C| = C)
N – set of communicating
pairs of devices (|N| = 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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p1
d2
d1
d5
d4
d3
Márk Félegyházi (EPFL)
p2
p3
d6
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System model (2/3)
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N communicating pairs of devices
C orthogonal channels
k radios at each device
ki , x→
number of radios
by sender i
on channel x
ki   ki , x
xC
k x   ki , x
example:
Intuition: ki , x  1
iN
kc2  3
Use multiple radios on one channel ? k p  4
3
k p3 ,c2  2
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System model (3/3)
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channels with the same properties
τ t(kx) – total throughput on any channel x
τ(kx) – throughput per radio
Márk Félegyházi (EPFL)
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Multi-radio channel allocation (MRCA) game
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selfish users (communicating pairs)
non-cooperative game GMRCA
– players → senders
– strategy → channel allocation
– payoff → total throughput
si  ki ,1 ,..., ki ,C 
►
strategy:
►
strategy matrix:
►
payoff:
 s1 
 
S    
s 
 N
ui   i    ki , x  (k x ) 
xC
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Márk Félegyházi (EPFL)
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Game-Theoretic Concepts
Best response: Best strategy of player i given the strategies of others.
bri ( si )  si  S : ui ( si , si )  ui ( si' , si ), si'  S 
Nash equilibrium: No player has an incentive to unilaterally deviate.
ui ( si* , s* i )  ui ( si , s* i ), si  S
Pareto-optimality: The strategy profile spo is Pareto-optimal if:
s ' : ui ( s ' )  ui ( s po ), i with strict inequality for at least one player i
Price of anarchy: The ratio between the total payoff of players playing a
socially-optimal (max. Pareto-optimal) strategy and a worst Nash
equilibrium.
POA 
so
u
i
i
w  NE
u
i
i
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Use of all radios
Lemma: If S* is a NE in GMRCA, then ki  k , i.
Each player should use all of his radios.
Intuition: Player i is always better off deploying unused
radios.
p4 p4
Lemma
all channel allocations
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Márk Félegyházi (EPFL)
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Load-balancing channel allocation
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Consider two arbitrary channels x and y in C, where kx ≥ ky
distance: dx,y = kx – ky
Proposition: If S* is a NE in GMRCA, then dy,x ≤ 1, for any
channel x and y.
Proposition
Lemma
all channel allocations
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Márk Félegyházi (EPFL)
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Nash equilibria (1/2)
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Consider two arbitrary channels x and y in C, where kx ≥ ky
distance: dx,y = kx – ky
Theorem 1: A channel allocation S* is a Nash equilibrium in GMRCA
if for all i:
► dx,y ≤ 1 and
► ki,x ≤ 1.
Nash
Equilibrium:
Use one radiop4per channel.
p2
Proposition
Lemma
all channel allocations
NE type 1
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Nash equilibria (2/2)
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Consider two arbitrary channels x and y in C, where kx ≥ ky
distance: dx,y = kx – ky
loaded and less loaded channels: C+ and C–
Theorem 2: A channel allocation S* is a Nash equilibrium in GMRCA if:
► dx,y ≤ 1,
 (k x  1)   (k x  1)
► for any player i who has ki,x ≥ 2, x in C, ki , x 
 (k x  1)   (k x )
+
► for any player i who has ki,x ≥ 2 and x in C , ki,y ≥ ki,x – 1, for all y in
C–
Nash
Equilibrium:
Proposition
Lemma
all channel allocations
Use multiple radios
on certain channels.
NE type 1
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NE type 2
– Félegyházi (EPFL)
CMárk
C+
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Efficiency (1/2)
Theorem: In GMRCA , the price of anarchy is:
POA 
 t 1
N k  t

t
t
k

1



k


k

1






 k x  1
 x

x
x
C 



 N k 
 N k 
, kx  1  
where k x  


C
C




Corollary: If τt(kx) is constant (i.e., ideal TDMA), then any
Nash equilibrium channel allocation is Pareto-optimal in
GMRCA.
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Márk Félegyházi (EPFL)
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Efficiency (2/2)
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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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Summary
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wireless networks with multi-radio devices
users of the devices are selfish players
GMRCA – multi-radio channel allocation game
results for a Nash equilibrium:
–
–
–
–
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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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Future work
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general scenario – conjecture: hard
approximation algorithms
extend model to mesh networks
(multihop communication)
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Extensions
Related work
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Channel allocation
–
–
–
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Multi-radio networks
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–
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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]
mesh networks [Adya et al. 2004, Alicherry et al. 2005]
cognitive radio [So et al. 2005]
Competitive medium access
–
–
–
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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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Fairness
Nash equilibria (fair)
Nash equilibria (unfair)
Theorem: A NE channel allocation S* is max-min fair iff
 ki, x   k j , x , i, j  N
xCmin
xCmin
Intuition: This implies equality: ui = uj, i,j  N
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Centralized algorithm
Assign links to the channels sequentially.
p
p
p
p
4
4
p
4
p
p
p
p
2
p
p
2
3
p
3
p
3
p
3
1
1
1
1
2
2
p
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Márk Félegyházi (EPFL)
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Convergence to NE (1/3)
Algorithm with imperfect info:
► move links from “crowded”
channels to other randomly
chosen channels
► desynchronize the changes
► convergence is not ensured
p5
p3
p2
p1
p5: c2→c5
c6→c4
p3: c2→c5
c6→c4
c1→c3
p2: c2→c5
p1: c2→c5
c6→c4
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p1
p4
N = 5, C = 6, k = 3
p
5
p1: c4→c6
c5→c2
p4: idle
time
Márk Félegyházi (EPFL)
p
p
p
4
5
p
4
p
3
p
3
p
2
p
p
5
p
3
1
1
2
2
c6 channels
p
p
4
p
c1 c2 c3 c4 c5
p
1
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Convergence to NE (2/3)
Algorithm with imperfect info:
► move links from “crowded”
channels to other randomly
chosen channels
► desynchronize the changes
► convergence is not ensured
 S   7
15  7 3
 S  

15  3 4
Balance:   S    k x 
xC
N k
C
best balance (NE):
unbalanced (UB):
 UB   3
 UB   15
Efficiency:   S  
 ( SUB )   ( S )
 ( SUB )   ( S NE )
0   S  1
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Convergence to NE (3/3)
N (# of pairs)
10
C (# of channels)
8
k (radios per device)
3
τ(1) (max. throughput) 54 Mbps
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