Computer Communications 32 (2009) 1058–1061

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Computer Communications

journal homepage: www.elsevier .com/locate /comcom

Perfectly fair channel assignment in non-cooperative multi-radiomulti-channel wireless networks q

Tingting Chen, Sheng Zhong *

Computer Science and Engineering, Department State University of New York at Buffalo, 201 Bell Hall, Amherst, NY 14260, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 September 2008Received in revised form 9 December 2008Accepted 21 December 2008Available online 30 December 2008

Keywords:Non-cooperative channel assignmentMulti-radio wireless networksFairnessGame theory

0140-3664/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.comcom.2008.12.030

q This work is supported by NSF CNS-0524030.* Corresponding author. Tel.: +1 716 645 3180; fax

E-mail address: szhong@cse.buffalo.edu (S. Zhong)

The channel assignment problem is very important in wireless networks. In this letter, we study the non-cooperative channel assignment problem in competitive multi-radio multi-channel wireless networks,with a focus on fairness issues. We propose a Nash equilibrium solution with stronger guarantee on fair-ness among players than existing works, by requiring a payment from each player. We show that, in ourscheme, when system converges to a stable status achieving Nash equilibrium, all players obtain thesame throughput. Simulation results verify that our scheme achieves perfect fairness.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

In some wireless networks (e.g. mesh networks [1]), each nodecan be equipped with multiple radios, which greatly enhances thechannel selection and thus increases the system throughput. Toachieve higher system performance in multi-radio communications,channel assignment becomes a crucial research topic (e.g. [2,3,1]).

Given that many nodes in wireless networks can be selfish, it isimportant for the channel assignment to be incentive-compatible,so that the system can converge to a stable status that all nodes aresatisfied with. Two recent works [4,5] studied this problem. Whiletheir solutions are interesting and practical, the fairness propertiesof their schemes are not perfect. In [4], the authors providedintriguing analysis of how to achieve Nash equilibria in a non-cooperative multi-radio multi-channel allocation game. However,only a small portion of the Nash equilibrium solutions can guaran-tee Max–Min fairness (defined in [6]). In [5], a strong solution con-cept called Strongly Dominant Strategy Equilibrium is used toachieve system-wide throughput optimality. However, the fairnessamong players can only be achieved in the long run. In a finiteamount of time, the assignment is not really fair for all players.Achieving fairness is very important in resource allocation in com-puter networks [7,8]. In many applications, fairness is even morecrucial than system-wide throughput optimization. Hence it isimportant to design an incentive-compatible multi-radio multi-channel assignment scheme with better fairness.

ll rights reserved.

: +1 716 645 3464..

In this letter, our objective is to achieve Nash equilibrium solu-tion with perfect fairness in terms of throughput in the non-cooper-ative multi-radio multi-channel assignment, where perfect fairness(see Section 2 for definition) is stronger than Max–Min fairnessused in [4]. The rest of this letter is organized as follows. In Section2 we introduce the system model, some concepts in game theoryand the game model of channel assignment. Then we present ourNash equilibrium solution with perfect fairness and give rigorousproofs of its properties. After the performance evaluation in Section3, we conclude this letter.

2. System model and concepts

2.1. System model

As in [4], we consider a wireless network in which each user hasa device with K radio transmitters. In this network, we assume thatusers communicate with each other in pairs, bidirectionally, andeach user only participates in one such communication. We alsoassume that users always have packets to transmit between them.The set of communicating pairs is written as N. The frequency bandavailable to the network is divided into orthogonal channels. Wedenote the channel set by C. We assume that the available channelshave the same characteristics. All users in the network are within asingle collision domain.

We denote the total throughput of channel c by Rc and thethroughput per radio on channel c by sc . We assume radios onthe same channel equally share the total throughput. For mediumaccess protocols like TDMA and CSMA/CA, Rc is a non-increasingfunction of the number of radios on channel c, as shown in [4].

T. Chen, S. Zhong / Computer Communications 32 (2009) 1058–1061 1059

So it is valid for us to assume that sc is a decreasing function of thenumber of radios on channel c.

2.2. Concepts and notations

Before modeling the multi-radio channel assignment game, wefirst review some important concepts in game theory. Also we willgive the formal definition of fairness in this subsection.

In a non-cooperative strategic game, any player i in player set Ntries to maximize its own utility (payoff), denoted by ui. We denoteone strategy of player i by si and the strategy space of i by Si. Theset of chosen strategies of all players constitutes a strategy profile,written as s ¼ fs1; s2; . . . ; sjNjg. Note that a strategy profile includesone and only one strategy for each player. s�i is the strategies setchosen by all the other players except player i. Formally,s�i ¼ fs1; . . . ; si�1; siþ1; . . . ; sjNjg. Every player prefers strategy si tos0i, if si brings higher utility, uiðsi; s�iÞ > uiðs0i; s�iÞ.

In game theory, Nash equilibrium is an important solution con-cept. The formal definition as in [9] is described below.

Definition 1. Nash equilibrium: The strategy profile s� constitutesa Nash equilibrium, if for each player i,

uiðs�i ; s��iÞP uiðsi; s��iÞ; 8si 2 Si: ð1Þ

A Nash equilibrium solution guarantees that no players canbenefit by deviating from it if other players do not change theirstrategies. Nevertheless, if the channel assignment scheme is notcarefully designed, Nash equilibrium solutions might be highly un-fair in which some players get very high utilities while others donot. In the following section we will present an approach toachieve Nash equilibrium and perfect fairness at the same time.

2.3. Strategic game model of multi-radio channel assignment

In a non-cooperative multi-radio channel assignment game,each player is a pair of communicating users in the network. Thestrategy of a player is the channel assignment of the radios thatit can manipulate. Formally, the strategy of player i issi ¼ fki;1; . . . ; ki;c; . . . ; ki;jCjg, where ki;c is the number of radios thatplayer i tunes to channel c and jCj is the number of channels. Notethat the total number of radios on channel c, kc ¼

Pi2Nki;c . We can

consequently have the rate that player i obtains on channel c, ri;c as

ri;c ¼ ki;csðkcÞ; ð2Þ

given that sðkcÞ is the throughput per radio on channel c.In this game, every player tries to maximize its total throughput

of its communication, which can be written asP

c2Cri;c .Recall that in the system model described above, each player

has K radios and always has packets to transmit. It means thatall players have the same demands for transmission rate. There-fore, the fairness in this game requires that all players obtain equaltotal throughputs.

Definition 2. Perfect fairness: In the channel assignment game,the strategy profile s is perfectly fair, if 8i; j 2 N,

uiðsÞ ¼ ujðsÞ andXc2C

ri;c ¼Xc2C

rj;c: ð3Þ

3. Nash equilibrium with perfect fairness

In the real world, network users usually need to pay to keepconnected and communicate with each other in the network. Thusit is fair for us to assume that there is some charge to players forutilizing the channels to transmit packets. More importantly, wecan achieve a Nash equilibrium with perfect fairness by carefully

designing the amount of payment that players should give basedon how they utilize the channels.

As in [5], we assume that there is a certain kind of virtual cur-rency in the system which represents the payment for using chan-nels. Each player pays some virtual currency to the systemadministrator based on the channels it has chosen to use.

We use pi;c to denote the payment of player i for using channel cand further define pi;c as

pi;c ¼ ri;c ki;c � a� 1c þ 1

� 1bþ 1

� ��������� ð4Þ

in which a ¼ KjCj

j k, b ¼ K modjCj.

Hence we define total utility of player i as

ui ¼Xc2C

ri;c �Xc2C

pi;c: ð5Þ

We also define a strategy profile s�: for every player i 2 N,

s�i ¼ fs�i;1; s�i;2; . . . ; s�i;jCjg ¼ faþ 1; . . . ; aþ 1|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}b

; a; . . . ; a|fflfflfflffl{zfflfflfflffl}jCj�b

g: ð6Þ

Note that ðaþ 1Þbþ aðjCj � bÞ ¼ K , which is the total number ofradios that each player has.

Theorem 3. In the channel assignment game described in Section 2.3,s� achieves a Nash equilibrium.

Proof. For any player i, as defined in (6),

k�i;c ¼aþ 1 if 1 6 c 6 b

a if bþ 1 6 c 6 jCj

�ð7Þ

Note that, for each channel c 2 C we also use c as the identificationnumber of that channel.

Also given b > 0, c > 0, we have

1c þ 1

� 1bþ 1

� �¼

1 if 1 6 c 6 b

0 if bþ 1 6 c 6 jCj

�ð8Þ

From (7) and (8), we can get 8c 2 C

k�i;c � a� 1c þ 1

� 1bþ 1

� ��������� ¼ 0 ð9Þ

so for s�i ,P

c2Cpi;c ¼ 0.Consequently,

uiðs�i ; s��iÞ ¼Xc2C

ri;c: ð10Þ

Now let us consider the strategies other than s�i . 8si–s�i , let C0i denotethe channel set on which the radio assignment of player i deviatesfrom s�i . Since si–s�i , 9c 2 C, ki;c–k�i;c . So jC0ij > 0.

Next we will show that for player i, 8c 2 C0i, pi;c P ri;c .Since ki;c is a non-negative integer and ki;c–k�i;c , 8c 2 C0i,

ki;c � a� 1c þ 1

� 1bþ 1

� ���������P 1: ð11Þ

Combining (4) and (11), we can obtain that 8c 2 C0i; pi;c P ri;c:

Finally we can get

uiðs�i ; s��iÞ � uiðsi; s��iÞ ¼Xc2C

r�i;c �XcRC0i

ri;c �XcRC0i

pi;c þXc2C0i

ri;c �Xc2C0i

pi;c

0@

1A

PXc2C

r�i;c �XcRC0i

ri;c � 0þXc2C0i

ri;c �Xc2C0i

ri;c

0@

1A

¼Xc2C

r�i;c �XcRC0i

ri;c ¼Xc2C

r�i;c �XcRC0i

r�i;c ¼Xc2C0i

r�i;c P 0:

Therefore, s� is a Nash equilibrium. h

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Number of Players

Max

−Min

Thr

ough

put D

iffer

ence

(Mbi

t/s)

Random Channel AssignmentNE in [5]NE with Perfect Fairness

Fig. 1. The difference of maximum and minimum throughput among players.

1060 T. Chen, S. Zhong / Computer Communications 32 (2009) 1058–1061

Next we will show that s� is perfectly fair as well.

Theorem 4. s� achieves perfect fairness.

Proof. From (2), we can see that the throughput of player i onchannel c depends on ki;c and the total number of radios on thechannel c, kc . It is easy to see that in s�i , the total number of radioson channel c (k�c) is

k�c ¼Xi2N

k�i;c ¼ðaþ 1ÞjNj if 1 6 c 6 bajNj if bþ 1 6 c 6 jCj

�ð12Þ

where jNj is the number of players.Given (10) and (12), 8i 2 N, we have

u�i ¼Xc2C

ri;c ¼Xc2C

k�i;csðk�cÞ

¼X

16c6b

ðaþ 1Þsððaþ 1ÞjNjÞ þX

b<c6jCjasðajNjÞ

¼ bðaþ 1Þsððaþ 1ÞjNjÞ þ ðjCj � bÞasðajNjÞ

So the value of u�i is independent from i. Hence, 8i; j 2 C;u�i ¼ u�j , andPc2Cri;c ¼

Pc2Crj;c . This completes the proof of the perfect fairness

of s�. h

Theorem 4 states that the strategy profile s� which achieves aNash equilibrium, also guarantees the prefect fairness in the sys-tem. In other words, each player can obtain the same totalthroughput, i.e. perfect fairness, when the system converges to astable state.

We note that Theorem 3 and 4 hold true for all positive valuesof K and jCj. For example, if we have three orthogonal channels asin traditional 802.11b/g and each node in the network has fourwireless interfaces, the strategy that each node takes in the strat-egy profile s� is f2;1;1g. If we have 10 players in the network, thenin this specific case, the total throughput that each node will obtainis 2 � sð2 � 10Þ þ sð10Þ þ sð10Þ ¼ 2sð20Þ þ 2sð10Þ. sð10Þ (resp.,sð20Þ) is the throughput per radio when there are 10 (resp., 20)radios in the channel. Consequently, the perfect fairness ofthroughput for each player is achieved.

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

Number of Players

Sta

ndar

d D

evia

tion

of T

hrou

ghpu

t

Random Channel AssignmentNE in [5]NE with Perfect Fairness

4. Evaluation

We simulate our scheme using MATLAB. We choose 11 orthog-onal channels as the set C (as in 802.11a protocol [10]) and we as-sume that the rate of each channel is decreasing as the number ofradios on that channel increases. We set K ¼ 5. The channel rate isset to 1 Mbit/s. Same system parameters are used as in [11] to sim-ulate varying rate channels. The results shown below are the aver-age of 100 runs.

To evaluate the fairness improvement of our scheme, we maketwo measurements in the stable system after it converges. First, wecompute the difference of throughputs between the players whogets maximum throughput and who gets the minimum. We callit max–min throughput difference for short. In Fig. 1, we comparethe max–min throughput difference of our scheme with Nash equi-librium (NE) solution in [4] and random assignment algorithm,varying the number of players in the game. We can see that themax–min throughput difference of our scheme remains 0 givendifferent number of players, which implies perfect fairness, whileNE solution in [4] can only improve the fairness by a limitedamount when jNj > 4.

In Fig. 2, we also compare the standard deviation of players’throughput for the three channel assignment schemes. Our schemeachieve 0 standard deviation of players’ throughputs no matterhow many players are in the network. It means every playerobtains the same total throughput when the system converges to

a stable status. In contrast, NE solution in [4] performs almostthe same as the random assignment, in terms of fairness.

5. Discussion

Our Nash equilibrium solution in this paper guarantees perfectfairness for the non-cooperative channel assignment in single col-lision domain. Non-cooperative channel assignment in multiplecollision domains is a much more complicated problem. We nowdiscuss the challenges in the problem of multiple collision domainchannel assignment with the objective of fairness. First of all, theremay be no perfect fair channel assignment possible for multiplecollision domains with some network topologies. A simple exam-ple is a network with three players A;B and C. B can hear both Aand C, while A and C are far from each other and thus no interfer-ence can be caused between them. There are two orthogonal chan-nels available in the network and each player has two interfaces. Inthis case, if all players want to use all of their devices, no matterhow the channels are allocated, equal total throughput amongA;B and C cannot be achieved. To deal with this difficulty, we canscale down the fairness criteria to looser ones (e.g, Max–Min fair-ness) and find solutions correspondingly. Second, as mentionedin [4], hidden terminal problem exists in multiple collision do-

Fig. 2. Standard deviation of players’ throughput.

T. Chen, S. Zhong / Computer Communications 32 (2009) 1058–1061 1061

mains. It places an obstacle for players in correctly learning howmany radios are in each channel and correspondingly adjust theirstrategies. As a result, it is more difficult for the system to convergeto a stable state by itself. If there is a system administrator whoknows the overall topology of the network, it can help the systemconvergence by interacting with players. A solution without cen-tral administration is even more attractive. We leave all these pos-sible solutions for future work.

Another possible extension of our work is to the scenarioswhere flows span multiple hops. For multiple hop cases, it maybe more appropriate to define the players in the game as flows,since essentially it is the flows that compete in the channel utiliza-tion. Achieving equal throughput for each flow is very challengingbecause multiple hop flows are usually across multiple collisiondomains. The nodes on the path of the same flow may actuallyinterfere with each other. To overcome this difficulty, we can inte-grate our scheme with scheduling. More specifically, we allownodes to place their radios on different channels in different timeslots. Correspondingly, the payment of using each channel for eachplayer can be made time-variant, such that the system can con-verge to a stable state in which overall system performance andfairness for each flow are guaranteed. As the details are beyondthe scope of this paper, we leave them in the future study.

6. Conclusion

In this letter, a Nash equilibrium solution with perfect fairnessis proposed for the non-cooperative multi-radio multi-channelassignment game. Compared with existing schemes, our solution

provides perfect fairness with a rigorous proof and guarantees thatthe system converges to the Nash equilibrium status.

References

[1] A. Raniwala, T.-C. Chiueh, Architecture and algorithms for an IEEE 802.11-based multi-channel wireless mesh network, in: Proc. IEEE INFOCOM 05, 2005,pp. 2223–2234.

[2] M. Alicherry, R. Bhatia, L. (Erran) Li, Joint channel assignment and routing forthroughput optimization in multi-radio wireless mesh networks, in: Proc. ACMMobicom 05, 2005, pp. 58–72.

[3] A. Adya, P. Bahl, J. Padhye, A. Wolman, L. Zhou, A multi-radio unificationprotocol for IEEE 802.11 wireless networks, in: Proc. Broadnets 04, 2004, pp.344–354.

[4] M. Felegyhazi, M. Cagalj, S.S. Bidokhti, J.-P. Hubaux, Noncooperative multi-radio channel allocation in wireless networks, in: Proc. IEEE INFOCOM 07,2007, pp. 1442–1450.

[5] F. Wu, S. Zhong, C. Qiao, Globally optimal channel assignment for non-cooperative wireless networks, in: Proc. IEEE INFOCOM 08, 2008, pp. 1543–1551.

[6] D. Bertsekas, R. Gallager, Data Networks, second ed., Prentice-Hall, 1992.[7] A. Mishra, V. Shrivastava, D. Agarwal, S. Banerjee, S. Ganguly, Distributed

channel management in uncoordinated wireless environments, in: Proc. ACMMobicom 06, 2006, pp. 170–181.

[8] J. Tang, G. Xue, W. Zhang, Maximum throughput and fair bandwidth allocationin multi-channel wireless mesh networks, in: Proc. IEEE INFOCOM 06, 2006,pp. 1–10.

[9] M.J. Osborne, A. Rubenstein, A Course in Game Theory, The MIT Press, Berlin,1994.

[10] IEEE 802.11a Working Group, Wireless LAN medium access control (MAC) andphysical layer (PHY) specifications – amendment 1: highspeed physical layerin the 5 GHz band, 1999.

[11] G. Bianchi, Performance analysis of the IEEE 802.11 distributed coordinationfunction, IEEE Journal on Selected Areas in Communications 18 (3) (2000)535–547.