Non-cooperative Multi-radio Channel Allocation inWireless Networks

Paper ID: 1568986796

Abstract— Channel allocation has been extensively studied inthe framework of cellular networks, but the emergence of newsystem concepts, such as cognitive radio systems, bring this topicinto the focus of research again. In this paper, we study theproblem of competitive multi-radio multi-channel allocation inwireless networks in detail. We characterize the Nash equilibriain a static game and we conclude that in spite of the non-cooperative behavior of such devices, their channel allocationresults in a system-efficient and load-balancing solution. Inaddition, we consider the fairness properties of the resultingchannel allocations and their resistance to the possible coalitionsof a subset of devices. Finally, we present three algorithms toachieve a load-balancing Nash equilibrium channel allocation,each of them using a different set of available information. Tothe best of our knowledge, our paper is the first contribution tothis important topic.

I. I NTRODUCTION

Wireless networks provide a flexible and cost-efficientmethod to establish communication between different parties.Each wireless network operates in a frequency band assignedby the authorities that regulate the frequency spectrum inthe given country. In general, the communication medium as-signed to a given network is shared among the communicationdevices using somemultiple accesstechnique.

Frequency Division Multiple Access (FDMA)is one of thewidely used techniques to enable several users to share acommunication medium that consists of a given frequencyband [17], [18]. The basic principle of FDMA is to split upthe available bandwidth to distinct sub-bands calledchannels.Assigning the radio transceivers to these channels is commonlyreferred to as thechannel allocationproblem1. Not surpris-ingly, an efficient channel allocation is a cornerstone of thedesign of existing wireless networks.

In this paper, we present a game-theoretic analysis of fixedchannel allocation strategies of devices using multiple radios.Using a static non-cooperative game, we analyze the scenarioof a single collision domain, i.e., if each of the devices caninterfere with a transmission of every other device. We derivethe Nash equilibria in this game and show that they are systemefficient and they achieve a load balancing solution. We alsostudy the fairness issues and the problem of coalition formingin the channel allocation problem. We show that a Nashequilibrium that resists to coalitions of users is necessarilyfair as well. Furthermore, we propose three algorithms toachieve the system-efficient Nash equilibrium solutions. Thefirst is a sequential algorithm that needs global coordination;

1In the literature, the terms channel assignment and frequency assignmentare also used for the channel allocation problem.

the second is a distributed algorithm that needs a local, butperfect information and the third is a distributed algorithmthat is based on imperfect local information. We provide theproof for the convergence properties of these algorithms.

This work is a first step towards the deeper understanding ofthe non-cooperative behavior of such devices and is applicablein a broad context of wireless communications, with particularattention to the emerging field of cognitive radio systems [10].To the best of our knowledge, our paper is the first to addressthe problem of multi-radio channel allocation in competitivenetworks, and hence it can give a guideline, how to study theimpact of selfishness in these novel technologies.

The paper is organized as follows. First, we present relatedwork on channel allocation and channel access in wirelessnetworks in Section II. In Section III, we introduce our systemmodel along with a game-theoretic description of competi-tive channel allocation. Section IV provides a comprehensiveanalysis of the Nash equilibria in the channel allocationgame. We study fairness issues in Section V. In addition,we provide some results on coalition-proof Nash equilibria inSection VI. In Section VII, we propose three simple algorithmsto reach the desired Nash equilibria. Finally, we conclude inSection VIII.

II. RELATED WORK

There has been a significant amount of work on channelallocation in wireless networks, notably for cellular networks.Channel allocation schemes in cellular networks can be di-vided into three categories: fixed channel allocation (FCA),dynamic channel allocation (DCA) and hybrid channel allo-cation (HCA), which combines the two former methods.

In a fixed channel allocation scheme, the same number ofchannels are permanently allocated to the radios at the basestations. To study fixed channel allocation, most authors usedgraph coloring / labelling techniques (e.g., in [19]). The FCAmethod performs very well under high traffic load, but itcannot adapt to changing traffic conditions or user distributions

To overcome the inflexibility of FCA, many authors pro-posed dynamic channel allocation (DCA) methods (e.g. aspresented in [6], [20]). In contrast to FCA, there is no constantrelationship between the base stations in a cell and theirrespective channels. All channels are available for each basestation and they are assigned dynamically as new users arrive.Typically, the available channels are evaluated according toa cost function and the one with the minimum cost is used[7]. Due to its dynamic property, the DCA can adapt tochanging traffic demand. Because adaptation implies some

cost, it performs worse in case of a heavy traffic load. Fora comprehensive survey on the topic, we refer the reader to[11].

Due to the emergence of alternative communication tech-nologies, channel allocation schemes became a focus of re-search again. Mishraet al. [14] propose a channel allocationmethod for wireless local area networks (WLANs) basedweighted graph coloring.

Recently, several researchers have considered devices usingmultiple radios, notably in mesh networks (for a survey onmesh networks, see [2]). In the multi-radio communicationcontext, channel allocation and access also became one of thecrucial topics. Related work on multi-radio medium accessincludes, but not restricted to [1], [3], [16].

In all the related work cited so far, their authors assumedthat the radio devices cooperate to achieve a high systemperformance. This assumption might not hold, as the users ofthese devices are usually selfish and they want to maximizetheir own performance without necessarily respecting thesystem objectives. Game theory provides a straightforwardtool to study medium access problems in competitive wirelessnetworks and has been applied to the CSMA/CA protocol[8], [12] and to the Aloha protocol [13]. Furthermore, a fixedchannel allocation game was presented in [9] based on graphcoloring. For cognitive radio networks, the authors of [15]propose a dynamic channel allocation schemes based on apotential game. In addition, they suggest another techniquebased on machine learning with different utility functions.

III. SYSTEM MODEL AND CONCEPTS

We assume an available frequency band divided into or-thogonal channels of the same bandwidth using the FDMAmethod (e.g., 11 orthogonal channels in case of the 802.11aprotocol). We also assume that these channels have the sameexpected channel characteristics. We denote the set of availableorthogonal channels byC.

In our model,pairs of userswant to communicate with eachother over a single hop. We assume that each user participatesin only one such communication session, hence we denotethe set of such communication links byN . Each user ownsa device equipped withk ≤ |C| radio transmitters, all havingthe same communication capabilities. The communication be-tween two devices is bidirectional and they always have somepackets to exchange. We assume that each communicating pairis a selfish player, whose objective is to maximize its totalrate or channel utilization. We will use this term to denoteboth the communicating pair of users and the communicationlink between them. In this paper, we further assume that eachdevice can hear the transmissions of every other device if theyare using the same channel. This means that the players residein asingle collision domain. We make this assumption to avoidthe hidden terminal problem described for example in [17],[18].

We assume that there is a mechanism that enables theplayers to use multiple channels to communicate at the sametime (as it is implemented in [1] for example). We denote the

number of radios of playeri using channelc by ki,c for everyc ∈ C. For the simplicity of presentation, let us denote theset of channels used by playeri by Ci, whereCi ⊂ C and0 ≤ |Ci| ≤ k. We further assume that there is no limitation onthe number of radios per channel.

We formulate the multi-radio channel allocation problem asa non-cooperative game as follows. We define thestrategyofplayer i as its channel allocation vector:

si = {ki,1, . . . , ki,|C|} (1)

Hence, its strategy consists in defining the number of radioson each of the channels2. The strategy vector of all playersdefines the strategy matrixS, where the rowi of the matrixcorresponds to the strategy vector of playeri:

S =

s1

. . .s|N |

(2)

Furthermore, we denote the strategy matrix except for thestrategy of playeri by S−i as shown in (3):

S−i =

s1

. . .si−1

si+1

. . .s|N |

(3)

Figure 1 presents an example channel allocation with sixavailable channels (|C| = 6), four players (|N | = 4) and eachuser device equipped by four radios (k = 4). Figure 2 presentsthe strategy matrix that corresponds to this example.

channels

c1 c2 c3 c4 c5

channels: c1 - c6number of radios

p1 p1 p1 p1

p2 p2 p2

p3 p3

p3

p3

p4

p4

players (pairs): p1 - p4

c6

p2

Fig. 1. An example for a channel allocation, where|C| = 6, |N | = 4 andk = 4.

The total number of radios employed by playeri canbe written aski =

∑c ki,c. Similarly, we can obtain the

number of radios using a particular channelkc =∑

i ki,c. InFigure 1, each player has a radio on channelc1, but channelc5 is occupied only by playerp2. Player p3 employs tworadios on channelc2 to get more bandwidth on that particularchannel. Regarding the number of radios per player, we havekp1 = kp2 = kp3 = 4 andkp4 = 2, meaning that playerp4 isnot using all of his radios.

2Note that this number can be zero.

1 0 0 1 0 01 2 0 0 0 11 0 1 1 1 0

play

ers

c1

channels

1 1 1 1 0 0p1

c2 c3 c4 c5

p2

p3

p4

c6

Fig. 2. Strategy matrix of the example in Figure 1.

We assume that the players are rational and their objective isto maximize theirutility in the network. We denote the utilityof player i by Ui. For simplicity, we assume that each playeri wants tomaximize his total rate (Ri) in the system and thusthe utility function is the achieved bitrate. We leave the studyof other utility functions for future work.

We assume that the total rate on channelc is shared equallyamong the radio transmitters using that channel. This fair rateallocation is achieved for example by using a reservation-basedTDMA schedule on a given channel. A similar result wasreported by Bianchi in [5] for the CSMA/CA protocol usingoptimal backoff window values. Even if the radio transmittersare controlled by selfish users in the CSMA/CA protocol,they can achieve this fair sharing as shown in [8]. We furtherassume that thetotal available bitrateRc(kc) on a channelc(i.e., the sum of the achieved bitrate of all players on channelc) is a non-increasing function of the number of radioskc

deployed on this channel. In factRc(kc) is independent ofkc

for a TDMA protocol and for the CSMA/CA protocol usingoptimal backoff window values [5]. In practice, the backoffwindow values used in the CSMA/CA protocol implementa-tion (e.g., in the 802.11 standard) are not optimal; and dueto packet collisionsRc(kc) becomes a decreasing functionfor kc > 1. Since we assume that channels have the samebandwidth and channel characteristics, the rate function doesnot depend on the channel and thus we can write thatR(kc)for any channelc ∈ C. If kc = 0, we defineR(0) = 0; notehowever that this case has no relevance in our model.

Figure 3 presents the total rateR(kc) as a function of thenumber of radios using channelc.

kc

R(kc)reservation TDMA

optimal CSMA/CA

practical CSMA/CA

Fig. 3. The total available rateR(kc) for different MAC protocols.

If player i chooses to operateki,c radios in a given channel,his rate on this channel can be written asRi,c = ki,c

kc·R(kc).

We assume that the players do not cheat at the MAC layeras in [8] for example. Thus, we can write thatRi,c > 0 forall c ∈ C, whereki,c > 0. Recall that in Figure 1, the higheris the number of radios in a given channel, the lower is therate per radio. Hence, for example for playerp2, we haveR2,1 < R2,4 < R2,3 < R2,5. We can obtain the total rateRi

for player i by Ri =∑

c Ri,c.In summary, we can write the utility function for playeri

as:

Ui(S) = Ri =∑

c∈CRi,c =

∑

c∈C

ki,c

kc·R(kc) (4)

We model the channel allocation problem with asingle stagegame, which corresponds to a fixed channel allocation amongthe players.

In order to study the strategic interaction of the players, wefirst introduce the concept of Nash equilibrium.

Definition 1: (Nash Equilibrium – NE): The strategy matrixS∗ = {s∗1, . . . , s∗|N|} defines a Nash Equilibrium (NE), if forevery playeri, we have:

Ui(s∗i , S∗−i) ≥ Ui(s

′i, S

∗−i) (5)

for every strategys′i.

In other words, in a NE none of the players can unilaterallychange its strategy to increase its utility. A NE solution is ofteninefficient from the system point of view. We characterize theefficiency of the solution by the concept of Pareto-optimality.

Definition 2: (Pareto-Optimality): The strategy matrixSpo

is Pareto-optimal if@S′ such that:

Ui(S′) ≥ Ui(Spo), ∀i (6)

with strict inequality for at least one playeri.This means that in a Pareto-optimal channel allocationSpo noplayer i can improve his utility without decreasing the utilityof at least one other playerj.

IV. NASH EQUILIBRIA

In this section, we study the existence of Nash equilibriain the single collision domain channel allocation game. Notethat we omit the proofs of many intermediate results due tospace limitations.

It is straightforward to see that if the total number of radiosis smaller than or equal to the number of channels, then a flatchannel allocation, in which the number of radios per channeldoes not exceed one, is a Nash equilibrium.

Fact 1: If |N | · k ≤ |C|, then any channel allocation, inwhich kc = 1,∀c ∈ C is a Pareto-optimal NE.

For the remainder of the paper, we assume that|N |·k > |C|,hence the devices have a conflict during the channel allocationprocess. In the following lemmas, we express necessary con-ditions for a Nash equilibrium. The first necessary conditionshows that the players should use all of their radios.

First we show that a selfish player should use all of hisradios in order to maximize his total rate.

Lemma 1: In a NE of the multi-radio channel allocationgame,ki = k,∀i.

In the example presented in Figure 1, Lemma 1 does nothold for playersp4, because it uses only two radios. Hence,the example cannot be a NE.

Proof: We can prove the lemma by contradiction. As-sume that there exists a NE, in which playeri uses onlyki < kradios. As mentioned previously, in our model we assume thatk ≤ |C| and we assumed that in the NE|Ci| ≤ ki < k, thuswe necessarily have|Ci| < |C|. This implies that there alwaysexists a channelc /∈ Ci. If the player deploys an additionalradio on this channelc, then he increases his utility due to thefact thatRi,c > 0 for ki,c = 1. Hence, we have a contradictionand the original allocation cannot be a NE.

Let us now consider a NE strategy matrix in the multi-radiochannel allocation game denoted byS∗, wheres∗i ∈ S∗ is theNE strategy of playeri (i.e., the i-th row of the matrix). Letus consider two arbitrary channelsb andc in this NE strategyallocation. Without loss of generality, we assume that thereare more radios using channelb, meaning thatkb ≥ kc, anddenote their difference by:

δb,c = kb − kc (7)

Assume that playeri moves one of his radios from channelb to c. Let us define thebenefit of change, i.e. the differencein the utility of playeri, as follows:

∆ = Ui(s′i, S

∗−i)− Ui(s∗i , S

∗−i)

=ki,b − 1kb − 1

·R(kb − 1) +ki,c + 1kc + 1

·R(kc + 1)

− ki,b

kb·R(kb)− ki,c

kc·R(kc) (8)

We can show a second necessary condition for a NE, namelythat playeri has a benefit of moving one radio to a channel,where he has no radios if the difference of the number ofradios deployed on the two channels exceeds one.

Lemma 2: If ki,b > 0, ki,c = 0 andδb,c > 1 for any playeri, thenS∗ is not a NE channel allocation.

In the example presented in Figure 1, Lemma 2 holds e.g.for playerp1 and the channelsb = c1 andc = c5. Hence, theexample cannot be a NE.

Proof: Assume thatS∗ is a NE channel allocation.Suppose that playeri moves one of his radios from channelb to c. Using the conditions in the lemma, we can write thebenefit of change defined in (8) as:

∆ =ki,b − 1kb − 1

R(kb − 1) +1

kc + 1R(kc + 1)− ki,b

kbR(kb)

=ki,b

kb − 1R(kb − 1)− 1

kb − 1R(kb − 1)

+1

kc + 1R(kc + 1)− ki,b

kbR(kb)

Let us notice that the sum of the first and last terms is alwaysstrictly greater than 0, becauseδb,c > 1 implies thatkb > 1.Hence, it is enough to investigate the sign of the sum of the

two other terms. Using (7), we can rewrite the sum of the twomiddle terms as:

R(kc + 1)kc + 1

− R(kc − 1)kc − 1

=R(kc + 1)

kc + 1− R(kc + δb,c − 1)

kc + δb,c − 1

Due to the assumptionδb,c > 1 and the non-increasing ratefunction R(·), we have:

R(kc + 1)kc + 1

− R(kc + δb,c − 1)kc + δb,c − 1

≥ 0

Hence, the benefit of change is positive and thusS∗ cannotbe a NE. This contradiction concludes the proof.

Let us now derive the third necessary condition. Thiscondition shows that playeri should again change the positionof one of his radios, if he has at least two radios more onchannelb than channelc and overall, there are more radioson channelb than on channelc. The rationale of the lemmais that playeri can decrease the imbalance of his radios byreallocating them between the channelsb andc.

Lemma 3: If ki,b > 1, ki,c = 0 andδb,c = 1 for any playeri, thenS∗ is not a NE.

In the example presented in Figure 1, the conditions ofLemma 3 hold for playerp3 and the channelsb = c2 andc = c3. Hence, the example cannot be a NE. The proof of thelemma is similar to the previous proof and hence we omit it.

Suppose now that we have two channelsb and c such thatkb = kc (which is equivalent toδb,c = 0), that means thatthe total number of radios is the same on the two channels.Assume that we have a playeri with ki,b > ki,c > 0. Let usdefine the integer valueγi,b,c as:

γi,b,c = ki,b − ki,c (9)

Using the valueγi,b,c introduced above, we can derive thefourth necessary condition. The lemma shows that if thereexist these two channels with an equal number of radios anda user has no radio on one of them and more than one radioon the other, than he should reallocate one of his radios to thechannel, in which he is not present yet.

Lemma 4: If γi,b,c ≥ 2, ki,c = 0 and δb,c = 0 for anyplayer i, thenS∗ is not a NE.

In Figure 1, the conditions of Lemma 3 hold for playerp3

and the channelsb = c2 and c = c4. Thus, this is not a NE.The proof of the lemma is similar to the proof of the previouslemmas.

Let us now consider a channel allocationS and let us dividethe channels into three sets. We define the set of channelsCmax with the maximum number of radios, i.e., whereb ∈Cmax has kb = maxl∈C kl. Similarly, let us define the setof the least occupied channelsCmin, where c ∈ Cmin haskc = minl∈C kl. We denote the set of the remaining channelsby Crem. In Figure 1,Cmax = {c1}, Cmin = {c5, c6} andCrem = {c2, c3, c4}.

Using Lemmas 1, 2, 3 and 4, we conclude on a fifthnecessary condition. This shows that in a Nash equilibrium,the difference in the total number of radios between any twochannels cannot exceed one.

Proposition 1: In a NE S∗ in the multi-radio channelallocation game, we haveδb,c ≤ 1 for all b, c ∈ C.Basically, in the proof we divide the possible scenarios intogroup and show that either of the lemmas apply to themembers of these groups.

Combining our results so far, we can establish a set ofnecessary and sufficient conditions for the NE.

Theorem 1:Assume that we have|N | · k > |C|. Then achannel allocationS∗ is a NE iff the two following conditionshold:• δb,c ≤ 1 for any b, c ∈ C and• ki,c ≤ 1 for any b, c ∈ C and i ∈ N except forplayers

j with @c ∈ Cmin such thatkj,c = 0. For such a playerj, the second condition changes as follows:kj,c ≤ 1 ifc ∈ Cmax andγi,a,c ≤ 1 for any channela, c ∈ Cmin.

An example of a NE channel allocation is shown in Figure 4,where the second condition of the theorem has an exceptionfor playerp1. Figure 5 presents an example with no exceptionon the second condition for any player.

channels

c1 c2 c3 c4 c5

p1

p1

p1 p2 p2p2

p6

p3 p3p3

p4p4 p4

p5

p5p5

p6p6

c6

p1

p2 p3

p4

p5 p6

p7 p7 p7 p7

Fig. 4. An example for a NE channel allocation. Here|C| = 6, |N | = 7and k = 4. Note that the second condition of Theorem 1 has an exceptionfor playerp1.

channelsc1 c2 c3 c4 c5

p1 p1p1

p2

p2

p2 p3

p3

p3

p4

p4 p4

c6

p1

p2 p3

p4

Fig. 5. A NE channel allocation with no exception of the second conditionof Theorem 1. Here|C| = 6, |N | = 4 andk = 4.

Proof: Let us show that the above conditions are nec-essary. Proposition 1 established that the first condition isnecessary. Lemmas 2, 3 and 4 make the second conditionnecessary.

Now we prove that these conditions are sufficient as well.According to the first condition of the theorem, the differencebetween the number of radios on any two channels cannotbe more than one, thus the setCrem does not exist. Thisalso means that any change consists in moving some radiosfrom channels inCmax to channels inCmin. Consequently,the moves can be considered separately.

Let us thus consider the moving of one radio from a channelb ∈ Cmax to a channelc ∈ Cmin which results in a strategy

s′i for player i. Substitutingγi,b,c = ki,b − ki,c ≤ 1, we can

write the benefit of change expressed in (8) as follows:

∆ =ki,c + γi,b,c − 1

kcR(kc) +

ki,c + 1kc + 1

R(kc + 1)

− ki,c + γi,b,c

kc + 1R(kc + 1)− ki,c

kcR(kc)

= (γi,b,c − 1)(

R(kc)kc

− R(kc + 1)kc + 1

)

Note that second factor is always positive and hence thedifference is non-positive forγi,b,c ≤ 1. For this value, thestrategy matrixS∗ defines a NE.

Theorem 1 establishes an interesting property about NE:In fact, all NE channel allocations achieve load-balancingover the channels inC. In the next theorem, we will showthat allowing selfish channel allocation over a wide band offrequencies results in an efficient spectrum utilization.

Theorem 2:Assume that we have|N | · k > |C|. Then anyNE channel allocationS∗ is Pareto-optimal.

Proof: The proof is straightforward. In a NE channelallocationS∗ we havekc > 0 for eachc ∈ C. Note that inS∗, the sum of the utility of all playersUtotal = maxi

∑i Ui

which implies both Pareto- and system-optimality.Usually, there is a way to improve the efficiency of the

non-cooperative NE solution using a cooperative solution,e.g., the Nash bargaining framework. Note that in this case,as shown in the proof of the theorem, the Pareto-optimalNE channel allocation is also system-optimal. All channelsare fully utilized, and hence the system efficiency cannot befurther improved. This implies that no cooperative solutioncan further increase the system’s efficiency.

V. FAIRNESS ISSUES

In this section, we study thefairness properties of theselfish multi-radio channel allocation game. Fairness is animportant aspect of resource allocation problems in general,and of computer networks in particular. We have seen inSection IV that in the selfish multi-radio channel allocationproblem, the NE solutions are Pareto-optimal and also system-efficient. Unfortunately, these Pareto-optimal allocations mightbe highly unfair by giving advantage to some players whileneglecting others. For example, in the channel allocationpresented in Figure 4 assuming that the rate functionR(·)is constant, playerp1 has the total rateU1 = 19

20 , while playerp4 has the total rateU4 = 16

20 . In order to study the fairnessproperties of the NE channel allocations, we use a particularmetric calledmax-min fairness (MMF)as defined in [4]:

Definition 3: (Max-Min Fairness – MMF): The strategymatrix Smmf is max-min fair if the utility of playeri cannotbe increased without decreasing the utility of another playerj for which Ui(Smmf ) ≥ Uj(Smmf ).

Using this concept, we identify the max-min fair NE channelallocations as expressed in Theorem 3.

Theorem 3:A NE channel allocationS∗ is max-min fairif and only if |Cmin| · kc ≡ 0 (mod |N |). This implies thatUi = Uj , ∀i, j ∈ N .

In other words, if the total number of radios in the leastallocated channels are equal for every player, the NE allocationis max-min fair. In the proof, we will show that the conditionimplies an equal utility for each player.

Proof: First, we will prove that |Cmin| · kc ≡ 0(mod |N |) implies Ui = Uj , ∀i, j ∈ N . Then we will showthat the latter condition implies max-min fairness.

Let us now express the utility of any playeri in a NEallocationS∗. In the equation below,kc denotes the numberof radios that use channelc ∈ Cmin.

Ui =

∑c∈Cmin

ki,c

kcR(kc) +

k −∑c∈Cmin

ki,c

kc + 1R(kc + 1)

=∑

c∈Cmin

ki,c(R(kc)

kc− R(kc + 1)

kc + 1) + k

R(kc + 1)kc + 1

Similarly, we can express the utility of another playerj:

Uj =∑

c∈Cmin

kj,c(R(kc)

kc− R(kc + 1)

kc + 1) + k

R(kc + 1)kc + 1

If |Cmin| ·kc = |N | ·κ, whereκ is an integer, then we havean equal number of radios in the channels inCmin, meaningthat

∑c∈Cmin

ki,c =∑

c∈Cminkj,c = κ and vice versa. If

and only if every player has an equal number of radios in thechannelsc ∈ Cmin, do we haveUi = Uj .

Second, we prove by contradiction that the equality of theutilities is necessary to max-min fairness.

Let us suppose that there exist a max-min fair NE channelallocation S∗ in which Ui < Uj for some i, j ∈ N . Weknow from Lemma 1 thatki = kj = k, and hence we caninterchange the radios of playeri with the radios of playerj.This results inUi > Uj while the utilities of other players donot change. Hence, the original channel allocationS∗ is notmax-min fair. Conversely, ifUi = Uj , this implies max-minfairness by definition.

From this theorem, we can immediately see that the per-fectly balanced channel allocation is also max-min fair.

Corollary 1: The NES∗ in whichCmin = Cmax (i.e.,kb =kc, ∀b, c ∈ C) is max-min fair as well.

We can easily check max-min fairness given the values of|C|, |N | andk.

kc =N · k − |Cmin|

C (10)

Due to the fact that|Cmin| ≤ |C|, we can write that:

kc =⌊N · k

C⌋

(11)

We can also compute|Cmin| as follows:

|Cmin| = N · k − kc · |C| (12)

If |Cmin| = 0, then every channel has the same number ofradios (i.e.,kb = kc, ∀b, c ∈ C). If |Cmin| > 0, then we caneasily verify the condition of Theorem 3.

VI. COALITION -PROOFNASH EQUILIBRIA

The definition of NE expresses the resistance to the devi-ation of a single player. In a realistic situation, it might bepossible that several players collude to increase their payoffat the expense of other players. Such a collusion is called acoalition. The problem of how these coalitions are formed isa separate research topic itself, thus in this paper we assumethat any players can form a coalition. We can generalize thenotion of NE for coalitions as follows.

Definition 4: (Coalition-Proof Nash Equilibrium – CPNE):The strategy matrixScpne defines a coalition-proof Nashequilibrium if theredoes not existany coalitionΓ ∈ N andany strategy of this coalitionS

′Γ such that the following set of

conditions is true:

Ui(S′Γ, Scpne

−Γ ) > Ui(ScpneΓ , Scpne

−Γ ), ∀i ∈ Γ (13)

This means that no coalition can deviate formScpne such thatthe utility of all of its members increases.

The definition of coalition-proof Nash equilibrium is veryrestrictive. We can define this notion in the broader sense:

Definition 5: (Strong Coalition-Proof Nash Equilibrium –SCPNE): The strategy matrixSscpne defines a coalition-proofNash equilibrium if theredoes not existany coalitionΓ ∈ Nand any strategy of this coalitionS

′Γ such that the following

set of conditions is true:

Ui(S′Γ, Sscpne

−Γ ) ≥ Ui(SscpneΓ , Sscpne

−Γ ), ∀i ∈ Γ (14)

with strict inequality for at least one playeri ∈ Γ.This means that no coalition can deviate formScpne such

that the utility of someof its members increases while theutility of other members do not change. From the definition,we can immediately see the following fact:

Fact 2: If in the NE S∗, we haveCmin = Cmax (i.e.,kb =kc, ∀b, c ∈ C), thenS∗ is (strong) coalition-proof.

The intuition is that in a channel allocationS∗ for whichkb = kc, ∀b, c ∈ C, any player that changes necessarilydecreases his utility, henceS∗ is a (strong) coalition-proofNE by definition.

In the remainder of this section, we assume thatCmin 6=Cmax and we derive results that highlight the strong coalition-proof NE. First, we prove a necessary condition that enablesa given NE allocation to be strong coalition-proof in addition.

Theorem 4:If NE channel allocationSscpne is strongcoalition-proof then theredoes not existtwo channelsb ∈Cmax and c ∈ Cmin and two playersi, j ∈ N such thatki,b > 0 andkj,b > 0 while ki,c = 0 andkj,c = 0.

To illustrate the condition of Theorem 4, let us emphasizethat the example shown in Figure 5 is not a strong coalition-proof NE.

We provide an example for a strong coalition-proof NE inFigure 6.

Proof: It is easy to see that if the conditions of thetheorem do not hold, theni and j can form a coalition andone of them (for example playeri) can increase the utilityof the other by moving one radio fromb to c. Hence it is anecessary condition.

channels

p4p3

p2p1

p3

p3

p1p1

p3

p4

c4c3c2c1 c5

p1

p2p2p2

p4

p4

c6

p2

p1

p3 p4

Fig. 6. An example for a strong coalition-proof NE channel allocation, where|C| = 6, |N | = 4 andk = 5.

We could not prove that the set of conditions in Theorem 4is sufficient to establish a strong coalition-proof NE, but wecould not find a counterexample, where the conditions holdand the channel allocation is not strong coalition-proof NE.Hence, we provide the following conjecture.

Conjecture 1: If there does not existtwo channelsb ∈Cmax and c ∈ Cmin and two playersi, j ∈ N such thatki,b > 0 and kj,b > 0 while ki,c = 0 and kj,c = 0 then theNE channel allocationSscpne is strong coalition-proof. Hencethe above condition is a sufficient condition.

Nonetheless, we can show that the set of strong coalition-proof NE channel allocations is a subset of the max-min fairchannel allocations.

To prove this result, we first prove the following lemmas.Due to lack of space, we omit their proof.

Lemma 5:For any NE channel allocationS∗ in whichCmax > 0, we havekc < |N |.

Note that Lemma 5 applies to any NE channel allocation,not only to the the max-min fair NE.

Lemma 6:For any NE channel allocationS in whichCmax > 0 andS is not MMF, we havekc + 1 < |N |.

Using Lemma 6, we can show in the following theorem thatall strong coalition-proof NE channel allocations are max-minfair as well.

Theorem 5:If NE channel allocationS is strong coalition-proof (strictly speaking if the necessary condition expressedin Theorem 4 holds) then it is max-min fair as well.

We again omit the proof of the theorem due to spacelimitations.

As a summary, Figure 7 shows all channel allocations byproperties.

all channel allocations

Pareto-optimal channel allocations

Nash equilibria

Max-min fair Nash equilibria

Strong coalition-proof Nash equilibria

Coalition-proof Nash equilibria

Fig. 7. Summary of channel allocations with different properties.

VII. C ONVERGENCE TO ANASH EQUILIBRIUM

We have demonstrated in Section IV that the non-cooperative behavior of the selfish players lead to Pareto-optimal, load balancing Nash equilibria. In this section, wepropose three different algorithms, each using a different set ofavailable information to enable the selfish players to convergeto one of these Nash equilibria from an arbitrary initial config-uration. The three algorithms are the following: 1) centralizedalgorithm using perfect information, 2) distributed algorithmusing perfect information, and 3) distributed algorithm usingimperfect (local) information, respectively.

A. Centralized Algorithm Using Perfect Information

We have proven in Theorem 1 that a Nash equilibriumchannel allocation has a load-balancing property. In addition,we have shown in Theorem 2 that all the Nash equilibriaare Pareto-optimal as well. Now, let us propose a simplecentralized algorithm to achieve one of these efficient Nashequilibria.

Algorithm 1 Pareto-optimal NE channel allocation with globalcoordination and perfect information

1: for i = 1 to |N | do2: for j = 1 to k do3: if kc = kl, ∀c, l ∈ C then4: use the radio on a channelc, whereki,c = 05: else6: use the radio on a channelc, wherekc = minl∈C kl

7: end if8: end for9: end for

Notice that the algorithm requires the sequential action ofthe players and hence it needs global coordination. In addition,the players have to have a perfect information about thenumber of radios on each of the channels. This can be achievedby the global coordination mentioned before or by havingan extra radio per device for scanning the channels. Globalcoordination is unlikely to exist in a wireless networkingscenario with selfish players. The second assumption aboutperfect information might not hold either, because selfishplayers should allocate all of their radios for communicationas shown in Lemma 1. It is possible to model the cost ofscanning with one radio instead of using it for communication.The investigation of this issue is part of our future work.

B. Distributed Algorithm Using Perfect Information

In order to overcome the limitations of the centralizedalgorithm proposed in Section VII-A, we suggest a secondalgorithm that does not require global coordination, but stillassumes a perfect information about the available channels.

We define around-baseddistributed algorithm that worksas follows. First we assume that there exist a random radioassignment of the players over the channels. For simplicity, weexclude the specific Nash equilibria that result in an exceptionof the second condition of Theorem 1. This means that we

assume that no players allocates more than one device onany channel. After the initial channel assignment, each playerevaluates the number of radios (which defines the approximatelength of the round) oneach of the channelsc ∈ C andtries to improve his total rate by reorganizing his radios.Unfortunately, this procedure might result in a continuousreallocation of the radios for all players. An example for such acontinuous reorganization is shown in Figure 8, where channelc6 is empty and thus each player moves his radio fromc1 toc6. In the next round, the same effect happens and they allmove their radios back fromc6 to c1.

channels

p3

p2

p1

p3

p1

p2

p1

p3

c4c3c2c1 c5

p2

c6

p1

p2 p3

p4 p4 p4

p4

Fig. 8. An example for a channel allocation which results in a continuousreallocation of the radios for all players (i.e., each player moves his radio fromc1 to c5 and back) if there is no randomized backoff mechanism implemented.In this example,|C| = 6, |N | = 4 andk = 4.

To avoid these instable channel allocations, we leverage thetechnique of backoff mechanism well known in the 802.11medium access technology [18]. We define abackoff win-dow W and each player chooses a random initial value forhis backoff counterwith uniform probability from the set{1, . . . ,W}. Then in every round each player decreases hisbackoff counter by one and applies the re-allocation of hisradios only when the backoff counter reaches zero. After thehe changed his channel allocation, he will reset the backoffcounter as described previously. We can notice that usingthe backoff mechanism, the players play a game in a quasi-sequential order.

We provide the pseudo-code for the distributed algorithmdescribed previously in Algorithm 2.

We can prove that Algorithm 2 stabilizes in one of the load-balancing Nash equilibrium channel allocations.

Theorem 6:Algorithm 2 converges to one of the NE chan-nel allocations.

To prove the theorem, let us introduce the notion ofstategraph G. We represent each possible channel allocation asa node inG. We call the NE channel allocations astablestateand the other states astransitional states. The transitionbetween any two states depends only on the set of nodes whichhave their backoff counter equal to zero. Hence, our algorithmhas the Markov-property.

Proof: To prove the theorem, one has to prove twoproperties: 1) The algorithm does not stabilize in a transitionalstate and 2) there exists a path with positive probability fromany transitional state to one of the stable states.

The first property is easy to prove, because in each transi-tional state (non-NE) there exists at least one player, who has

Algorithm 2 Distributed Pareto-optimal NE channel allocationalgorithm using perfect information

1: RandomChannelAllocation()2: while there is changedo3: ChannelUpdate()4: for i = 1 to |N | do5: if backoff counter is0 then6: reorganize the radios ofi in the to maximize the

total rate:7: for j = 1 to k do8: assume that radioj uses channelb9: move the radioj from b to channelcmin if

∃cmin ∈ C, cmin = arg minc kc such thatki,cmin = 0 andkcmin < kb − 1

10: end for11: reset the backoff counter to a new value from the

set{1, . . . , W}12: else13: decrease the backoff counter value by one14: end if15: end for16: end while

the motivation to change. The second property holds as well,because we have a positive probability that only one playerchanges in each round with no repetition of the players untila NE is reached. This special case is exactly the proceduredescribed in Algorithm 1.

C. Distributed Algorithm Using Local Information

The distributed algorithm presented in Section VII-B usedperfect information. This assumption requires that the eitherthe players share their local information about the channelallocation, or that each of them uses a separate radio forscanning the channels he is not using at the moment. Theseassumptions might not hold in a selfish networking context.The first assumption requires that the players collaborate,which might not be their best interest. The second assumptioncontradicts with Lemma 1, which proves that selfish playersshould use all of their radios for communication3.

In this subsection, we assume that players have only alocal information about the channels, on which they operatea radio. In order to improve their performance, they proceedas follows. The players apply the random backoff mechanismwe introduced in Section VII-B. In each round where playeri’s backoff counter is equal to zero, he calculates the averagenumber of devices on the channels he knows (recall that wedenote this set byCi). We denote the average number ofdevices on the channels inCi by mi. For each channelb ∈ Ci

with kb − mi ≥ 1 player i moves his radio with a uniformrandom probability to another channelc /∈ Ci. This is the firstproperty of the algorithm with imperfect information.

3In our future work, we will model the cost of scanning, which might resultin a temporary decrease of the total rate, but with the promise of better totalrate in the future.

Similarly to Theorem 6, one can show that the aboveprocedure reaches a stable state. Unfortunately, the availablelocal information might be insufficient for the players toidentify if the achieved stable state is Nash equilibrium. Weshow an example for such a “false Nash equilibrium” inFigure 9.

channels

p4

p3

p2

p1

p3

p5 p1

p3

p4

c4c3c2c1 c5

p1 p5

p2p2

p4

c6

p5

Fig. 9. An example for a stability state using the distributed algorithm withimperfect information. Each player believes that this is a Nash equilibriumdue to the insufficient local information. Here|C| = 6, |N | = 5 andk = 3.

In order to solve the problem of inefficient stable states,we introduce the following mechanism: playeri checks thenumber of radios for each of the channelsb ∈ Ci as suggestedabove andwith a small probabilityε he moves his radio toanother channelc /∈ Ci even if 0 < kb − mi < 1. Hechooses the new channelc with a uniform random probabilityas presented before. This second property allows us to resolvethe inefficient stability states, but in the same time, it will alsocause the instability of the Nash equilibria.

We provide the description of our algorithm below. Notethat this algorithm now includes both properties: 1) the backoffmechanism and 2) the mechanism to break inefficient stablestates.

Due to the second property of our algorithm, its doesnot perfectly converge to the existing Nash equilibria (moreprecisely, it converges there with high probability, but it doesnot stay in a Nash equilibrium solution). Nevertheless, we canobserve that the algorithm remains in states that are “close”to Nash equilibria in terms of load-balancing. We demonstratethis intuition by the simulations presented in Section VII-D.

D. Simulation Results for Algorithm 3

We implemented Algorithm 3 in MATLAB 7.0.0.1 and witha special focus on wireless 802.11a protocol (meaning that wehave chosen 11 orthogonal channels as a default value forN )In this subsection, we present our simulation results that wewe will investigate the performance of algorithm 3 in terms ofconvergence time and efficiency. In each of the simulations,we assumed a constant rate functionR(·), note however, thatthe algorithm shows similar properties for any decreasing ratefunction introduced in Section III.

Let us first highlight the best and worst case in terms ofthe desired load-balancing. The best case is one of the NEchannel allocations, while the worst case is characterized bythe fact that there existk channels where each of the playershave a radio, while the rest of the channels have no radios atall. In Figure 10, we present the worst case channel allocation

Algorithm 3 Distributed Pareto-optimal NE channel allocationalgorithm using local information

1: RandomChannelAllocation()2: while there is changedo3: ChannelUpdate()4: for i = 1 to |N | do5: if backoff counter is0 then6: if (maxc∈Ci(kc)−minc∈Ci(kc) > 1) then7: for j = 1 to k do8: assume that radioj uses channelb9: if kb > mi then

10: move the radioj from b to c /∈ Ci, wherecis chosen with uniform random probabilityfrom the setC\Ci

11: end if12: end for13: else14: for j = 1 to k do15: assume that radioj uses channelb16: if kb ≥ mi then17: move the radioj from b to c /∈ Ci with

probability ε, wherec is chosen with uni-form random probability from the setC\Ci

18: end if19: end for20: end if21: else22: decrease the backoff counter value by one23: end if24: end for25: end while

that is opposed to the best case NE in Figure 5 and we referto it asunbalanced (UB).

channels

p4

p3

p2

p1

p3 p3

p1p1

p3

p4

c4c3c2c1 c5

p1

p2p2p2

p4p4

c6

Fig. 10. An example for a worst case channel allocation that is completelyunbalanced, as opposed to a NE (best case) shown in Figure 5. Here|C| = 6,|N | = 4 andk = 4.

We calculate the average number of radios per channel asm = |N |·k

|C| . We can compare the utilization of every channelc to the average to achieve the total balance of the channelallocationS:

Definition 6: (Balance:) Thebalanceβ of a channel allo-cationS is defined as the sumβ(S) =

∑c∈|C| |kc −m|.

The notion of balance allows us to define the efficiency ofa given channel allocation as a proportion between the worstcase and the best case channel allocations.

Definition 7: (Efficiency:) The efficiencyφ of a channelallocationS is defined asφ(S) = β(SUB)−β(S)

β(SUB)−β(SNE) .Let us emphasize that for any channel allocationS, we have

0 ≤ φ(S) ≤ 1. Furthermore,φ(SNE) = 1 andφ(SUB) = 0 asdesired by this measure.

Let us now we define theaverage efficiencyover time andefficiency ratio. To this end, we denote the efficiency in roundt by φ(t, S).

Definition 8: (Average efficiency and efficiency ratio:) Theaverage efficiencyφ at roundT is defined as the sumφ(T ) =∑T

t=1 φ(t, S). From this definition, we derive theefficiency

ratio asΦ = limT→∞φ(T )

T

Note that the efficiency ratio expresses the performance ofthe distributed channel allocation algorithm per round over along period of time. In our simulations, we applied a finitesimulation time, hence we measured the efficiency ratio atT = 10000.

Finally, let us define theconvergence timeof Algorithm 3as follows.

Definition 9: (Convergence Time): We define theconver-gence timeof Algorithm 3, as the time when the channelallocation efficiency first reaches the value of one (i.e., theefficiency of a NE,φ(SNE)).

Suppose that there are a few players in the game and theyuse the 802.11a medium access protocol. We assume that thelength of one round in the updating algorithm is 10ms. Thisduration of one round corresponds roughly to the time untilthese devices all transmit one MAC layer packet, i.e., the timethat the devices can learn about other devices in the channel.

For each simulation result, where we present average val-ues, we have performed 100 simulation runs. As mentionedpreviously, we run each simulation for 10000 rounds thatcorresponds to 100s according to the assumption above. Forthe convergence time simulations, we present our results witha 0.95 confidence level on the mean value.

Let us first present an example run for our distributedalgorithm with imperfect information in Figure 11. One cannotice that the algorithm quickly reaches the NE state andthus the average efficiency converges to one. Also, one canobserve that the system sometimes leaves the NE state due tothe second property (change a radio on a channelc ∈ Cmax

in a stable state with probabilityε), but the system quicklyreturns to a NE.

First, we investigate the effect of the number of radios perdevice on the efficiency ratio (shown in Figure 12a) and on theconvergence time of the algorithm (presented in Figure 12b).We can observe on the figures that the efficiency is basicallyone if there are more than two radios per device. This meansthat the algorithm drives the system into one of the NE stateand it stays in these NE for most of the time. For two radiosper devices, the effect of changing the channel for even oneradio has a significant impact that undermines the stability ofthe NE more easily. Note however, that even in this case, theefficiency ratio is very high (close to the efficiency of the NE).It is also worth to mention that the higher the number of radios

1 2 3 4 5 6 7 8 9 100.5

0.6

0.7

0.8

0.9

1

Time (Sec.)

Effi

cien

cy

EfficiencyAveraged Efficiency

Fig. 11. One simulation run: Efficiency and averaged efficiency vs. timeusing the values|C| = 11, |N | = 10, k = 3 andW = 15.

per device, the more channels the players know. Hence, theincreasing amount information helps their decisions.

Next, we investigate the effect of the number of players,each device having three radios and present our results inFigures 12c and 12d. We can see that our distributed algorithmkeeps the system in an efficient state, although the efficiency isslightly lower for particular values of|N |. The reason is thatour algorithm is designed to lead the players toany possibleNE. Indeed, the number of possible NE depends on the valueof |Cmin| as derived in (12). If|Cmin| = 0, there existsonly one NE, which is perfectly load-balanced. The higherthe absolute value||Cmin| − |Cmax|| = | |C|2 − |Cmin||, thelower the number of possible NE, hence it is more difficultto reach one of them. This explains the higher convergencetime in Figure 12d for|N | = 11, 15, 22. One can notice thatfor |N | = 11, 22, we have|Cmin| = 0, thus the relativelyhigh convergence time. Furthermore, if there exist only a fewNE, then the algorithm is likely to break them due to theproperty caused byε. This results in a lower efficiency forthese values. Let us emphasize that even for these cases, ouralgorithm performs very well resulting in a high efficiencyratio.

In the second set of simulations, we study the effect of thetwo parameters that introduce the randomness to Algorithm 3.First, we show the effect ofε on the efficiency ratio and theconvergence time in Figures 13a and 13b. One can observethat the efficiency ratio is very high, but slightly decreasesas ε increases. The reason is that with a higherε value it ismore likely that the algorithm does not stay in a NE, once ithas reached it. As a tradeoff, for lowε values, the algorithmconverges slower, because it might stabilize in “false stablestates” for a longer time. It is interesting to observe that somevalues of ε achieve a good balance between efficiency andconvergence time, which motivated our choiceε = 0.0001 forthe default value.

Finally, we study the effect of the size of the backoff win-dow in Figures 13c and 13d. Intuitively, efficiency increaseswith the backoff window size, because of the decreasing

2 3 4 50.5

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2 3 4 50

10

20

30

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50

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70

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ec.)

(a) (b)

10 12 14 16 18 20 22 24 250.5

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10 12 14 16 18 20 22 24 260

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20

30

40

50

60

70

80

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Number of Devices

Con

verg

ence

Tim

e (S

ec.)

(c) (d)

Fig. 12. The effect of the total number of radios: (a) The efficiency ratio and (b) the convergence time as a function of the number of radios per devicek.Similarly, we show (c) the efficiency ratio and (d) the convergence time as a function of the number of players|N |. The simulation parameters are|C| = 11,ε = 10−4 andW = 15. In addition, we used the following default values|N | = 10 andk = 3, where they did not correspond to the measured parameter.

number of simultaneous channel changes. Interestingly, thisincrease is very rapid and the algorithm is very efficient forquite small backoff window values. With a larger backoffwindow, there is a high chance that our algorithm realizesa sequential procedure similar to the centralized algorithmwith perfect information described in Algorithm 1. Notehowever that setting a very high backoff window value isnot reasonable, because it makes the players waiting for anunnecessary long time. Due to the same reason, convergencetime drops quickly as the backoff window value increases.

In summary, we can observe that, in spite of the factthat convergence is not theoretically ensured, the proposeddistributed algorithm based on imperfect information ensureshigh system performance and good convergence time.

VIII. C ONCLUSION

In this paper, we have considered the problem of compet-itive channel allocation among devices using multiple radios.Our main conclusion is that in spite of the non-cooperativebehavior of such devices, their Nash equilibrium channelallocations result in a Pareto-optimal solution, which meansthat they are system-efficient as well. These Nash equilibria arecharacterized by the fact that the devices occupy the available

channels almost evenly. We have also studied fairness andcoalitions in the selfish context. Finally, we have providedthree algorithms to achieve this efficient Nash equilibriumchannel allocation and we study their convergence propertieseither theoretically or numerically.

In terms of future work, we will pursue our theoreticalinvestigations of selfish multi-radio channel allocation in mul-tiple collision domains. We will pay a particular attention tothe application of study of existing fairness metrics in thecompetitive context. In addition, we will take the cost ofchannel scanning into consideration. Last but not least, wewill study the convergence properties of our algorithms thatachieve Nash equilibria in the single collision domain and wewill extend them (or design new algorithms) to the generalcase of multiple collision domains.

REFERENCES

[1] A. Adya, P. Bahl, J. Padhye, A. Wolman, and L. Zhou. A multi-radiounification protocol for IEEE 802.11 wireless networks. InProc. ofBroadnets’04, pages 344–354, 2004.

[2] I. F. Akyildiz, X. Wang, and W. Wang. Wireless mesh networks: Asurvey.Computer Networks, Elsevier North-Holland, Inc., 47:445–487,March 2005.

1E−8 1E−7 1E−6 1E−5 1E−4 1E−3 1E−2 1E−10.5

0.6

0.7

0.8

0.9

1

Epsilon Probability

Effi

cien

cy R

atio

0 1E−8 1E−7 1E−6 1E−5 1E−4 1E−3 1E−2 1E−10

5

10

15

20

25

30

35

40

45

50

Epsilon Probability

Con

verg

ence

Tim

e (S

ec.)

(a) (b)

0 5 10 15 20 25 30 35 40 450.5

0.6

0.7

0.8

0.9

1

Backoff Window Size

Effi

cien

cy R

atio

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

50

Backoff Window Size

Con

verg

ence

Tim

e (S

ec.)

(c) (d)

Fig. 13. The effect of randomness parameters: (a) Efficiency ratio and (b) convergence time as a function ofε. Furthermore, we present (c) the efficiencyratio and (d) the convergence time as a function of the backoff window sizeW . The simulation parameters are|C| = 11, |N | = 10 andk = 3. In addition,we used the following default valuesε = 10−4 andW = 15, where they did not correspond to the measured parameter.

[3] M. Alicherry, R. Bhatia, and L. (Erran) Li. Joint channel assignmentand routing for throughput optimization in multi-radio wireless meshnetworks. InProc. of Mobicom’05, pages 58–72, Aug. 28–Sep. 2 2002.

[4] D. Bertsekas and R. Gallager.Data Networks. Prentice-Hall, 2nd ed.,1992.

[5] G. Bianchi. Performance analysis of the IEEE 802.11 distributed coor-dination function.IEEE Journal on Selected Areas in Communication,18(3), March 2000.

[6] M. M. L. Cheng and J. C. I. Chuang. Performance evaluation ofdistributed measurement-based dynamic channel assignment in localwireless communications.IEEE Journal on Selected Areas in Com-munications, 14(4):698–710, May 1996.

[7] D. C. Cox and D. O. Reudink. Dynamic channel assignment in twodimension large-scale mobile radio systems.Bell Sys. Tech., 51:1611–1628, 1972.

[8] M. Cagalj S. Ganeriwal, I. Aad, and J.-P. Hubaux. On selfish behaviorin CSMA/CA networks. InProceedings of the IEEE Conference onComputer Communications (INFOCOM ’05), March 13–17 2005.

[9] M. M. Halldorsson, J. Y. Halpern, L. (Erran) Li, and V. S. Mirrokni.On spectrum sharing games. InProceedings of the 23rd annual ACMsymposium on Principles of Distributed Computing (PODC 2004), pages107–114, July 25–28 2004.

[10] S. Haykin. Cognitive radio: Brain-empowered wireless communications.IEEE Journal on Selected Areas in Communications (JSAC), 23(2):201–220, Feb. 2005.

[11] I. Katzela and M. Naghshineh. Channel assignment schemes for cellularmobile telecommunication systems: a comprehensive survey.IEEEPersonal Communications, 3(3):10–31, Jun 1996.

[12] J. Konorski. Multiple access in ad-hoc wireless LANs with noncoop-

erative stations. InProceedings of Networking 2002, pages 1141–1146,2002.

[13] A. B. MacKenzie and S. B. Wicker. Stability of multipacket slotted alohawith selfish users and perfect information. InProceedings of the IEEEConference on Computer Communications (INFOCOM ’03), March 30– April 3 2003.

[14] A. Mishra, S. Banerjee, and W. Arbaugh. Weighted coloring based chan-nel assignment for WLANs.Mobile Computing and CommunicationsReview (MC2R), 9(3), 2005.

[15] N. Nie and C. Comaniciu. Adaptive channel allocation spectrumetiquette for cognitive radio networks. InProceedings of IEEE DynamicSpectrum Access Networks (DySPAN’05), Nov. 2005.

[16] A. Raniwala and T.-C. Chiueh. Architecture and algorithms for an IEEE802.11-based multi-channel wireless mesh network. InProceedings ofthe IEEE Conference on Computer Communications (INFOCOM ’05),March 13–17 2005.

[17] T. S. Rappaport. Wireless Communications: Principles and Practice(2nd Edition). Prentice Hall, 2002.

[18] M. Schwartz.Mobile Wireless Communications. Cambridge Univ. Press,2005.

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

[20] J. Zander and H. Eriksson. Asymptotic bounds on the performance ofa class of dynamic channels assignment algorithms.IEEE Journal onSelected Areas in Communications (JSAC), 11(6):926–933, Aug 1993.