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Int. J. Electron. Commun. (AEÜ) 65 (2011) 320–330

Contents lists available at ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

journa l homepage: www.e lsev ier .de /aeue

chieving the Nash bargaining solution in OFDMA uplink using distributedcheduling with limited feedback

lias Yaacoub ∗, Zaher Dawyepartment of Electrical and Computer Engineering, American University of Beirut, P.O. Box 11-0236, Beirut, Lebanon

r t i c l e i n f o

rticle history:eceived 8 September 2009ccepted 28 March 2010

a b s t r a c t

The Nash bargaining solution in OFDMA uplink scheduling is investigated. The problem is equivalent toproportional fair scheduling. The logarithm of throughput, when used as a utility function, ensures pro-

eywords:ash bargaining solutionroportional fairFDMA

portional fairness and thus is equivalent to the Nash bargaining solution. An algorithm to implement thesolution in the centralized and distributed scenarios is proposed. In the centralized scheduling scenario,the base station is assumed to enforce the cooperative solution. In the distributed scheduling scenario,cooperation between mobile users is implemented using limited feedback of channel state information.A quantization scheme for channel state information is proposed and shown to achieve results close to

mberesult

plinkcheduling

optimal with a limited nuto achieve near-optimal r

. Introduction

The Nash bargaining problem (NBP) [1] is a well known scenarion game theory. Players in the NBP negotiate to maximize their pay-ffs. The optimal solution of the NBP, the Nash bargaining solutionNBS), consists of distributing the resources in a way to maximizehe product of the payoffs [2]. It was shown that proportional fairPF) scheduling is equivalent to the implementation of the NBS inhe resource allocation of wireless communication systems, theayoff of each user being its throughput [3,4]. PF scheduling isidely investigated in the literature, mainly in the framework of

entralized resource allocation [5–7]. With Orthogonal Frequencyivision Multiple Access (OFDMA) adopted as the accessing schemef next generation cellular systems, e.g., 3GPP Long-Term EvolutionLTE) and mobile WiMAX (IEEE 802.16e), several applications of PFo OFDMA were studied [8–10].

Conversely to centralized resource allocation, mobile users haveore autonomy in making transmission decisions in distributed

chemes. The benefits of distributed resource allocation are beingidely investigated in the context of ad-hoc networks, relay-based

etworks, and sensor networks [11–13], in addition to wireless

ocal area networks (WLANs) [14,15] and cognitive radio (CR)etworks [16–19]. In this paper, we propose a distributed PFcheduling approach that leads to results close to the centralizedase. The PF solution is desirable in distributed scenarios since it is

∗ Corresponding author.E-mail addresses: eey00@aub.edu.lb (E. Yaacoub), zd03@aub.edu.lb (Z. Dawy).

434-8411/$ – see front matter © 2010 Elsevier GmbH. All rights reserved.oi:10.1016/j.aeue.2010.03.007

of feedback bits. In fact, only one bit feedback per subcarrier was sufficients with proportional fair scheduling.

© 2010 Elsevier GmbH. All rights reserved.

equivalent to the NBS, and hence it is in the benefit of the users tocooperate in order to implement it.

The main contribution of this paper is the investigation of col-laborative distributed scheduling in OFDMA uplink and proposinga distributed scheduling scheme that can be easily implementedby the users with a limited exchange of channel state information(CSI). A CSI quantization technique is proposed in order to achievea near optimal performance with one bit feedback per subcarrier.Other contributions of the paper include proposing a low complex-ity scheduling algorithm by extending an algorithm proposed bythe authors in [20] for the full CSI case so that it can be implementedby the users in a distributed collaborative scenario with limitedfeedback. Furthermore, the algorithm is shown to be efficientlyapplicable with various utility functions in both the centralized anddistributed scenarios.

The paper is organized as follows. An overview of proportionalfair scheduling and its equivalence to the Nash bargaining solu-tion are presented in Section 2. The proposed scheduling algorithmis described in Section 3. The application of the algorithm to cen-tralized scheduling is presented in Section 4, and its applicationto distributed scheduling is discussed in Section 5. The simulationresults are presented and analyzed in Section 6. Section 7 presentsa comparison of the proposed approach to the existing literature.Finally, Section 8 concludes the paper.

2. Proportional fairness and the Nash bargaining solution

2.1. PF scheduling methods

In [21], proportional fairness was defined as follows: a feasiblerate vector R is proportional fair if for any other feasible rate vector

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∗, the aggregate of proportional changes satisfies

K

k=1

R∗k

− Rk

Rk≤ 0 (1)

here the summation is over the number of users K .In CDMA systems, proportional fairness consists of allocating

ll the available resources to a single user for a given transmissionime interval (TTI). The user k∗ selected is the one satisfying [22]

∗ = argmaxkR(n)k

Dtot(2)

ith R(n)k

the rate achievable at TTI n and Dtot the total achievedhroughput in a previous time window of fixed duration, e.g., theast 1000 TTIs. This approach is known as proportional fairness inime (PFT).

Applying the previous scheme in OFDMA consists of allocatingll the subcarriers to a single user at a given scheduling interval. Thisolution is inefficient, since subcarrier allocation plays an importantole in the scheduling process in OFDMA. Hence, at a given TTI n,2) is applied on a subcarrier basis, i.e., subcarrier i is allocated toser k∗ satisfying [9]

∗ =argmaxkR(n)

i,k

Dtot(3)

ith R(n)i,k

the rate achievable by user k over subcarrier i at TTI n. Aore accurate approach would be to allocate subcarrier i to user k∗

atisfying [9]

∗ = argmaxk

R(n)i,k

Dtot +∑

j ∈ I(n)sub,k

,j /= i

R(n)j,k

T(4)

ith I(n)sub,k the set of subcarriers already allocated to user k during

he scheduling process for TTI n, and T is the time duration of aTI. The difference between (3) and (4) is that in (4), the rates onhe subcarriers already allocated to a user at the current schedul-ng interval are taken into account before allocating the remainingubcarriers at the same interval.

Scheduling according to (3) and (4) is known as proportionalairness in time and frequency (PFTF) since it involves frequencycheduling on a per subcarrier basis while taking into account thechieved data rate at previous TTIs.

Considering, in (4), the current TTI only and neglecting thehroughput achieved from previous allocations, then subcarrier is allocated to user k∗ satisfying [23]

∗ = argmaxk

R(n)i,k∑

j ∈ I(n)sub,k

,j /= i

R(n)j,k

T(5)

he approach of (5) is referred to as proportional fairness in fre-uency (PFF) since the time dependence is neglected and only therequency dimension is used in the resource allocation process.n the sequel, the superscript (n) will be dropped to simplify theotations when no confusion can occur.

It was shown in [21] that maximizing the logarithmic utility, i.e.,

axK∑

ln(R ) (6)

k=1

k

chieves proportional fairness. This result is widely used in the lit-rature, e.g., [6–8]. Using the achievable rate at each schedulingnstant in (6) achieves PFF, e.g., [7], whereas including the previous

mmun. (AEÜ) 65 (2011) 320–330 321

scheduling instants by using the average rate achieves PFTF, e.g.,[10].

It should be noted that PFF allocates at least one subcarrier toeach user as long as the number of users does not exceed the num-ber of subcarriers. In fact, it is clear that to avoid a −∞ value in (6),no user should have a zero throughput. Hence, letting K be the num-ber of users and M the number of subcarriers, PFF allocates at leastone subcarrier to each user when K ≤ M. When K = M, exactly onesubcarrier is allocated to each user. However, when K > M, the Ksubcarriers are allocated to the K users having the best channel con-ditions (one for each), and there are no sufficient resources for theremaining users. The case K > M does not represent a problem forPFTF where the time dimension is used in the scheduling process,in addition to the frequency and multiuser diversity dimensions.Hence, the users alternate on the available subcarriers so that theaverage allocation is fair.

2.2. Equivalence of PF and NBS

In this section, in order to confirm the equivalence of PF and NBS,we model the resource allocation problem in the OFDMA uplinkas a bargaining game. We consider that each user is a player (inthis section, both terms user and player are used interchangeably)who wants to maximize its payoff, considered to be its throughput.Cooperation is assumed between players. Consequently, playersshould share the resources in an optimal way, i.e., a way they cannotjointly improve on. The resources to be shared are the subcarriersin each TTI. Allocating the shared resources in a way to maximizethe users’ payoffs is equivalent to allocating subcarriers to users ina way to maximize each user’s throughput, given the shares allo-cated to the other users. It is a well known result in game theorythat the solution to the cooperative bargaining problem maximizesthe Nash product NP [2]:

NP =K∏

k=1

(Wk(xk) − Fk) (7)

where xk represents the fraction of resources allocated to playerk, Wk(xk) corresponds to the payoff of player k when xk is allo-cated to it, and Fk is the payoff of player k in the case where noagreement is reached in the bargaining problem. In the OFDMAscheduling problem, the player payoff is the rate achieved, i.e.,Wk(xk) = R(Pk, Isub,k), with Pk the total transmission power of userkand Isub,k the set of subcarriers allocated to user k. In addition,Fk = 0 since no transmission occurs if no agreement on subcarrierallocation is reached. Hence, the optimization problem becomes

maxK∏

k=1

R(Pk, Isub,k) (8)

Since the logarithm is a continuous strictly increasing function,solving the problem in (8) is equivalent to finding the solution ofthe following problem:

ln

(max

(K∏

k=1

R(Pk, Isub,k)

))= max ln

(K∏

k=1

R(Pk, Isub,k)

)

= maxK∑

k=1

ln(

R(Pk, Isub,k)) (9)

Hence, this proves that the logarithm of the throughput, whenused as a utility, ensures proportional fairness, since maximizingit leads to maximizing the Nash product thus leading to the Nashbargaining solution (NBS).

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. Proposed scheduling algorithm

.1. Constrained optimization problem

Letting Isub be the set of subcarriers, Isub,k the set of subcarriersllocated to user k, M the number of subcarriers, K the number ofsers, Pi,k the transmission power of user k over subcarrier i, Pk =

Mi=1Pi,k the instantaneous transmission power of user k, Pk,max its

aximum transmission power, and Rk its achievable throughput,he maximization of the sum of user utilities can be formulated as

axK∑

k=1

U(Rk(Pk, Isub,k)|H) (10)

ubject to

≤ Pk ≤ Pk,max; ∀k = 1,...,K (11)

K

k=1

˛i,k = 1; ∀i = 1,...,Nsub (12)

here U(Rk|H) is the utility of user k as a function of the throughputk given the matrix H of the quantized CSI values Hi,k correspondingo user k over subcarrier i, and ˛i,k = 1 if subcarrier i is allocatedo user k, i.e., i ∈ Isub,k. Otherwise, ˛i,k = 0. Hence, (12) representshe constraint that each subcarrier can be allocated to a single usernly during one TTI.

.2. Scheduling algorithm

In [20], the authors presented a centralized scheduling algo-ithm for the full CSI case. In this section, we present a lowomplexity scheduling algorithm that can be implemented in cen-ralized and distributed scenarios with full or quantized CSI. Thelgorithm is an extension of the work in [20] to the quantized CSIase. It should be noted that the algorithm can be used with anytility function while having the property of ensuring proportionalairness when the utility used is the logarithm of the throughput.he proposed algorithm consists of allocating subcarrier j to user kn a way to maximize the difference

�j,k = U(Rk(Pk, Isub,k ∪ {j})|H)−U(Rk(Pk, Isub,k)|H)

(13)

here the marginal utility, �j,k, represents the gain in the utilityunction U when subcarrier j is allocated to user k, compared to thetility of user k before the allocation of j. The algorithm is describeds follows:

Consider the set of available subcarriers Iavail sub ⊆ {1, 2,...,M}. Atthe start of the algorithm, Iavail sub = {1, 2,...,M}. The algorithmstarts the search with the subcarrier with index 1 in Iavail sub andcontinues till reaching the last subcarrier index. The sorting ofsubcarriers is assumed fixed, e.g., in increasing order of frequency,and known beforehand by the users.Consider the set of users Iusers = {1, 2,...,K} sorted from 1 to Kaccording to IDs allocated sequentially by the BS when each userjoins the network.Step 1: For each user k in the cell, allocate power equally overthe subcarriers in Isub,k ∪ {j}, with subcarrier j the first availablesubcarrier in Iavail sub.

Step 2: Find the user that has the highest marginal utility definedin (13) among all users when subcarrier j is allocated to it. In otherwords, for each subcarrier j, find the user k∗ such that:

k∗ = argmaxk�j,k (14)

mun. (AEÜ) 65 (2011) 320–330

• Step 3: Allocate subcarrier j to user k∗: Isub,k∗ = Isub,k∗ ∪ {j}. Ifmore than one user satisfy (14), allocate subcarrier j to the userhaving the lowest ID in Iusers. This convention is applied insteadof arbitrary allocation, in order to ensure a consistent applicationof the algorithm by all users when the algorithm is implementedin a distributed way. In the centralized case, the BS can break tiesarbitrarily.

• Step 4: For all users k /= k∗, set Pj,k = 0. This step is necessary toreverse the effect of Step 1 for users to which subcarrier j was notallocated.

• Step 5: Delete the subcarrier from the set of available subcarriers:

Iavail sub = Iavail sub − {j} (15)

• Repeat Steps 1–5 until all subcarriers are allocated.

3.2.1. Complexity analysis of the proposed algorithmThe proposed algorithm allocates each subcarrier after perform-

ing a linear search on the users in order to find the user thatmaximizes the marginal utility. Consequently, the total complexityof the algorithm is O(MK), i.e., the algorithm has linear complexityin the number of users and in the number of subcarriers, and thuscould be implemented in real-time.

3.2.2. Properties of the proposed algorithmThe following lemmas highlight some important properties of

the proposed algorithm.

Lemma 1. The proposed algorithm achieves proportional fairness infrequency when the utility considered is the logarithm of the through-put at the current scheduling TTI.

Proof. Let C(n)j

be the throughput of a user at TTIn when a given

subcarrier j is allocated to it and C(n) its throughput at TTI n with-out subcarrier j. When the utility is the throughput, the schedulerallocates j to the user that maximizes the difference (C(n)

j− C(n)),

which is the user having the best channel conditions, i.e., in mostcases the user closest to the BS. On the other hand, when the utilityis the logarithm of the throughput, the scheduler allocates j to theuser that maximizes the difference (ln(C(n)

j) − ln(C(n))), or, equiva-

lently, the ratio C(n)j

/C(n), thus achieving (5) by allocating j to theuser that achieves the largest proportional increase in throughput.�

Lemma 2. The proposed algorithm achieves proportional fairness intime and frequency when the utility considered is the logarithm of theaverage throughput of each user, i.e., when the throughput achievedat previous TTIs is included in the scheduling process.

Proof. The proof is similar to that of Lemma 1. In fact, lettingC = Dtot + C(n) and Cj = Dtot + C(n)

j, the scheduler maximizes the

difference (ln(Cj) − ln(C)), or, equivalently, the ratio Cj/C, whichcorresponds to achieving (3). �

The proof of Lemmas 1 and 2 used the fact that the algorithmmaximizes the marginal utility. However, the use of the differ-ence in (13) is not only applicable to PF. It leads to better resultsregardless of the utility (e.g., sum-throughput utility). In fact, defin-ing the full utility of userk when subcarrierj is allocated to itby U(Rk|Isub,k ∪ {j}), marginal utility is capable of achieving bet-ter performance than full utility. In other words, finding the userk∗

1 that maximizes the marginal utility �j,k leads to a sum-utility

greater than finding the user k∗

2 that maximizes the full utilityU(Rk|Isub,k ∪ {j}). This corresponds to approximating the steepestascent maximization method (the maximization equivalent of thesteepest descent minimization method described, for example, in[24]). In (13), the number of subcarriers is discrete, conversely to

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he steepest ascent method where the gradient is computed overn infinite set of optimization variables.

As an example, we consider the following case: U(Rk∗1|Isub,k∗

1∪

j}) = X − ε, U(Rk∗1|Isub,k∗

1) = X/2, U(Rk∗

2|Isub,k∗

2∪ {j}) = X + ε, and

(Rk∗2|Isub,k∗

2) = X , with X ε. In this case, the proposed algo-

ithm allocates j to k∗1, leading to a sum-utility increase of �j,k∗

1=

/2 − ε. On the other hand, an algorithm maximizing the full util-ty would allocate j to k∗

2 since U(Rk∗2|Isub,k∗

2∪ {j}) = X + ε is greater

han U(Rk∗1|Isub,k∗

1∪ {j}) = X − ε. However, in this case, the increase

n the sum-utility by allocating subcarrier j to user k∗2 is only

(Rk∗2|Isub,k∗

2∪ {j}) − U(Rk∗

2|Isub,k∗

2) = ε. Since X/2 − ε > ε, the max-

mization of marginal utility leads to better results.

.2.3. Practical implementation aspects of the proposed algorithmThe use of logarithmic utilities in the proposed algorithm poses

ertain limitations during the implementation of the algorithm. Inact, when a user has zero throughput, the logarithm of its through-ut will be equal to −∞. Hence, to avoid dealing with −∞ values,e set the throughput of that user to a very low value � instead. Inatlab for example, the default value is � = 2.2 × 10−16. Using, in

he algorithm, ln(R + �) instead of ln(R), we can define the utilityhreshold at zero throughput as Xth = ln(�) = −36.0437.

With PFF scheduling for example, at the start of the algorithm,hen Isub,k = ∅ for all k, the first subcarrier is allocated to the user

1 maximizing:

(Rk(Pk, Isub,k = {1})|H) − Xth

ince Xth is a finite constant, k1 is the user having the highesthroughput on the first subcarrier. When the turn comes for theecond subcarrier j = 2, we have:

�2,k1= U(Rk1

(Pk1, Isub,k1

∪ {j = 2})|H)−U(Rk1

(Pk1, Isub,k1

)|H)

ith Isub,k1= {1}.

For any user k /= k1, we will have �2,k > �2,k1. In fact,

U(Rk1(Pk1

, Isub,k1)|H) < 0, whereas −Xth > 0. Hence, in order to

llocate subcarrier 2 to user k1, we must have �2,k1> �2,k for all

/= k1, or:

2,k1> ln(Rk(Pk, Isub,k = {2})|H) − Xth

ubstituting �2,k1by its expression and Xth by its value, we obtain:

(ln(Rk1(Pk1

, Isub,k = {1, 2})|H) − ln(Rk1(Pk1

, Isub,k = {1})|H)) > 36.04

ince ln(Rk(Pk, Isub,k = {2})|H) > 0. Consequently,

Rk1(Pk1

, Isub,k = {1, 2})Rk1

(Pk1, Isub,k = {1}) > exp(+36.0437)

inally,

Rk1(Pk1

, Isub,k = {1, 2}) > 4.5038 × 1015Rk1(Pk1

, Isub,k = {1})n other words, subcarrier 2 is allocated to k1 if it leads to an increasen its throughput on the order of 4.5038 × 1015 compared to itshroughput when only subcarrier 1 is allocated to it. This result isbviously near to impossible. Hence subcarrier 2 will be allocatedo the user k2 /= k1 maximizing

(Rk(Pk, Isub,k = {2})|H) − Xth

nd the process continues. Consequently, the use of � avoids dealingith ∞ values, while at the same time allowing the algorithm to

chieve PFF scheduling by allocating at least one subcarrier per users long as K ≤ M.

mmun. (AEÜ) 65 (2011) 320–330 323

The same concept applies with PFTF utilities: we use ln(R + �)instead of ln(R) where R is the average throughput (as opposed toinstantaneous throughput for PFF). In addition, for utilities involv-ing division by Dtot, we use (Dtot + �) to avoid division by zero. By areasoning similar to the one presented above, it can be easily shownthat the algorithm allocates at least one subcarrier per user whenDtot = 0 and K ≤ M.

3.3. Throughput calculations

The throughput achieved by userk is expressed as:

R(Pk, Isub,k) =M∑

i=1

˛i,kB

M· log2(1 + ˇ�i,k) (16)

where B is the total bandwidth and ˇ is called the SNR gap. Itindicates the difference between the SNR needed to achieve a cer-tain data transmission rate for a practical M-QAM system andthe theoretical limit (Shannon capacity) [25]. It is given by ˇ =−1.5/ ln(5Pb), where Pb denotes the bit error rate (BER). Each user isassumed to transmit at the maximum power (Pk = Pk,max), and thepower is assumed to be subdivided equally among all the subcar-riers allocated to that user. �i,k is the SNR of userk over subcarrieri. It is given by:

�i,k = Pi,kHi,k

�2i

(17)

where �2i

is the noise power.The power of user k is subdivided equally over the subcarriers

as follows:

Pi,k = Pk

|Isub,k| (18)

Subdividing the power equally over the subcarriers is justified in[26] by the fact that the achieved gains are negligible comparedto the increase in complexity when optimal power allocation isperformed. In addition, it was shown in [27], via simulations, thatoptimal power allocation using water-filling and equal power allo-cation over subcarriers lead to approximately the same results inthe uplink.

For the throughput calculations with quantized CSI in the pro-posed algorithm, the following expression is used:

R(Pk, Isub,k|H) =M∑

i=1

˛i,kB

M· log2(1 + ˇ�i,k) (19)

�i,k is the estimated SNR of user k over subcarrier i in the case ofquantized CSI Hi,k. It is obtained by replacing, in (17)�i,k by �i,k andHi,k by Hi,k.

In the cooperative distributed scenario described in Section 5,users use the expression in (19) while implementing the schedulingalgorithm since they only know the quantized CSI of other users.However, after subcarrier allocation, the throughput achieved byeach user is a function of its actual (not quantized) CSI, given by(16). Conversely, in the centralized case, the BS uses the expressionin (16) while implementing the scheduling algorithm, or, equiva-lently, (19) with Hi,k = Hi,k since the BS is assumed to know theexact CSI values.

4. Centralized scheduling scenario

The NBS in cooperative game theory assumes the presence ofsome authority to enforce the agreement between the differentplayers [2]. In a centralized scheduling scenario, no negotiationstake place between the players. The BS is responsible for making

3 n. Commun. (AEÜ) 65 (2011) 320–330

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24 E. Yaacoub, Z. Dawy / Int. J. Electro

he scheduling decisions. However, since the NBS in a bargainingcenario is equivalent to proportional fairness, it could be assumedhat virtual negotiations take place between the players that reachhe optimal NBS, and that the BS is the authority assumed to enforcehe cooperative solution.

To investigate the performance of the proposed algorithm in theentralized case, we consider the following schemes:

PFF: this corresponds to using, in the algorithm, Uk = ln(R(n)k

). Itcan be easily shown that using this utility in (13) is equivalent toapplying (5).PFTF1: This corresponds to using, in the algorithm, Uk =ln(∑

nR(n)k

). It can easily be shown that using this utility in (13) isequivalent to applying (6) with average throughput.PFTF2: this corresponds to using, in the algorithm,

Uk =

∑j ∈ I(n)

sub,k

R(n)j,k

T

Dtot +∑

j ∈ I(n)sub,k

,j /= i

R(n)j,k

T(20)

It can easily be shown that using this utility in (13) is equivalentto applying (4).PFTF3: this corresponds to using, in the algorithm,

Uk =∑

j ∈ I(n)sub,k

R(n)j,k

T

Dtot(21)

It can easily be shown that using this utility in (13) is equivalentto applying (3).Greedy scheduling: this corresponds to using, in the algorithm,Uk = R(n)

k. This scheme maximizes the sum-throughput and is

unfair to edge users. It is considered for comparison purposes.

PFT is not used since it is known to perform worse than the otherF schemes [9]. PFTF1, PFTF2, and PFTF3 are variants of the samebjective function, with different degrees of accuracy. They are pre-ented to test the performance of the algorithm in the differentcenarios and to select the scheme achieving the best results. Theelected PFTF scheme represents a benchmark to which distributedcheduling results are compared.

. Distributed scheduling scenario

In a game theoretical formulation, it is implicity implied thatsers have more independence, which could be applicable in ourase to a distributed scenario, where users take part in the schedul-ng decisions. In such a scenario, the BS would allocate a subsetf the subcarriers to a group of users that are close enough toommunicate with each other. The users would then perform aargaining game to share the allocated subcarriers, with each userrying to maximize its share. Obviously, the solution to this problems the NBS, which is the optimal (Pareto efficient) solution. The chal-enge in such a scenario is to determine an efficient low-overheadommunication protocol between the users that allows them toxchange the information necessary to perform the schedulingperation (e.g., broadcast of channel state information of each usern each subcarrier).

.1. Cooperative distributed scheduling model

The system studied consists of a single BS covering an area ofimited size. The reduced coverage area is assumed because thesers are required to be in a relatively close proximity so that they

Fig. 1. System model of the distributed collaborative scheduling scenario.

can collaborate by exchanging information. Although in order tobe consistent with the case of centralized scheduling, we will usethe term BS in the discussion of distributed scheduling, a BS canrepresent in practice: a BS serving a small coverage area, a remoteantenna in a distributed BS system, an access point in a local areanetwork, a central controller in a cognitive radio (CR) network, ora femto BS in an indoor scenario. The model is shown in Fig. 1. Thescheduling process takes place as follows:

• The BS indicates to the users the available subcarriers for uplinktransmission, by transmitting appropriate pilot signals on thesesubcarriers. In a cellular system with full reuse, the users aregenerally aware of these subcarriers and use the pilot signals forchannel estimation. In other cases, e.g., in a CR network, the BSindicates to the users the free subcarriers available for transmis-sion in order to avoid interference to primary users.

• Each user estimates its CSI on each of the available subcarriers.The CSI is estimated on the channel between the BS and user. In atime division duplex (TDD) system, the users can determine theiruplink CSI from downlink signals. In a frequency division duplex(FDD) system, the subcarriers used for the uplink are differentfrom those used for the downlink. In this case, each uplink TTIshould start with a pilot transmission phase where the BS sendspilot signals on the subcarriers used for the uplink. This will notinterfere with uplink transmission since the users are forbiddenfrom transmitting during this phase. This method can be used tosynchronize the users with the BS in an FDD system, since usershave to wait for the pilot transmission phase in order to start theuplink scheduling process.

• After calculating the CSI on each subcarrier, each user broadcastsits CSI so that the other users can use it while implementing thescheduling algorithm. Several techniques can be used to performthe CSI broadcast: the broadcast can take place over dedicatedsubcarriers not included in the scheduling process, one for eachuser. Another approach would be to encode the CSI informationusing CDMA and transmit it over a single subcarrier dedicated forthe exchange of CSI by all users. Another solution would be to usea completely different system, e.g., Bluetooth, to exchange the CSI

information between users. In all cases, the exchanged informa-tion should be reduced as much as possible without affecting theefficiency of the scheduling process.

• After receiving the CSI of all other users, each user implements thesame low complexity scheduling algorithm described in Section

on. Commun. (AEÜ) 65 (2011) 320–330 325

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E. Yaacoub, Z. Dawy / Int. J. Electr

3.2. Hence, each user will know which subcarriers are allocatedto it and which are allocated to other users. Finally, each usercommunicates with the BS over its allocated subcarriers until thenext scheduling interval starts.

Although greedy scheduling achieves a higher system through-ut than PF, PF scheduling is more justified when collaborativeistributed resource allocation is performed. In fact, edge userso not have any incentive to cooperate when greedy scheduling ispplied, since they will spend resources in sending feedback infor-ation and then they will be unfavored by the scheduling process.owever, with PF scheduling, they are motivated to cooperate since

hey will be achieving their Pareto optimal throughput, as deter-ined by the Nash bargaining solution. In the above scheme, only

he exchange of the necessary CSI information is needed to imple-ent the low complexity proposed algorithm with quantized CSI.Clearly, for a given number of subcarriers, the complexity of the

lgorithm (of order O(MK) as shown in Section 3.2) and the signal-ng overhead increase with the number of users. A major advantagef the proposed algorithm is that this increase is linear (as opposedo polynomial or exponential complexity). In a centralized schedul-ng scenario where the BS has perfect or near-perfect knowledge ofhe CSI in addition to considerable computational power, the com-lexity of the algorithm does not present significant performance

imitations. Conversely, in a distributed scenario where users haveimited computational power, an increase in the number of userseads to an increase of the CSI signaling information that shoulde received by each user. In addition, the computational powerequired for each user to implement the distributed schedulinglgorithm increases linearly. However, since the scheme is pro-osed for scenarios with reduced coverage areas (so that the usersan collaborate successfully), the number of users accommodatedn such areas will be relatively limited. Hence, the signaling com-lexity/overhead of the proposed distributed collaborative schemeould still be within the practical limitations of the mobile devices.

.2. CSI quantization method

It is essential to have a reduced feedback rate between users.ence, an efficient CSI quantization scheme must be used, sincexchanging full CSI will lead to prohibitive feedback rates. Given Nb

s the number of feedback bits per subcarrier (Nb ≥ 1), and �l andh as the lower and upper allowed CSI values in dB, respectively,he step size (dB) is determined by:

s = (�h − �l)2Nb

(22)

sers exchange the index of each quantized CSI value Hi,k selectedrom the set Hq defined by:

q =[

�l + �s

2, �l + 3�s

2, · · ·, �h − �s

2

](23)

uch that

ˆ i,k = arg minH ∈Hq

(|Hi,k − H|) (24)

n example is shown in Fig. 2. The uniform quantization is per-ormed on the CSI values in dB. This is justified by the fact thatsers are usually assumed to be uniformly distributed over the BSoverage area, and thus their distance to the BS is uniformly dis-ributed. Furthermore, The fading distribution has a mean equal to

he pathloss, which is generally proportional to d−�, i.e., the means proportional to a power of the distance. Hence, its expressionn dB is proportional to the distance (due to the logarithm oper-tion). Consequently, the value of the pathloss in dB is uniformlyistributed in the range [�l�h].

Fig. 2. Proposed quantization method. (a) 1-bit CSI and (b) 2-bit CSI. The symbol“↔” corresponds to the mapping of the quantization bits to the appropriate valueHi,k in Hq .

5.2.1. Logarithmic vs. linear quantizationIt is important to note that uniform quantization in the dB scale

corresponds to logarithmic quantization in the linear scale. Forexample, assuming 1-bit quantization, and considering the typicalvalues discussed in Section 6.3, �l = −124 dBm and �h = −46 dBm,the values Hi,k = �h/2 − ε and Hi,k = �l would both be mapped toa binary 0 with uniform quantization in the linear scale. However,with the proposed scheme, Hi,k = �l would be mapped to a binary 0whereas Hi,k = �h/2 − ε corresponds approximately to �h − 3 dBmin the dB scale and hence would be mapped to a binary 1.

5.2.2. Differences with optimal quantizationThe optimal quantization scheme would be to apply the Lloyd-

Max quantization algorithm described in [28]. However, in thisalgorithm, the boundaries of the quantization intervals and therepresentative value of each interval depend on the channel prob-ability density function (pdf) of each user. Even if all users followthe same fading type (e.g., Rayleigh fading), the mean of the pdfof each user will be different since users will be generally locatedat different distances from the BS. Hence, in order to implementthe Lloyd-Max algorithm in a distributed way, each user needs toknow the pdf of all other users. Furthermore, it needs to perform aseries of integrations to determine different quantization intervalsand quantization values for every other user. In addition, when theposition of a user varies, its pdf will change (due to the change in thedistance and possibly shadowing variance). Consequently, all usersneed to repeat the process for that user. It is needless to say thatsuch a scheme presents a large amount of complexity preventingits dynamic implementation in a distributed way by mobile users.Furthermore, this scheme assumes that a user is able to know itschannel pdf in order to communicate it to other users.

Conversely, the proposed quantization scheme does not requirethe users to know their channel pdf, does not depend on the userpositions, and does not depend on the channel pdf. It gains thisflexibility from the fact that all known channel models depend ona power of the user distances from the BS (e.g., see the channelmodels discussed in [29,30]). Hence, the logarithm of the channelgain depends linearly on the distance. Although it has much lesscomplexity than the optimal Lloyd-Max quantizer, the proposedscheme is sufficient to achieve near optimal performance with alimited number of feedback bits, as shown in Section 6.3.

5.3. Price of anarchy

A unified metric for quantifying the performance of the dis-tributed scheduling schemes achieving the NBS compared to thebest achievable result (centralized scheduling) is desirable. Hence,

to compare the distributed schemes to centralized schemes, theprice of anarchy (PA), a frequently used metric in the literature,is applied. It is usually defined as the ratio of the cost of the sub-optimal case to the cost of the optimal solution, e.g., [31], or asthe ratio of the maximum achievable utility to the utility achieved

3 n. Commun. (AEÜ) 65 (2011) 320–330

bsPstiupt

P

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Fig. 3. Throughput and subcarrier allocation results as a function of the distancefrom the base station: four users and eight subcarriers.

Table 1Scheduling results with different utility functions: four users case.

Method∑K

k=1Rk (Mbps)

∏K

k=1Rk

∑K

k=1ln(Rk)

Greedy 19.247 21.4178 58.3263PFF 13.024 25.9986 58.5201

Centralized scenario 3: the number of users is greater than thenumber of subcarriers. This scenario is shown in Fig. 5 for 16users and eight subcarriers. The sum-utility results are displayedin Table 3.

Table 2Scheduling results with different utility functions: eight users case.

Method∑K

k=1Rk (Mbps)

∏K

k=1Rk

∑K

k=1ln(Rk)

26 E. Yaacoub, Z. Dawy / Int. J. Electro

y the suboptimal solution, e.g., [32]. The definition of [32] is bestuited to the proposed approach. For PF, the utility to be used inA is the logarithm of the throughput, since it is also used in thecheduling algorithm. However, when the schemes are very closeo optimal, the differences are not well expressed due to the slowncrease of the logarithm operation. Hence, we consider the prod-ct of the throughput as an additional metric for the PA, since theroduct is considerably more sensitive to the variations betweenhe schemes. Hence, the two metrics are:

A1 =

∏k

(R(c)k

(Pk, Isub,k))

∏k

(R(d)k

(Pk, Isub,k) − Ok(Isub))(25)

A2 =

∑k

ln(R(c)k

(Pk, Isub,k))

∑k

ln(R(d)k

(Pk, Isub,k) − Ok(Isub))(26)

here R(c) is the throughput in the centralized case, R(d) is thehroughput in the distributed case, and Ok is the overhead in bpsncurred by user k to transmit its CSI over the subcarriers in Isub. Its given by: Ok = MNb/T . As the ratios in (25) and (26) become closeo 1, the performance becomes closer to optimal. The two metricsill have different numerical values, but both will lead to similar

onclusions regarding the performance of the different schemes.

. Results and discussion

In this section, after presenting the simulation model in Section.1, the results of the centralized scheduling scenario with per-ect CSI are discussed in Section 6.2. They constitute a benchmarko which the results of distributed scheduling with quantized CSI,resented in Section 6.3, are compared.

.1. Simulation model

The simulation model consists of a single cell with a BS equippedith an omnidirectional antenna. The throughput is averaged over

00 TTIs, with the duration of a TTI being 1 msec. Then the simula-ion is repeated over 100 iterations. The total bandwidth considereds B = 5 MHz. We consider a target BER of 10−6. The maximum userransmit power is considered to be 125 mW. All users are assumedo transmit at the maximum power, and the power is subdividedqually among all subcarriers allocated to the user. The channelain over subcarrier i corresponding to user k is given by:

i,k,dB = (− − �log10dk) − i,k + 10log10Fi,k (27)

n (27), the first factor captures propagation loss, with a constanthosen to be 128.1 dB, dk the distance in km from user k to theS, and � the path loss exponent, which is set to a value of 3.76.he second factor, i,k, captures log-normal shadowing with an 8B standard deviation, whereas the last factor, Fi,k, corresponds toayleigh fading with a Rayleigh parameter a such that E[a2] = 1.

The fairness of the algorithm is measured by maximizing theroduct of the user throughputs or, equivalently, the sum of the

ogarithms of the user throughputs. Lower values of these met-ics indicate poor fairness, whereas increased values indicate better

airness performance. In fact, assuming a user has zero throughput,he product of the throughputs will be equal to zero and the sum ofhe logarithms of the user throughputs will be equal to −∞. Hence,he value of these metrics is highly affected by the performancef the users that are worst served and thus by the fairness in the

PFTF1 14.763 46.6128 59.1039PFTF2 14.719 45.8719 59.0879PFTF3 14.703 45.9718 59.0901

system. Consequently, these metrics are included in the tabulatedresults of Sections 6.2 and 6.3.

6.2. Centralized scheduling results

To illustrate the performance of the different PF schemes in thecentralized scheduling case, we consider three scheduling scenar-ios:

• Centralized scenario 1: the number of users is less than the num-ber of subcarriers. This scenario is shown in Fig. 3 for four usersand eight subcarriers. The sum-utility results are displayed inTable 1.

• Centralized scenario 2: the number of users is equal to the num-ber of subcarriers. This scenario is shown in Fig. 4 for eight usersand eight subcarriers. The sum-utility results are displayed inTable 2.

•

Greedy 37.273 8.1008 112.6160PFF 18.277 93.0743 115.0575PFTF1 22.908 524.9444 116.7874PFTF2 22.823 509.4447 116.7574PFTF3 22.730 491.8786 116.7223

E. Yaacoub, Z. Dawy / Int. J. Electron. Co

Fig. 4. Throughput and subcarrier allocation results as a function of the distancefrom the base station: eight users and eight subcarriers.

Ff

acBruipo

TS

ig. 5. Throughput and subcarrier allocation results as a function of the distancerom the base station: 16 users and eight subcarriers.

In all scenarios, we consider a cell radius of 1 km. The users aressumed to be located at fixed distances equally spaced from theell center to the cell edge. Perfect CSI estimation is assumed at theS. It should be noted that the number of users and subcarriers iselatively low in order to visualize the performance for the various

ser locations in Figs. 3–5, but the algorithm is easily applicable

n other scenarios. In fact, results from its full CSI version wereresented in [20] for up to 64 users and 300 subcarriers in the casef greedy and PFF scheduling.

able 3cheduling results with different utility functions: 16 users case.

Method∑K

k=1Rk (Mbps)

∏K

k=1Rk

∑K

k=1ln(Rk)

Greedy 55.591 0 –PFF 36.432 29.0824 224.4183PFTF1 27.470 86.0293 225.5029PFTF2 27.335 80.0188 225.4304PFTF3 27.191 72.4726 225.3314

mmun. (AEÜ) 65 (2011) 320–330 327

Figs. 3 and 4 show that the three PFTF schemes outperform thePFF scheme in terms of throughput for all user positions. However,Fig. 5 shows that the throughput of PFF is higher for the nearesteight users. Figs. 3–5 show that all three PFTF schemes tend toallocate on average an approximately equal number of subcarri-ers to each user, and this trend becomes clearer as the number ofusers increases. In this regard, the PFF scheme outperforms the PFTFschemes when the number of subcarriers is greater than or equalto the number of users. In fact, when the number of subcarriersis equal to the number of users, Fig. 4 shows that the PFF schemeallocates exactly one subcarrier to each user, as discussed in Sec-tion 2.1. When the number of users increases, it can be seen fromFig. 5 that the PFF scheme allocates most resources to the nearestusers, since instantaneous subcarrier allocation mandates provid-ing M subcarriers to the M (out of K > M) users having the bestchannel conditions, because the PFF scheme does not keep track ofthe previously achieved data rate of a given user.

From Figs. 3–5, it can be seen that the PFTF schemes have aclose performance, and it is difficult to determine which schemeperforms best. However, from Tables 1–3, it can be concluded thatPFTF1 achieves the best performance in all three scenarios. PFTF2outperforms PFTF3 in the eight and 16 users scenarios, but theyhave an almost equal performance with four users (with a negligiblesuperiority for PFTF3). All three PFTF schemes outperform PFF interms of the PF utilities (product and sum of logarithms) in all cases,but also in the sum-throughput utility when the number of users isless than or equal to the number of subcarriers (Tables 1 and 2). Asthe number of users increases, the performance gap between PFFand the PFTF schemes increases, and the behavior of PFF becomescloser to greedy scheduling, as seen from Table 3. It should be notedthat the throughput used in the product utility is in Mbps in orderto avoid excessively large numbers, whereas the throughput usedin the logarithmic utility is in bps.

6.3. Distributed scheduling results

In the distributed scheduling scenario, we consider a single BSwith users located within a radius of 100 meters from the BS. Thisradius is the largest range that allows the signals transmitted byusers on the cell edge to be within the reference sensitivity level ofusers at the diametrically opposed edge (the reference sensitivitylevel is discussed in [33]). Users are located at fixed distances fromthe BS, such that they are uniformly distributed in the interval [0100] meters. The total bandwidth of 5 MHz is subdivided into 16subcarriers as in [34]. The threshold �l is selected such that Pk10�l/10

corresponds to the sensitivity level in [33], and �h is selected suchthat Pk10�h/10 corresponds to the maximum receiver power in [33],as shown in Table 4.

Due to the low complexity of the proposed algorithm, it can beeasily implemented by the users. However, only the PFF schemeis used in the distributed scheduling scenario, since it is not prac-tical to assume that each user is capable of keeping track of thepreviously achieved throughput of all other users in order to imple-

ment PFTF scheduling. In fact, the users have an estimation of thethroughput of other users according to (19) in order to imple-ment the scheduling algorithm, but only the user knows its actualachieved throughput according to (16). Hence, the only exchanged

Table 4Threshold values.

Receiver sensitivity level −103 dBmMax. receiver power −25 dBmPk,max 21 dBm�l −124 dBm�h −46 dBm

328 E. Yaacoub, Z. Dawy / Int. J. Electron. Commun. (AEÜ) 65 (2011) 320–330

Ff

icoiaig

uscottcrbfv

a(q

TP

TP

ig. 6. Throughput and subcarrier allocation results as a function of the distancerom the base station: distributed scheduling with four users and 16 subcarriers.

nformation is a fixed reduced number of feedback bits per sub-arrier. This requirement is considerably less stringent than manyther contributions in the literature, e.g., in [19], where each users required to know the full CSI on all links between all users overll the subcarriers in order to implement a scheduling algorithmnvolving dual optimization and subgradient computations of Lan-rangian parameters.

The results are displayed in Figs. 6 and 7 for four and eightsers, respectively. The results of the centralized PFF and PFTF1chemes (which achieved the best performance in the centralizedase) are shown for comparison. These figures show that near-ptimal results can be achieved with only one bit feedback. Ashe number of feedback bits increases, the results become closero optimal. However, the very slight increase in throughput indi-ates that one bit feedback can be used efficiently. The near-optimalesults do not justify the investigation of more than three feedbackits per subcarrier. The achieved results with such a low number ofeedback bits are due to the efficient mapping of bits to quantized

alues as described by the approach of Section 5.2.

Tables 5 and 6 show the price of anarchy results for the fournd eight users case, respectively. We use the metrics defined in25) and (26). Since only the PFF scheme was implemented withuantized CSI, a fair comparison should be with the centralized

able 5rice of anarchy results for distributed scheduling: four users case.

Method∑K

k=1Rk (Mbps) 10−4

∏K

k=1Rk

∑K

k=1ln(Rk)

PFTF-Full CSI 70.73 8.9188 66.6605PFF-Full CSI 66.98 6.9784 66.4152PFF-1 bit CSI 64.66 5.8919 66.2460PFF-2 bits CSI 65.52 6.2665 66.3076PFF-3 bits CSI 66.43 6.7171 66.3770

able 6rice of anarchy results for distributed scheduling: eight users case.

Method∑K

k=1Rk (Mbps) 10−7

∏K

k=1Rk

∑K

k=1ln(Rk)

PFTF-Full CSI 80.67 8.7427 128.8104PFF-Full CSI 76.87 6.0392 128.4405PFF-1 bit CSI 74.30 4.3478 128.1119PFF-2 bits CSI 75.06 4.8779 128.2269PFF-3 bits CSI 76.02 5.4874 128.3446

Fig. 7. Throughput and subcarrier allocation results as a function of the distancefrom the base station: distributed scheduling with eight users and 16 subcarriers.

PFF scheme with full CSI. Furthermore, since the logarithmic utilityis used in the algorithm, (26) is more representative of the actualperformance. However, a comparison with the PFTF1 scheme is alsopresented for reference since it represents the long-term optimalNBS. In addition, since the variations of the ln function are smallwhen the performance is close between the different scenarios, (25)is also used in the comparisons. Although different values of theprice of anarchy are yielded by the two metrics, they both lead tothe same conclusions: the quantized CSI schemes perform closer toPFF than PFTF1, and the performance becomes closer to the full CSIcase as the number of feedback bits increases.

Although the results of Figs. 6 and 7 correspond to a comparisonof the different cases with the centralized schemes at each userlocation and hence are more representative than PA1 and PA2, theprice of anarchy results represent a unified metric for quantifyingthe performance of each approach.

7. Comparison to the existing literature

In this paper, several novel contributions were presented:proposing a distributed scheduling algorithm applicable withquantized CSI, applying the algorithm with various utility func-tions achieving proportional fairness, and deriving a distributed

Overhead (kbps) P(PFF)A1 P(PFF)

A2 P(PFTF)A1 P(PFTF)

A2

– – – – –– – – – –16 1.1895 1.0026 1.5202 1.006332 1.1230 1.0017 1.4352 1.005448 1.0518 1.0008 1.3442 1.0045

Overhead (kbps) P(PFF)A1 P(PFF)

A2 P(PFTF)A1 P(PFTF)

A2

– – – – –– – – – –16 1.4095 1.0027 2.0404 1.005632 1.2742 1.0019 1.8446 1.004848 1.1483 1.0011 1.6623 1.0040

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E. Yaacoub, Z. Dawy / Int. J. Electr

esource allocation approach using the algorithm in conjunctionith an efficient CSI quantization scheme. Results illustrated theifferences between PFF and PFTF scheduling. In addition, it washown that distributed scheduling with 1-bit feedback using theroposed quantization scheme leads to near-optimal PF through-ut, although the performance was enhanced when the number ofits increased. Although both centralized and distributed schedul-

ng were investigated, the major contributions of this work areelated to distributed scheduling.

Centralized PF scheduling was studied extensively in the lit-rature, e.g., [5–10]. In this paper, we studied the problem ofistributed PF scheduling where mobile users actively participate

n the scheduling process. Although in [20] we presented a lowomplexity centralized scheduling algorithm for the perfect CSIase, the algorithm in this paper was presented in the general casef quantized CSI (with perfect CSI being a special case), power allo-ation was explicitly included in the algorithm, and the propertiesf the algorithm were discussed (Lemmas 1 and 2). Furthermore,lthough the proposed algorithm was implemented in the dis-ributed scenario of Section 5.1, it also has practical applicationsn relay networks and CR networks.

In relay-based techniques (e.g., [12,13]), certain users carry partf the information destined to other users. For example, in WiMAXesh mode, a user plays the role of a mesh BS with which other

sers communicate, and the mesh BS relays the information to theentral BS [35]. Most works on relaying and multihop schedulingonsider maximizing the achieved throughput, regardless of fair-ess. Hence, the relay sacrifices a portion of its power for the benefitf other users. From a fairness perspective, PF scheduling is opti-al according to the proposed approach. However, the BS in the

istributed scheduling model could play a similar role to the meshS in WiMAX: users perform distributed PF scheduling using theroposed approach and the mesh BS receives the information fromhe users and relays it to a central BS.

In CR networks, secondary users access the spectrum dedicatedo primary users in an opportunistic way, while causing minimumnterference to the primary network. Optimal scheduling of sec-ndary users is an active research topic. Most papers treat theroblem of sensing time (to determine when the primary network

s idle) vs. transmission time [16–18]. In the proposed distributedpproach, the BS could play the role of a central controller in aR network. The role of the CR controller would be to sense thehannel and determine the subcarriers free for transmission thenndicate those subcarriers to secondary users who can perform theroposed distributed scheduling approach. Thus, with the proposedpproach, an important contribution is presented to the CR litera-ure: secondary users would achieve their NBS while the BS ensureshat no interference is caused to the primary network.

In [19], distributed scheduling for cognitive OFDMA radios wasnvestigated. However, weighted sum-rate maximization (greedycheduling) was considered. In [11], utility maximization in generalas treated in a distributed ad-hoc network. Hence, the solutionresented in [11] could be applied to PF scheduling, although theresented results are limited to greedy scheduling. In [36], a greedycheduling scheme was applied in a multicell OFDMA network,here users in each cell implement distributed scheduling withower control used according to a pricing game in order to reduce

nterference. The results of this work are more justified than thosef [11,19,36], since in a distributed scenario PF scheduling achieveshe Pareto efficient NBS solution. The methods of [11,19], and [36]ssume that the users are aware of the perfect CSI of all the links

ver all subcarriers, and that they perform iterative schedulingechniques using subgradients in [19] and pricing-based powerontrol in [11] and [36]. In this paper, we presented a more prac-ical approach via the CSI quantization scheme and showed thatear optimal results can be achieved with only 1-bit feedback. In

[

[

mmun. (AEÜ) 65 (2011) 320–330 329

addition, the scheduling algorithms of [11] and [36] do not explic-itly enforce exclusive subcarrier allocation (i.e., several users maytransmit on the same subcarrier), which may lead to an increasein the interference in the network. The proposed scheduling algo-rithm of Section 3.2 ensures, via the subcarrier allocation variables(˛i,k) that each subcarrier is exclusively allocated to a unique userduring each TTI.

8. Conclusions

Uplink scheduling in OFDMA was considered. The problem wasformulated as a cooperative bargaining problem, where each useraims to maximize its own utility. A Nash bargaining solution wasderived for the resource allocation problem, and a heuristic algo-rithm to implement the solution was proposed. The logarithm ofthroughput, when used as a utility function ensures proportionalfairness, and thus is equivalent to the Nash bargaining solution.

The proposed game theoretical model was applied in boththe centralized and the distributed scenarios. In the centralizedscheduling scenario, the BS was assumed to enforce the coopera-tive solution. However, a game theoretical formulation insinuates amore distributed scenario, where users take part in the schedulingdecisions. In the distributed OFDMA uplink scheduling scenario,cooperation between mobile users was implemented using lim-ited feedback of channel state information. Results close to optimalwere achieved with a reduced number of feedback bits. Further-more, only one bit feedback per subcarrier was sufficient to achievenear-optimal results with proportional fair scheduling.

Acknowledgements

The authors would like to thank the anonymous reviewers fortheir comments that helped in enhancing the quality of the paper.

This work was supported by the American University of Beirut(AUB), the AUB Research Board, Dar Al-Handassah (Shair and Part-ners) Research Fund, and the Rathman (Kadifa) Fund.

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Elias Yaacoub received the B.E. degree in Electrical Engi-neering from the Lebanese University in 2002, and theM.E. degree in Computer and Communications Engineer-ing from the American University of Beirut in 2005. Heworked as a Research Assistant in the American Universityof Beirut from 2004 to 2005, and in the Munich Universityof Technology in Spring 2005. From 2005 to 2007, he wasa Telecommunications Engineer with Dar Al-Handasah,Shair and Partners. He is currently a Ph.D. student atthe American University of Beirut. His research interestsinclude wireless communications and antenna theory.

Zaher Dawy received the B.E. degree in Computer andCommunications Engineering from the American Univer-sity of Beirut in 1998. He received his M.Sc. and Dr.-Ing.degrees in Electrical Engineering from Munich Univer-sity of Technology (TUM) in 2000 and 2004, respectively.Between 1999 and 2000, he worked as a part-time Com-munications Engineer at Siemens AG research labs inMunich focusing on the development of enhancementtechniques for UMTS. At TUM, between 2000 and 2003 hemanaged and developed a research project with SiemensAG where he designed advanced multiuser receiver struc-

ests include Cooperative Communications, Cellular Technologies (WCDMA, HSPA,LTE), Radio Network Planning and Optimization, Multiuser Information Theory,Multimedia over IP Networks, and Computational Biology.