Abstract— Orthogonal frequency division multiple access (OFDMA) systems emerge as potential choices for the development of the next generation (4G) mobile communications system mainly due to their bandwidth efficiency and capability of inter symbol interference (ISI) combat. Recently, dynamic resource allocation (DRA) technique is intensively studied to enhance system performance such as throughput maximization, power minimization, fairness and so forth. However, tradeoffs between maximal throughput and fairness get more attention since it adapts to varying wireless network conditions as well as provides reasonable fairness among users. In this paper, we focus on achieving tradeoffs in the downlink case. A comparison between several DRA schemes is given to illustrate how to get tradeoffs in a downlink OFDMA system.

I. I NTRODUCTION Orthogonal frequency division multiple access (OFDMA) systems are emerging as potential choices for the next generation radio interface in mobile wireless networks because of their robustness to combat inter-symbol interference (ISI) and high bandwidth efficiency [1]. In a OFDMA system, high rate data can be transmitted simultaneously in a bundle on a number of lower data rate streams on orthogonal subcarriers. Also, there are diverse channel patterns between the Base-Station (BS) and users. The probability that all users experience a deep fade in a particular subcarrier is very low. Thus, dynamic resource allocation (DRA), i.e. assigning subcarriers and power to different users, plays a very important role in making further efficient use of scarce radio resource and thus maximizes system performance [2], [3]. Most of the previous approaches deal with maximization of system throughput or minimization of the total transmitted power under user quality-of-service (QoS) constraints. Solutions of these approaches benefit the users closer to the base station or with a higher power capability. Another common problem in OFDMA systems is starvation of getting services due to the lack of system fairness. Users who have constantly good channels will be provided more chance while others who have constantly bad channels will have less chance to be served. However, schemes which provide quasi-perfect system fairness such as the max-min approach by Kelly in [9] deal with the worst case of the system, penalizing users with better condition and reducing the system efficiency. Consequently, tradeoffs between maximal throughput and fairness becomes the most important issue in OFDMA systems [5], [7]. It can be obtained by a proportional fairness (PF) scheme since a resource allocation is fair in proportion to user conditions [4], [7].

Proportional Fairness Definition (see also in [4] and [7]): A scheduler P is a PF if the sum of any relative change of a feasible resource allocation vector uk of user k is not positive, or mathematically ∆P =

K X uk (S) − uk (P ) k=1

uk (P )

≤ 0,

(1)

where uk (S) is user k’s data rate (Rk ) or average data rate (Rkavg ) by any feasible scheduler S. The physical meaning of proportional fairness is that any positive change of a user in allocation must result in a negative average change for a system [7]. Another interpretation is that a resource allocation is fair if it is in proportion to the user conditions. In [4], the authors has proved that a PF should maximize the sum of logarithmic of uk which can be formulated as follows P F = arg max

K X

ln uk .

(2)

k=1

The PF scheme proposed by Qualcomm, well known as the PF scheduling, in high data rate (HDR) (single-carrier) systems is to assign one user taking turn to transmit at each epoch decision time [8]. The user k ∗ is selected according to k ∗ = arg max k

Rk Rkavg

(3)

However, in downlink multicarrier systems, a PF is a two dimensional, power and subcarrier, allocation scheme which finds best assignments of subcarriers and power to different users in order to maximize P F at each epoch decision time. In addition, uk includes Shannon capacity term which results in the fact that PF contains a non-linear term ln(ln(x)) which is unable to convert into linear problem even if fixed power is assumed to be allocated to each subcarrier. Therefore, PF is NP-hard which requires large complexity to get optimum solutions. A PF scheme proposed in [7] requires an exhaustive search with exponential complexity which is impossible to implement when the number of users or the number of subcarrier increases. In this paper, we introduce two different reducedcomplexity PF schemes to achieve tradeoffs between maximal throughput and fairness in an OFDMA system. The performance comparison of these schemes will be also given.

Channel State Information TX AMC 1 User 1, R1 User 2, R2

Subcarrier and bit allocation

AMC 2 Add cyclic prefix

IFFT AMC K

User K, RK

RX

ADMC 1 Remove cyclic prefix

ADMC 2 Extract bits for user k

FFT

User k, Rk

ADMC K

Subcarrier and bit allocation information Fig. 1.

Downlink configuration of an OFDMA system with subcarrier and bit allocation

The rest of paper is organized as follows: In section II, system model is described and problem formulation is introduced. Two PF schemes are given in section III. The evaluation of the system performance and discussions on numerical results are drawn in section IV. Concluding remarks are provided in section V. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model The system under considerations is downlink multiuser OFDMA shown in Fig. 1 (see also in [2]). The complex symbols at the output of the modulators are transformed into the time domain samples by inverse fast Fourier transform (IFFT). Cyclic extension of the time domain samples, known as the guard interval, is then added to ensure orthogonality between the subcarriers, provided that the maximum time dispersion is less than the guard interval. The transmit signal is then passed through different frequency selective fading channels to different users. At the receiver, the guard interval is removed to eliminate the ISI, and the time samples of user k are transformed by the FFT block into modulated symbols. The bit allocation information is used to configure the demodulators while the subcarrier allocation information is used to extract the demodulated bits from the subcarriers assigned to user k. To support user QoS, Bit-Error-Rate (BER) and minimum required rate must be satisfied. In addition, the complexity of subcarrier-and-bit allocation algorithm should be fast enough to assign resources within the period of OFDM symbol. Quantitatively, we assume that there are K users moving in a cell, sharing N subcarriers. Each subcarrier is slowly time-varying, perfectly orthogonal to one another, narrowband channel, and has the same bandwidth of W . Channel state

information is perfectly estimated at both the transmitter and receiver ends. A subcarrier-and-power-allocation scheduler is performed at the BS. Each user can be assigned a set of the subcarriers (a subcarrier set) to transmit data, and each subcarrier is assigned exclusively to one another at any given time. Let gk,n be the signal-to-noise ratio (SNR), pk,n be the power of user k on subcarrier n. The channel capacity of user k on subcarrier n with BER is given by rk,n = f (BER, gk,n , pk,n ),

(4)

where f (.) is a nonlinear function that depends on the type of constellation used. Define an assignment indicator variable wk,n = 1 if subcarrier n is assigned to user k, and wk,n = 0 otherwise. Instantaneous data rate of user k can be written as Rk =

N X

wk,n rk,n .

(5)

n=1

The average data rate, or long term throughput of user k at current time instant t is updated based on the following rule ¶ µ 1 1 ∆ Rkavg (t − 1) + Rk , (6) Rkavg = Rkavg (t) = 1 − Tc Tc where Rkavg (t−1) is the average data rate of user k at previous time instant and Tc is the average window size. And system throughput is R=

K X N X k=1 n=1

wk,n rk,n .

(7)

n∗ = arg max rk,n

B. Problem Formulation A PF scheduler P is to find the best assignments for wk,n and pk,n to maximize the sum of logarithmic of uk under total transmitted power constraint PF =

max

wk,n ,pk,n

K X

ln uk

(8)

k=1

where uk can be Rk or Rkavg . subject to K X N X pk,n ≤ PT ,

B. Scheme 2

(9)

k=1 n=1 K X

wk,n ≤ 1 ∀ n,

n∈S

→ wk,n∗ = 1, and S = S\{n∗} Rk = Rk + rk,n∗ if (S = ∅) then break End for End while

(10)

k=1

pk,n ≥ 0, wk,n ≥ 0 ∀ k and ∀ n,

(11)

Rk ≥ Rkmin ∀ k,

(12)

where PT is the total transmit power and Rkmin is the minimum required rate of user k. III. P ROPOSED S CHEMES

In [6], the PF problem is formulated under user data rate consideration, i.e uk = Rk . Using optimization technique, it is claimed that: To maximize PF, subcarrier n should be allocated to user k ∗ by, rk,n k ∗ = arg max , (13) k Rk the power of user k ∗ allocated to subcarrier n is, · ¸+ 1 pk∗ ,n = λk∗ − , gk∗ ,n

(14)

and λk∗ is the water-filling level of user k ∗ is given by µ ¶ X 1 1 Pk + , (15) λk = n∈Sk gk,n |Sk | where [x]+ = max{x, 0}, and |Sk | is the cardinality of Sk .

A. Scheme 1 In [5], the authors proposed an heuristic PF algorithm based on a PF scheduling adopted for an HDR system. The priority of each user to select subcarriers to transmit data is given based on the utility function derived from PF scheduling in an HDR system with the assumption that equal power pk,n = PNT is applied to every subcarrier since it has proved that equal power allocation is negligible in system throughput. Subcarrier allocation process is done by user-by-user basis. The algorithm is divided into two stages. In the first stage, an efficient method is provided to accommodate as many users as possible while guaranteeing user QoS. And in the second stage, remaining subcarriers are further assigned to users whose QoS are satisfied in the first stage to achieve tradeoffs between maximal throughput and fairness. Denote U and S as the sets of users’ index and subcarriers’ index respectively; UA is the set of serviced users’ index (whose QoS is satisfied). The algorithm is elaborated as follows 1. Initialize: U = {1,2,...,K}, UA = ∅, S = {1,2,...,N}, and Rk = 0 n min P Rk and sort descending gk,n ) Ravg 2. Calculate uk = ( N1 n=1

k

3. For each user k from highest to lowest uk n∗ = arg minn abs(Rk + rk,n − Rkmin ) wk,n∗ = 1; S = S\{n∗ } Rk = Rk + rk,n∗ if S = ∅ or Rk ≥ Rkmin then UA = UA ∪ {k} and break End for 4. While (S 6= ∅) For each user k ∈ UA

According to above-mentioned proportion, subcarrier and power allocation are intertwined to each other. Hence, achieving the optimal solution requires large complexity. Moreover, it is also difficult to implement optimal subcarrier allocation described by (13) because the possibility for user k ∗ to obtain subcarrier n depends on other subcarrier n0 6= n which will be assigned simultaneously to user k ∗ . Therefore, a reduced-complexity, sub-optimal algorithm that performs the joint subcarrier and power allocation is proposed by separating subcarrier and power allocation. Subcarrier allocation is performed first with the assumption of equal power to determine subcarrier sets Sk (∀k). Later, water-filling algorithm is implemented individually on each subcarrier set. The algorithm also involves two stages. In the first stage, assume that equal power pk,n = PNT is applied to every subcarrier. First of all, user QoS is guaranteed by the priority down from users having the highest average channel SNR. Later, once subcarrier n is assigned to user k if the following utility function is the largest among all user ∆

uk,n (τ ) =

rk,n rk,n = , ∀k, ∀n, Rk (τ ) Rk (τ − 1) + rk,n

(16)

where Rk (τ − 1) is user k’s data rate at previous decision epoch (τ − 1). The utility function is an increasing function of rk,n . This implies that subcarrier n prefers to be assigned to user k ∗ whose SNR on this channel is the best. However, Rk∗ will be larger and at the next decision epoch (τ + 1), for any n0 6= n, rk∗ ,n0 (17) uk∗ ,n0 (τ + 1) = Rk∗ (τ ) + rk∗ ,n0

will be decreasing. As a result, user k ∗ will have lower priorities to select subcarriers. Therefore, both throughput and fairness are enhanced. In the second stage, the subcarrier set of each user Sk (∀k) is known. We allocate transmitted power to each subcarrier set proportional to the number of subcarriers, Pk = |Sk | PNT . The power allocation of each user on each assigned subcarrier can be calculated by Eqs. (14) and (15). The proposed algorithm is elaborately presented as follows 1. Initialize: Sk = ∅, Rk = 0, ∀k. 2. Step 1: ∀ user k from highest to lowest average SNR do k ) do While (Rk < Rmin ∗ n = arg maxn rk,n ; wk,n∗ = 1, Sk = Sk ∪ {n∗ } Rk = Rk + rk,n∗ 1 λk = λk + PNT + gk,n ∗ End while 3. Step 2: ∀ available subcarrier n = 1 to N do rk,n k ∗ = arg maxk Rk +r ; wk,n∗ = 1, Sk∗ = Sk∗ ∪ {n} k,n Rk∗ = Rk∗ + rk∗ ,n λk∗ = λk∗ + PNT + gk1∗ ,n 4. Power allocation: ∀ subcarrier n = i1+to N : h λk 1 if wk , n = 1, then pk,n = |Sk | − gk,n

0.5km

Fig. 2.

0.5km

User behavior in the center cell of 7-cell environment

IV. P ERFORMANCE EVALUATION In this section, we evaluate the long-term (averaged over all simulation time) system throughput and fairness. We compare the proposed algorithm with maximum throughput performed by water-filling algorithm proposed in [3], quasi-perfect fairness performed by max-min algorithm proposed in [9]. We use the Fairness Index (FI) proposed by Jain et al. [10] to measure system fairness µK ¶2 P xk F I = k=1 (18) ¶, µK P 2 xk K

Fig. 3.

System fairness vs. number of users

given as follows ½ 31.5 + 3.5 log(d), if d > 0.035km , 31.5 + 3.5 log(0.035), if d < 0.035km

(19)

k=1

where xk is the resource portion allocated to user k. In this paper, we define xk = Rk − Rkmin which is a surplus rate to user k. If all users get the same surplus rate, then xk ’s are all equal to let FI be 1, and the system is 100% fair. As the disparity increases, the fairness index decreases to 0. A. Simulation Environment and Parameters Assume that the proposed algorithms are performed at the center cell of 7-cell environment. Consider a single cell (BS) with the radius of 1km. We generated two classes of users by dividing the cell into two zones: inner zone (0 < d ≤ 0.5km) and outer zone (0.5 < d ≤ 1km), where d is the distance from the center (or BS) to a user. And the number of users in each zone is the same, i.e. 50% of the total users. Users are assumed to be uniformly distributed in each zone and their moving speed is uniformly distributed over the range [0-100] km/h. The typical scenario is shown in Fig. 2. The COST 231 Hata suburban propagation model is used for the link gain between BS and users [11]. The pathloss is

where d (in kilometers) is the distance between BS and a user. Shadowing components is lognormal distribution with mean value of 0 and standard deviation of 8dB. Other parameters to be mentioned are: BER of 10−4 , system bandwidth of 10[MHz] the number of subcarriers of 512, carrier frequency of 1.9 [GHz], total transmit power of BS 10[W], average window size Tc of 100, cochannel probability (the probability of existing the same subcarrier n on the neighbor cells) of 90%, number of users varies in [5, 10, 15, 20, 25] and number of trials of 10,000. B. Performance Evaluation Fig. 3 shows the performance of system fairness versus the number of users. In this figure, the Fairness Index of max-min algorithm is almost near 1 for all cases which means that this algorithm achieves quasi-perfect system fairness. However, water-filling algorithm provides low system fairness. Although system fairness of the two schemes is less than that of maxmin, it is much more better than that of water-filling, especially

ACKNOWLEDGMENT This work was supported in part by the Institute of Information Technology Assessment (IITA) through the Ministry of Information and Communication (MIC), Korea. R EFERENCES

Fig. 4.

System throughput vs. number of users

for larger number of users. Particularly, the fairness of scheme 1 is better than that of scheme 2. In Fig. 4, system throughput is shown versus the number of users. The system throughput of the scheme 2 is slightly degraded comparing to maximal throughput and significantly enhanced comparing to scheme 1 and max-min algorithm. Also, the system throughput of scheme 1 is much more better than that of max-min algorithm. In fact, as shown in the figures, it does not need to provide totally fair data rate to all users in the scenario. It is because that we form two classes of users: users in inner zone having more better subcarriers should have more data throughput while others in outer zone having more worse subcarriers should have less data throughput. Two schemes achieve different tradeoffs between maximal throughput and fairness. They all achieve high throughput while provide reasonable fairness among users. However, scheme 1 has better fairness but worse throughput performance compared to scheme 2. Scheme 1 and 2 can be chosen for the demand of high fairness system or high throughput, respectively, depending on the goal of system design. V. C ONCLUSION In this paper, tradeoffs between maximal throughput and fairness in OFDMA systems is mainly addressed which can be achieved by solving PF optimization problem. It can be shown that optimum solution of this optimization problem is NP-hard which requires large complexity. Therefore, to reduce the complexity, two efficient schemes are introduced. The first scheme is an heuristic derived from PF scheduling adopted from HDR systems. The second one is obtained by a suboptimal algorithm that performs a joint subcarrier and power allocation. Both are divided into two stages: In the first stage, to accommodate more users is implemented; and in the second stage, remaining resources are assigned to achieve a tradeoff. Simulation results have shown that both schemes can achieve good tradeoffs in throughput and fairness performance.

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