Define channel gain as. Hk,n, total noise power spectral density as Nk,n, and pk,n as ... nications University, Daejon,
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 11, NOVEMBER 2006
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A Proportional Fairness Algorithm with QoS Provision in Downlink OFDMA Systems Tien-Dzung Nguyen and Youngnam Han, Senior Member, IEEE
Abstract— In this letter, we formulate a downlink packet scheduling problem for proportional fairness in orthogonal frequency division multiple access with frequency division multiple access (OFDMA) systems to derive necessary conditions for optimality, which results in efficient subcarrier and power allocation algorithms. Simulation results reveal that our proposed algorithm achieves the tradeoff between system throughput and fairness. Index Terms— OFDMA, proportional fairness, channel allocation, power allocation, scheduling.
allocated power of user k to subcarrier n. Assume that MQAM modulation is applied with a BER requirement, then |Hk,n |2 the signal to noise ratio (SNR) is gk,n = Nk,n Γ , where Γ = −ln(5BER)/1.5 [4]. The capacity of user k on subcarrier n is normalized by rk,n = ln(1 + pk,n gk,n ). An indicator variable is defined as follows: wk,n = 1 if subcarrier n is allocated to user k, and wk,n = 0, otherwise. Instantaneous data rate of user k can be written as
I. I NTRODUCTION
Rk =
P
ACKET scheduling has been one of the most important issues in the context of radio resource management, where proportional fairness (PF) scheduling is considered to achieve tradeoff between system throughput and fairness, i.e., decrease in system throughput while improving fairness among users and vice versa[1], [2]. It is known that a PF problem is NP-hard which requires high complexity. Cooperative game theoretic approach was taken in [2] to solve PF problem in uplink OFDMA systems. In downlink case, however, previous works are mostly heuristics derived from a PF scheduling adopted for single carrier systems such as high data rate (HDR) [3]. In [1], they extended and derived the PF scheduling for multi-carrier systems, without providing an efficient subcarrier and power allocation algorithm. In this letter, we formulate an optimal PF problem as the maximization of the sum of logarithmic user data rate in downlink OFDMA systems with constraints on total transmit power and user data rate. Using Karush-Kuhn-Tucker (KKT) conditions for optimality [7], we propose an efficient subcarrier and power allocation algorithm based on water filling method for the set of subcarriers allocated to each user. It should be mentioned that the same analysis can be applied to the PF problem formulated as maximization of the sum of the logarithmic function of user average data rate. II. P ROBLEM F ORMULATION AND O PTIMALITY
A. Problem Formulation In this section, we provide a problem formulation, from which our algorithm can be derived. Define channel gain as Hk,n , total noise power spectral density as Nk,n , and pk,n as Manuscript received August 25, 2006. The associate editor coordinating the review of this letter and approving it for publication was Prof. Costas Georghiades. This work was supported in part by the Institute of Information Technology Assessment (IITA) through the Ministry of Information and Communication (MIC), Korea. The authors are with the School of Engineering, Information and Communications University, Daejon, Korea (email: {dungnt, ynhan}@icu.ac.kr). Digital Object Identifier 10.1109/LCOMM.2006.060750.
N
wk,n ln(1 + pk,n gk,n ).
(1)
n=1
A PF optimization problem can be formulated as maximizing the sum of logarithmic user data rate: Maximize P F (w, p) =
max
wk,n ,pk,n
K
ln Rk
(2)
k=1
subject to K N
pk,n ≤ PT ,
(3)
k=1 n=1 K
wk,n ≤ 1 ∀ n,
(4)
k=1
pk,n ≥ 0, wk,n ≥ 0 ∀ k and ∀ n,
(5)
Rk ≥ Rkmin ∀ k,
(6)
where w = [wk,n ]K×N , p = [pk,n ]K×N , PT is the total transmit power and Rkmin is the minimum required data rate of user k. And it is assumed that there are N subcarriers and K users in the network. Note that P F (w, p) is neither convex nor concave with respect to (w, p). Although wk,n takes the value of either 0 or 1, it can be relaxed to a real number in [0,1] to make the problem tractable. B. Optimality Conditions Considering PF problem with no data rate constraint (6), we have the following proposition. Proposition (Subcarrier and Power Allocation): To maximize PF(w, p), subcarrier n should be allocated to user k ∗ , where rk,n , (7) k ∗ = arg max k Rk
c 2006 IEEE 1089-7798/06$20.00
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 11, NOVEMBER 2006
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and the power of user k ∗ allocated to subcarrier n is given as, + 1 (8) pk∗ ,n = λk∗ − gk∗ ,n where [x]+ = max{x, 0} and λk∗ is the water-filling level of user k ∗ . Proof : Using Lagrangian, we have N K L(wk,n , pk,n ) = ln wk,n ln(1 + pk,n gk,n ) −λ
k=1 K N k=1 n=1
pk,n − PT
n=1
−
N
µn
n=1
K
wk,n − 1 ,
(9)
k=1
where λ and µn are non-negative Lagrangian multipliers. After differentiating L(w, p) with respect to wk,n and pk,n , one can obtain necessary conditions for optimal solution according to KKT conditions ∂L(w, p) ln(1 + pk,n gk,n ) rk,n = − µn = − µn ≤ 0 (10) ∂wk,n Rk,n Rk ∂L(w, p) wk,n gk,n −λ≤0 = ∂pk,n Rk (1 + pk,n gk,n ) rk,n wk,n − µn = 0 Rk wk,n gk,n −λ =0 pk,n Rk (1 + pk,n gk,n )
(11) (12) (13)
From Eqs. (10) and (12), we have two cases: (1) if subcarrier n is not allocated to user k (wk,n = 0), then rk,n = 0 and rk,n /Rk − µn ≤ 0 and (2) if subcarrier n is allocated to user k (wk,n > 0), then rk,n > 0 and rk,n /Rk − µn = 0. This implies that subcarrier n should be allocated to user k ∗ by Eq. (7). Similarly, from Eqs. (11) and (13), power allocated on subcarrier n to user k ∗ is wk∗ ,n gk∗ ,n − λ = 0. (14) Rk∗ (1 + pk∗ ,n gk∗ ,n ) Following the same implication as in [5], we conclude that the problem (2) always has an optimal solution where wk,n ’s are binary. Thus, wk∗ ,n = 1 and from Eq. (14), 1 1 ∆ + pk∗ ,n = = λk ∗ , gk∗ ,n λRk∗
(15)
Notice that pk,n ≥ 0, implies Eq. (8). III. P ROPOSED A LGORITHM Firstly, we propose a subcarrier allocation algorithm based on Eq. (7). Then, a power allocation algorithm is determined by Eq. (8). And finally, a joint subcarrier and power allocation algorithm is presented to reduce computational complexity. Subcarrier Allocation: To make Eq. (7) tractable, we assume that equal power is distributed to all subcarriers, i.e. pk,n = PT /N for all k and n. At a decision epoch t, if subcarrier n is assigned to user k, then user k’s data rate is updated by Rk (t) = Rk (t − 1) + rk,n ,
(16)
where Rk (t − 1) is user k’s data rate at (t − 1). The right hand side of Eq. (7) becomes ∆
uk,n (t) =
rk,n rk,n = , ∀k, ∀n, Rk (t) Rk (t − 1) + rk,n
(17)
which 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 (t + 1), for any n = n, uk∗ ,n (t + 1) =
rk∗ ,n R (t) + rk∗ ,n k∗
(18)
will be decreasing. As a result, once user k ∗ has been selected, his chance of selecting other subcarriers n (= n) will be decreased, or other users have higher priorities to select subcarriers. Therefore, both throughput and fairness are enhanced. The subcarrier allocation algorithm is divided into two steps: Step 1 : Assign the smallest number of subcarriers to each user to guarantee the minimum required rate. Step 2 : The rest of subcarriers are assigned further to users in the first step by Eq. (7), where Rk can be calculated recursively by Eq. (16). Power Allocation: Assume that Sk is the set of subcarriers allocated to user k. From (8), power allocation to user k on subcarriers in Sk is by water filling with total transmit power Pk = |Sk |PT /N , where |Sk | is the cardinality of Sk . Since λ k = pk,n + 1/gk,n , ∀n ∈ Sk , we have 1 1 ) = Pk + n∈Sk gk,n |Sk |λk = n∈Sk (pk,n + gk,n
1 . ⇒ λk = |S1k | Pk + n∈Sk gk,n Substituting λk into Eq. (8),we can calculate pk,n . Joint subcarrier and power allocation algorithm is then elaborately presented as follows; 1. Initialize: Sk = ∅, Rk = 0, λk = 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 k,n wk∗ ,n = 1, Sk∗ = Sk∗ ∪ {n} Rk∗ = Rk∗ + rk∗ ,n λk∗ = λk∗ + PNT + gk1∗ ,n 4. Power allocation: ∀ subcarrier n = 1 to +N : λk 1 if wk , n = 1, then pk,n = |Sk | − gk,n IV. P ERFORMANCE E VALUATION We compared our proposed algorithm with following algorithms: (1) water filling (WF) algorithm proposed in [5] which provides near optimal system throughput; and (2) max-min algorithm proposed in [6] which provides nearly full system fairness.
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 11, NOVEMBER 2006
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propagation model is used for the channel:
31.5 + 3.5 log(d), if d > 0.035km , 31.5 + 3.5 log(0.035), if d < 0.035km
(20)
Shadowing is assumed to be lognormally distributed with mean 0 dB and standard deviation 8 dB. Other parameters to be mentioned are: BER of 10−4 , minimum required rate of 2[Mbps], modulation constellation of QPSK, system bandwidth of 10[MHz] the number of subcarriers of 512, carrier frequency of 1.9 [GHz], total transmit power of BS 10[W] and noise power of -100[dB]. B. Simulation Results
Fig. 1.
Fig. 1 shows the performance of system fairness with respect to the number of users. In this figure, FI of maxmin algorithm is almost 1 for all cases, which means that this algorithm achieves nearly a perfect fairness. On the contrary, water filling algorithm provides the lowest fairness. Although the fairness of our proposed algorithm is lower than that of max-min optimization, it is much higher than that of water filling, especially when the system is heavily loaded. In Fig. 2, system throughput is shown with respect to the number of users. The throughput of our proposed algorithm is slightly degraded comparing with maximum throughput algorithm and significantly enhanced over max-min algorithm. Therefore it can be said that the proposed algorithm provides a good tradeoff for system throughput and fairness.
System fairness vs. number of users.
V. C ONCLUSION
Fig. 2.
System throughput vs. number of users.
Fairness Index (FI) proposed in [5] is adopted for comparison. 2 K K xk / K x2k , (19) FI = k=1
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 FI decreases to 0. A. Simulation Environment and Parameters Consider a single cell (BS) with the radius of 1km. We generate 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 to a user. And the number of users are equally divided into each zone, i.e., 50% of the total users. Users are assumed to be uniformly distributed in each zone. The COST 231 Hata urban
In this letter, we formulate a PF optimization problem as a maximization of the sum of logarithmic user data rate in downlink OFDMA systems. From necessary conditions on optimality obtained analytically by KKT condition, we propose an efficient subcarrier and power allocation algorithm based on water filling. It is shown through simulation that our proposed algorithm is slightly degraded in throughput and provides improved fairness comparing with the best schemes in throughput and perfect fairness, especially when there are large number of users. R EFERENCES [1] H. Kim and Y. Han, “A proportional fair scheduling for multicarrier transmission systems,” IEEE Commun. Lett. vol. 9, no. 3, pp. 210-212, Mar. 2005. [2] Z. Han et al., “Fair multiuser channel allocation for OFDMA networks using Nash bargaining solutions and coalitions,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1366-1376, Aug. 2005. [3] T.-D. Nguyen and Y. Han, “A dynamic channel assignment algorithm for OFDMA systems,” appears in Proc. IEEE Vehicular Technology Conference ’06. [4] J. Jang, K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171-178, Feb. 2003. [5] K. Kim, H. Kim, and Y. Han, “Subcarrier and power allocation in OFDMA systems,” in Proc. IEEE VTC2004-Fall, vol. 2, pp. 1058-1062. [6] R. Jain, D-M. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimination for resource allocation inshared computer systems,” Technical Report TR-301, DECResearch Report, 1984. [7] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004
