462 IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 6, JUNE 2006 Proportional Fairness in Multi–Carrier System: Upper Bound and Approximation Algorithms Megumi Kaneko, Petar Popovski, and Joachim Dahl Abstract— The solution for the optimal Multi-Carrier (MC) Proportional Fair Scheduling (PFS) is prohibitively complex to obtain. In this letter we obtain an upper bound on the achievable proportional fairness (PF) performance in the case of finite PFS window size. Next, three approximation algorithms are proposed. The first one is computationally complex, but achieves near– optimal PF. The two others achieve a good tradeoff between throughput and PF with low complexity. Index Terms— Multi-carrier system, Orthogonal Frequency Division Multiple Access (OFDMA), Proportional Fair Schedul- ing (PFS). I. I NTRODUCTION R ADIO resource allocation for Orthogonal Frequency Division Multiple Access (OFDMA)-based 4G wireless system is a key design issue. While large throughput gains can be achieved by exploiting multi-user diversity, fairness should be guaranteed at the same time. In the Single Car- rier (SC) case, this was achieved by the PFS implemented for Qualcomms’ HDR system [1]. A transmission is made when the user’s channel condition is favorable, while the past throughput is tracked to ensure PF. This was adapted to the MC case in [2] and the utility-based optimization approach of [3] derives the same solution. This is a suboptimal solution as each subcarrier allocation is made independently from each other. The optimal PF solution is derived in [4], which shows that each subcarrier allocation depends on the other subcarrier allocations. The optimal dependency can be approximated by updating the average user rates in a partial manner, i.e. after every subcarrier allocation, as done in [5]. However, algorithm in [5] can result in different user mappings depending on the order of allocated subcarriers, as this order is chosen arbitrarily. To avoid this, we define a priority rule for determining the order of subcarriers to be allocated, which results in a unique user mapping. In this work, we focus on the single cell downlink (DL) transmissions with OFDMA, where users feed back to the Base Station (BS) their Channel State Information (CSI) as a per–subcarrier Signal–to–Noise–Ratio (SNR). As the optimal solution for MC-PFS of [4] is prohibitively complex to obtain, we derive an upper bound on PF for a finite window size. We propose an approximation algorithm which achieves a near–optimal PF, but still with high complexity. We have also designed two low–complexity algorithms which improve throughput and PF compared to existing algorithms. Manuscript received January 27, 2006. The associate editor coordinating the review of this letter and approving it for publication was Dr. Jaap van de Beek. The authors are with the Center for TeleInFrastruktur (CTIF), Dept. of Communication Technology, Aalborg University, Niels Jernes Vej 12, DK- 9220 Aalborg, Denmark (e-mail: {mek, petarp, joachim}@kom.aau.dk). Digital Object Identifier 10.1109/LCOMM.2006.06022. II. PROPORTIONAL FAIR SCHEDULING FOR MULTI -CARRIER SYSTEM The condition for achieving optimal MC-PFS is [4]: a scheduler P is proportionally fair in a MC system if and only if it maximizes the sum of logarithmic average user rates R (S) k P = arg max S K  k=1 log R (S) k . (1) A scheduler S is defined as an entity that selects a set of users for a time frame and K is the total number of users. We define as the PF metric of a scheduler S the General Proportional Fairness (GPF) parameter, Γ (S) , as Γ (S) = K  k=1 log R (S) k . (2) In the suboptimal algorithm of [2], referred as Conventional MC-PFS, each subcarrier n is allocated to the user k ∗ (n) satisfying k ∗ (n) = arg max k ρ k,n (t), where the user PFS metric ρ k,n is equal to r k,n (t) R ′ k (t) . In frame t, r k,n (t) is the achievable rate of user k in subcarrier n and R ′ k (t) is the past average rate of user k, summed over all subcarriers and averaged over a window of length T . After all subcarriers are allocated, the average user rates are updated as R k (t) = (1 − 1 T )R ′ k (t)+ 1 T N  n=1 c k,n r k,n (t). (3) N is the total number of subcarriers. c k,n =1 if subcarrier n is allocated to user k, otherwise c k,n =0. In the following frame, the past average rate is updated as: R ′ k (t + 1) = R k (t). In our proposal, the average rates undergo virtual partial updates. Initially is set c k,n =0 for all n, k. The allocation is done in α ≤ N iterations and at l−th iteration, for each subcarrier n ∗ allocated to each user k ∗ , we set c k ∗ ,n ∗ =1. After the l−th iteration, the allocated users’ average rates are partially updated as R l k ∗ (t) = (1 − 1 T )R ′ k ∗ (t)+ 1 T N  n=1 c k ∗ ,n r k ∗ ,n (t). (4) These updates are virtual, since they are used only to facilitate the decisions for subcarrier allocation. The actual update is still done per-frame, using (3), after all subcarriers are allocated and data is transmitted. III. UPPER BOUND OF THE OPTIMAL MC-PFS We can rewrite the MC-PFS problem as minimize − ∑ K k=1 log(a (k)T x + b (k) ) subject to h (n)T x =1, n =1,...,N x i ∈{0, 1}, i =1,...,KN (5) 1089-7798/06$20.00 c  2006 IEEE