A Game Theoretical Formulation for Proportional Fairness in LTE Uplink Scheduling Elias Yaacoub and Zaher Dawy Department of Electrical and Computer Engineering, American University of Beirut, Beirut, Lebanon Email: {eey00, zd03}@aub.edu.lb Abstract—Uplink scheduling in LTE systems is considered. A game theoretical formulation is derived where the scheduling problem is represented as a Nash bargaining solution. An algo- rithm to implement the proposed scheduling scheme is presented. Throughput and fairness analysis are performed via simulations. Results show that channel aware scheduling schemes outperform the round-robin scheme, but a tradeoff must be made between the increase of total throughput and fairness towards the different users. Index Terms— Uplink scheduling, LTE, SCFDMA, game the- ory, Nash bargaining solution I. I NTRODUCTION The ‘Long Term Evolution’ (LTE) of UMTS will stretch the performance of 3G technology, in order to meet user expectations in a 10-year perspective and beyond [1]. The objectives of LTE include reduced latency, higher user data rates, improved system capacity and coverage, and reduced overall cost for the operator. Data rates of 100 Mbps in the downlink and 50 Mbps in the uplink are expected. To meet these requirements, orthogonal frequency division multiple access (OFDMA) based on OFDM is the key technique of LTE because it has the immunity to inter-symbol interference and frequency selective fading [1]. However, in spite the numerous advantages of OFDM and OFDMA, their major disadvantage is that their waveforms have high peak to average power ratio (PAPR). High PAPR is problematic for uplink transmission where the mobile transmission power is usually limited. For PAPR reduction, 3GPP-LTE agreed on using single carrier frequency division multiple access (SCFDMA) transmission with cyclic prefix in the uplink where frequency domain generation of the signal by a DFT precoding followed by an IFFT structure was assumed [2]. SCFDMA is a modified form of OFDMA. As in OFDMA, the transmitters in SCFDMA use different orthogonal frequencies (subcarriers) to transmit information symbols. However, they transmit the subcarriers sequentially, rather than in parallel [3]. Relative to OFDMA, SCFDMA reduces considerably the envelope fluctuations in the transmitted waveform. Therefore, SCFDMA signals have inherently lower PAPR than OFDMA signals [3]. SCFDMA has two types of sub-carrier mapping [3]: Lo- calized FDMA (LFDMA) and Interleaved FDMA (IFDMA). In LFDMA, the scheduler assigns consecutive subcarriers to convey information from a particular user. In IFDMA, users are assigned sub-carriers that are distributed over the entire frequency band in order to avoid allocating adjacent subcarriers that are simultaneously in a deep fade. IFDMA was not included into the LTE standard due to slight perfor- mance disadvantages caused by the requirements of channel estimation accuracy [4]. Hence, in the remainder of this paper, we will only consider LFDMA. Consequently, the available spectrum is divided into resource blocks (RB) consisting of 12 adjacent subcarriers. The duration of a single RB is 1 ms, agreed to be the duration of one transmission time interval (TTI) [4]. With the high bit rates expected for LTE, it would be inter- esting to investigate efficient scheduling of users, especially in the uplink. Work in this direction was performed in [1], where link and system throughput are estimated by simulations, with a simple round-robin scheduling policy. More efficient scheduling algorithms were developed in [3], where the utility functions to maximize are the sum of user throughput or the sum of the logarithm of user throughput. Fairness issues are also discussed. In this work, we present a game theoretical formulation that links proportional fairness to the logarithm of the throughput as a utility function to be maximized. The proposed game theoretical formulation applies in both centralized and distributed scenarios. A heuristic scheduling algorithm achieving this proportional fairness with low com- plexity is then designed and tested in a simulation environment consistent with the current LTE standards. It should be noted that the proposed algorithm has the flexibility to accommodate various utility functions. Thus, it is used to compare the performance of sum-throughput maximization to that of the proposed proportional fairness scheme. The paper is organized as follows. The problem formulation and the game theoretical modeling of the uplink scheduling so- lution are presented in Section II. The sub-optimal scheduling algorithm to implement the proposed solution is described in Section III. The simulation results are presented and discussed in Section IV. Finally, in Section V, conclusions are drawn and extensions for future research are described. II. PROBLEM FORMULATION We define the RB allocation problem as a constrained utility maximization problem first, then we present a game theoretical approach that leads to the same initial model while at the same time taking proportional fairness into account. A. Constrained Optimization Problem Letting I RB,k be the set of RBs allocated to user k, I sub,k the set of subcarriers allocated to user k, N sub the number of subcarriers, P k the instantaneous transmission power of user k, P k,max its maximum transmission power, and R k its achievable throughput, the maximization of the sum of user utilities U can be formulated as 978-1-4244-2948-6/09/$25.00 ©2009 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.