Int. J. Electron. Commun. (AEÜ) 65 (2011) 320–330 Contents lists available at ScienceDirect International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue Achieving the Nash bargaining solution in OFDMA uplink using distributed scheduling with limited feedback Elias Yaacoub ∗ , Zaher Dawy Department of Electrical and Computer Engineering, American University of Beirut, P.O. Box 11-0236, Beirut, Lebanon article info Article history: Received 8 September 2009 Accepted 28 March 2010 Keywords: Nash bargaining solution Proportional fair OFDMA Uplink Scheduling abstract The Nash bargaining solution in OFDMA uplink scheduling is investigated. The problem is equivalent to proportional fair scheduling. The logarithm of throughput, when used as a utility function, ensures pro- portional fairness and thus is equivalent to the Nash bargaining solution. An algorithm to implement the solution 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 optimal with a limited number of feedback bits. In fact, only one bit feedback per subcarrier was sufficient to achieve near-optimal results with proportional fair scheduling. © 2010 Elsevier GmbH. All rights reserved. 1. Introduction The Nash bargaining problem (NBP) [1] is a well known scenario in game theory. Players in the NBP negotiate to maximize their pay- offs. The optimal solution of the NBP, the Nash bargaining solution (NBS), consists of distributing the resources in a way to maximize the product of the payoffs [2]. It was shown that proportional fair (PF) scheduling is equivalent to the implementation of the NBS in the resource allocation of wireless communication systems, the payoff of each user being its throughput [3,4]. PF scheduling is widely investigated in the literature, mainly in the framework of centralized resource allocation [5–7]. With Orthogonal Frequency Division Multiple Access (OFDMA) adopted as the accessing scheme of next generation cellular systems, e.g., 3GPP Long-Term Evolution (LTE) and mobile WiMAX (IEEE 802.16e), several applications of PF to OFDMA were studied [8–10]. Conversely to centralized resource allocation, mobile users have more autonomy in making transmission decisions in distributed schemes. The benefits of distributed resource allocation are being widely investigated in the context of ad-hoc networks, relay-based networks, and sensor networks [11–13], in addition to wireless local area networks (WLANs) [14,15] and cognitive radio (CR) networks [16–19]. In this paper, we propose a distributed PF scheduling approach that leads to results close to the centralized case. 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). equivalent to the NBS, and hence it is in the benefit of the users to cooperate in order to implement it. The main contribution of this paper is the investigation of col- laborative distributed scheduling in OFDMA uplink and proposing a distributed scheduling scheme that can be easily implemented by the users with a limited exchange of channel state information (CSI). A CSI quantization technique is proposed in order to achieve a 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 by the authors in [20] for the full CSI case so that it can be implemented by the users in a distributed collaborative scenario with limited feedback. Furthermore, the algorithm is shown to be efficiently applicable with various utility functions in both the centralized and distributed scenarios. The paper is organized as follows. An overview of proportional fair scheduling and its equivalence to the Nash bargaining solu- tion are presented in Section 2. The proposed scheduling algorithm is described in Section 3. The application of the algorithm to cen- tralized scheduling is presented in Section 4, and its application to distributed scheduling is discussed in Section 5. The simulation results are presented and analyzed in Section 6. Section 7 presents a 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 feasible rate vector R is proportional fair if for any other feasible rate vector 1434-8411/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2010.03.007