Joint Opportunistic Scheduling and Network Coding for Bidirectional Relay Channel Mohammad Shaqfeh ⋆ , Ammar Zafar ‡ , Hussein Alnuweiri ⋆ , and Mohamed-Slim Alouini ‡ ⋆ Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar Email: {Mohammad.Shaqfeh, Hussein.Alnuweiri}@qatar.tamu.edu ‡ CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia Email:{ammar.zafar, slim.alouini}@kaust.edu.sa Abstract—In this paper, we consider a two-way communication system in which two users communicate with each other through an intermediate relay over block-fading channels. We investi- gate the optimal opportunistic scheduling scheme in order to maximize the long-term average transmission rate in the system assuming symmetric information flow between the two users. Based on the channel state information, the scheduler decides that either one of the users transmits to the relay, or the relay transmits to a single user or broadcasts to both users a combined version of the two users’ transmitted information by using linear network coding. We obtain the optimal scheduling scheme by using the Lagrangian dual problem. Furthermore, in order to characterize the gains of network coding and opportunistic scheduling, we compare the achievable rate of the system versus suboptimal schemes in which the gains of network coding and opportunistic scheduling are partially exploited. Index Terms—Bidirectional relay, block-fading, linear network coding, multi-user diversity, opportunistic scheduling. I. I NTRODUCTION Network coding (NC) [1]–[3] can considerably enhance the utilization efficiency of the channels’ resources in multi- direction information flow over multi-hop wireless networks due to the exploitation of the broadcast nature of the wireless channel as well as the available side information at the users about their own transmitted information. Consequently, NC is considered for application in next generation wireless networks [4]. Also, channel-aware opportunistic scheduling (OS) is a useful way to exploit multiuser diversity in wireless networks, e.g. [5], by allocating the channel resources to the users dynamically such that each user transmits when its channel is in a good state. By combining NC and OS together, we can obtain the gains of both schemes jointly. This is the main objective of this paper. In [6], [7], the authors considered optimizing the transmis- sion from a central relay in a multiuser network that has a star topology, and they discussed the optimal number of information flows to be combined together using network coding. However, they did not consider the uplink from the This paper was made possible by YSREP grant # 2-011-2-002 from the Qatar National Research Fund (a member of Qatar Foundation). Furthermore, KAUST funded the efforts of A. Zafar partially and the efforts of M.- S. Alouini. The statements made herein are solely the responsibility of the authors. users to the relay in the scheduler design as we do in this paper. In [8], the authors considered a large wireless cell with many users and relays and discussed when to apply network coding. However, they divided each channel block into three orthogonal phases making the users and relay transmit on every channel block. However, in our paper, we let a single node transmit per channel block and we demonstrate its gains over the other scheme. In our paper, we characterize the optimal scheduling scheme for a two-user system that is described in details in Sec- tion II in order to maximize the symmetric long-term average transmission rate in the system. We assume that the system applies linear network coding using the conventional xor (⊕) operation on the source information bits 1 . The optimal scheduler, which is derived in Section III, decides which node 2 (i.e. user or relay) should transmit in a given channel block based on the channel state information. We characterize the performance of the optimal scheme analytically. Furthermore, we compare the system performance with suboptimal schemes that apply either network coding or opportunistic scheduling, but not both of them, leading to partial exploitation of the multiuser diversity and side information gains. We provide numerical results in Section V and we summarize the main conclusions in Section VI. II. SYSTEM MODEL AND PROBLEM FORMULATION System Setup: We consider the three-node system that is shown in Fig. 1. Two users, denoted N 1 and N 2 , want to communicate with each other with the assistance of a third node, which is the relay (R). There is no direct link between the two users, and the three nodes operate in half-duplex mode. We assume that there are no delay constraints on the two-way communication, and hence opportunistic communication can be applied. 1 We know from information theory that xor-based network coding is suboptimal and it can be enhanced by using a special channel coding technique [9], which is out of the scope of this paper. 2 We know from information theory that for the uplink (i.e. multiple access channel), single user selection is in general suboptimal and the optimal scheme involves non-orthogonal transmission from multiple users. However, this is out of the scope of this paper.