Optimizing completion time and energy consumption in a bidirectional relay network

Consider a wireless network with multiple sources and destinations, where the amount of data of each source node is finite. An interesting question is what is the shortest completion time, i. e. the time required that all data from the sources gets to the respective destinations. A similar question arises for the minimal required energy. While the requirement for minimal energy consumption is obvious, the shortest completion time is relevant when certain multi-node network needs to reserve the wireless medium in order to carry out the data exchange among its nodes. The completion time/energy consumption required for multiple flows depends on the current channel realizations, transmission methods used and, notably, the relation between the data sizes of different source nodes. In this paper we investigate the shortest completion time and minimal energy consumption in a two-way relay wireless network. The system applies optimal time multiplexing of several known transmission methods, including one-way relaying and wireless network coding (WNC). We show that when the relay applies Amplify-and-Forward (AF), both minimizations are linear optimization problems. On the other hand, when the relay uses Decode-and-Forward (DF), each of them is a quadratic optimization problem. The results show that, for given channel realizations, there is an optimal ratio of the data packets at the sources to obtain minimal completion time or energy consumption. This can be used as a guidance for the nodes to apply traffic shaping. In most cases, DF leads to shorter completion time and energy consumption compared to AF.