Utility Optimal Coding for Packet Transmission over Wireless Networks - Part I: Networks of Binary Symmetric Channels

We consider multi--hop networks comprising Binary Symmetric Channels ($\mathsf{BSC}$s). The network carries unicast flows for multiple users. The utility of the network is the sum of the utilities of the flows, where the utility of each flow is a concave function of its throughput. Given that the network capacity is shared by the flows, there is a contention for network resources like coding rate (at the physical layer), scheduling time (at the MAC layer), etc., among the flows. We propose a proportional fair transmission scheme that maximises the sum utility of flow throughputs subject to the rate and the scheduling constraints. This is achieved by {\em jointly optimising the packet coding rates of all the flows through the network}.