Routing versus energy optimization in a linear network

In wireless networks, devices (or nodes) often have a limited battery supply to use for the sending and reception of transmissions. By allowing nodes to relay messages for other nodes, the distance that needs to be bridged can be reduced, thus limiting the energy needed for a transmission. However, the number of transmissions a node needs to perform increases, costing more energy. Defining the lifetime of the network as the time until the first node depletes its battery, we investigate the impact of routing choices on the lifetime. In particular we focus on a linear network with nodes sending messages directly to all other nodes, or using full routing where transmissions are only sent to neighbouring nodes. We distinguish between networks with nodes on a grid or uniformly distributed and with full or random battery supply. Using simulation we validate our analytical results and discuss intermediate options for relaying of transmissions.