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This paper presents optimum Multi-Hop Transmission Strategies (MHTS) for energy constrained Wireless Sensor Networks (WSNs). Nodes in a multi-hop WSN need to transmit their own information and to relay each other's information to a Base Station (BS), and there are usually multiple available paths between a node and the BS. Two different scheduling strategies are considered, the fair equal information strategy that requires all the nodes deliver the same amount of information to the BS, and the unfair maximum information strategy that maximizes the total amount of information delivered to the BS. The optimum MHTS for these two strategies are derived, with either convex optimization or analytical expressions, under a per node energy constraint and a total energy constraint, respectively.
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