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We consider a two-tier wireless sensor network and focus on the flow routing problem for the upper tier aggregation and forwarding nodes (AFNs). Assuming each AFN is equipped with directional antennas for transmission, we are interested in how to perform flow routing at each node such that the network lifetime is maximized. We present a flow routing algorithm that provably has the following properties: (1) when the average source rate of each AFN is known a priori, the flow routing algorithm is optimal and gives maximum network lifetime performance; (2) when the average source rate of each AFN is unknown but is within a fraction, ε, of an estimated rate value, then the network lifetime given by the proposed flow routing algorithm is no more than 2ε/(1-ε) from optimal. As a result, the proposed flow routing algorithm can provide predictable lifetime performance, even when the source bit rate can be time-varying.