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Recently directional sensors have been widely deployed as they are more practicable under constraints of manufacture, size or cost. One common functionality of networks formed by such directional sensors is to monitor a set of discrete targets continuously. Sensors are often unattended after deployment making recharging impossible. By abundant deployment, it is reasonable and necessary to select subsets of sensors to operate successively so as to prolong the network lifetime. Furthermore, distance between sensor and target usually dominates accuracy of gathered data. This paper first formulates the utility-based lifetime of directional sensor network monitoring target set, and proves such problem is NP-complete. Next, network lifetime is approximated by proposed randomized algorithm. Simulation results reveal different factors impacting on network lifetime.