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A reinforcement learning (RL) method for quality of service (QoS)-aware power management (PM) of an energy harvesting wireless sensor network, named QoS-aware RLPM, is proposed in this paper. The RL environment for each sensor node is represented by the observable measurements of harvesting energy and residual energy stored in the battery. To achieve QoS-awareness, the proposed QoS-aware RLPM attempts to satisfy various QoS demands by adjusting the duty-cycle rate of sensor node according to the observable measurements. The outcomes of these interactions are evaluated by rewards that express how well the duty-cycle rate adjustments satisfy QoS requests under energy neutrality criteria. The QoS-aware RLPM learns the adjustment strategy of the duty-cycle rate by interacting with the environment and at the same time the QoS of the sensor network is maintained. Experiment results show that the proposed method satisfies the QoS requests under the energy neutrality criteria and performs better than the adaptive duty-cycling method.