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Considering physical sensors with certain sensing capabilities in an Internet-of-Things (IoT) sensory environment, in this paper, we propose an efficient energy management framework to control the duty cycles of these sensors under quality-of-information (QoI) experience in a multi-task-oriented IoT sensory environment. Contrary to past research efforts, our proposal is transparent and compatible both with the underlying low-layer protocols and diverse applications, and preserving energy-efficiency in the long run without sacrificing the QoI levels attained. Specifically, we first introduce the novel concept of QoI-aware “sensor-to-task relevancy” to explicitly consider the sensing capabilities offered by an sensor to the IoT sensory environments, and QoI requirements required by a task. Second, we propose a novel concept of the “critical covering set” of any given task in selecting the sensors to service a task over time. Third, energy management decision is made dynamically at runtime, to reach the optimum for long-term application arrivals and departures under the constraint of their service delay. Finally, an extensive case study based on utilizing the sensing sensors to perform water quality monitoring is given to demonstrate the ideas and algorithms proposed in this paper, and a complete simulation is made to support all performance analysis.