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In this work, we assume a geographic area populated by tiny sensors, each perhaps no larger than a dime. In order to save their energy, the sensors spend most of their lifetime in sleep mode and wake up for short periods of time to participate in various tasks supportive of the overall mission of the network. We assume that the tasks to be performed stipulate QoS parameters expressed in terms of the minimum number of sensors that need to monitor their sensing area. Since only awake sensors participate in tasks, the Effective Sensor Density (ESD), defined as the density of awake sensors, is an important network parameter that obviously depends on the sleep schedules adopted in the network. The first main contribution of this work is to provide a mathematical analysis of ESD from the perspective of the monitored events. We also provide design guidelines to determine deployment-time sensor density and an associated sleep schedule that probabilistically keeps the ESD at a level needed by QoS requirements. We also propose a fully distributed sleep schedule which adaptively adjusts the duty cycles of sensors within the same sensing area based on the relative difference in their remaining energy budget. The main advantage of the proposed adaptive scheme is to balance energy consumption among sensors, thus promoting the functional longevity of the sensor network without adversely affecting the ESD.