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In wireless sensor networks, some sensor nodes are put in sleep mode while other sensor nodes are in active mode for sensing and communication tasks in order to reduce energy consumption and extend network lifetime. This approach is a special case (k=2) of a randomized scheduling algorithm, in which k subsets of sensors work alternatively. In this paper, we first study the randomized scheduling algorithm via both analysis and simulations in terms of network coverage intensity, detection delay, and detection probability. We further study asymptotic coverage and other properties. Finally, we analyze a problem of maximizing network lifetime under quality of service constraints such as bounded detection delay, detection probability, and network coverage intensity. We prove that the optimal solution exists, and provide conditions of the existence of the optimal solutions.