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In this paper, we investigate and analyze the expected per-node throughput in a wireless sensor network under a randomized sleep scheduling framework with a connectivity constraint. The unique optimal sleep probability maximizing the per-node throughput in the network is found. A sleep probability range in which the throughput monotonically increases along the sleep probability does exist, and it becomes a large portion of the total range when a network is heavy-traffic. This finding contradicts the common belief that throughput decreases along with the frequency of sleep of nodes. In addition, the connectivity confidence level for the optimal sleep probability is derived and proved to be bounded, and a distributed algorithm that makes each node achieve the maximum throughput is provided. Simulation results verify the aforementioned findings as well. For sensors allowed to choose their own sleep probability, an advisable policy is to choose as high probability as the connectivity constraint allows.