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There is significant interest in battery-powered sensor networks to be used for detection in a wide variety of applications, from surveillance and security to health and environmental monitoring. Severe energy and bandwidth constraints at each sensor node demand system-level approaches to design that consider detection performance jointly with system-resource constraints. Our approach is to formulate detection problems with constraints on the expected cost arising from transmission (sensor nodes to a fusion node) and measurement (at each sensor node) to address some of the system-level costs in a sensor network. For a given resource constraint, we find that randomization over the choice of measurement and over the choice of when to transmit achieves the best performance (in a Bayesian, Neyman-Pearson, and Ali-Silvey sense). To facilitate design, we describe performance criteria in the send/no-send transmission scenario, where the joint optimization over the sensor nodes decouples into optimization at each sensor node.