Power efficiency is a capital issue in the study of mobile wireless nodes owing to constraints on their battery size and weight. In practice, especially for low-power nodes, it is often the case that the power consumed for non-transmission processes is not always negligible. In this paper, we consider the channels with a special form of overhead: a processing energy cost whenever a non-zero signal is transmitted. We show that under certain conditions, achieving the capacity of such channels requires intermittent, or `bursty', transmissions. Thus, an optimal sleeping schedule can be specified for wireless nodes to achieve the optimal power efficiency. We show that in the low SNR regime, there is a simple relation between the optimal burstiness and the overhead cost: one should use a fraction of the available degrees of freedom at an SNR level of 2 , where is the normalized overhead energy cost. We extend this result to use bursty Gaussian transmissions in multiple parallel channel...