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Duty-cycling is the primary technique to save energy in wireless sensor networks (WSNs). A fundamental new theory of duty-cycling in WSNs with preamble sampling and random medium access control (MAC) is proposed. The theory is based on accurate modeling of delay, packet reception probabilities, and energy consumption as function of the relevant parameters of the network. The challenges for modeling are the random access MAC and the sleep policy of the receivers that makes it impossible to determine the exact time of data packet transmission. The analysis developed in this paper gives expressions of the delay, reliability and energy consumption as a function of sleep time, listening time, traffic rate and MAC parameters. The theory allows to design duty-cycling protocols and algorithms to implement these protocols. As illustrative examples, we use the new modelling for an unconstrained minimization of the energy consumption of the network, and a constrained minimization subject to delay and packet loss probability requirements. The optimization ensures a significant reduction of the energy consumption compared to the popular X-MAC protocol previously proposed in the literature. It is concluded that duty-cycling protocols must be build by considering simultaneously the energy consumption, the delay and the probability of successful packet reception to achieve maximum efficiency.