To effectively conserve energy in wireless sensor networks (WSNs) and reduce packet delay, we propose a ()-policy sleep scheme for each sensor node, functioning in four distinct states. We model the sensor node, which incorporates the sleep mechanism, as a discrete-time vacation queueing system that accounts for startup times and an activation threshold. We first employ a probabilistic analysis technique to conduct a transient analysis of the system, aiming to derive recursive formulas for the steady-state distribution of the number of packets. We further obtain explicit expressions for several essential system performance metrics, including the expected number of packets, mean delay, and average energy cost of the node. The simulation experiments on models with various service time distributions confirm the analytical results, and extensive numerical experiments evaluate the sensitivity of system performance to several parameters. A weighted-sum cost function integrating mean delay and average energy consumption is formulated, and optimal sleep-wake strategies that minimise the weighted sum cost are evaluated across diverse sleep time distributions, service time distributions, weight coefficients, and delay constraints. The results demonstrate the advantages of the -policy in achieving an ideal equilibrium between energy efficiency and mean delay in WSNs.
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