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The main problem in dealing with energy-harvesting (EH) sensor nodes is represented by the scarcity and non-stationarity of powering, due to the nature of the renewable energy sources. In this work, the authors address the problem of task scheduling in processors located in sensor nodes powered by EH sources. Some interesting solutions have appeared in the literature in the recent past, as the lazy scheduling algorithm (LSA), which represents a performing mix of scheduling effectiveness and ease of implementation. With the aim of achieving a more efficient and conservative management of energy resources, a new improved LSA solution is here proposed. Indeed, the automatic ability of foreseeing at run-time the task energy starving (i.e. the impossibility of finalizing a task due to the lack of power) is integrated within the original LSA approach. Moreover, some modifications have been applied in order to reduce the LSA computational complexity and thus maximizing the amount of energy available for task execution. The resulting technique, namely energy-aware LSA, has then been tested in comparison with the original one, and a relevant performance improvement has been registered both in terms of number of executable tasks and in terms of required computational burden. [ABSTRACT FROM AUTHOR]