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Generalized processor sharing (GPS) is a fluid scheduling policy providing perfect fairness over both constant-rate and variable-rate links. The minimum deviation (lead/lag) with respect to the GPS service achievable by a packet scheduler is one maximum packet size. To the best of our knowledge, the only packet scheduler guaranteeing the minimum deviation is worst-case fair weighted fair queueing , which requires on-line GPS simulation. Existing algorithms to perform GPS simulation have worst-case computational complexity per packet transmission (being the number of competing flows). Hence, has been charged for complexity too. However it has been proven that the lower bound complexity to guarantee deviation is, yet a scheduler achieving such a result has remained elusive so far. In this paper, we present L-GPS, an algorithm that performs exact GPS simulation with worst-case complexity and small constants. As such it improves the complexity of all the packet schedulers based on GPS simulation. We also present , an implementation of based on L-GPS. has complexity with small constants, and, since it achieves the minimum possible deviation, it does match the aforementioned complexity lower bound. Furthermore, both L-GPS and comply with constant-rate as well as variable-rate links. We assess the effectiveness of both algorithms by simulating real-world scenarios.