Exact GPS simulation with logarithmic complexity, and its application to an optimally fair scheduler

Generalized Processor Sharing (GPS) is a fluid scheduling policy providing perfect fairness. The minimum deviation (lead/lag) with respect to the GPS service achievable by a packet scheduler is one packet size. To the best of our knowledge, the only packet scheduler guaranteeing such minimum deviation is Worst-case Fair Weighted Fair Queueing (WF(2)Q), that requires on-line GPS simulation. Existing algorithms to perform GPS simulation have O(N) complexity per packet transmission (N being the number of competing flows). Hence WF(2)Q has been charged for O(N) complexity too. Schedulers with lower complexity have been devised, but at the price of at least O(N) deviation from the GPS service, which has been shown to be detrimental for real-time adaptive applications and feedback based applications. Furthermore, it has been proven that the lower bound complexity to guarantee O(1) deviation is Omega(log N), yet a scheduler achieving such result has remained elusive so far. In this paper we present an algorithm that performs exact GPS simulation with Omega(log N) worst-case complexity and small constants. As such it improves the complexity of all the packet schedulers based on GPS simulation. In particular, using our algorithm within WF(2)Q, we achieve the minimum deviation from the GPS service with O(log N) complexity, thus matching the aforementioned lower bound. Furthermore, we assess the effectiveness of the proposed solution by simulating real-world scenarios.