Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold

We prove a surprising lower bound for resource augmented nonclairvoyant algorithms for scheduling jobs with sublinear nondecreasing speed-up curves on multiple processors with the objective of average response time. Edmonds in STOC99 shows that the algorithm Equi-partition is a (2+ǫ)-speed Θ ( 1 ǫ)-competitive algorithm. We define its speed threshold to be 2 because it is constant competitive when given speed 2+ǫ but not when given speed 2. (Its load threshold is the inverse of its speed threshold.) The optimal speed threshold is 1 because then the algorithm is constant competitive no matter how little extra resources it is given. Edmonds and Pruhs in SODA09 imply that they have found such an algorithm. (They use the term scalable.) We, however, rebut that their algorithm only accomplishes this nondeterministically. They prove that for every ǫ> 0, there is an algorithm Alg ǫ that is (1+ǫ)-speed O ( 1 ǫ 2)-competitive. A problem, however, is that this algorithm Alg ǫ depends on ǫ. Hence, to have one algorithm it would have to runs Alg ǫ after nondeterministically guessing the correct ǫ. We prove that like Equi-partition