Scheduling heterogeneous processors isn't as easy as you think

We consider preemptive online scheduling algorithms to minimize the total weighted/unweighted flow time plus energy for speed-scalable heterogeneous multiprocessors. We show that the well-known priority scheduling algorithms Highest Density First, Weighted Shortest Elapsed Time First, and Weighted Late Arrival Processor Sharing, are not O(1)-speed O(1)-competitive for the objective of weighted flow even in the special case of fixed variable speed processors (aka the related machines setting). This illustrates that scheduling heterogeneous multiprocessors is a different, and algorithmically more challenging problem, than scheduling homogeneous multiprocessors. We then show that a variation of the non-clairvoyant algorithm Late Arrival Processor Sharing coupled with a non-obvious speed scaling algorithm is scalable for the objective of unweighted flow plus energy on speed-scalable multiprocessors. This is the first provably scalable non-clairvoyant algorithm on heterogeneous multi-processors, even in the related machines setting, for the objective of total (unweighted) flow time.