Online scheduling of parallelizable jobs in the directed acyclic graphs and speed-up curves models

This paper considers scheduling jobs online on m identical machines such that the jobs can be parallelized across the machines. Two models of parallelizability are considered, one is the speed-up curves model, and the other is the directed-acyclic-graph (DAG) model. For both models, the objectives considered are the average, maximum, and ℓ_k-norms of flow time for k≥1. 
We establish an Ω(m) lower bound on the competitive ratio of any algorithm for optimizing average flow time in both models without resource augmentation. With resource augmentation, we give a (1+ϵ)-speed O(1/ε^2k+1)-competitive algorithm in the DAG model for the ℓ_k-norms of flow time. This essentially matches the best-known result in the speed-up curve model for the ℓ_k-norms of flow time. Finally, we show an O(1)-competitive algorithm for minimizing the maximum flow time in the speed-up curves model.