Online load balancing for heterogeneous machines aims to minimize the makespan (maximum machine workload) by scheduling arriving jobs with varying sizes on different machines. In the adversarial setting, where an adversary chooses not only the collection of job sizes but also their arrival order, the problem is well-understood and the optimal competitive ratio is known to be $\Theta(\log m)$ where $m$ is the number of machines. In the more realistic random arrival order model, the understanding is limited. Previously, the best lower bound on the competitive ratio was only $\Omega(\log \log m)$. We significantly improve this bound by showing an $\Omega( \sqrt {\log m})$ lower bound, even for the restricted case where each job has a unit size on two machines and infinite size on the others. On the positive side, we propose an $O(\log m/\log \log m)$-competitive algorithm, demonstrating that better performance is possible in the random arrival model.

翻译：异构机器上的在线负载均衡旨在通过在不同机器上调度不同大小的工作来最小化最大完工时间（即机器最大负载）。在对抗性设置中，对手不仅选择工作大小的集合，还选择它们的到达顺序，该问题已被充分研究，且已知最优竞争比为$\Theta(\log m)$，其中$m$为机器数量。在更符合实际的随机到达顺序模型中，相关理解仍然有限。此前，竞争比的下界仅为$\Omega(\log \log m)$。我们显著改进了这一下界，证明了即使对于每个工作在两台机器上具有单位大小、在其他机器上具有无限大小的受限情况，下界也达到$\Omega( \sqrt {\log m})$。在积极方面，我们提出了一种$O(\log m/\log \log m)$竞争比的算法，表明在随机到达模型中可以实现更优的性能。