Online makespan minimization is a classic model in the field of scheduling. In this paper, we consider the over-time version, where each job is associated with a release time and a processing time. We only know a job after its release time and should schedule it on one machine afterward. The Longest Processing Time First (LPT) algorithm, as proven by Chen and Vestjens in 1997, achieves a competitive ratio of 1.5. However, for the case of two machines, Noga and Seiden introduced the SLEEPY algorithm, which achieves a competitive ratio of 1.382. Unfortunately, for the case of $m\geq 3$, there has been no convincing result that surpasses the performance of LPT. we propose a natural generalization that involves locking all the other machines for a certain period after starting a job, thereby preventing them from initiating new jobs. We show this simple approach can beat the $1.5$ barrier and achieve $1.482$-competitive when $m=3$. However, when $m$ becomes large, we observe that this simple generalization fails to beat $1.5$. Meanwhile, we introduce a novel technique called dynamic locking to overcome the new challenge. As a result, we achieve a competitive ratio of $1.5-\frac{1}{O(m^2)}$, which beats the LPT algorithm ($1.5$-comeptitive) for every constant $m$.

翻译：在线时间跨度最小化是调度领域中的一个经典模型。本文考虑的是带释放时间的版本，其中每个作业都关联一个释放时间与处理时间。我们仅能在作业释放后知晓其信息，并需随后将其调度至某台机器上。Chen和Vestjens于1997年证明，最长处理时间优先（LPT）算法可实现1.5的竞争比。然而，对于双机情形，Noga与Seiden提出的SLEEPY算法达到了1.382的竞争比。遗憾的是，当机器数$m\geq 3$时，尚未有令人信服的结果能超越LPT的性能。我们提出一种自然的泛化方法：在启动一个作业后，将其他所有机器锁定一段特定时间，从而阻止它们开启新作业。我们证明了这种简单方法能在$m=3$时突破1.5的界限，实现1.482的竞争比。然而，当$m$增大时，我们观察到这一简单泛化无法突破1.5。与此同时，我们引入一种称为动态锁定的新技术来应对这一新挑战。最终，我们取得了$1.5-\frac{1}{O(m^2)}$的竞争比，对于每个常数$m$均优于LPT算法（竞争比1.5）。