We study an interval job scheduling problem in distributed systems. We are given a set of interval jobs, with each job specified by a size, an arrival time and a processing length. Once a job arrives, it must be placed on a machine immediately and run for a period of its processing length without interruption. The homogeneous machines to run jobs have the same capacity limits such that at any time, the total size of the jobs running on any machine cannot exceed its capacity. Launching each machine incurs a fixed cost. After launch, a machine is charged a constant cost per time unit until it is terminated. The problem targets to minimize the total cost incurred by the machines for processing the given set of interval jobs. We focus on the algorithmic aspects of the problem in this article. For the special case where all the jobs have a unit size equal to the machine capacity, we propose an optimal offline algorithm and an optimal 2-competitive online algorithm. For the general case where jobs can have arbitrary sizes, we establish a non-trivial lower bound on the optimal solution. Based on this lower bound, we propose a 5-approximation algorithm in the offline setting. In the non-clairvoyant online setting, we design a $O(\mu)$O(μ)-competitive Modified First-Fit algorithm which is near optimal ($\mu$μ is the max/min job processing length ratio). In the clairvoyant online setting, we propose an asymptotically optimal $O(\sqrt{\log \mu })$O(logμ)-competitive algorithm based on our Modified First-Fit strategy.

中文翻译：

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具有机器启动成本的间隔作业调度

我们研究分布式系统中的间隔作业调度问题。我们得到一组间隔作业，每个作业由大小、到达时间和处理长度指定。作业到达后，必须立即放置在机器上并不间断地运行其处理长度的一段时间。运行作业的同类机器具有相同的容量限制，因此在任何时候，任何机器上运行的作业的总大小都不能超过其容量。启动每台机器都会产生固定成本。启动后，机器按每单位时间的固定成本收费，直到它被终止。该问题旨在最小化机器处理给定间隔作业集所产生的总成本。我们在本文中关注问题的算法方面。对于所有作业的单位大小等于机器容量的特殊情况，我们提出了最优离线算法和最优2-竞争在线算法。对于作业可以具有任意大小的一般情况，我们在最佳解决方案上建立了一个非平凡的下界。基于这个下界，我们提出了一种离线设置中的 5-近似算法。在非透视网络环境中，我们设计了一个$O(\mu)$哦(μ)- 接近最优的竞争性改进的首次拟合算法（$\亩$μ是最大/最小作业处理长度比）。在千里眼的在线设置中，我们提出了一个渐近最优$O(\sqrt{\log \mu })$哦(日志μ)-基于我们修改后的 First-Fit 策略的竞争算法。