This paper studies a scheduling problem in a parallel machine setting, where each machine must adhere to a predetermined fixed order for processing the jobs. Given $n$ jobs, each with processing times and deadlines, we aim to minimize the number of machines while ensuring deadlines are met and the fixed order is maintained. We show that the first-fit algorithm solves the problem optimally with unit processing times and is a 2-approximation in the following four cases: (1) the order aligns with non-increasing slacks, (2) the order aligns with non-decreasing slacks, (3) the order aligns with non-increasing deadlines, and (4) the optimal solution uses at most 3 machines. For the general problem we provide an $O(\log n)$-approximation.

翻译：本文研究并行机器环境下的调度问题，其中每台机器必须按照预定的固定顺序处理作业。给定$n$个作业，每个作业具有处理时间和截止期限，我们的目标是在确保满足截止期限且维持固定顺序的前提下，最小化机器使用数量。我们证明：在处理时间均为单位时间的情况下，首次适应算法可求得最优解；并在以下四种情形下具有2-近似比：(1) 顺序与松弛时间非增序一致，(2) 顺序与松弛时间非减序一致，(3) 顺序与截止期限非增序一致，(4) 最优解最多使用3台机器。针对一般性问题，我们提出了$O(\log n)$-近似算法。