In this paper, we study the following batch scheduling model: find a schedule that minimizes total flow time for $n$ uniform length jobs, with release times and deadlines, where the machine is only actively processing jobs in at most $k$ synchronized batches of size at most $B$. Prior work on such batch scheduling models has considered only feasibility with no regard to the flow time of the schedule. However, algorithms that minimize the cost from the scheduler's perspective -- such as ones that minimize the active time of the processor -- can result in schedules where the total flow time is arbitrarily high \cite{ChangGabowKhuller}. Such schedules are not valuable from the perspective of the client. In response, our work provides dynamic programs which minimize flow time subject to active time constraints. Our main contribution focuses on jobs with agreeable deadlines; for such job instances, we introduce dynamic programs that achieve runtimes of O$(B \cdot k \cdot n)$ for unit jobs and O$(B \cdot k \cdot n^5)$ for uniform length jobs. These results improve upon our modification of a different, classical dynamic programming approach by Baptiste. While the modified DP works when deadlines are non-agreeable, this solution is more expensive, with runtime $O(B \cdot k^2 \cdot n^7)$ \cite{Baptiste00}.
翻译:在本文中,我们研究了以下批次排期模式:找到一个能够最大限度地减少美元统一长度工作总流量时间的时间表,该时间表的发布时间和截止时间,即机器仅积极处理最多为美元同步数量的工作,最多为美元同步数量,最多为B$美元。以前关于这类批次排期模式的工作只考虑可行性,而没有考虑到时间表的流时间。然而,从调度员的角度出发,将成本最小化的算法 -- -- 例如尽量减少处理器的有效时间 -- -- 可能导致总流量时间任意高\cite{ChangGabowKhuller}的时间表。从客户的角度看,这些时间表并不宝贵。作为回应,我们的工作提供了动态程序,最大限度地减少流动时间,但受积极时间限制。我们的主要贡献侧重于与可商定的最后期限有关的工作;然而,我们引入了能够实现O$运行时间(B\cdott k k\cdot n) 的动态程序,而对于统一长度工作的O$O$(Basticrial\cal-deal-ral-trading the dival programamatial romatial romatial) rograal rodutional rodutional rodutional rodutional rodutional rodutional rodutional routd routdlexxxxxxxxxxxxxxxxxxxxx。