We study a class of scheduling problems, where each job is divided into a batch of unit-size tasks and these tasks can be executed in parallel on multiple servers with New-Better-than-Used (NBU) service time distributions. While many delay optimality results are available for single-server queueing systems, generalizing these results to the multi-server case has been challenging. This motivated us to investigate near delay-optimal scheduling of batch jobs in multi-server queueing systems. We consider three lowcomplexity scheduling policies: the Fewest Unassigned Tasks first (FUT) policy, the Earliest Due Date first (EDD) policy, and the First-Come, First-Served (FCFS) policy. We prove that for arbitrary number, batch sizes, arrival times, and due times of the jobs, these scheduling policies are near delay-optimal in stochastic ordering for minimizing three classes of delay metrics among all causal and non-preemptive policies. In particular, the FUT policy is within a constant additive delay gap from the optimum for minimizing the mean average delay, and the FCFS policy within twice of the optimum for minimizing the mean maximum delay and the mean p-norm of delay. The key proof tools are several novel samplepath orderings, which can be used to compare the sample-path delay of different policies in a near-optimal sense.

翻译：我们研究了一类调度问题，其中每个作业被划分为一批单位大小的任务，这些任务可在具有“新优于旧”（NBU）服务时间分布的多台服务器上并行执行。尽管单服务器排队系统已获得许多延迟最优性结果，但将这些结果推广到多服务器场景仍具有挑战性。这促使我们研究多服务器排队系统中批处理作业的近延迟最优调度问题。我们考虑了三种低复杂度调度策略：最少未分配任务优先（FUT）策略、最早截止时间优先（EDD）策略和先来先服务（FCFS）策略。我们证明：对于任意数量、批大小、到达时间和截止时间的作业，这些调度策略在随机排序意义下是所有因果非抢占策略中最小化三类延迟指标的最优解，且接近最优。特别地，FUT策略在最小化平均延迟方面与最优解的延迟偏差为常数加性常数，而FCFS策略在最小化平均最大延迟和延迟的p-范数方面与最优解的偏差不超过两倍。关键证明工具是若干新颖的样本路径排序方法，可用于从近最优意义下比较不同策略的样本路径延迟。