The efficient scheduling of multi-task jobs across multiprocessor systems has become increasingly critical with the rapid expansion of computational systems. This challenge, known as Multiprocessor Multitask Scheduling (MPMS), is essential for optimizing the performance and scalability of applications in fields such as cloud computing and deep learning. In this paper, we study the MPMS problem under both deterministic and stochastic models, where each job is composed of multiple tasks and can only be completed when all its tasks are finished. We introduce $\mathsf{NP}$-$\mathsf{SRPT}$, a non-preemptive variant of the SRPT algorithm, designed to accommodate scenarios with non-preemptive tasks. Our algorithm achieves a competitive ratio of $\ln α+ β+ 1$ for minimizing response time, where $α$ represents the ratio of the largest to the smallest job workload, and $β$ captures the ratio of the largest non-preemptive task workload to the smallest job workload. We further establish that this competitive ratio is order-optimal when the number of processors is fixed. For the stochastic $\mathsf{M}$/$\mathsf{G}$/$\mathsf{N}$ system, we prove that $\mathsf{NP}$-$\mathsf{SRPT}$ achieves asymptotically optimal mean response time as the traffic intensity approaches $1$, assuming task size distribution with finite support. Moreover, the asymptotic optimality extends to infinite task size distributions under mild probabilistic assumptions, including the standard $\mathsf{M}$/$\mathsf{M}$/$\mathsf{N}$ model. Finally, we extend the analysis to the setting of unknown job sizes, proving that non-preemptive adaptations of the $\mathsf{M\text{-}Gittins}$ and $\mathsf{M\text{-}SERPT}$ policies achieve asymptotic optimality and near-optimality, respectively, for a broad class of job size distributions. Experimental results validate the effectiveness of $\mathsf{NP}$-$\mathsf{SRPT}$.

翻译：随着计算系统的快速扩张，多处理器系统中多任务作业的高效调度变得日益关键。这一挑战被称为多处理器多任务调度，对于优化云计算和深度学习等领域应用的性能和可扩展性至关重要。本文研究了确定性和随机模型下的MPMS问题，其中每个作业由多个任务组成，且仅当其所有任务完成时作业才算完成。我们提出了$\mathsf{NP}$-$\mathsf{SRPT}$，这是SRPT算法的一种非抢占式变体，旨在适应任务不可抢占的场景。我们的算法在最小化响应时间方面实现了$\ln α+ β+ 1$的竞争比，其中$α$表示最大与最小作业工作负载之比，$β$表示最大非抢占任务工作负载与最小作业工作负载之比。我们进一步证明，当处理器数量固定时，该竞争比是阶次最优的。对于随机$\mathsf{M}$/$\mathsf{G}$/$\mathsf{N}$系统，我们证明在任务规模分布具有有限支撑的假设下，当流量强度趋近于$1$时，$\mathsf{NP}$-$\mathsf{SRPT}$实现了渐近最优的平均响应时间。此外，在温和的概率假设下（包括标准的$\mathsf{M}$/$\mathsf{M}$/$\mathsf{N}$模型），该渐近最优性可扩展至无限任务规模分布。最后，我们将分析扩展到作业规模未知的场景，证明对于广泛的作业规模分布，$\mathsf{M\text{-}Gittins}$和$\mathsf{M\text{-}SERPT}$策略的非抢占式改编分别实现了渐近最优性和接近最优性。实验结果验证了$\mathsf{NP}$-$\mathsf{SRPT}$的有效性。