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 Shortest Remaining Processing Time (SRPT) algorithm, designed to accommodate scenarios with non-preemptive tasks. Our algorithm achieves a competitive ratio of $\ln \alpha + \beta + 1$ for minimizing response time, where $\alpha$ represents the ratio of the largest to the smallest job workload, and $\beta$ 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 stochastic systems modeled as M/G/N queues, where job arrivals follow a Poisson process and task workloads are drawn from a general distribution, we prove that $\mathsf{NP}$-$\mathsf{SRPT}$ achieves asymptotically optimal mean response time as the traffic intensity $\rho$ approaches $1$, assuming the task size distribution has finite support. Moreover, the asymptotic optimality extends to cases with infinite task size distributions under mild probabilistic assumptions, including the standard M/M/N model. Experimental results validate the effectiveness of $\mathsf{NP}$-$\mathsf{SRPT}$, demonstrating its asymptotic optimality in both theoretical and practical settings.

翻译：随着计算系统的快速扩展，在多处理器系统上高效调度多任务作业变得日益关键。这一被称为多处理器多任务调度（MPMS）的挑战，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，仅当其所有任务完成时作业才算完成。我们提出了$\mathsf{NP}$-$\mathsf{SRPT}$算法，作为最短剩余处理时间（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。该算法在最小化响应时间方面实现了$\ln \alpha + \beta + 1$的竞争比，其中$\alpha$表示最大与最小作业工作负载之比，$\beta$表示最大非抢占式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，该竞争比是阶数最优的。对于建模为M/G/N队列的随机系统，其中作业到达服从泊松过程且任务工作负载取自一般分布，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$\rho$趋近于$1$时，$\mathsf{NP}$-$\mathsf{SRPT}$算法能实现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M/M/N模型），该渐近最优性可推广至任务规模分布无限的情况。实验结果验证了$\mathsf{NP}$-$\mathsf{SRPT}$的有效性，证明了其在理论和实际场景中的渐近最优性。