In this paper we study the multiple-processor multitask scheduling problem in both deterministic and stochastic models. We consider and analyze Modified Shortest Remaining Processing Time (M-SRPT) scheduling algorithm, a simple modification of SRPT, which always schedules jobs according to SRPT whenever possible, while processes tasks in an arbitrary order. The M-SRPT algorithm is proved to achieve a competitive ratio of $\Theta(\log \alpha +\beta)$ for minimizing response time, where $\alpha$ denotes the ratio between maximum job workload and minimum job workload, $\beta$ represents the ratio between maximum non-preemptive task workload and minimum job workload. In addition, the competitive ratio achieved is shown to be optimal (up to a constant factor), when there are constant number of machines. We further consider the problem under Poisson arrival and general workload distribution (\ie, $M/GI/N$ system), and show that M-SRPT achieves asymptotic optimal mean response time when the traffic intensity $\rho$ approaches $1$, if job size distribution has finite support. Beyond bounded job workload, the asymptotic optimality of M-SRPT also holds for unbounded job size distributions with certain probabilistic assumptions, for example, $M/M/N$ system with upper bounded task workload. An byproduct of our analysis is a tight characterization of the heavy traffic behavior of work-conserving algorithms in single-task job scheduling. We prove that the average response time in $GI/GI/1$ scales with $1/(1-\rho)$, if the job size distribution has finite support, which generalizes the growth rate in [Lin, Wierman and Zwart, 2011] to general arrival processes and all work-conserving algorithms.

中文翻译：

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改进的SRPT在多处理器多任务调度中的性能分析

在本文中，我们研究了确定性和随机模型中的多处理器多任务调度问题。我们考虑并分析了修改后的最短剩余处理时间 (M-SRPT) 调度算法，这是 SRPT 的简单修改，它总是尽可能根据 SRPT 调度作业，同时以任意顺序处理任务。M-SRPT 算法被证明在最小化响应时间方面实现了 $\Theta(\log \alpha +\beta)$ 的竞争比率，其中 $\alpha$ 表示最大工作负载和最小工作负载之间的比率，$\ beta$ 表示最大非抢占式任务工作量与最小作业工作量之间的比率。此外，当机器数量不变时，所达到的竞争比率被证明是最佳的（达到一个常数因子）。我们进一步考虑了泊松到达和一般工作负载分布（\即 $M/GI/N$ 系统）下的问题，并表明当流量强度 $\rho$ 接近 $1$ 时，M-SRPT 实现了渐近最优平均响应时间，如果工作规模分布的支持有限。除了有界工作量之外，M-SRPT 的渐近最优性也适用于具有某些概率假设的无界工作量分布，例如，具有上限任务工作量的 $M/M/N$ 系统。我们分析的一个副产品是对单任务作业调度中工作节约算法的高流量行为的严格表征。我们证明了 $GI/GI/1$ 中的平均响应时间与 $1/(1-\rho)$ 成比例，如果作业规模分布具有有限支持，则概括了 [Lin、Wierman 和 Zwart，