On Resolving Non-Preemptivity in Multitask Scheduling: An Optimal Algorithm in Deterministic and Stochastic Worlds
Wenxin Li
TL;DR
This work tackles the hard online problem of scheduling multitask jobs on multiple processors without preemption, aiming to minimize total response time. It introduces NP-SRPT, a non-preemptive adaptation of SRPT, and proves a competitive ratio of $\ln \alpha + \beta + 1$, with order-optimality when the number of machines is fixed; it also shows asymptotic mean response-time optimality in heavy traffic for Poisson arrivals with finite-task distributions, and extends the analysis to infinite-task distributions under mild conditions. The paper further extends these ideas to unknown job sizes via NP-M-Gittins and NP-M-SERPT policies, establishing asymptotic optimality or near-optimality in broad distribution classes, and validates the results with experiments (notably Weibull-task simulations). Overall, the work advances online MPMS theory by delivering strong worst-case and stochastic guarantees, plus practical adaptations for unknown sizes, with demonstrated impact on cloud and DL workloads.
Abstract
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 $\mathsf{NP}$-$\mathsf{SRPT}$.