On Resolving Non-Preemptivity in Multitask Scheduling: An Optimal Algorithm in Deterministic and Stochastic Worlds

TL;DR

This paper introduces an optimal non-preemptive scheduling algorithm, NP-SRPT, for multiprocessor multitask jobs, achieving near-optimal response times in deterministic and stochastic models, with proven theoretical bounds and experimental validation.

Contribution

It develops NP-SRPT, a non-preemptive algorithm with proven optimality bounds for multitask scheduling in both deterministic and stochastic environments.

Findings

Achieves competitive ratio of ln α + β + 1 for response time minimization.

Proves asymptotic optimality in stochastic M/G/N systems as traffic approaches capacity.

Extends to unknown job sizes with asymptotic optimality of adapted policies.

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 $\ln \alpha + \beta + 1$ 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 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 $\mathsf{NP}$-$\mathsf{SRPT}$.