Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems

TL;DR

This paper investigates the problem of scheduling sporadic real-time tasks on identical multiprocessor systems by allocating additional processors, providing polynomial-time algorithms with instance-dependent approximation factors, and establishing complexity bounds.

Contribution

It introduces new polynomial-time algorithms for processor allocation in real-time scheduling with approximation guarantees dependent on task parameters, and proves the non-existence of APTAS under certain conditions.

Findings

Algorithms with approximation factors related to task period-to-deadline ratios.

Proved no APTAS exists for constrained deadline tasks unless P=NP.

Algorithms cannot achieve constant approximation factors in general cases.

Abstract

In real-time systems, in addition to the functional correctness recurrent tasks must fulfill timing constraints to ensure the correct behavior of the system. Partitioned scheduling is widely used in real-time systems, i.e., the tasks are statically assigned onto processors while ensuring that all timing constraints are met. The decision version of the problem, which is to check whether the deadline constraints of tasks can be satisfied on a given number of identical processors, has been known ${\cal NP}$-complete in the strong sense. Several studies on this problem are based on approximations involving resource augmentation, i.e., speeding up individual processors. This paper studies another type of resource augmentation by allocating additional processors, a topic that has not been explored until recently. We provide polynomial-time algorithms and analysis, in which the approximation factors are dependent upon the input instances. Specifically, the factors are related to the maximum ratio of the period to the relative deadline of a task in the given task set. We also show that these algorithms unfortunately cannot achieve a constant approximation factor for general cases. Furthermore, we prove that the problem does not admit any asymptotic polynomial-time approximation scheme (APTAS) unless ${\cal P}={\cal NP}$ when the task set has constrained deadlines, i.e., the relative deadline of a task is no more than the period of the task.