Scheduling Periodic Tasks in a Hard Real-Time Environment

Abstract

We give a rigorous account on the complexity landscape of an important real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks τ _i  = (c _i ,p _i ) on a minimum number of identical machines and to compute offsets a _i for the tasks such that no collision occurs. A task τ _i releases a job of running time c _i at each time $a_i + k\cdot p_i, \, k \in {\mathbb N}_0$ and a collision occurs if two jobs are simultaneously active on the same machine. Our main results are as follows: (i) We show that the minimization problem cannot be approximated within a factor of n ^1 − ε for any ε> 0. (ii) If the periods are harmonic (for each i,j one has p _i |p _j or p _j |p _i ), then there exists a 2-approximation for the minimization problem and this result is tight, even asymptotically. (iii) We provide asymptotic approximation schemes in the harmonic case if the number of different periods is constant.

This work was partially supported by Berlin Mathematical School, by DFG research center Matheon, by the DFG Focus Program 1307 within the project “Algorithm Engineering for Real-time Scheduling and Routing”, and by the Swiss National Science Foundation.