Scheduling periodic tasks in a hard real-time environment

Abstract

We consider a real-time scheduling problem that occurs in the design
of software-based  aircraft control. The goal is to distribute tasks
$	au_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 $	au_i$ releases a job of running time $c_i$ at each time $a_i +
kcdot p_i, , k in mathbb{N}_0$ and a collision occurs if two jobs are
simultaneously active on the same machine.  

We shed some light on the complexity and approximability landscape of this problem.
Although the problem  cannot be approximated
 within a factor of $n^{1-varepsilon}$ for any $varepsilon>0$, an interesting restriction 
is much more tractable: If the periods are dividing (for each $i,j$ one has $p_i |
 p_j$ or $p_j | p_i$), the problem allows for a better structured representation of solutions, which leads
to a 2-approximation. This result is tight, even asymptotically.