### Article

## Non-conflict scheduling criterion for strict periodic tasks

In the paper, we address mission critical systems, such as automobile, avionic, mobile robotic, telecommunication, etc. Such systems must meet hard real-time constraints in order to avoid catastrophic consequences. To meet the real-time constraints, strict periodicity is used (i.e. for any periodic task, time between release points is constant). Sensors, actuators and feedback control functions are typical examples of strict periodic tasks. We study a monoprocessor preemptive scheduling problem for arbitrary number of strict periodic tasks. In this context, we focus on the following problem: how to find non-conflict set of task release points (i.e. sequences of instance release points for different tasks must not intersect). First, as a preliminaries, we introduce some fundamental definitions and prove several elementary schedulability conditions. Next, we investigate the correlation between the scheduling problem and a graph coloring problem for graphs of some special kinds. The graphs under consideration are built on the basis of the tasks' period values. We introduce a notion of divisibility graph for tasks' periods, and study compatibility of graphs' coloring with respect to the schedulability problem. At last, we prove a theorem that provides necessary and sufficient graph coloring conditions for schedulability of given strict periodic tasks. This theorem allows either to find non-conflict set of task release points directly, or to determine quickly that scheduling is impossible.