A graph‐based analysis of the cyclic scheduling problem with time constraints: schedulability and periodicity of the earliest schedule

We consider in this paper a set of generic tasks constrained by a set of uniform precedence constraints corresponding to a natural generalization of the basic cyclic scheduling problem. The two parameters of any uniform constraint (namely the value and the height) between two tasks may be negative, which allows one to tackle a larger class of practical applications. We firstly study the structure and the existence of a periodic schedule. A necessary and sufficient condition for the existence of a schedule is then deduced. As there are no resource constraints, tasks following the earliest schedule have minimum average cycle times. The permanent state of the earliest schedule is characterized and we point out that the minimum average cycle times of tasks are equal for the earliest schedule and an optimal periodic schedule. An algorithm to check the existence of a schedule and to compute these minimum average cycle times using linear programming is lastly presented.