Graph models for scheduling systems with machine saturation property

Let $$T \, =\, \{T_1, T_2, \ldots, T_n\}$$ be a set of n independent tasks and $$\mathcal{P}=\{P_1, P_2,\ldots, P_m\}$$ a set of m processors. During each time instant, each processor can be used by a single task at most. A schedule is for each task an allocation of one or more time intervals to one or more processors. A schedule is said to be optimal if it minimizes the maximum completion time. We say a schedule S has the machine saturation property (MS property) if, at any time instant of task execution, all the machines are simultaneously busy. In this paper, we analyze the conditions under which a parallel scheduling system allows a schedule with the MS property. While for some simple models the analytical conditions can be easily stated, a graph model approach is required when conflicts of processor usage are present. For this reason, we define the class of saturated graphs that correspond to scheduling systems with the MS property. We present efficient graph recognition algorithms to verify the MS property directly on some classes of saturated graphs