Lower bound on the number of processors and time for scheduling precedence graphs with communication costs

A lower bound on the number of processors and finish time for the problem of scheduling precedence graphs with communication costs is presented. The notion of the earliest starting time of a task is formulated for the context of lower bounds. A lower bound on the completion time is proposed. A task delay which does not increase the earliest completion time of a schedule is defined. Each task can then be scheduled within a time interval without affecting the lower bound performance on the finish time. This leads to definition of a new lower bound on the number of processors required to process the task graph. A derivation of the minimum time increase over the earliest completion time is also proposed for the case of a smaller number of processors. A lower bound on the minimum number of interprocessor communication links required to achieve optimum performance is proposed. Evaluation had been carried out by using a set of 360 small graphs. The bound on the finish time deviates at most by 5% from the optimum solution in 96% of the cases and performs well with respect to the minimum number of processors and communication links