Scheduling tasks with exponential duration on unrelated parallel machines

AbstractThis paper introduces a stochastic scheduling problem. In this problem a directed acyclic graphs (DAG) represents the precedence relations among m tasks that n workers are scheduled to execute. The question is to find a schedule Σ such that if tasks are assigned to workers according to Σ, the expected time needed to execute all the tasks is minimized. The time needed to execute task t by worker w is a random variable expressed by a negative exponential distribution with parameter λwt and each task can be executed by more than one worker at a time. In this paper, we will prove that the problem in its general form is NP-hard, but when the DAG width is constant, we will show that the optimum schedules can be found in polynomial time