Scheduling tasks with exponential duration on unrelated parallel machines

This 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 $\Sigma$ such that if tasks are assigned to workers according to $\Sigma$, 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 ${\lambda }_{wt}$ 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.