On Stochastic Scheduling with In-Tree Precedence Constraints

We consider the problem of optimal scheduling of a set of jobs obeying in-tree precedence constraints, when a number M of processors is available. It is assumed that the service times of different jobs are independent identically distributed random variables. Subject to a minor assumption on the service time distribution, we show that policies of the "Highest Level First" type are optimal asymptotically, as the number of jobs tends to infinity.