Makespan Minimization of task graphs with random task running times

The problem of scheduling a set of tasks on two parallel and identical processors is considered. The executions of tasks are constrained by precedence relations. The running times of the tasks are independent random variables with a common exponential distribution. The goal of scheduling is to minimize the makespan, i.e. the maximum task completion time. A simple optimal preemptive policy is proven to stochastically minimize the makespan when the precedence graph belongs to a class of forest-cut graphs. 1. Introduction Parallel programs are usually represented by task graphs which are directed acyclic graphs where vertices represent tasks and arcs represent precedence relations between tasks. The executions of these tasks have to satisfy these precedence constraints in such a way that a task can start execution only when all its predecessor tasks have completed execution. For any given task graph, the scheduling problem consists in assigning tasks to a set of processors in such a way that the makespan, i.e. the maximum task completion time, is minimized. Due to the partial order relation deened on the set of tasks, such a scheduling problem is NP-hard in general (see Ullman 15]), even when the task running times are equal. The reader is referred to Liu and Sanlaville 10] for a survey of complexity results and optimal polynomial algorithms for special cases. Most scheduling literature is concerned with the deterministic scheduling problem , where task running times are assumed to be known constants. However, in practice, task running times are diicult to obtain in advance. For example, when a task contains loops, the number of iterations can depend on the input data. Even if a task contains a xed number of executions of several instructions, the number of memory cycles per instruction can depend on the contentions on shared memory. In this paper, we consider the scheduling problem where task running times are random variables. More speciically, we assume that task running times are independent and identically distributed (i.i.d.) random variables with a common