This paper is concerned with the problem of scheduling preemptive tasks subject to precedence constraints in order to minimize the maximum lateness and the makespan. The number of available parallel processors is allowed to vary in time. It is shown that when an Earliest Due Date first algorithm provides an optimal nonpreemptive schedule for unitexecution -time (UET) tasks, then the preemptive priority scheduling algorithm, referred to as Smallest Laxity First, provides an optimal preemptive schedule for real-execution-time (RET) tasks. When the objective is to minimize the makespan, we get the same kind of result between Highest Level First schedules solving nonpreemptive tasks with UET and the Longest Remaining Path first schedule for the corresponding preemptive scheduling problem with RET tasks. These results are applied to four specific profile scheduling problems and new optimality results are obtained. Keywords: Preemptive Scheduling, List Schedule, Priority Schedule, Variable P...

We study the problem of scheduling unit-length tasks with release dates and deadlines subject to precedence constraints and unit communication delays. Two polynomial-time algorithms are defined: one constructs schedules for graphs with uniform release dates, the other for graphs with arbitrary release dates. They have a special structure: unlike most scheduling algorithms, they do not consider individual tasks, but pairs of tasks. It is proved that the algorithms find minimum-lateness schedules for interval orders on an arbitrary number of processors. 1 Introduction Finding a shortest schedule for a precedence graph is a very difficult problem: deciding whether a schedule for a set of tasks of length at most D exists on an arbitrary number of processors is an NP-complete problem, even without precedence constraints. The problem remains NP-complete if all task lengths are equal and the number of processors is arbitrary [11]. Only for special classes of precedence relations there are po...

In this paper, we consider the stochastic profile scheduling problem of a partially ordered set of tasks on uniform processors. The set of available processors varies in time. The running times of the tasks are independent random variables with exponential distributions. We obtain a sufficient condition under which a list policy stochastically minimizes the makespan within the class of preemptive policies. This result allows us to obtain a simple optimal policy when the partial order is an interval order, or an in-forest, or an out-forest. Keywords: Stochastic Scheduling, Profile Scheduling, Makespan, Precedence Constraint, Interval Order, In-Forest, Out-Forest, Uniform Processors, Stochastic Ordering. 1 Introduction Consider the following scheduling problem. We are given a set of tasks to be run in a system consisting of uniform processors (i.e., processors having different speeds). The executions of these tasks must satisfy some precedence constraints which are described by a dire...

: The notion of profile scheduling was first introduced by Ullman in 1975 in the complexity analysis of deterministic scheduling algorithms. In such a model, the number of processors available to a set of tasks may vary in time. Since the last decade, this model has been used to deal with systems subject to processor failures, multiprogrammed systems, or dynamically reconfigured systems. The aim of this paper is to overview optimal polynomial solutions for scheduling a set of partially ordered tasks in these systems. Particular attentions are given to a class of algorithms referred to as list scheduling algorithms. The objective of the scheduling problem is to minimize either the maximum lateness or the makespan. Results on preemptive and nonpreemptive deterministic scheduling, and on preemptive stochastic scheduling, are presented. Keywords: Deterministic Scheduling, Stochastic Scheduling, Profile Scheduling, List Schedule, Priority Schedule, Precedence Constraints, Lateness, Makespan...

. 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 wa...