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...

We investigate the problem of scheduling a sequence of jobs running in a centralized parallel processing system with identical processors. The jobs represent parallel programs that contain probabilistic loops of tasks that can be simultaneously executed. We show that the Smallest Phase first policy is optimal within the class of nonpreemptive policies when the task processing times are identical and independently distributed random variables with an increasing likelihood ratio distribution. The optimality extends to the class of preemptive policies when the task processing times have an exponential distribution. The optimality is understood to be the stochastic minimization of the process of the numbers of jobs in the system and the minimization of mean response times of the jobs. Stronger optimality results on the minimization of job response time are obtained for a simpler job model.

: 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...

We consider preemptive and nonpreemptive scheduling of partially ordered tasks on parallel processors, where the precedence relations have an interval order, an in-forest, or a uniform out-forest structure. Processing times of tasks are random variables with an ILR (increasing in likelihood ratio) distribution in the nonpreemptive case and an exponential distribution in the preemptive case. We consider a general cost that is a function of time and of the uncompleted tasks and show that the Most Successors policy (MS) stochastically minimizes the cost function when it satisfies certain agreeability conditions. A consequence is that MS stochastically minimizes makespan, weighted flowtime, and weighted number of late jobs.