Minimizing Mean Flow Time with Error Constraint

Abstract.

We consider the problem of minimizing mean flow time for the Imprecise Computation Model introduced by Lin et al. A task system TS=({T _i },{r(T _i )},{d(T _i )},{m(T _i )},{o(T _i )}) consists of a set of n independent tasks, where r(T _i ),d(T _i ),m(T _i ) , and o(T _i ) denote the ready time, deadline, execution time of the mandatory part, and execution time of the optional part of T _i , respectively. Given a task system TS and an error threshold K , our goal is to find a preemptive schedule on one processor such that the average error is no more than K and the mean flow time of the schedule is minimized. Such a schedule is called an optimal schedule. In this article we show that the problem of finding an optimal schedule is NP-hard, even if all tasks have identical ready times and deadlines. A pseudopolynomial-time algorithm is given for a set of tasks with identical ready times and deadlines, and oppositely ordered mandatory execution times and total execution times (i.e., there is a labeling of tasks such that m(T _i )≤ m(T _i+1 ) and m(T _i )+o(T _i )≥ m(T _i+1 )+o(T _i+1 ) for each 1≤ i<n ). Finally, polynomial-time algorithms are given for (1) a set of tasks with identical ready times, and similarly ordered mandatory execution times and total execution times and (2) a set of tasks with similarly ordered ready times, deadlines, mandatory execution times, and total execution times.