Throughput Maximization in Multiprocessor Speed-Scaling

We are given a set of n jobs that have to be executed on a set of m speed-scalable machines that can vary their speeds dynamically using the energy model introduced in [Yao et al., FOCS’95]. Every job j is characterized by its release date rj , its deadline dj , its processing volume pi,j if j is executed on machine i and its weight wj . We are also given a budget of energy E and our objective is to maximize the weighted throughput, i.e. the total weight of jobs that are completed between their respective release dates and deadlines. We propose a polynomial-time approximation algorithm where the preemption of the jobs is allowed but not their migration. Our algorithm uses a primal-dual approach on a linearized version of a convex program with linear constraints. Furthermore, we present two optimal algorithms for the non-preemptive case where the number of machines is bounded by a fixed constant. More specifically, we consider: (a) the case of identical processing volumes, i.e. pi,j = p for every i and j, for which we present a polynomial-time algorithm for the unweighted version, which becomes a pseudopolynomial-time algorithm for the weighted throughput version, and (b) the case of agreeable instances, i.e. for which ri ≤ rj if and only if di ≤ dj , for which we present a pseudopolynomial-time algorithm. Both algorithms are based on a discretization of the problem and the use of dynamic programming.