On multi-processor speed scaling with migration

Abstract

We investigate a very basic problem in dynamic speed scaling where a sequence of jobs, each specified by an arrival time, a deadline and a processing volume, has to be processed so as to minimize energy consumption. We study multi-processor environments with m parallel variable-speed processors assuming that job migration is allowed, i.e. whenever a job is preempted it may be moved to a different processor. We first study the offline problem and show that optimal schedules can be computed efficiently in polynomial time, given any convex non-decreasing power function. In contrast to a previously known strategy, our algorithm does not resort to linear programming. For the online problem, we extend two algorithms Optimal Available and Average Rate proposed by Yao et al. [15] for the single processor setting. Here we concentrate on power functions P (s) = s α , where s is the processor speed and α > 1 is a constant.

Green scheduling, flows and matchings

“…Polynomial-time algorithms for finding an optimal solution for S|pmtn, r j , d j |E, known as the multiprocessor speed-scaling problem with migration, have been proposed by Bingham and Greenstreet [10], Albers et al [5] and Angel et al [7]. The algorithm in [10] is based on the use of the Ellipsoid method.…”

“…the possibility to interrupt the execution of a job and resume it later, is allowed. Since then, different problems have been studied taking into account the energy consumption, mainly in the single processor case (e.g., [6,20]), but more recently in the multiprocessor case as well (e.g., [5,7,10,17]). Different algorithmic techniques have been used in order to optimally solve different speed-scaling scheduling problems, including the use of greedy algorithms, dynamic programming, convex programming, and more recently, maximum flows.…”

“…This adds a new tool for solving speed-scaling scheduling problems. More precisely, we first revisit the multiprocessor speed-scaling problem with migrations studied in [5,7,10], and we show that it can be solved easily using a convex cost flow formulation, simplifying both the existing algorithms and their proofs of optimality. This problem is the same as the one considered in [21], except that now there are m processors on which the jobs have to be executed and that the execution of an interrupted job may be continued on the same or on another processor (i.e., the migration of jobs is allowed).…”

“…In [10], Bingham and Greenstreet presented an optimal algorithm for this problem based on the use of the Ellipsoid method. In [5], Albers et al proposed an O(n 2 f (n))-time combinatorial algorithm, where f (n) is the time to compute a maximum flow in a graph. This algorithm is based on a series of maximum flow computations each one of them determining a set of jobs that will be executed with the same speed.…”

“…Independently, in [7] a similar approach has been presented. Our method simplifies the proof of optimality of the algorithm proposed in [5,7] in the price of the use as black box of a more sophisticated flow algorithm, namely a computation of a convex cost flow. This flow provides the speeds for all jobs in an optimal solution, i.e., the speeds that minimize the energy consumption which corresponds to the convex cost of the flow.…”