The bell is ringing in speed-scaled multiprocessor scheduling

Abstract

This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

On multi-processor speed scaling with migration

“…which is again the bound of(8).Completion/arrival case When OPT finishes a job J k the potential does not change because in the terms W OPT (i) or W OPT (i ) the remaining processing volume of J k simply goes to 0. Similarly if OA(m) completes a job already finished by OPT, the potential does not change.…”

“…as stated in(8).Next consider the case that the job J k executed on processor l in OPT's schedule is not executed by OA(m) on processor l and hence is not executed on any of its processors at time t. If J k is still unfinished by OA(m), then let J i be the set containing J k . Recall that OA(m)'s schedule is optimal.…”

Throughput maximization in multiprocessor speed-scaling

“…Albers et al proposed a 2(2 − 1/m) α -approximation algorithm for instances with common release dates, or common deadlines, and an (α α 2 4α )-approximation algorithm for instances with agreeable deadlines. Greiner et al [21] proposed a B α -approximation algorithm for general instances, where B α is the α-th Bell number. Recently, the approximation ratio for agreeable instances has been improved to (2 − 1/m) α−1 in [9].…”

Scheduling for energy minimization on restricted parallel processors

“…They reduced the problem to Qm|prec|C max and obtained a poly-log(m)-approximation algorithm assuming that the processors can change speed continuously over time. Greiner et al [12] presented research on the trade-off between energy and delay; i.e., their objective was to minimize the sum of the energy cost and delay cost. They suggested a randomized algorithm RA for multiple processors: each task was assigned uniformly at random to a processor, and then the single-processor algorithm A was applied separately to each processor.…”

“…The number of CPU cycles w j executed in a time interval is the speed integrated over time, and the energy consumption E j is the power integrated over time; that is, w j =  s jt dt and E j =  (s jt ) α dt, following the classical models in the literature [4,27,16,15,8,1,23,12,3]. Note that in this work we focus on speed scaling and all processors are alive during the whole execution, and so we do not take static energy into account [4,1,23,12]. Let c j be the time when the task J j finishes its execution.…”