Energy-Efficient Multiprocessor Scheduling for Flow Time and Makespan

We consider energy-efficient scheduling on multiprocessors, where the speed of each processor can be individually scaled, and a processor consumes power $s^{\alpha}$ when running at speed $s$, for $\alpha>1$. A scheduling algorithm needs to decide at any time both processor allocations and processor speeds for a set of parallel jobs with time-varying parallelism. The objective is to minimize the sum of the total energy consumption and certain performance metric, which in this paper includes total flow time and makespan. For both objectives, we present instantaneous parallelism clairvoyant (IP-clairvoyant) algorithms that are aware of the instantaneous parallelism of the jobs at any time but not their future characteristics, such as remaining parallelism and work. For total flow time plus energy, we present an $O(1)$-competitive algorithm, which significantly improves upon the best known non-clairvoyant algorithm and is the first constant competitive result on multiprocessor speed scaling for parallel jobs. In the case of makespan plus energy, which is considered for the first time in the literature, we present an $O(\ln^{1-1/\alpha}P)$-competitive algorithm, where $P$ is the total number of processors. We show that this algorithm is asymptotically optimal by providing a matching lower bound. In addition, we also study non-clairvoyant scheduling for total flow time plus energy, and present an algorithm that achieves $O(\ln P)$-competitive for jobs with arbitrary release time and $O(\ln^{1/\alpha}P)$-competitive for jobs with identical release time. Finally, we prove an $\Omega(\ln^{1/\alpha}P)$ lower bound on the competitive ratio of any non-clairvoyant algorithm, matching the upper bound of our algorithm for jobs with identical release time

Speed Scaling for Energy Aware Processor Scheduling: Algorithms and Analysis

We present theoretical algorithmic research of processor scheduling in an energy aware environment using the mechanism of speed scaling. We have two main goals in mind. The first is the development of algorithms that allow more energy efficient utilization of resources. The second goal is to further our ability to reason abstractly about energy in computing devices by developing and understanding algorithmic models of energy management. In order to achieve these goals, we investigate three classic process scheduling problems in the setting of a speed scalable processor. Integer stretch is one of the most obvious classical scheduling objectives that has yet to be considered in the speed scaling setting. For the objective of integer stretch plus energy, we give an online scheduling algorithm that, for any input, produces a schedule with integer stretch plus energy that is competitive with the integer stretch plus energy of any schedule that finishes all jobs. Second, we consider the problem of finding the schedule, S, that minimizes some quality of service objective Q plus B times the energy used by the processor. This schedule, S, is the optimal energy trade-off schedule in the sense that: no schedule can have better quality of service given the current investment of energy used by S, and, an additional investment of one unit of energy is insufficient to improve the quality of service by more than B. When Q is fractional weighted flow, we show that the optimal energy trade-off schedule is unique and has a simple structure, thus making it easy to check the optimality of a schedule. We further show that the optimal energy trade-off schedule can be computed with a natural homotopic optimization algorithm. Lastly, we consider the speed scaling problem where the quality of service objective is deadline feasibility and the power objective is temperature. In the case of batched jobs, we give a simple algorithm to compute the optimal schedule. For general instances, we give a new online algorithm and show that it has a competitive ratio that is an order of magnitude better than the best previously known for this problem

Nonmigratory multiprocessor scheduling for response time and energy

Energy usage has been an important concern in recent research on online job scheduling, where processors are allowed to vary the speed dynamically so as to save energy whenever possible. Notice that providing good quality of service such as response time (flow time) and conserving energy are conflicting objectives. An interesting problem for scheduling is how to optimize an economic tradeoff of flow time and energy. To this end, the past two years have witnessed significant progress in the single-processor setting, and online algorithms with performance close to optimal have been obtained. In this paper we extend the study of optimizing the tradeoff between flow time and energy to the multi-processor setting. We derive and analyze a simple non-migratory online algorithm that makes use of the classified-round-robin (CRR) strategy to dispatch jobs. Even in the worst case, its performance is within O(log P) times of the optimal migratory offline algorithm, where P is the ratio of the maximum job size to the minimum job size. Technically speaking, this online result stems from a non-trivial solution to an offline problem of eliminating migration, which is also interesting by itself. © 2008 IEEE.link_to_subscribed_fulltex