Speed-scaling with no Preemptions

We revisit the non-preemptive speed-scaling problem, in which a set of jobs have to be executed on a single or a set of parallel speed-scalable processor(s) between their release dates and deadlines so that the energy consumption to be minimized. We adopt the speed-scaling mechanism first introduced in [Yao et al., FOCS 1995] according to which the power dissipated is a convex function of the processor's speed. Intuitively, the higher is the speed of a processor, the higher is the energy consumption. For the single-processor case, we improve the best known approximation algorithm by providing a $(1+\epsilon)^{\alpha}\tilde{B}_{\alpha}$-approximation algorithm, where $\tilde{B}_{\alpha}$ is a generalization of the Bell number. For the multiprocessor case, we present an approximation algorithm of ratio $\tilde{B}_{\alpha}((1+\epsilon)(1+\frac{w_{\max}}{w_{\min}}))^{\alpha}$ improving the best known result by a factor of $(\frac{5}{2})^{\alpha-1}(\frac{w_{\max}}{w_{\min}})^{\alpha}$. Notice that our result holds for the fully heterogeneous environment while the previous known result holds only in the more restricted case of parallel processors with identical power functions

Energy Efficient Scheduling and Routing via Randomized Rounding

We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same quality as for the homogeneous environment. By exploiting the result for the preemptive-non-migratory variant, we are able to improve the best known approximation ratio for the single processor non-preemptive problem. Furthermore, we show that our approach allows to obtain a constant-factor approximation algorithm for the power-aware preemptive job shop scheduling problem. Finally, we consider the min-power routing problem where we are given a network modeled by an undirected graph and a set of uniform demands that have to be routed on integral routes from their sources to their destinations so that the energy consumption is minimized. We improve the best known approximation ratio for this problem.Comment: 27 page

Energy-efficient algorithms for non-preemptive speed-scaling

We improve complexity bounds for energy-efficient speed scheduling problems for both the single processor and multi-processor cases. Energy conservation has become a major concern, so revisiting traditional scheduling problems to take into account the energy consumption has been part of the agenda of the scheduling community for the past few years. We consider the energy minimizing speed scaling problem introduced by Yao et al. where we wish to schedule a set of jobs, each with a release date, deadline and work volume, on a set of identical processors. The processors may change speed as a function of time and the energy they consume is the $\alpha$th power of its speed. The objective is then to find a feasible schedule which minimizes the total energy used. We show that in the setting with an arbitrary number of processors where all work volumes are equal, there is a $2(1+\varepsilon)(5(1+\varepsilon))^{\alpha -1}\tilde{B}_{\alpha}=O_{\alpha}(1)$ approximation algorithm, where $\tilde{B}_{\alpha}$ is the generalized Bell number. This is the first constant factor algorithm for this problem. This algorithm extends to general unequal processor-dependent work volumes, up to losing a factor of $(\frac{(1+r)r}{2})^{\alpha}$ in the approximation, where $r$ is the maximum ratio between two work volumes. We then show this latter problem is APX-hard, even in the special case when all release dates and deadlines are equal and $r$ is 4. In the single processor case, we introduce a new linear programming formulation of speed scaling and prove that its integrality gap is at most $12^{\alpha -1}$. As a corollary, we obtain a $(12(1+\varepsilon))^{\alpha -1}$ approximation algorithm where there is a single processor, improving on the previous best bound of $2^{\alpha-1}(1+\varepsilon)^{\alpha}\tilde{B}_{\alpha}$ when $\alpha \ge 25$

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Non-clairvoyant Scheduling Games

In a scheduling game, each player owns a job and chooses a machine to execute it. While the social cost is the maximal load over all machines (makespan), the cost (disutility) of each player is the completion time of its own job. In the game, players may follow selfish strategies to optimize their cost and therefore their behaviors do not necessarily lead the game to an equilibrium. Even in the case there is an equilibrium, its makespan might be much larger than the social optimum, and this inefficiency is measured by the price of anarchy -- the worst ratio between the makespan of an equilibrium and the optimum. Coordination mechanisms aim to reduce the price of anarchy by designing scheduling policies that specify how jobs assigned to a same machine are to be scheduled. Typically these policies define the schedule according to the processing times as announced by the jobs. One could wonder if there are policies that do not require this knowledge, and still provide a good price of anarchy. This would make the processing times be private information and avoid the problem of truthfulness. In this paper we study these so-called non-clairvoyant policies. In particular, we study the RANDOM policy that schedules the jobs in a random order without preemption, and the EQUI policy that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time

Adaptive Dispatching of Tasks in the Cloud

The increasingly wide application of Cloud Computing enables the consolidation of tens of thousands of applications in shared infrastructures. Thus, meeting the quality of service requirements of so many diverse applications in such shared resource environments has become a real challenge, especially since the characteristics and workload of applications differ widely and may change over time. This paper presents an experimental system that can exploit a variety of online quality of service aware adaptive task allocation schemes, and three such schemes are designed and compared. These are a measurement driven algorithm that uses reinforcement learning, secondly a "sensible" allocation algorithm that assigns jobs to sub-systems that are observed to provide a lower response time, and then an algorithm that splits the job arrival stream into sub-streams at rates computed from the hosts' processing capabilities. All of these schemes are compared via measurements among themselves and with a simple round-robin scheduler, on two experimental test-beds with homogeneous and heterogeneous hosts having different processing capacities.Comment: 10 pages, 9 figure