Energy Efficient Scheduling via Partial Shutdown

Abstract

Motivated by issues of saving energy in data centers we define a collection of new problems referred to as "machine activation" problems. The central framework we introduce considers a collection of m machines (unrelated or related) with each machine i having an activation cost of ai. There is also a collection of n jobs that need to be performed, and pi,j is the processing time of job j on machine i. Standard scheduling models assume that the set of machines is fixed and all machines are available. However, in our setting, we assume that there is an activation cost budget of A -we would like to select a subset S of the machines to activate with total cost a(S) ≤ A and find a schedule for the n jobs on the machines in S minimizing the makespan (or any other metric).We consider both the unrelated machines setting, as well as the setting of scheduling uniformly related parallel machines, where machine i has activation cost ai and speed si, and the processing time of job j on machine i is pi,j = p j s i , where pj is the processing requirement of job j.For the general unrelated machine activation problem, our main results are that if there is a schedule with makespan T and activation cost A then we can obtain a schedule with makespan (2+ )T and activation cost 2(1+ 1 )(ln n OP T +1)A, for any > 0. We also consider assignment costs for jobs as in the generalized assignment problem, and using our framework, provide algorithms that minimize the machine activation and the assignment cost simultaneously. In addition, we present a greedy algorithm which only works for the basic version and yields a makespan of 2T and an activation cost A(1 + ln n).For the uniformly related parallel machine scheduling problem, we develop a polynomial time approximation scheme that outputs a schedule with the property that the activation cost of the subset of machines is at most A and the makespan is at most (1 + )T for any > 0. For the special case of m identical speed machines, the machine activation problem is trivial, since the cheapest subset of k machines is always the best choice if the optimal solution activates k machines. In addition, we consider the case when some jobs can be dropped (and are treated as outliers).

Online Mixed Packing and Covering

“…In addition to its theoretical significance, one of the motivations for this problem comes from the need to minimize energy consumption in large data centers, such as those used by Google and Amazon (see [12,29] for the practical significance of this problem). This problem has been studied earlier in both the offline [19,29] and online [17,27,28] models. While near-optimal algorithms were known in the offline model, we give the first near-optimal online algorithm for this problem.…”

“…The analysis for this algorithm involves proving a bound on the value of the cumulative potential function over all the facilities. We then adapt randomized rounding techniques used for offline machine scheduling with startup costs [29] and online set cover [15] to round the fractional solution and obtain an integral assignment of jobs to machines.…”

“…The (offline) UMSC problem was first studied in [19,29]. They give offline algorithms for the problem using different techniques.…”

A Min-Edge Cost Flow Framework for Capacitated Covering Problems

“…al. [22], in which jobs must be assigned to machines, each machine has an associated cost and capacity, and subject to staying within a budget, the goal is to purchase a set of machines and assign jobs to them to minimize the maximum load on any machine. This problem is clearly Set Cover-hard.…”

Bi-criteria scheduling with machine assignment costs

“…T'kindt et al (2001) considered the problem to minimize the sum of machine cost plus balancedness defined by the maximum difference among the total processing times on individual machines. Most related work was done by Khuller et al (2010). They considered the offline scheduling problem with machine activation cost as well as with machine cost.…”