Online multidimensional load balancing

Abstract

Energy efficient algorithms are becoming critically important, as huge data centers and server farms have increasing impact on monetary and environmental costs. Motivated by such issues, we study online load balancing from an energy perspective. Our framework extends recent work by Khuller, Li, and Saha (SODA 2010) to the online model. We are given m machines, each with some energy activation cost ci and d dimensions (i.e., components). There are n jobs which arrive online and must be assigned to machines. Each job induces a load on its assigned machine along each dimension. We must select machines to activate so that the total activation cost of the machines falls within a budget B and the largest load over all machines and dimensions (i.e., the makespan) by assigning jobs to active machines is at most Λ. We first study the model in which machines are unrelated and can have arbitrary activation cost. In this problem, which we call Machine Activation, we extend previous work to handle jobs which arrive online. We consider a variant where the target makespan Λ and budget B are given. The first main result is an online algorithm which is O(log(md) log(nm))-competitive on the load Λ and O(d log 2 (nm))-competitive on the energy budget B. We also address cases where one parameter is given and we are asked to minimize the other, or where we want to minimize a convex combination of the two. Running our previous algorithm in phases gives results for these variants. We prove lower bounds indicating that the effect on the competitive ratio due to multiple phases is necessary. Our second main result is in the same setting except all machines are identical and have no activation cost. We call this problem Vector Load Balancing, our objective is to minimize the largest load induced over all machines and dimensions (makespan) and the sum of the largest induced load on each machine (energy). We give an online algorithm that is O(log d)-competitive on makespan, which improves even on the best prior offline result, and O(log d)-competitive on the sum of induced loads if the target makespan is given; without this knowledge we show that it is impossible to get a competitive ratio independent of m. © 2013 Springer-Verlag.