Online busy time scheduling with flexible jobs
Susanne Albers, G. Wessel van der Heijden
TL;DR
The paper studies online busy time scheduling for flexible jobs, seeking to minimize the total span of busy times across machines under release/deadline constraints. It develops tight online algorithms for both unbounded and bounded parallelism, including a $2$-competitive approach for unbounded parallelism with uniform jobs and for unbounded parallelism with agreeable deadlines, and a $9$-competitive approach for bounded parallelism with arbitrary processing times and $p_{\max}$ lookahead. Key techniques include flag-based scheduling and interval expansions, as well as track/bundle decompositions and a greedy tracking algorithm that leverages lookahead. The results clarify how parallelism constraints and lookahead affect achievable competitive ratios, providing both lower bounds and constructive algorithms with practical implications for energy-aware cloud scheduling.
Abstract
We consider the online busy time scheduling problem motivated by energy and cost minimization in cloud computing systems. The input is a set of jobs $J=\{1,\dots,n\}$ where each job $j\in J$ has a release time $r_j$, deadline $d_j$, and processing time $p_j$. $m$ homogeneous machines are given with a parallelism parameter $g\geq 1$, which is the maximal number of jobs that can be processed simultaneously on a machine. A machine is called \emph{busy} when at least one job is being processed. The objective is to find a feasible schedule for all jobs such that the sum of busy times over all machines is minimized. We consider the online setting, where a job $j\in J$ is revealed at its release time $r_j$. We show multiple algorithms in different problem variants that have a tight competitive ratio. For the busy time scheduling problem, uniform processing time jobs, and where the parallelism is unbounded ($g=\infty$), we show a $2$-competitive algorithm and an online adversary that shows that the algorithm is tight. For the setting where jobs have arbitrary processing time, agreeable deadlines, and the parallelism is unbounded, we show a different tight $2$-competitive algorithm. For machines with bounded parallelism, we show lower bounds on the competitive ratio of any online algorithm when $g$ is small. Furthermore, we improve the setting with arbitrary jobs where the algorithm is allowed lookahead.