Online busy time scheduling with flexible jobs

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.