An Optimal Control Framework for Online Job Scheduling with General Cost Functions

We consider the problem of online job scheduling on a single machine or multiple unrelated machines with general job/machine-dependent cost functions. In this model, each job $j$ has a processing requirement (length) $v{ij}$ and arrives with a nonnegative nondecreasing cost function $g{ij}(t)$ if it has been dispatched to machine $i$, and this information is revealed to the system upon arrival of job $j$ at time $rj$. The goal is to dispatch the jobs to the machines in an online fashion and process them preemptively on the machines so as to minimize the generalized completion time $\sum{j}g{i(j)j}(Cj)$. Here $i(j)$ refers to the machine to which job $j$ is dispatched, and $Cj$ is the completion time of job $j$ on that machine. It is assumed that jobs cannot migrate between machines and that each machine can work on a single job at any time instance. In particular, we are interested in finding an online scheduling policy whose objective cost is competitive with respect to a slower optimal offline benchmark, i.e., the one that knows all the job specifications a priori and is slower than the online algorithm. We first show that for the case of a single machine and special cost functions $gj(t)=wjg(t)$, with nonnegative nondecreasing $g(t)$, the highest-density-first rule is optimal for the generalized fractional completion time. We then extend this result by giving a speed-augmented competitive algorithm for the general nondecreasing cost functions $gj(t)$ by utilizing a novel optimal control framework. This approach provides a principled method for identifying dual variables in different settings of online job scheduling with general cost functions. Using this method, we also provide a speed-augmented competitive algorithm for multiple unrelated machines with convex functions $g{ij}(t)$, where the competitive ratio depends on the curvature of cost functions $g{ij}(t)$.