Online Job Scheduling on a Single Machine with General Cost Functions

We consider the problem of online job scheduling on a single machine with general job-dependent cost functions. In this model, each job j has a processing requirement (length) vj and arrives with a nonnegative nondecreasing cost function gj(t), and this information is revealed to the system upon arrival of job j at time rj. The goal is to schedule the jobs preemptively on the machine in an online fashion so as to minimize the generalized completion time $\sum\limits_{} {_j{g_j}} \left({{C_j}}\right)$, where Cj is the completion time of job j on the machine. It is assumed that the machine has a unit processing speed that 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. Under some mild assumptions, we provide a speed-augmented competitive algorithm for general nondecreasing cost functions gj(t) by utilizing a novel optimal control framework.