Online parallel machine scheduling to maximize the number of early jobs

We study a maximization problem: online scheduling on m identical machines to maximize the number of early jobs. The problem is online in the sense that all jobs arrive over time. Each job's characteristics, such as processing time and due date, become known at its arrival time. We consider the preemption-restart model, in which preemption is allowed, while once a job is restarted, it loses all the progress that has been made on this job so far. If in some schedule a job is completed before or at its due date, then it is called early (or on time). The objective is to maximize the number of early jobs. For m identical machines, we prove an upper bound 1 - (1 / 2 m) of competitive ratio and show that ECT (earliest completion time) algorithm is 1 / 2 -competitive.