Online scheduling to minimize modified total tardiness with an availability constraint

We consider online scheduling problems to minimize modified total tardiness. The problems are online in the sense that jobs arrive over time. For each job Jj, its processing time pj, due date dj and weight wj become known at its arrival time (or release time) rj. Preemption is not allowed. We first show that there is no finite competitive ratio for problem 1|online,rj,dj|∑wjTj. So we focus on problem 1|online,rj,dj|∑wj(Tj+dj) and show that D-SWPT (Delayed Shortest Weighted Processing Time) algorithm is 3-competitive. We further study two problems 1|online,rj,dj,h(1),res|∑wj(Tj+dj) and 1|online,rj,dj,h(1),N−res|∑wj(Tj+dj), where res and N−res denote resumable and non-resumable models respectively, and h(1) denotes a non-available time interval [s,αs] with s>0 and α≥1. We give a lower bound of 1+α for both problems and prove that M−D−SWPT (Modified D-SWPT) is 3α and 6α-competitive in the resumable and non-resumable models, respectively. Moreover, we extend the upper bounds to the scenario of parallel machine scheduling with uniform job weight and an assumption that all machines have the same non-available time interval [s,αs]. A lower bound of is given as well for the scenario.