Online Scheduling on Two Parallel Identical Machines Under a Grade of Service Provision

In this paper, we investigate online scheduling problems on two parallel identical machines under a grade of service provision with makespan as the objective function. The jobs arrive over time and each job can be scheduled only when it has already been arrived. We consider three different versions: (i) the two machines cannot be idle at the same time until all arrived jobs have been processed; (ii) further to the first problem, jobs are processed on a first-come, first-serviced basis; (iii) each job must be assigned to one of the two machines as soon as it arrives. Respectively for three online scheduling problems, optimal online algorithms are proposed with competitive ratio $$ (\sqrt 5 + 1)/2 $$ , $$ (\sqrt 5 + 1)/2 $$ and $$ 5/3 $$ .