Online scheduling of two identical machines under a grade of service provision

In this paper, we consider the problem of two identical machines scheduling that jobs are processed under a grade of service (GoS) provision. The jobs arrive over time and the two machines can be idle at the same time only when all the jobs arrived are completed. To our best knowledge, this problem has never been studied so far. For the problem with objective makespan minimization, we prove that any online algorithm without delay has a competitive ratio of at least(root 5+1)/2. Finally, we develop an algorithm with competitive ratio 5/3, which is a very effective online algorithm.