Online Scheduling on Parallel Machines with Two GoS Levels

This paper investigates the online scheduling problem on parallel and identical machines with a new feature that service requests from various customers are entitled to many different grade of service (GoS) levels. Hence each job and machine are labeled with the GoS levels, and each job can be processed by a particular machine only when the GoS level of the job is not less than that of the machine. The goal is to minimize the makespan. In this paper, we consider the problem with two GoS levels. It assumes that the GoS levels of the first k machines and the last m− k machines are 1 and 2, respectively. And every job has a GoS level of 1 alternatively or 2. We first prove the lower bound of the problem under consideration is at least 2. Then we discuss the performance of algorithm AW presented in Azar et al. (J. Algorithms 18:221–237, 1995) for the problem and show it has a tight bound of 4−1/m. Finally, we present an approximation algorithm with competitive ratio 12+4 √ 2 7 ≈ 2.522.