Online scheduling on an unbounded parallel-batch machine and a standard machine to minimize makespan

We consider the online scheduling on an unbounded parallel-batch machine and a standard machine to minimize makespan. In the problem, the jobs arrive online over time and to be processed on two machines M1 and M2. M1 is an unbounded parallel-batch machine and M2 is a standard machine. The objective is to minimize the makespan, i.e., the maximum completion time of all jobs. For this problem, we present an online algorithm of competitive ratio 5+54≈1.809.