Online scheduling with migration on two hierarchical machines

We consider online scheduling with migration on two hierarchical machines, with the goal of minimizing the makespan. In this model, one of the machines can run any job, while the other machine can only receive jobs from a subset of the input jobs. In addition, in this problem, there is a constant parameter $$M \ge 0$$ M ≥ 0 , called the migration factor. Jobs are presented one by one, and every arrival of a new job of size x does not only require the algorithm to assign the job to one of the machines, but it also allows the algorithm to reassign any subset of previously presented jobs, whose total size is at most $$M \cdot x$$ M · x . We show that no algorithm with a finite migration factor has a competitive ratio below $$\frac{3}{2}$$ 3 2 , and design an algorithm with this competitive ratio and migration factor 1. We prove that this is the best possible result, in the sense that no algorithm with a smaller migration factor can have a competitive ratio of $$\frac{3}{2}$$ 3 2 . This provides tight bounds on the competitive ratio for all values $$M\ge 1$$ M ≥ 1 . We also find tight bounds on the competitive ratio for many other values of M.