Why is the Greedy Algorithm for Online Scheduling on uniform machines NOT competitive?


Consider the Online Scheduling Problem with m machines and different speeds.

Each instance $ \sigma$ consists of a sequence of jobs with different sizes $ j_i\in\mathbb{R^+}$ . At each step in time any algorithm gets to know the one new job of $ \sigma$ which needs to be assigned to one machine. The goal is to minimize the makespan (by the end of the online sequence \sigma$ ).

I want to find an instance such that the Greedy Algorithm, which assigns each new job to the machine which finishes it first, is only $ \Theta(\log m)$ -competitive.

Any ideas? I can’t find any articles regarding my problem.