Consider the Online Scheduling Problem with $ 3$ identical machines. Jobs, with arbitrary size, arrive online one after another and need to be scheduled on one of the $ 3$ machines without ever moving them again.

How can I show, that there can’t be any deterministic Online-Algorithm which achieves a competitive ratio of $ c<\frac{5}{3}$ .

This should be solved by just giving some instance $ \sigma$ and arguing that no det. algorithm can do better. Same can easily be done for $ 2$ machines and $ c<\frac{3}{2}$ . Sadly I can’t find any solution to this (more or less) textbook answer.