Semi-online scheduling with non-increasing job sizes and a buffer

This work considers a semi-online version of scheduling on m identical machines, where the objective is to minimize the makespan. In the variant studied here, jobs are presented sorted by non-increasing sizes, and a buffer of size k is available for storing at most k jobs. Every arriving job has to be either placed into the buffer until its assignment, or else it has to be assigned immediately to a machine. We prove a lower bound greater than 1 on the competitive ratio of the problem for any m and any buffer size. To complement this negative result, we design a simple algorithm for any m whose competitive ratio tends to 1 as the buffer size grows. Using those results, we show the best possible competitive ratio is \(1+\Theta (\frac{m}{k})\). We provide additional bounds for small values of m. In particular, we show that for \(m=2\) the case \(k=1\) is not different from the case without a buffer, while \(k=2\) admits an improved competitive ratio.