Online and semi-online hierarchical scheduling for load balancing on uniform machines

Abstract

In this paper, we consider online and semi-online hierarchical scheduling for load balancing on m parallel uniform machines with two hierarchies. There are k machines with speed s and hierarchy 1 which can process all the jobs, while the remaining m − k machines with speed 1 and hierarchy 2 can only process jobs with hierarchy 2. For the model with the objective to maximize the minimum machine completion time, we show that no online algorithm can achieve bounded competitive ratio, i.e., there exists no competitive algorithm. We overcome this barrier by way of fractional assignment, where each job can be arbitrarily split between the machines. We design a best possible algorithm with competitive ratio 2 k s + m − k k s + m − k for any 0 < s < ∞ . For the model with the objective of minimizing makespan, we give a best possible algorithm with competitive ratio ( k s + m − k ) 2 k 2 s 2 + k ( m − k ) s + ( m − k ) 2 for any 0 < s < ∞ . For the semi-online model with the objective to minimize makespan, where the total job size ( s u m ) is known in advance, we present an optimal algorithm (i.e., an algorithm of competitive ratio 1).