Fair Online Scheduling for Selfish Jobs on Heterogeneous Machines

Scheduling jobs on multiple machines has numerous applications and has been a central topic of research in the scheduling literature. Recently, much progress has been made particularly in online scheduling with the development of powerful analysis tools. In this line of wok a centralized scheduler typically dispatches jobs to machines to exploit the given resources the best to achieve the best system performance which is measured by a certain global scheduling objective. While this approach has been very successful in attacking scheduling problems of growing complexity, the underlying assumption that jobs follow a centralized scheduler may not be realistic in certain scheduling settings. In this paper we initiate the study of online scheduling for selfish jobs in the presence of multiple machines. Selfish behavior of jobs is a common aspect observed in the absence of a centralized scheduler. We explore this question in the unrelated machines setting, arguably one of the most general multiple machine models. In this model each job can have a completely different processing time on each machine. Motivated by several practical scenarios, we assume that when a job arrives it chooses the machine that completes the job the earliest i.e. minimizes the flow time of the job. The goal is to design a local scheduling algorithm on each machine with the goal of minimizing the total (weighted) flow time. We show that the algorithm Smoothed Latest Arrival Processor Sharing, which was introduced in a recent work by Im et al. [27,28], yields an O(1 / ε2)-competitive schedule when given (1 + ε) speed. We also extend our result to minimize total flow-time plus energy consumed. To show this result we establish several interesting properties of the algorithm which could be of potential use for other scheduling problems.