  We consider the following scheduling problem. There is a single machine and the jobs will arrive for completion online. Each job j is preemptive and, upon its arrival, its other characteristics are immediately revealed to the machine: the deadline requirement, the workload and the value. The objective is to maximize the aggregate value of jobs completed by their deadlines. Using the minimum of the ratios of deadline minus arrival time to workload over all jobs as the slackness s, a non-committed and a committed online scheduling algorithm is proposed in [Lucier et al., SPAA'13; Azar et al., EC'15], achieving competitive ratios of 2+f(s), where the big O notation f(s)=\mathcal{O}(\frac{1}{(\sqrt[3]{s}-1)^{2}}), and (2+f(s*b))/b respectively, where b=\omega*(1-\omega), \omega is in (0, 1), and s is no less than 1/b. In this paper, without recourse to the dual fitting technique used in the above works, we propose a simpler and more intuitive analytical framework for the two algorithms, improving the competitive ratio of the first algorithm by 1 and therefore improving the competitive ratio of the second algorithm by 1/b. As stated in [Lucier et al., SPAA'13; Azar et al. EC'15], it is justifiable in scenarios like the online batch processing for cloud computing that the slackness s is large, hence the big O notation in the above competitive ratios can be ignored. Under the assumption, our analysis brings very significant improvements to the competitive ratios of the two algorithms: from 2 to 1 and from 2/b to 1/b respectively. 
Improved Competitive Analysis of Online Scheduling Deadline-Sensitive   Jobs