Search

Online scheduling is one of the most basic online problems. We are given a sequence of jobs in an online fashion, and they must be assigned to one of m ( ≥ 1 ) equivalent processors. Each job j with the processing time p ( j ) is given at the release time r ( j ) . An algorithm decides which processor is used to process j and must schedule it for time p ( j ) after time r ( j ) on the chosen processor. Many criteria have been proposed to evaluate the performance of scheduling algorithms, and Muthukrishnan et al. (FOCS 1999 and SIAM Journal on Computing 34 (2), 2005) used the stretch of a job j in an objective function, which is defined as the ratio of the amount of time which j spends in the system to its processing time p ( j ) . The cost of an algorithm is the total stretch to schedule all the given jobs, and our goal is to minimize it. Muthukrishnan et al. considered the case in which preemption is allowed, that is, any job can be stopped during the process and after a while the process can resume. In this paper, we show how to construct instances to obtain various lower bounds on the competitive ratio for each m . In the instances, m jobs are given regularly for a sufficiently large time span after a maliciously chosen time. The processing times of the given jobs are taken depending on the remaining processing times of uncompleted jobs at the start time of the “burst” of jobs. We prove that for the instances, the stretch of a job completed before the burst hardly affects the evaluation of a competitive ratio. Further, we provide a job sequence given before the burst for each m . Then, we can improve the previous lower bounds for any deterministic online algorithm for each m using the instances. For example, we obtain lower bounds of 1.228, 1.257 and 21 / 17 ≈ 1.235 for m = 1 , 2 and ∞, respectively. Moreover, we obtain the first non-trivial lower bounds for any randomized online algorithm for all m . [ABSTRACT FROM AUTHOR]