The expected asymptotical ratio for preemptive stochastic online problem

This paper considers the preemptive stochastic online scheduling problem on m uniform parallel machines with the objective to minimize the total weighted completion times, where the processing times of jobs are independent random variables. The actual processing time of each job is only learned upon its completion. In online environment, the existence and all parameters of each job, including its weight, release date and the distribution of processing time are unknown until its release date. During the processing of each job, it could be preempted and resumed later without extra cost. For this problem, a scheduling decision has to be made each time as the information of the processing time of the job is released step by step. Building on the Gittins index, we devise the Semi-Preemptive Policy for Uniform Machine Problem (SP) policy. With the assumption that all weights and processing times are bounded, we prove the expected asymptotical ratio of SP is one, which means approaches to one as the number of jobs increases to infinity, where Cj denotes the completion time of job Jj and OPT the offline optimum value.