Performance Analysis of Work-Conserving Schedulers in Minimizing the Total Flow Time with Phase Precedence

We consider the problem of minimizing the total flow time of multiple jobs in a pool of multiple homogenous machines, where the jobs arrive over time and have to be served with phase precedence. This is a common occurrence in job scheduling for the increasingly popular data center oriented systems, where jobs need to be processed through a MAP and Reduce procedure before leaving the system. For this problem, one can construct an arrival pattern such that no scheduler can achieve a constant competitive ratio. However, what we find is that by using a slightly weaker metric of performance, which we call the efficiency ratio. We say that a scheduler achieves an efficiency ratio of γ when the flow time incurred by that scheduler divided by the minimum flow time achieved over all possible schedulers is less than or equal to γ almost surely, when the time slots or job arrivals go to infinity. We show a surprising property that any work-conserving scheduler for the flow time problem with phase precedence has a constant efficiency ratio in both preemptive and non-preemptive scenarios under some weak assumptions. We provide numerical results to support our analysis.