Minimizing $\ell_2$ Norm of Flow Time by Starvation Mitigation
Tung-Wei Kuo
TL;DR
The paper addresses online single-machine scheduling to minimize the $\ell_2$ norm of flow time, balancing SRPT's low average flow time with FCFS's low maximum flow time through a starvation-mitigating hybrid policy. It introduces BAL$(\tilde{n}^{2/3})$, where $\tilde{n}$ is a bounded-error estimate of the number of jobs, and proves a competitive ratio of $O(n^{1/3})$ when $\tilde{n}=\Theta(n)$, with a robust bound $O\left(\tilde{n}^{1/3} + n^{1/2}/\tilde{n}^{1/6}\right)$ for general $\tilde{n}$. The contributions include the first theoretical demonstration that mitigating SRPT starvation improves the $\ell_2$ performance, a concrete online scheduling algorithm, and a majorization-based analytical framework supporting the guarantees. The results have practical implications for data-center peak-hour scheduling where the exact job count is not known but can be estimated with bounded error.
Abstract
The assessment of a job's Quality of Service (QoS) often revolves around its flow time, also referred to as response time. This study delves into two fundamental objectives for scheduling jobs: the average flow time and the maximum flow time. While the Shortest Remaining Processing Time (SRPT) algorithm minimizes average flow time, it can result in job starvation, causing certain jobs to experience disproportionately long and unfair flow times. In contrast, the First-Come-First-Served (FCFS) algorithm minimizes the maximum flow time but may compromise the average flow time. To strike a balance between these two objectives, a common approach is to minimize the $\ell_2$ norm of flow time. SRPT and FCFS are $O(n^{\frac{1}{2}})$-competitive for this problem, where $n$ is the number of jobs. Prior to this work, no algorithm is known to achieve a competitive ratio better than SRPT and FCFS. In this paper, we use FCFS to mitigate the starvation caused by SRPT. Given a good estimate of $n$, we prove that this approach achieves a much better competitive ratio of $O(n^{\frac{1}{3}})$. Our results provide the first theoretical evidence that mitigating starvation in SRPT leads to a provable improvement in scheduling performance.