Recently it was shown that the response time of First-Come-First-Served (FCFS) scheduling can be stochastically and asymptotically improved upon by the {\it Nudge} scheduling algorithm in case of light-tailed job size distributions. Such improvements are feasible even when the jobs are partitioned into two types and the scheduler only has information about the type of incoming jobs (but not their size). In this paper we introduce Nudge-$M$ scheduling, where basically any incoming type-1 job is allowed to pass any type-2 job that is still waiting in the queue given that it arrived as one of the last $M$ jobs. We prove that Nudge-$M$ has an asymptotically optimal response time within a large family of Nudge scheduling algorithms when job sizes are light-tailed. Simple explicit results for the asymptotic tail improvement ratio (ATIR) of Nudge-$M$ over FCFS are derived as well as explicit results for the optimal parameter $M$. An expression for the ATIR that only depends on the type-1 and type-2 mean job sizes and the fraction of type-1 jobs is presented in the heavy traffic setting. The paper further presents a numerical method to compute the response time distribution and mean response time of Nudge-$M$ scheduling provided that the job size distribution of both job types follows a phase-type distribution (by making use of the framework of Markov modulated fluid queues with jumps).

翻译：近期研究表明，在作业尺寸分布呈轻尾特征时，通过"轻推"调度算法可在随机意义和渐近意义上改进先来先服务（FCFS）调度的响应时间。即使将作业分为两类且调度器仅知晓到达作业的类型（而非尺寸），这种改进依然可行。本文提出Nudge-M调度，其中任一到达的类型-1作业，若其在最后M个到达作业之列，则允许其优先于仍在队列中等待的类型-2作业。我们证明，在作业尺寸呈轻尾分布时，Nudge-M在较大一类轻推调度算法族中具有渐近最优响应时间。推导出Nudge-M相对于FCFS的渐近尾部改进比率的简洁显式结果，以及最优参数M的显式表达式。在重流量场景下，给出仅依赖于类型-1与类型-2平均作业尺寸及类型-1作业占比的ATIR表达式。本文进一步提出一种数值方法，在两种作业类型的尺寸分布服从相型分布时（利用带跳跃的马尔可夫调制流体队列框架），可计算Nudge-M调度的响应时间分布与平均响应时间。