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 ad 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：最近研究表明，在作业大小分布呈轻尾特征的情况下，Nudge调度算法能够在随机性和渐近性上改进先来先服务（FCFS）调度的响应时间。即使作业被分为两类且调度器仅获知新到作业的类型（而非其大小），这种改进依然可行。本文提出Nudge-M调度算法，其核心机制是允许任何新到的类型1作业，只要它属于最近到达的M个作业之一，即可超越仍在队列中等待的类型2作业。我们证明，在作业大小呈轻尾分布时，Nudge-M在Nudge调度算法大类中具有渐近最优的响应时间。给出了Nudge-M相较于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，提出了一个仅依赖于类型1和类型2平均作业大小及类型1作业占比的ATIR表达式。本文进一步提出了一种数值方法，在两类作业大小分布均服从相位型分布时（利用带跳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。