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，在作业尺寸分布为轻尾分布的情况下，{\it Nudge}调度算法能够在随机性和渐近性意义上改进先来先服务（FCFS）调度的响应时间。即使作业被划分为两种类型，且调度器仅了解到达作业的类型信息（而非作业尺寸），这种改进仍然是可行的。本文提出了Nudge*$(M)$调度算法，其核心机制是：任意到达的类型-1作业，若在最后到达的$M$个作业之列，则允许其超越仍在队列中等待的所有类型-2作业。我们证明，当作业尺寸为轻尾分布时，在Nudge调度算法的大类族中，Nudge*$(M)$具有渐近最优的响应时间。本文推导了Nudge*$(M)$相对于FCFS的渐近尾部改进比（ATIR）的显式简洁结果，并给出了最优参数$M$的显式结果。在高负载场景下，我们给出了一个仅依赖于类型-1和类型-2的平均作业尺寸以及类型-1作业占比的ATIR表达式。此外，本文提出了一种数值方法，通过利用马尔可夫调制流体队列与跳跃模型框架，在两种作业类型的尺寸分布均服从相型分布的条件下，计算Nudge*$(M)$调度的响应时间分布与平均响应时间。