We consider a discrete-time system comprising a first-come-first-served queue, a non-preemptive server, and a stationary non-work-conserving scheduler. New tasks arrive at the queue according to a Bernoulli process. At each instant, the server is either busy working on a task or is available, in which case the scheduler either assigns a new task to the server or allows it to remain available (to rest). In addition to the aforementioned availability state, we assume that the server has an integer-valued activity state. The activity state is non-decreasing during work periods, and is non-increasing otherwise. In a typical application of our framework, the server performance (understood as task completion probability) worsens as the activity state increases. In this article, we expand on stabilizability results recently obtained for the same framework to establish methods to design scheduling policies that not only stabilize the queue but also reduce the utilization rate, which is understood as the infinite-horizon time-averaged expected portion of time the server is working. This article has a main theorem leading to two main results: (i) Given an arrival rate, we describe a tractable method, using a finite-dimensional linear program (LP), to compute the infimum of all utilization rates achievable by stabilizing scheduling policies. (ii) We propose a tractable method, also based on finite-dimensional LPs, to obtain stabilizing scheduling policies that are arbitrarily close to the aforementioned infimum. We also establish structural and distributional convergence properties, which are used throughout the article, and are significant in their own right.
翻译:我们考虑的是一个离散时间系统, 由先到先到先到的队列、 非先发制人服务器和固定的非工作保存排程组成。 新的任务根据伯努利进程到达队列。 服务器每时每刻都忙于执行任务或可用, 调度员要么给服务器分配新的任务, 要么允许它继续提供( 休息 ) 。 除了上述可用状态之外, 我们假设服务器有一个整数- 高度的活动状态。 活动状态在工作期间没有停止, 并且没有增加。 在我们框架的典型应用中, 服务器的性能( 任务完成概率不足) 随着活动状态的增加而恶化。 在此文章中, 我们扩展了最近为同一框架获得的可稳定性结果, 以制定不仅稳定队列, 而且降低利用率的方法。 我们的理解是, 服务器运行的时间是无限- 高度平均时间, 并且不会增加活动状态 。 文章的主要导向两个主要结果 : (i) 服务器的结构性性能随着活动状态的完成概率, 也随着活动状态的增加时间递增的递增的进度, 。 (i) 使用一个固定程序使用一个固定程序使用的方法, 我们使用一个稳定的程序使用一个稳定的递增的周期 。