Consider a system of identical server pools where tasks with exponentially distributed service times arrive as a time-inhomogenenous Poisson process. An admission threshold is used in an inner control loop to assign incoming tasks to server pools while, in an outer control loop, a learning scheme adjusts this threshold over time to keep it aligned with the unknown offered load of the system. In a many-server regime, we prove that the learning scheme reaches an equilibrium along intervals of time where the normalized offered load per server pool is suitably bounded, and that this results in a balanced distribution of the load. Furthermore, we establish a similar result when tasks with Coxian distributed service times arrive at a constant rate and the threshold is adjusted using only the total number of tasks in the system. The novel proof technique developed in this paper, which differs from a traditional fluid limit analysis, allows to handle rapid variations of the first learning scheme, triggered by excursions of the occupancy process that have vanishing size. Moreover, our approach allows to characterize the asymptotic behavior of the system with Coxian distributed service times without relying on a fluid limit of a detailed state descriptor.
翻译:考虑一个完全相同的服务器共享库系统, 即使用指数分布服务时间的任务到达时是一个时间不相容的 Poisson 进程。 在内部控制循环中, 使用入门阈值向服务器共享库分配即将到来的任务, 在外部控制循环中, 一个学习计划随着时间的推移调整这个阈值, 使其与未知的系统提供的负载保持一致。 在许多服务器系统中, 我们证明学习计划与每个服务器集合的正常提供载荷被适当捆绑的时间间隔达到平衡, 从而导致负载的均衡分布。 此外, 当 Coxian 分布服务时间达到一个恒定率, 并且仅使用系统中的任务总数来调整阈值时, 我们设定了一个类似的结果 。 本文开发的新证据技术与传统的液限分析不同, 能够处理第一个学习计划因占用过程的外推而发生快速变化, 其规模已经消失 。 此外, 我们的方法可以将系统与Coxian 分布服务时间不依赖于详细状态脱压器的流值限制, 来描述系统无症状的行为。