发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究部分队列信息下的负载均衡问题，将其建模为RMAB，通过定义新状态验证Whittle可索引性，并推导出低复杂度的Whittle索引策略，分别处理有限队列和阻塞队列模型。
AI 中文摘要
我们考虑一个包含一个调度器和$N$个并行服务器的系统中的负载均衡问题。调度器必须在每个时间步选择一个服务器来分发新作业，并且每个服务器在队列中缓冲到达的作业。然而，调度器不知道服务器的积压情况，必须基于先前的观察做出调度决策。调度器的目标是将作业分发到最短的队列。这个问题可以表述为一个 restless 多臂老虎机（RMAB）问题，其中每个臂的状态是其对应的信念向量。我们的目标是验证该问题的 Whittle 可索引性，并推导出一种低复杂度的 Whittle 索引策略。由于信念向量的多维性，先前的 Whittle 可索引性结果不能直接应用。为了克服这个问题，我们将 RMAB 状态定义为最近一次积压观察与自该观察以来的时间的元组。我们考虑两种模型变体：标准有限队列模型和阻塞队列模型。我们证明了这两种模型的单臂解耦问题在某些假设下具有阈值最优解。对于标准有限队列模型，我们证明了可索引性，并推导出了闭式形式的 Whittle 索引策略。对于阻塞队列模型，我们推导出了在阈值最优性下可索引性的充分条件，并用它来证明某些特殊情况下的可索引性。
英文摘要
We consider the problem of load balancing in a system with one dispatcher and $N$ parallel servers. The dispatcher must select one server to dispatch new jobs at every time-step and each server buffers incoming jobs in a queue. However, the dispatcher does not know the servers' backlogs and must make dispatching decisions based on previous observations. The dispatcher's objective is to dispatch jobs to the shortest queue. This problem can be formulated as a restless multi-arm bandit (RMAB) problem where each arm's state is its corresponding belief vector. Our goal is to verify Whittle indexability for this problem and derive a low complexity Whittle index policy. Previous Whittle indexability results cannot be directly applied due to the multi-dimensional nature of the belief vector. To overcome this issue, we define the RMAB state as the tuple of the most recent backlog observation and the time since this observation. We consider two model variations, a standard finite queue model and a blocking queue model. We show that the single-arm decoupled problems of both these models have threshold optimal solutions under some assumptions. For the standard finite queue model, we prove indexability and derive the Whittle index policy in closed form. For the blocking queue model, we derive a sufficient condition for indexability under threshold optimality and use it to show indexability for some special cases.
