AI 中文总结
研究具有一个快速服务器和两个相同慢速服务器的排队系统最优控制问题，证明阈值策略最优，核心技术思想由GPT-5.5 Pro生成，作者验证证明并改写论文，关键引理在Lean 4中得到验证。
AI 中文摘要
本文研究了具有三个服务器（一个快速服务器和两个相同慢速服务器）的排队系统的最优控制问题。Lin和Kumar（1984）曾引入过该问题的双服务器版本（一个快速服务器和一个慢速服务器），其最优策略已被证明是阈值类型。然而，将此结果推广到双服务器设置之外一直是个开放问题。本文通过证明所考虑的三服务器系统的阈值策略是最优的，解决了第一个非平凡情况。本文的核心技术思想由GPT-5.5 Pro生成，作者还包含了一份描述与GPT-5.5 Pro交互的简短报告，并对证明进行了验证，为使论文更严谨、清晰和易读进行了重写。此外，三个关键引理也已在Lean 4中得到验证。
英文摘要
This paper studies the optimal control problem of a queueing system with three servers: one fast server and two identical slow servers. The two-server version of this problem, with one fast server and one slow server, was introduced by Lin and Kumar (1984), and the optimal policy has been shown to be of threshold type. However, generalizing this result beyond the two-server setting has been considered an open problem. In this paper, we resolve the first nontrivial case by proving that a threshold policy is optimal for the three-server system considered. The core technical ideas in this paper are generated by GPT-5.5 Pro. We have included a short report describing the authors' interactions with GPT-5.5 Pro. The authors verified the proofs and rewrote the paper for better rigor, clarity, and exposition. In addition, three key lemmas have also been verified in Lean 4.
