This paper analyzes the online facility assignment problem in a geometric setting where facilities with unit capacity are positioned at the vertices of a regular $n$-gon. Customers arrive sequentially at uniformly random positions along the edges. They must be assigned immediately to the nearest available facility, with ties broken by coin toss. The sequential nature and unknown future arrivals require a probabilistic analysis of the expected assignment cost. Our main contribution is a recursive characterization of the expected cost: for any occupancy state $S$, the expected remaining cost $V(S)$ equals the average over all edge positions of the immediate assignment cost plus the expected future cost after assignment. We prove that this integral equation can calculate a solution and provide the expected value for small $n$ ($n = 3, 4, 5$). For larger values of $n$ and expected cost, we develop efficient numerical methods, including a discretized dynamic programming approach and Monte Carlo simulation. The work establishes a fundamental probabilistic approach for online assignment in polygonal environments.

翻译：本文分析了一种几何设定下的在线设施分配问题，其中容量为单位的设施位于正$n$边形的顶点上。顾客按顺序均匀随机地抵达各条边上的位置，必须立即被分配至最近的可用设施，若距离相等则通过抛硬币决定。这种顺序特性及未来到达的未知性要求对期望分配成本进行概率分析。我们的主要贡献是期望成本的递归刻画：对于任意占用状态$S$，期望剩余成本$V(S)$等于所有边位置上立即分配成本与分配后期望未来成本的平均值。我们证明了该积分方程可计算解，并给出了小$n$（$n = 3, 4, 5$）时的期望值。针对更大的$n$值与期望成本，我们开发了高效的数值方法，包括离散化动态规划方法与蒙特卡洛模拟。该工作为多边形环境中的在线分配问题建立了一种基础的概率分析方法。