While the Euclidean distance characteristics of the Poisson line Cox process (PLCP) have been investigated in the literature, the analytical characterization of the path distances is still an open problem. In this paper, we solve this problem for the stationary Manhattan Poisson line Cox process (MPLCP), which is a variant of the PLCP. Specifically, we derive the exact cumulative distribution function (CDF) for the length of the shortest path to the nearest point of the MPLCP in the sense of path distance measured from two reference points: (i) the typical intersection of the Manhattan Poisson line process (MPLP), and (ii) the typical point of the MPLCP. We also discuss the application of these results in infrastructure planning, wireless communication, and transportation networks.

中文翻译：

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曼哈顿泊松线 Cox 过程中的最短路径距离

虽然文献中已经研究了泊松线 Cox 过程 (PLCP) 的欧几里德距离特征，但路径距离的分析特征仍然是一个悬而未决的问题。在本文中，我们解决了固定曼哈顿泊松线 Cox 过程 (MPLCP) 的这个问题，它是 PLCP 的变体。具体来说，我们推导出从两个参考点测量的路径距离意义上到 MPLCP 最近点的最短路径长度的精确累积分布函数 (CDF)：(i) 曼哈顿泊松线过程的典型交点(MPLP) 和 (ii) MPLCP 的典型点。我们还讨论了这些结果在基础设施规划、无线通信和交通网络中的应用。