Shortest Path Distance in Manhattan Poisson Line Cox Process
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Shortest Path Distance in Manhattan Poisson Line Cox Process Vishnu Vardhan Chetlur1
· Harpreet S. Dhillon1 · Carl P. Dettmann2
Received: 7 April 2020 / Accepted: 12 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Stochastic geometry · Manhattan Poisson line process · Manhattan Poisson line Cox Process · Path distance · Shortest path
1 Introduction The study of random spatial patterns, formally called stochastic geometry, has played an important role in statistical physics. Some of the well known examples include the study of percolation over both lattices and random sets of points, referred to as point processes [1–4], as well as the characterization of the properties of tessellations formed by point processes and random sets of lines called line processes [5,6]. In fact, as will be discussed shortly, the modern treatment of line processes was inspired by the study of particle trajectories in
Communicated by Eric A. Carlen.
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Vishnu Vardhan Chetlur [email protected] Harpreet S. Dhillon [email protected] Carl P. Dettmann [email protected]
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Wireless@VT, Department of ECE, Virginia Tech, Blacksburg, VA, USA
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School of Mathematics, University of Bristol, Bristol, UK
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V. V. Chetlur et al.
a cloud-chamber experiment [7]. The Poisson line process (PLP), which will be defined formally in Sect. 2, is often the preferred choice for analysis in this line of work due to its tractability [8–10]. Given its rich history, a lot is already known about the distributional properties of a PLP [11,12]. However, there has been a growing interest in a doubly stochastic point process that is constructed by defining a random set of points on each line of a PLP in R2 , which is relatively less understood and is the focus of this paper. Specifically, we focus on the distribution of the shortest distance between two points of this point process when traveling only along the random lines. This distance, which will henceforth be referred to as the shortest path distance, has not been analytically characterized in the literature yet. Before formulating the problem mathematically, it is instructive to discuss the rich history of PLP and the context in which this new doubly stochastic point proc
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