Shortest Path Lengths in Poisson Line Cox Processes: Approximations and Applications

Abstract

We derive exact expressions for the shortest path length to a point of a Poisson line Cox process (PLCP) from the typical point of the PLCP and from the typical intersection of the underlying Poisson line process (PLP), restricted to a single turn. For the two turns case, we derive a bound on the shortest path length from the typical point and demonstrate conditions under which the bound is tight. We also highlight the line process and point process densities for which the shortest path from the typical intersection under the one turn restriction may be shorter than the shortest path from the typical point under the two turns restriction. Finally, we discuss two applications where our results can be employed for a statistical characterization of system performance: in a re-configurable intelligent surface (RIS) enabled vehicle-to-vehicle (V2V) communication system and in electric vehicle charging point deployment planning in urban streets.

Figures (10)

• Figure 1: Illustration of the nearest $\ell_2$ vs $\ell_1$ distances. From the perspective of the typical point (black), the red point is the nearest point of the PLCP in the Euclidean plane, while, the green point is the nearest point from a path length perspective.
• Figure 2: Approximation using recursive equations
• Figure 3: All the green colored points are within a distance $t$ by taking at most one turn from the origin (black point) when (a) the origin is the typical point of the PLCP and (b) the origin is the typical intersection of the PLP.
• Figure 4: The single turn case starting from the typical intersection.
• Figure 5: The two turns case. $L_1, L_2, \ldots$ intersect the line $L_{\rm x}$.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Corollary 1: Zero Turn Case
• Corollary 2: Upper bound
• Theorem 3