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

Abstract

We study street-constrained ($\ell_1$) shortest paths in a Poisson line Cox process (PLCP), where Poisson points of linear intensity $\mu$ lie on the lines of an underlying Poisson line process (PLP) of density $\lambda$. Under a one-turn restriction, we derive closed-form expressions for the distribution of the nearest-neighbor path length from (i) the typical PLCP point and (ii) the typical PLP intersection, by explicitly evaluating the relevant void probabilities via a geometric decomposition of the feasible path-length set. For the intersection case, we further provide analytically tractable upper and lower bounds that capture the impact of $\lambda$ and $\mu$. Allowing two turns from the typical point, we obtain a computable upper bound using a feasible-set shrinking argument and identify regimes in which it is tight. We also delineate parameter ranges where a one-turn route from a typical intersection can outperform a two-turn route from a typical point. Finally, we discuss how the results enable statistical performance characterization of ride-hailing services in terms of service guarantee, trip time, and consequently, derive dimensioning insights. We also illustrate qualitatively, how the results can be employed to study vehicle-to-vehicle communication broadcast messages near intersections.