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

Gourab Ghatak; Martin Haenggi; Sanjoy Kumar Jhawar

REVIEW 1 major objections 1 minor 20 references

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

T0 review · 1 major / 1 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives exact one-turn shortest-path distributions for Poisson line Cox processes, plus a computable two-turn bound, and applies them to V2V broadcasting and ride-hailing service guarantees.

desk verdict The paper tackles a real open problem and the proof strategy is sound, but the main exact formulas contradict the proofs in a load-bearing way, so as written the results are unsupported. read the letter →

arxiv 2411.16441 v2 pith:QORRAXAF submitted 2024-11-25 cs.IT math.ITstat.AP

Gourab Ghatak , Sanjoy Kumar Jhawar , Martin Haenggi This is my paper · ORCID

classification cs.ITmath.ITstat.AP MSC 60D0560G55

keywords PoissonlineprocessCoxshortestpathlengthvoidprobabilitystochasticgeometryvehicle-to-vehiclecommunicationride-hailingreconfigurableintelligentsurface

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Closed-form distributions for street-constrained paths have been missing for the isotropic Poisson line Cox process, where random streets are drawn as a Poisson line process and users sit on each street as a one-dimensional Poisson process. This paper tries to close that gap. Under a one-turn restriction, it derives exact complementary CDFs for the nearest-neighbor path length from the typical point and from the typical intersection; under a two-turn restriction, it derives a computable upper bound that is tight in identified regimes. The formulas depend directly on the street density $\lambda$ and the user density $\mu$, and the paper uses them to evaluate the probability that a ride-hailing vehicle or a vehicle-to-vehicle link meets a service threshold, and to dimension electric-vehicle charging-point placement. A reader should care because these closed forms convert a quantity that was previously available only as a mean or through asymptotics into an explicit distribution that can be evaluated and compared across network parameters.

What carries the argument

The load-bearing mechanism is the void-probability decomposition. The event $D>t$ is rewritten as the event that no point of the PLCP lies on the initial street within path length $t$, and no point lies on any crossing street within the remaining budget $t-s_i$ from its intersection with the initial street. Since the crossings of a Poisson line process with a fixed line form a one-dimensional Poisson process, the number of crossings is Poisson and their positions are uniform, so the union over all crossing lines becomes a product of exponentials that evaluates in closed form. For the typical intersection, the decomposition demands a case analysis of where a third line intersects the two streets through the typical intersection; the cases are summarized by the void segment length $Z(x,\omega_1,\omega)$ and averaged inside the integrals $T_x,T_y$. For the two-turn case, the proof shrinks the feasible set of line configurations and applies a conditional-expectation argument through $T(w,u)$, producing a nested exponential bound from the typical point.

What would settle it

Run a Monte Carlo simulation of a stationary Poisson line Cox process with fixed $\lambda$ and $\mu$, measure the empirical CCDF of the one-turn shortest path from a typical point, and compare it against Eq. (1) and against the same expression with $4\lambda t$ in place of $2\lambda t$; the matching variant identifies the correct intersection-count normalization.

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Extended reading notes

Core claim

The paper's central claim is that the one-turn nearest-neighbor path length in a Poisson line Cox process has an exact, closed-form void probability. From the typical point, Theorem 1 states the CCDF $$\bar F_D(t) = \exp\!\left(-2\mu t - 2\$\lambda$ t + \frac{\$\lambda$}{\mu}\bigl(1-$e^{{-2\mu t}}$\bigr)\right),$$ so that $F_D(t)=1-\bar F_D(t)$. From the typical intersection, Theorem 2 gives a similar exact expression, $$F_D(t)=1-\exp\!\left(-4\mu t - 2\$\lambda$(2t-T_x-T_y)\right),$$ where $T_x,T_y$ are triple integrals over the angles and crossing positions of a line that meets the two streets through the intersection, with integrand $\exp(-\mu Z(x,\omega_1,\omega))$ and $Z$ the length of the segment that must be empty of points. Theorem 3 bounds the two-turn CCDF from the typical point by a nested exponential whose innermost kernel $T(w,u)$ again enforces empty segments on the second street. If these expressions are correct, they give the first exact path-length distributions for an isotropic Poisson line Cox process, replacing earlier mean-value and asymptotic results with explicit functions of the line density $\lambda$ and point density $\mu$.

Load-bearing premise

The proof of Theorem 1 assumes that the number of street crossings on the initial street within distance $t$ of the typical point is Poisson with mean $2\lambda t$; if both sides of the point should be counted, the correct mean is $4\lambda t$, and the exponents in the main formulas must be rescaled.

Editorial extensions

If this is right

With known street and user densities, the probability that the nearest reachable user lies within a target trip time is $F_D(t)$ from Theorem 1, giving a direct service-guarantee curve for ride-hailing.
In the RIS-assisted V2V application, the probability that the nearest vehicle decodes the safety message is $F_D$ evaluated at the SNR-derived path-length threshold, so link reliability becomes a closed-form function of $\lambda$ and $\mu$.
The parameter study shows regimes of dense users and sparse streets where the one-turn route from a typical intersection is stochastically shorter than the two-turn route from a typical point; this favors intersection-anchored placement of charging points and reflectors.
The two-turn upper bound, although an inequality, is tight in those regimes and provides a conservative reachability estimate for routing with at most two turns.
The lower and upper bounds for the typical-intersection case offer computationally cheap surrogates for system performance, with the lower bound tight for large $\mu$ and the upper bound tight for large $\lambda$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the paper's approach, the same void-probability recursion should apply to any fixed number of turns as a deeper nesting of the same exponential factors; the general $k$-turn distribution is left open by the paper.
Fitted to a real street network, the two-parameter model provides a null benchmark: systematic deviations between measured trip-length distributions and these formulas would quantify anisotropy or non-Poissonian structure in the city grid.
The appearance of the ratio $\lambda/\mu$ in the exponents suggests that the shape of the distribution, not just its mean, changes across density regimes; the paper's own numerical comparison of the one-turn and two-turn curves makes these regime boundaries visible and testable.
In the V2V near-field setting, the additive RIS model turns the path-length CDF directly into a coverage probability, which implies a placement rule: favor intersection-mounted RIS when the one-turn intersection distribution dominates the two-turn point distribution.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this paper aims at a real open problem—exact one-turn shortest-path distribution in an isotropic Poisson line Cox process—and the authors correctly note that prior work only covers Manhattan, means, or asymptotics. The claimed closed forms are new, and the proof strategy (void probability via geometric decomposition) is reasonable. But the central formulas as stated do not follow from the proofs, and the discrepancies are load-bearing.

What's good: The problem is well-motivated; the comparison with the trivial recursive approximation in Sec. II is instructive; the applications to RIS and EV charging are plausible illustrations. The independence steps in the proofs are standard and mostly sound in spirit.

The soft spots are serious. Theorem 1's statement (Eq. 1) has exponent -2μt - 2λt + (λ/μ)(1-e^{-2μt}), but the proof immediately yields -2μt - 4λt + (2λ/μ)(1-e^{-2μt}). Remark 1 itself describes the term exp(-4λt) as the 'no line intersects' probability, consistent with the proof and not with Eq. (1). The proof also says N_t is Poisson with mean 2λt, but with the two-sided segment [-t,t] under the paper's Definition 1 the mean should be 4λt; either way, the stated formula is not what the proof gives. Theorem 2 has a similar mismatch: the piecewise Z in the statement (e.g., first case uses (sin ω - sin ω1)/sin(ω1-ω)) doesn't match the proof's Eq. (9), which uses (sin θ + sin θ1)/sin(θ1-θ). Theorem 3's bound is too under-specified to verify from the text. Since the applications in Section VI use Eq. (1) directly, the quantitative claims inherit the error.

This is not a case of a missing minor derivation; it's an internal contradiction between theorem statements and their own proofs. The paper could be salvageable—if the proof version is correct, the results would still be new—but as written the main exact results are unsupported. I would not cite the formulas in their current form. Still, the topic and approach merit a serious referee, because the errors are identifiable and potentially fixable, and the underlying problem is genuinely open.

Recommendation: send to peer review only if the authors are given the opportunity to fix the discrepancies; otherwise, a desk reject with a clear explanation is defensible. My call: would accept for review, with the expectation of major revision.

Referee Report

1 major / 1 minor

Summary. The paper studies shortest path (ℓ1) distances in Poisson line Cox processes (PLCPs), where points of intensity μ sit on the lines of a Poisson line process (PLP) of density λ. Under a one-turn restriction, it claims exact closed-form complementary cumulative distribution functions (CCDFs) for the nearest-neighbor path length from the typical PLCP point (Theorem 1) and from the typical PLP intersection (Theorem 2), plus an upper bound for the two-turn case from the typical point (Theorem 3). The results are then applied to RIS-assisted V2V communication and electric-vehicle charging placement. The central claims are presented as exact, parameter-free characterizations of a quantity whose distribution has been open.

Significance. If the claimed results were correct, they would constitute a substantial contribution to stochastic geometry: exact path-length distributions for an isotropic Poisson line Cox process have been open, and the applications to vehicular networks and infrastructure planning are relevant and well motivated. The paper is self-contained in its derivations, does not fit parameters to match simulations, and explicitly identifies the gap in prior work. However, the central theorems contain internal inconsistencies with their own proofs and with the model definitions, and these inconsistencies invalidate the quantitative statements that are subsequently used in the applications. The paper therefore cannot be accepted in its present form.

major comments (1)

[Section IV, Theorem 3] The proof of Theorem 3 contains a Poisson-parameter mismatch. After stating that the number n1 of lines intersecting Lx in [0,s1] is Poisson with parameter 2s1λ, the expectation is written with exp(-2λs1)(λs1)^k/k! instead of exp(-2λs1)(2λs1)^k/k!. Consequently the displayed simplification to exp(-λ∫[2 - T(w,u) f(w)] dw) corresponds to a Poisson mean of λs1, not 2λs1. This affects the bound in Eq. (14) and the subsequent numerical evaluation in Fig. 8, even if the geometric construction of the event Ei is accepted.

minor comments (1)

[Remark 1 vs Theorem 1] Remark 1 states that exp(-4λt) corresponds to the event that no line intersects within distance t, but Theorem 1 uses -2λt in the exponent; resolve this inconsistency and align the remark with the corrected theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained, and the cited external results are independent support rather than recycled conclusions.

full rationale

The paper's central claims are derived directly from standard Poisson line process properties and void-probability calculations. Theorem 1 conditions on the PPP of intersections along the typical line and averages the void probability over intersecting lines; Theorem 2 performs a geometric decomposition of feasible one-turn paths from the typical intersection; Theorem 3 uses a feasible-set shrinking argument. No fitted parameter is renamed as a prediction, and no target distribution is inserted into the derivation by definition. The cited result [4] that the number of intersections within distance t is Poisson with mean 2λt is external support, not a self-citation; even if that intensity premise is questionable, relying on an external result is not circularity. The self-citations (Refs. [7] and [8]) concern distance distributions and small-cell coverage and are not load-bearing for the shortest-path derivations. The proof of Theorem 1 as printed evaluates to exp(-2μt - 4λt + (2λ/μ)(1 - exp(-2μt))) while the theorem statement gives exp(-2μt - 2λt + (λ/μ)(1 - exp(-2μt))); this is an internal inconsistency or normalization issue, not a circular reduction, and therefore does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The model inputs lambda and mu are physical parameters, not fitted values. The main burden is the standard PLP intersection intensity and Palm conditioning. No invented entities are introduced. The derivations rely on standard results, with the noted misapplication of the intersection intensity.

assumptions (3)

standard math Number of lines intersecting a segment of length ell in a PLP of density lambda is Poisson with mean 2 lambda ell.
Invoked in the proof of Theorem 1 via [4] for N_t, and used in Theorem 2 and Corollaries. The paper applies it as mean 2 lambda t for a segment of length 2t, which is inconsistent unless lambda is redefined.
domain assumption Palm conditioning of the PLCP typical point leaves the line process as a PLP with the same density lambda and independent point processes on distinct lines.
Used implicitly throughout Section III to factor the void probability into independent line intensities.
domain assumption In the near-field ORIS model, received power is proportional to 1/(d1+di)^2 and additive over paths.
Application Section VI.A relies on this physical model from [18], not derived here.

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Cite this review

Pith. "Pith review of Shortest Path Lengths in Poisson Line Cox Processes: Approximations and Applications." pith.science (2026). https://pith.science/paper/QORRAXAF

@misc{pith2026241116441,
  author       = {Pith},
  title        = {Pith review of: Shortest Path Lengths in Poisson Line Cox Processes: Approximations and Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QORRAXAF}},
  note         = {Machine review of arXiv:2411.16441}
}

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.

Figures

Figures reproduced from arXiv: 2411.16441 by the authors.

**Figure 1.** Illustration of the nearest ℓ2 vs ℓ1 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. difference between the nearest neighbor with respect to the Euclidean distance in contrast to the same in terms of the path length. Although the nearest ℓ2 distance is simpl… view at source ↗

**Figure 2.** Approximation using recursive equations II. BACKGROUND AND NOTATION A line process in R 2 is a set of points in R 2 that constitute a set of lines. Each line Li is uniquely characterized by its signed distance ri from the origin and the angle θi that the normal to the line makes with the x−axis. The parameter pair (θi , ri) thus resides as a point in a parameter space [0, π)×(−∞, ∞) that generates the line Li in R 2… view at source ↗

**Figure 3.** All the green colored points are within a distance [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The single turn case starting from the typical intersection. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: The two turns case. L1, L2, . . . intersect the line Lx. denote by xi the distance between the intersection Ii1 and I1X. Similarly, yi denotes the distance between Ii1 and IiX. Naturally, we have xi = yi sin θi sin θ1 , and yi = di cos θi − sin θi cot θ1 . From the int…

**Figure 6.** Figure 6: Single turn case from the typical point and the typical [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: Single turn case from a typical intersection along with the [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 10.** Figure 10: Illustration of V2V communication using ORIS. The trans [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

The foundations of stochastic geometry,

B. Ripley, “The foundations of stochastic geometry,” The Annals of Probability, vol. 4, no. 6, pp. 995–998, 1976

work page 1976

[2] [2]

Stochastic geometry and architecture of communica- tion networks,

F. Baccelli et al., “Stochastic geometry and architecture of communica- tion networks,” Telecommunication Systems, vol. 7, no. 1, pp. 209–227, 1997

work page 1997

[3] [3]

Poisson Cox point processes for vehicular networks,

C.-S. Choi and F. Baccelli, “Poisson Cox point processes for vehicular networks,” IEEE Transactions on Vehicular Technology, vol. 67, no. 10, pp. 10 160–10 165, 2018

work page 2018

[4] [4]

Poisson line Cox process: Foundations and applications to vehicular networks,

H. S. Dhillon and V . V . Chetlur, “Poisson line Cox process: Foundations and applications to vehicular networks,” Synthesis Lectures on Learning, Networks, and Algorithms , vol. 1, no. 1, pp. 1–149, 2020

work page 2020

[5] [5]

Cox models for vehicular networks: SIR performance and equivalence,

J. P. Jeyaraj and M. Haenggi, “Cox models for vehicular networks: SIR performance and equivalence,” IEEE Transactions on Wireless Communications, vol. 20, no. 1, pp. 171–185, 2020

work page 2020

[6] [6]

The transdimensional Poisson process for vehicular network analysis,

J. P. Jeyaraj et al., “The transdimensional Poisson process for vehicular network analysis,” IEEE Transactions on Wireless Communications , vol. 20, no. 12, pp. 8023–8038, 2021

2021

[7] [7]

Binomial line processes: Distance distributions,

G. Ghatak, “Binomial line processes: Distance distributions,” IEEE Transactions on Vehicular Technology , vol. 71, no. 2, pp. 2176–2180, 2021

work page 2021

[8] [8]

Small cell deployment along roads: Coverage analysis and slice-aware RAT selection,

G. Ghatak et al., “Small cell deployment along roads: Coverage analysis and slice-aware RAT selection,”IEEE Transactions on Communications, vol. 67, no. 8, pp. 5875–5891, 2019

work page 2019