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arxiv: 2606.21946 · v1 · pith:F656YXZEnew · submitted 2026-06-20 · 📡 eess.SY · cs.SY· eess.SP

Joint Visibility Analysis of RIS in Non-Terrestrial Networks through Stochastic Geometry

Ashutosh Balakrishnan , Junse Lee , Francois Baccelli This is my paper

Pith reviewed 2026-06-26 11:50 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP
keywords reconfigurable intelligent surfacesnon-terrestrial networksstochastic geometryjoint visibilityPoisson point processline-of-sight probability
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The pith

RISs modeled as a Poisson point process with exponential heights yield a closed-form expected count of surfaces jointly visible to both user and non-terrestrial base station, maximized at twice the Basel number in 2D.

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

The paper derives the expected number of reconfigurable intelligent surfaces that maintain line-of-sight links to both a ground user and a non-terrestrial base station when rooftops are modeled as a Poisson point process with random heights. This count is obtained exactly for the unconditional geometry and also conditional on whether the direct user-to-base-station path is line-of-sight or blocked. A reader cares because the result supplies an exact planning quantity for choosing surface density and placement to restore connectivity in cities where buildings obstruct signals.

Core claim

Accounting for the dual stochasticity from RIS locations and heights, the expected number of jointly visible RISs is derived in closed form for a homogeneous Poisson point process with exponentially distributed heights; in the 2D setting this expectation reaches a maximum value of twice the Basel number π²/6.

What carries the argument

The joint visibility probability obtained by integrating the void probability of the Poisson point process of building locations together with the independent exponential height distribution.

If this is right

  • The unconditional maximum expected count equals 2*(π²/6).
  • Separate closed-form expressions exist for the count conditioned on the user-to-base-station link being line-of-sight or non-line-of-sight.
  • The expressions depend explicitly on building density, mean height, base-station altitude, and horizontal position.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The factor of two that appears in the 2D maximum may be replaced by a different constant when the same model is lifted to three dimensions.
  • The probability heatmaps supplied in the paper can be used to derive location-specific placement rules rather than relying solely on the average count.
  • Economic cost-benefit calculations for surface density follow immediately from the closed-form expectation once installation and maintenance costs per surface are known.

Load-bearing premise

Building rooftops form a homogeneous Poisson point process with independent exponentially distributed heights and the non-terrestrial base station sits at a fixed deterministic location.

What would settle it

Measure the average number of jointly visible rooftop surfaces in a real urban block with recorded building positions and heights, then compare the empirical average against the closed-form expression evaluated at the measured density and mean height.

Figures

Figures reproduced from arXiv: 2606.21946 by Ashutosh Balakrishnan, Francois Baccelli, Junse Lee.

Figure 1
Figure 1. Figure 1: System model when the RIS is located on the left of the U [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: System models when the RIS is located on the RHS to the U [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Variation of expected number of RISs jointly visible [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of Expected RIS jointly visible in a rural s [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of Expected RIS jointly visible in an urban [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of Expected RIS jointly visible in a dense u [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: For a UAV based BS deployed at a height of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: For a UAV based BS deployed at a height of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: For a high altitude NTN-BS deployed at a height of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Non-Terrestrial networks (NTNs) are a key theme in upcoming 6G communications, especially for ubiquitous coverage. Urban environments, comprising of high rise buildings often result in blocking the line of sight (LoS) path between the user equipment (UE) and the NTN base station (NTN-BS). In this paper we investigate the situation where reconfigurable intelligent surfaces (RIS) are deployed on the building roof-tops to ensure multi-hop connectivity between the UE and the NTN-BS. In such a scenario, it becomes crucial to statistically study the LoS visibility of the RIS from the UE as well as from the NTN-BS, hence termed as joint visibility. In this work, accounting for the dual stochasticity arising from the locations of the RIS deployed buildings and the respective random building heights, we statistically study the probability of joint RIS visibility in a two-dimensional (2D) scenario considering a deterministic location of the NTN-BS. Further, we study the joint RIS visibility statistics conditional on the UE-NTN link being LoS or non-LoS. For the RISs deployed as a point point process (PPP) having exponentially distributed heights, the expected RISs jointly visible under the unconditional and conditional geometric settings are derived in closed form. Interestingly, in the 2D setting, the maximum expected RISs jointly visible, unconditionally, is twice the Basel number $(\pi^2/ 6)$. The simulated results are analyzed over building density, average building height, the altitude and position of the NTN-BS. We also illustrate probability heatmaps, demonstrating the strongest chance to have a RIS used conditioned on the system geometry. This study is expected to be useful in planning the deployment of RIS in urban areas, improving the signal and for assessing economic aspects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper models RIS locations on building rooftops as a homogeneous Poisson point process (PPP) in 2D with i.i.d. exponential heights and a deterministic NTN-BS location. It derives closed-form expressions for the expected number of jointly visible RISs (visible to both UE and NTN-BS) in the unconditional case and conditional on the UE-NTN link being LoS or NLoS. The unconditional expectation is shown to attain a maximum value of twice the Basel number (π²/6). Simulation results and visibility heatmaps are presented over parameters including building density, average height, NTN-BS altitude, and position.

Significance. If the derivations hold, the work supplies exact analytical expressions for joint visibility statistics under standard stochastic-geometry assumptions, directly linking PPP properties and exponential height distributions to expected counts. The explicit connection to the Basel number is a notable mathematical feature, and the conditional results are relevant for NTN link-budget planning and RIS deployment economics in urban settings.

minor comments (4)
  1. [Abstract] Abstract: the phrase 'point point process (PPP)' is a typographical error and should read 'Poisson point process'.
  2. [Derivation sections (e.g., §III or §IV)] The manuscript states that closed forms are obtained from PPP properties and exponential heights, but the main text should explicitly flag any steps that rely on the memoryless property or on independence between horizontal locations and heights.
  3. [Numerical results] Simulation section: the parameter ranges used for Monte-Carlo validation (density, height mean, altitude) should be listed in a table so readers can directly compare against the closed-form expressions.
  4. [Figures] Figure captions for the heatmaps should state the exact conditioning (unconditional vs. LoS/NLoS) and the fixed values of all other parameters.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations follow directly from PPP and exponential assumptions

full rationale

The paper's central results are closed-form expectations for the count of jointly visible RISs (unconditional and conditional on UE-NTN LoS/NLoS) obtained by integrating over a homogeneous PPP of building locations with i.i.d. exponential heights under a fixed NTN-BS position in 2D. These follow from standard stochastic-geometry properties (void probabilities, Campbell's theorem) applied to the visibility indicators; the Basel-number maximum is an extremum of one such integral, not a redefinition or fit. No self-citation is invoked as a load-bearing uniqueness theorem, no parameter is fitted then relabeled a prediction, and no ansatz is smuggled via prior work. The derivation chain is therefore self-contained against the modeling premises stated in the abstract.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard stochastic-geometry modeling assumptions for point processes and height distributions that are common in wireless literature but are domain-specific choices for this NTN-RIS setting; no free parameters are fitted inside the derivations themselves.

axioms (2)
  • domain assumption RIS locations form a homogeneous Poisson point process
    Invoked when modeling buildings that host RIS in the 2D urban scenario.
  • domain assumption Building heights are independent and exponentially distributed
    Used to capture random building heights when computing joint visibility probabilities.

pith-pipeline@v0.9.1-grok · 5874 in / 1454 out tokens · 19792 ms · 2026-06-26T11:50:20.202512+00:00 · methodology

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