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arxiv: 2605.16560 · v1 · pith:33TMPEEAnew · submitted 2026-05-15 · 🧮 math.PR · cs.NI

Seasonal Statistics of Shannon Capacity in a Dynamical Poisson-Voronoi Cellular Network

Sanjoy Kumar Jhawar , Fran\c{c}ois Baccelli This is my paper

Pith reviewed 2026-05-19 21:06 UTC · model grok-4.3

classification 🧮 math.PR cs.NI
keywords Poisson point processShannon capacityhandovercellular networkdynamical networkVoronoi tessellationsignal to interference ratioseasonal statistics
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The pith

In dynamical cellular networks with moving base stations, Shannon capacity fluctuates in recurring seasonal patterns based on signal and interference strengths at key events.

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

This paper studies a cellular network where base stations move continuously at constant speeds in random directions, modeled as a Poisson point process. A user always connects to the nearest base station, experiencing changing signal and interference as positions shift. The analysis focuses on specific time epochs, including handovers when the serving station changes, times of closest approach to the serving station, and times when the nearest interferer is closest or farthest. At these epochs, the paper compares quality of service and Shannon capacity, identifying patterns of good and bad performance that recur over time. These patterns are likened to seasons, with strong or mild signal and interference powers defining periods of high or low capacity along the system's evolution.

Core claim

The Shannon capacity in this dynamical Poisson-Voronoi network shows distinct statistics at different event times, such as those inducing handovers or maximal proximity of the serving station, and at times of nearest or farthest nearest-interferer distance. These lead to recurring good or bad scenarios, with an analogy to seasons based on the fluctuations of signal and interference power, where strong or mild conditions correspond to different capacity seasons.

What carries the argument

The time-dependent nearest-neighbor connection in an evolving Poisson-Voronoi tessellation generated by base stations moving at constant random velocities, used to evaluate capacity at specific event epochs.

If this is right

  • Quality of service and capacity differ systematically between handover events and proximity extrema.
  • Recurring good and bad capacity scenarios emerge from the periodic-like fluctuations in distances.
  • Seasonal analogy allows grouping of epochs into categories based on signal and interference strength.
  • Performance evaluation at typical time epochs provides a baseline for comparison with event-driven times.

Where Pith is reading between the lines

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

  • This framework suggests that network operators could anticipate capacity variations by tracking base station positions over time.
  • Similar seasonal effects might be observable in other mobile ad-hoc or sensor networks with random motion.
  • Extensions could include incorporating user mobility or more complex velocity distributions to test robustness of the seasonal patterns.

Load-bearing premise

Base stations are distributed as a homogeneous Poisson point process and each moves independently at a constant speed in a uniformly random direction, while the user connects exclusively to the nearest base station at every moment.

What would settle it

A simulation or measurement where the Shannon capacity does not exhibit distinguishable recurring patterns or seasonal variations when sampled at handover times, maximal proximity times, or nearest-interferer extremal times.

Figures

Figures reproduced from arXiv: 2605.16560 by Fran\c{c}ois Baccelli, Sanjoy Kumar Jhawar.

Figure 1
Figure 1. Figure 1: Trajectory of BSs in a finite space and time window with respect to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time series distance to the nearest (in red) and second nearest (in blue) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The blue nodes (•), black nodes (•), red nodes (•), cyan nodes (•) and magenta nodes (•) respectively on individual vertical blue, black, red, cyan and magenta lines, form the point processes corresponding to the distance of all the mobile BSs at min-signal max-interference, max-signal, typical time, max-interference and min￾interference epochs, respectively. C. The k-th lower envelope and its characteriza… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: (s, h) ∈ C(T ,H) ⇔ (T, H) ∈ ∂Us h . t (s, 0) hˆ • (s, h) • U s h • (T, H) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: All birds with head point inside and outside the half-ball of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The radial bird in red correspond to a BS that partly [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Non-tropical case (with fading): Comparison of coverage probabilities at typical min-signal max-interference (handover), typical time, typical max [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Non-tropical case (with fading): Comparison of coverage probabilities at typical max-interference, typical min-interference, with respect to SINR [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Non-tropical case (without fading): Comparison of coverage probabilities at typical min-signal max-interference (handover), typical time and typical [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Non-tropical case (without fading): Comparison of coverage probabilities at typical max-interference and typical min-signal, with respect to SINR [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Tropical case (with fading): Comparison of coverage probabilities at typical min-signal max-interference (handover), typical time and typical max [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Tropical case (with fading): Comparison of coverage probabilities at typical max-interference and typical min-interference, with respect to STINR [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Tropical case (without fading): Comparison of coverage probabilities at typical min-signal max-interference (handover), typical time and typical [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Tropical case (without fading): Comparison of coverage probabilities at typical max-interference, typical min-interference, with respect to STINR [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
read the original abstract

In this work we consider a dynamical cellular communication network in which mobile BSs are modeled as a homogeneous Poisson point process on $\mathbb R^2$. Each base station moves at a constant speed in a random direction. A typical user connects to the nearest base station and it experiences variable signal and interference powers depending on the distance of all the stations. Along the motion of the stations, the user swaps its serving station, and such an event is called a handover. We are interested in the performance evaluation of the system under some classical and tropical metrics of interest at different time of events, inducing handovers, maximal proximity of serving station, nearest interferer at closest or farthest distance with respect to the user or at any typical time epoch. A comparison study of quality of service and Shannon capacity at these epochs is also provided, among the recurrence of such ``good'' or ``bad'' scenarios. We can make an analogy with seasons based on the fluctuations of signal and interference power. Strong or mild signal or interference power correspond to different seasons of Shannon capacity along the evolution of the system.

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

2 major / 2 minor

Summary. The manuscript models a cellular network with mobile base stations distributed as a homogeneous Poisson point process on the plane, each moving at constant speed in a uniformly random direction. A typical user connects to the nearest base station at every instant. The authors analyze the Shannon capacity and quality of service at specific epochs defined by geometric events: handover times, times of maximal proximity to the serving base station, times when the nearest interferer is closest or farthest, and arbitrary typical times. They compare these metrics across epochs and invoke the recurrence of 'good' or 'bad' scenarios to draw an analogy with seasons based on fluctuations in signal and interference power.

Significance. If the comparisons and recurrence properties are rigorously established, the work contributes to temporal analysis in stochastic-geometry models of mobile networks by linking geometric epochs to capacity variations. The seasonal analogy offers an intuitive lens for interpreting fluctuations, provided it is supported by interval statistics rather than isolated snapshots. The model setup (PPP with independent constant-velocity motions) is standard and allows well-defined nearest-BS processes.

major comments (2)
  1. [§4] §4 (Epoch-based capacity analysis): The central seasonal claim requires showing that the geometrically defined epochs delimit intervals whose lengths and capacity statistics differ systematically. The manuscript compares marginal Shannon-capacity distributions at handover instants, maximal-proximity times, and extremal-interferer distances, but does not derive Palm distributions or renewal-type results for inter-epoch times. Without these, the recurrence of good/bad scenarios reduces to a sequence of snapshot comparisons whose continuous-time interpretation remains heuristic.
  2. [§5.1] §5.1 (Numerical comparisons): The reported capacity values at the various epochs exhibit differences, yet the paper does not quantify sojourn-time distributions or provide confidence intervals on the frequency of recurrence. This leaves the seasonal analogy without direct statistical support for the claim that good or bad periods recur in a manner analogous to seasons.
minor comments (2)
  1. [Abstract] The abstract refers to 'tropical metrics' without definition; a brief clarification or reference in §2 would improve readability.
  2. [§2] Notation for the serving distance process and aggregate interference at a typical time t should be introduced once in §2 and used consistently thereafter to avoid ambiguity in the epoch definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below, indicating where we agree with the need for clarification and where we maintain the scope of our contribution.

read point-by-point responses

  1. Referee: [§4] §4 (Epoch-based capacity analysis): The central seasonal claim requires showing that the geometrically defined epochs delimit intervals whose lengths and capacity statistics differ systematically. The manuscript compares marginal Shannon-capacity distributions at handover instants, maximal-proximity times, and extremal-interferer distances, but does not derive Palm distributions or renewal-type results for inter-epoch times. Without these, the recurrence of good/bad scenarios reduces to a sequence of snapshot comparisons whose continuous-time interpretation remains heuristic.

    Authors: We appreciate the referee's observation that a complete continuous-time analysis would ideally include renewal-type results on inter-epoch intervals. Our contribution centers on the capacity statistics evaluated precisely at the geometrically defined epochs, which mark the transitions between regimes of signal and interference strength. The systematic differences observed in the marginal distributions at these epochs, combined with the stationarity and ergodicity of the underlying Poisson-Voronoi motion process, support the recurrence of good and bad scenarios. While deriving Palm distributions for the epochs would constitute a valuable extension, it is not required to establish the snapshot comparisons that underpin the seasonal analogy. In the revised manuscript we will add a clarifying paragraph in §4 that explicitly states the heuristic nature of the continuous-time interpretation in the absence of interval statistics and positions the analogy as an intuitive interpretation of the recurrent geometric events rather than a fully quantified seasonal model. revision: partial

  2. Referee: [§5.1] §5.1 (Numerical comparisons): The reported capacity values at the various epochs exhibit differences, yet the paper does not quantify sojourn-time distributions or provide confidence intervals on the frequency of recurrence. This leaves the seasonal analogy without direct statistical support for the claim that good or bad periods recur in a manner analogous to seasons.

    Authors: The numerical results in §5.1 are obtained from Monte Carlo simulations that demonstrate clear differences in Shannon capacity across the defined epochs. We agree that bootstrap confidence intervals would strengthen the presentation of these differences and will include them in the revised version. Full sojourn-time distributions are not derived because the focus remains on capacity at the event instants; however, average inter-epoch times can be estimated from the known intensity of the Poisson point process and the constant velocity, and we will add a short discussion of these estimates together with the confidence intervals to provide additional quantitative backing for the recurrence claim. revision: partial

Circularity Check

0 steps flagged

Derivation chain is self-contained with no identified circular reductions

full rationale

The paper defines epochs via independent geometric events (handovers, maximal proximity, extremal interferer distances) in a homogeneous PPP with constant-velocity motions. Capacity and QoS metrics are evaluated directly at these points using standard stochastic geometry tools. No equations or claims reduce a derived statistic to a fitted parameter or self-referential definition by construction. The seasonal analogy is presented as an interpretive description of fluctuations rather than a load-bearing mathematical result. The model assumptions (PPP, nearest-BS association, independent motions) are stated upfront and do not embed the target recurrence statistics. This is the common case of an honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the Poisson point process model for base-station locations and the constant-speed random-direction mobility assumption; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Base stations form a homogeneous Poisson point process on R^2.
    Stated in the abstract as the model for mobile BS locations.
  • domain assumption Each base station moves at constant speed in a random direction.
    Stated in the abstract as the dynamical component of the network.

pith-pipeline@v0.9.0 · 5725 in / 1239 out tokens · 42545 ms · 2026-05-19T21:06:42.418350+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We are interested in the performance evaluation of the system under some classical and tropical metrics of interest at different time of events, inducing handovers, maximal proximity of serving station, nearest interferer at closest or farthest distance... We can make an analogy with seasons based on the fluctuations of signal and interference power.

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The non-periodicity of the seasons and their sequential but random appearance are also evident even in the simple tropical model without noise and fading

What do these tags mean?
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The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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