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Kinetic Handover Modeling in Nonstationary Cellular Networks with Fluid Spatiotemporal Stochastic Geometry

Wen-Yu Dong, Zheng Lin, Jinfeng Xu, Yuxin Zhang, Zihan Fang, Qi Bi, Sheng Chen

arXiv:2610.34211v1cs.NI

TL;DR

Aggregate density evolution does not uniquely reveal physical pairwise transport or gross crossings from counter-moving users. The paper connects boundary-normal velocity statistics to F-STSG density-snapshot reconstruction, distinguishing net transfer from gross traffic and deriving recovery conditions, closures, and bounds. In held-out vehicular simulations, F-STSG has lower net-flow reconstruction errors than three baselines, while auxiliary velocity statistics improve directional estimates under matched Gaussian mobility.

  • Problem

    Density evolution does not uniquely determine physical cell-pair transport, and first-order flux can hide counter-moving crossings that contribute to gross traffic.

    Source: §I · p. 1

  • Method

    The paper defines directional crossings through boundary-normal velocity moments and connects them to density-snapshot F-STSG transport reconstruction, with recovery conditions, mobility closures, and moment-constrained bounds.

    Source: §I · p. 1, B. Contributions and Organization · p. 2, A. Cellular Geometry and Population… · p. 4

  • Results

    On held-out vehicular simulations, F-STSG achieves lower net-flow error than the considered baselines, while auxiliary velocity statistics improve directional estimates under matched Gaussian mobility.

    Source: §VI · p. 13

  • Takeaways & Limitations

    Potential transport permits exact canonical flux recovery, while the stationary Poisson–Voronoi crossing law is recovered as a special case.

    Source: §VI · p. 12

    Substantial absolute error remains in the held-out vehicular simulations.

    Source: §VI · p. 13

Abstract

from arXiv · show

Inferring directional handover traffic from aggregate mobility observations remains challenging because population density evolution does not fully describe the movement of individual users across cell boundaries. This paper develops a kinetic framework for geometry-induced serving-region crossings in nonstationary cellular networks, connecting boundary-normal velocity statistics with fluid spatiotemporal stochastic geometry (F-STSG) for density-based transport reconstruction. The framework distinguishes signed net population transfer from gross bidirectional crossings and reveals two information limits: density evolution does not uniquely determine physical pairwise transport, while first-order transport alone cannot resolve counter-moving crossings. We establish exact recovery conditions for potential transport and develop coherent and Gaussian mobility closures for estimating directional crossing rates. Sharp moment-constrained bounds further characterize the remaining uncertainty under bounded normal velocities without assuming a specific velocity distribution. An error bound links directional estimation accuracy to transport reconstruction and unresolved mobility information. The formulation also recovers the classical stationary Poisson--Voronoi handover law. Numerical experiments examine the information limits, reconstruction sensitivity, and closure accuracy. On 47 held-out vehicular simulation windows, F-STSG achieves lower net-flow reconstruction errors than three baselines, while auxiliary velocity statistics improve directional estimates under matched Gaussian mobility.

I. INTRODUCTION

Aggregate density evolution leaves physical pairwise transport and gross crossings unresolved, motivating a kinetic model that connects boundary-normal mobility to density-based reconstruction. The paper situates this model among trajectory-based, protocol-level, and measurement-based approaches, then states its recovery, bounding, and evaluation contributions.

I. INTRODUCTION

  • Density evolution constrains flux divergence but not unique cell-pair transfer, and counter-moving users can cancel in signed flux while still crossing boundaries.The model therefore separates physical mobility flux from reconstructed transport, and signed net transfer from gross crossing traffic.Source: §I · p. 1
  • Directional crossing rates are defined from population density and positive boundary-normal UE velocity, distinguishing signed net transfer from gross bidirectional traffic.F-STSG reconstructs a canonical first-order transport field from successive density snapshots; it need not equal the physical flux, and executed-HO protocol mechanisms are outside the crossing model.Source: §I · pp. 1–2

A. Related Work

  • Prior mobility models generally specify individual trajectories or mobility distributions, while measurement studies observe HO events rather than infer pre-event directional cell-pair crossings from population fields.F-STSG field-level studies also do not formulate directional crossings between neighboring serving regions.Source: A. Related Work · p. 2

B. Contributions and Organization

  • The paper identifies two information limits, exact-recovery conditions for potential transport, sharp distribution-free crossing bounds, and coherent and Gaussian closures under additional mobility information.The bounds use the first two normal-velocity moments and a bounded-speed constraint.Source: B. Contributions and Organization · p. 2
  • The contributions connect density-snapshot F-STSG reconstruction to directional crossing estimates, bound estimation error, and evaluate reconstruction and auxiliary velocity statistics in controlled and held-out vehicular simulations.The held-out comparison uses three baselines; the passage does not report a winning method or numerical values.Source: B. Contributions and Organization · p. 2

C. Notation

The section sets notation and a closed-network model in which UE density and physical flux describe population transport across nearest-BS serving regions. It distinguishes that physical flux from the canonical F-STSG field reconstructed from density evolution.

C. Notation

  • Hats mark reconstructed or estimated quantities; [·]+, E[·], ∇, and ∇· denote positive part, expectation, gradient, and divergence.Scalars use italic letters, vectors bold lowercase letters, and sets or spatial regions calligraphic letters.Source: C. Notation · p. 3

A. Cellular Geometry and Population Transport

  • The model uses a bounded planar service region with fixed base stations and nearest-BS serving regions; neighboring cells share a boundary of nonzero length.The common-boundary normal points from cell i to cell j, and the baseline excludes radio fluctuations and protocol filtering.Source: A. Cellular Geometry and Population… · pp. 3–4
  • UE density defines cell population, while physical flux equals density times conditional mean velocity and retains first-order transport, not the full local velocity distribution.The conditional velocity distribution is assumed to have a finite first moment.Source: A. Cellular Geometry and Population… · p. 4
  • The closed-system continuity model imposes no flux at the outer boundary, conserving total UE population; open boundaries and arrivals or departures are outside the baseline.The text notes that those cases could instead be represented by boundary fluxes or source terms.Source: A. Cellular Geometry and Population… · p. 4
  • F-STSG selects a canonical density-consistent flux by minimizing weighted kinetic energy, but that reconstructed field need not equal the physical flux across each cell boundary.The potential and reconstructed flux satisfy the stated boundary condition under regularity conditions; the potential is defined up to an additive constant that does not affect the flux.Source: A. Cellular Geometry and Population… · p. 4

B. Boundary-Crossing Model

The model defines directional serving-region crossings from boundary-normal UE motion, separating signed net transfer from gross crossings. Density evolution and first-order transport each leave distinct ambiguities, with exact recovery available for a specified potential-transport class.

B. Boundary-Crossing Model

  • A geometry-induced crossing occurs when a continuous UE trajectory crosses a shared boundary and changes serving region; only normal motion contributes to its directional rate.These crossings characterize mobility-induced association changes, not protocol-level handover execution.Source: B. Boundary-Crossing Model · p. 4

A. Net and Gross Crossing Flows

  • Signed net flow is antisymmetric between cells, whereas gross crossing traffic is symmetric and can include simultaneous traffic in both directions.Net flow reflects first-order population transport; gross traffic depends on the one-sided normal-velocity distribution.Source: A. Net and Gross Crossing Flows · p. 5
  • Divergence-free transport components can change physical pairwise flows without changing the observed density evolution.This separates cell population balance, which density evolution fixes, from individual pairwise transport.Source: A. Net and Gross Crossing Flows · p. 5
  • For the stated potential-transport class, the F-STSG reconstructed pairwise flow equals the physical pairwise flow for every neighboring cell pair.Outside this class, matching cell population balance does not establish matching pairwise flows.Source: A. Net and Gross Crossing Flows · p. 5
  • Even when density and physical first-order flux are known, they do not generally determine gross crossing rates.Different zero-mean normal-velocity distributions can preserve the same conditional mean velocity and flux while producing different gross rates.Source: A. Net and Gross Crossing Flows · pp. 5–6

B. Closure for Unresolved Normal Mobility

The closure section addresses directional crossings when local normal-velocity variation is unresolved. One-sided motion supports recovery from first-order transport; Gaussian closure uses additional mobility information, while bounded-speed moment constraints yield sharp distribution-free intervals.

Coherent closure

  • When conditional normal velocity is supported on one side of zero, first-order transport suffices for the coherent crossing result; violation of this condition requires additional local velocity information.The one-sided condition need not imply identical UE velocities.Source: B. Closure for Unresolved Normal… · p. 6

Gaussian closure

  • The Gaussian closure preserves prescribed first-order transport, including zero net flux with positive gross crossing traffic when the normal mean is zero and variance is positive.As the variance tends to zero, it approaches the coherent result.Source: B. Closure for Unresolved Normal… · p. 6
  • The Gaussian closure requires covariance information that density snapshots cannot infer, so it must be supplied through auxiliary mobility statistics, a trajectory subsample, or a residual-mobility model.The Gaussian assumption applies to boundary-normal velocity, not the full two-dimensional UE velocity distribution.Source: B. Closure for Unresolved Normal… · p. 6

Moment bounds

  • With prescribed moments and bounded normal speed, distribution-free crossing-rate intervals are sharp, and their width vanishes when variance is zero or the second moment reaches the squared speed bound.The endpoints are attainable within the class of boundary-local velocity distributions satisfying the specified moments and support.Source: B. Closure for Unresolved Normal… · pp. 6–7

C. Stationary Isotropic Limit

The stationary isotropic Poisson–Voronoi setting recovers the classical mean handover rate. It also illustrates that a stationary density and zero first-order flux can coexist with nonzero crossing traffic.

Stationary isotropic case

  • For homogeneous Poisson base stations, constant UE density, and constant-speed motion in independent uniform directions, nearest-BS association induces a stationary Poisson–Voronoi tessellation.The result is a consistency check and does not use the Gaussian closure.Source: C. Stationary Isotropic Limit · p. 7
  • The derived crossing rate is the classical mean handover rate for isotropic mobility and nearest-BS association in a homogeneous Poisson cellular network.The calculation uses the mean Voronoi edge length per unit area and mean absolute normal speed.Source: C. Stationary Isotropic Limit · p. 7
  • Isotropy makes the first-order flux and minimum-energy F-STSG reconstruction zero, although velocity variation produces a nonzero crossing rate.The crossing rate therefore reflects mobility information absent from the first-order population flux.Source: C. Stationary Isotropic Limit · p. 7

IV. HANDOVER ESTIMATION FROM DENSITY SNAPSHOTS

The estimation procedure uses successive density snapshots to reconstruct a minimum-energy transport field, then applies local mobility closure to estimate directional crossings. Continuous reconstruction requires a spatial density field rather than cell counts alone.

IV. HANDOVER ESTIMATION FROM DENSITY SNAPSHOTS

  • F-STSG selects the minimum-energy transport field consistent with observed population evolution before local mobility closure converts transport into directional crossing rates.Propositions 2 and Corollary 1 characterize when the reconstructed field can represent physical pairwise transport.Source: §IV · p. 7
  • Density snapshots may be constructed from gridded positioning estimates, spatially aggregated measurement reports, or other location-aware population measurements.A snapshot is a spatial estimate of UE population over the service region at an observation time.Source: §IV · p. 7
  • Cell-level UE counts provide only serving-region population integrals, so continuous reconstruction requires additional spatial interpolation or a density model.The effect of density observation errors is examined separately.Source: §IV · p. 7

A. Population Flux Reconstruction

F-STSG reconstructs a first-order transport field from successive density snapshots, then derives signed cell flows consistent with the observed population changes. The reconstruction requires compatibility and regularity conditions, and population consistency alone does not identify physical pairwise flow.

A. Population Flux Reconstruction

  • Centered differences of uniformly spaced density snapshots estimate temporal derivatives at interior snapshots; midpoint experiments retain second-order temporal accuracy.For the closed system, the weighted Neumann problem requires a zero-integral temporal derivative; small compatibility errors are corrected.Source: A. Population Flux Reconstruction · p. 7
  • The small compatibility correction is not appropriate for real changes in total UE population, which must instead be modeled with an open boundary or source term.The correction is intended only for numerical compatibility errors.Source: A. Population Flux Reconstruction · p. 7
  • F-STSG obtains the reconstructed field from density and the corrected temporal derivative, using a weighted elliptic formulation with a no-flux boundary condition.The density must be sufficiently regular and bounded away from zero on the reconstruction domain; zero-density regions may be excluded or regularized.Source: A. Population Flux Reconstruction · p. 7
  • Spatial discretization produces a sparse system with one potential unknown per active grid point, while reconstructions at different snapshot times can run in parallel.The local finite-volume matrix has a number of nonzeros linear in active grid points, but this does not guarantee linear solution time.Source: A. Population Flux Reconstruction · p. 7
  • Equation (33) makes reconstructed cell flows reproduce density-implied cell population changes, but does not guarantee equality with physical pairwise flows.Exact recovery holds for the potential-transport class identified in Corollary 1 when the exact density derivative is available.Source: A. Population Flux Reconstruction · p. 8

B. Directional Handover Estimation

The reconstructed signed transport field is mapped to directional crossing rates under a local mobility closure. Coherent motion needs no additional velocity statistic, whereas unresolved counterflow requires velocity-dispersion information and introduces separate sources of estimation error.

B. Directional Handover Estimation

  • The reconstructed field determines signed first-order transport, but directional rates additionally depend on the unresolved local normal-velocity distribution.The transport reconstruction and the mapping to directional mobility-induced crossing rates are separate steps.Source: B. Directional Handover Estimation · pp. 8–9
  • A coherent directional estimate depends on reconstructed boundary-normal transport, and its error is bounded by the difference from physical boundary-normal flux.Convergence of reconstructed boundary-normal transport in L1 on the shared boundary implies convergence of the corresponding coherent crossing estimate.Source: B. Directional Handover Estimation · p. 8
  • The error bound separates snapshot reconstruction, canonical-to-physical flux mismatch, auxiliary-statistic error, and distributional closure mismatch.These terms bound distinct sources of error and are not additive components of the realized estimation error.Source: B. Directional Handover Estimation · p. 8
  • The estimated neighboring-cell rates form a geometric crossing-rate matrix, with zero diagonal and zero entries for nonneighboring cell pairs.Neighboring-pair entries use either the coherent or dispersion-aware closure, according to the adopted local mobility model.Source: B. Directional Handover Estimation · pp. 8–9
  • Under unresolved counterflow, the dispersion-aware Gaussian closure uses local normal-velocity variance, which cannot be inferred from density snapshots alone.The covariance may come from auxiliary mobility measurements, a trajectory subsample, or a prescribed residual-mobility model.Source: B. Directional Handover Estimation · pp. 8–9

V. NUMERICAL RESULTS

The numerical evaluation combines controlled mobility experiments with held-out vehicular simulation, using distinct measures for pairwise reconstruction, flux accuracy, and population-balance consistency.

V. NUMERICAL RESULTS

  • Controlled experiments use prescribed physical mobility fields as references, while external evaluation measures window-integrated counts.The study evaluates flux-field error and a cell-level population-balance residual in addition to pairwise errors.Source: §V · p. 9
  • Normalized pairwise error is evaluated over ordered neighboring cell pairs, with directed quantities defined for handover or net flow.For external evaluation, the quantities are window-integrated counts.Source: §V · p. 9
  • The population-balance residual measures conservation consistency, not pairwise reconstruction accuracy.Reconstructed-flux and physical-flux residuals use the same cell-population changes but their respective boundary fluxes; pooled errors also differ from mean paired window differences.Source: §V · p. 9

A. Nonstationary Coherent Mobility

In the potential-flow experiment, the physical and F-STSG gross crossing-rate curves nearly coincide as the UE distribution moves across fixed serving regions. F-STSG has lower reported error than trajectory counts in this setup, which uses analytical density inputs.

A. Nonstationary Coherent Mobility

  • The experiment tests a potential-transport field satisfying the exact-recovery condition and coherent boundary-normal mobility.The setup uses a horizontal potential flow with maximum speed 15 m/s and a no-flux outer boundary.Source: A. Nonstationary Coherent Mobility · p. 9, Figure 2 · p. 9
  • 7.32 × 10−4 for F-STSG versus 9.60 × 10−3 for trajectories (ϵH = ϵF); physical and F-STSG gross-rate curves nearly coincide.The F-STSG reconstruction uses noise-free analytical density snapshots, so the smaller error does not establish a general advantage over trajectory observations.Source: A. Nonstationary Coherent Mobility · p. 9, Figure 2 · p. 9
  • The gross crossing rate rises from 0.064 to 0.073 s−1, then falls to 0.061 s−1 as the UE distribution moves across fixed serving regions.The cell-balance residual is 1.19 × 10−2.Source: A. Nonstationary Coherent Mobility · p. 9

B. Information Limits of First-Order Population Fields

First-order population fields leave two distinct uncertainties: density evolution does not uniquely recover physical pairwise transport, and net transport alone does not determine gross crossings. The stationary Poisson–Voronoi limit is recovered, while reconstruction sensitivity depends on observation resolution and density accuracy.

B. Information Limits of First-Order Population Fields

C. Stationary Isotropic Limit

  • The stationary Poisson–Voronoi rate scales linearly with UE speed and with the square root of BS density; trajectory-to-theory deviation is at most 1.02%.The largest trajectory-to-kinetic-integral difference is 0.184% of the theoretical rate.Source: Figure 5 · p. 10, C. Stationary Isotropic Limit · p. 11

D. Snapshot Resolution and Observation Error

  • For snapshot intervals from 0.25 to 2 s, flux error stays below 1.18 × 10−3 and directional-crossing error below 8.12 × 10−4; at 8 s they rise to 9.33 × 10−3 and 4.25 × 10−3.The sensitivity comparison holds spatial grid spacing at h = 20 m.Source: Figure 6 · p. 11, D. Snapshot Resolution and Observation… · p. 11
  • Reducing grid spacing from 40 to 10 m decreases flux error from 4.64 × 10−3 to 2.61 × 10−4 and directional-crossing error from 2.87 × 10−3 to 1.86 × 10−4.Mean solve time rises from 8.37 to 457.52 ms per snapshot across those grid spacings.Source: Figure 6 · p. 11, D. Snapshot Resolution and Observation… · p. 11
  • At 1%, 2%, and 10% relative RMS density error, (ϵJ, ϵH) is (0.120, 0.0175), (0.243, 0.0555), and (1.219, 0.825), respectively.The reported growth reflects error amplification in temporal differentiation, even for potential-form physical transport.Source: Figure 6 · p. 11, D. Snapshot Resolution and Observation… · p. 11
  • Under noisy potential transport, reconstruction discrepancies are 0.305 for coherent motion and 0.222 for Gaussian motion; for nonpotential transport, canonical-to-physical discrepancy is 0.633.These native-interval stage discrepancies are not additive and differ from errors evaluated after temporal aggregation.Source: Table III · p. 11, D. Snapshot Resolution and Observation… · p. 11

E. External Validation and Baseline Comparison

On held-out vehicular windows, F-STSG has lower net-flow error than the three baselines, though its absolute error remains substantial. Directional estimates are sensitive to temporal aggregation and improve under matched Gaussian mobility when auxiliary velocity statistics are supplied.

E. External Validation and Baseline Comparison

  • The held-out comparison uses 47 windows; F-STSG and three baselines are evaluated on the same windows and directed cell pairs.Density snapshots are estimated without serving-cell records or crossing labels.Source: E. External Validation and Baseline… · pp. 11–12
  • Mean paired net-flow error differences favor F-STSG by 0.224, 0.231, and 0.123 against Graph-MinNorm, Graph-Weighted, and Continuity-OF, respectively, with all paired 95% confidence intervals above zero.Pooled normalized net-flow error on the same subset remains 0.585.Source: Figure 7 · p. 12, E. External Validation and Baseline… · p. 12
  • At K = 3, Native aggregation raises high-crossing-edge recall from 0.418 to 0.539 and reduces directional error from 0.772 to 0.611 versus Common aggregation.The two variants use the same reconstructed flux but differ in whether the positive part is taken before or after pairwise aggregation.Source: Figure 7 · p. 12, E. External Validation and Baseline… · p. 12

F. Temporal Aggregation and Kinetic Closure

VI. CONCLUSION

  • The formulation distinguishes signed net transfer from gross crossings, establishes two information limits, and recovers the stationary Poisson–Voronoi law as a special case.Potential transport permits exact canonical flux recovery; coherent and Gaussian closures characterize directional rates under specified mobility conditions.Source: §VI · pp. 12–13

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