PaperlayerView exact revision on arXiv

Source-linked AI summary

Distributed Power Control for Equal-Priority NGSO Mega-Constellation Coexistence

Hao Yuan, Xing Zhang

arXiv:2609.38935v1eess.SP

TL;DR

Co-frequency interference among independent NGSO operators motivates modeling their downlink power choices as an equal-priority non-cooperative game. The paper develops equilibrium analysis and distributed water-filling, then reports gains over uncoordinated transmission across nominal and near-inline geometries and at larger K.

  • Problem

    Co-frequency interference among independent NGSO operators motivates direct coexistence methods beyond resource-management and avoidance approaches, including an operator-level continuous power game with equilibrium guarantees.

    Source: §I · p. 1

  • Method

    The paper models two equal-priority NGSO operators’ per-beam power choices as a non-cooperative game, establishes a weak-coupling potential approximation and equilibrium results, and derives distributed water-filling.

    Source: §I · p. 1, A. Potential Structure · p. 2

  • Results

    0.06% nominally, 3.91% in the 2°–3° near-inline window, and 10.41% at K=50 are NE gains over uncoordinated transmission; all 50 near-inline runs converge.

    Source: §IV · p. 4, §V · p. 5

  • Takeaways & Limitations

    The geometry sweep localizes useful adaptation, while centralized and lower-tail gaps expose an efficiency–fairness cost of independent sum-utility maximization.

    Source: §V · p. 5

    In the near-inline case, the additive potential approximation is not asserted tight, with mean η = 13.195.

    Source: §IV · p. 4

Abstract

from arXiv · show

Non-geostationary orbit (NGSO) mega-constellations sharing spectrum create dynamic co-frequency interference. We model downlink power control between two independent, equal-priority NGSO operators as a non-cooperative game. We establish an additive weak-coupling potential approximation, prove pure-strategy Nash equilibrium existence, and derive distributed water-filling with a weighted spectral-norm convergence condition. A spherical-Earth model applies off-axis discrimination at both link ends and distinguishes nominal satellite-selection-aware geometry from an unavoided $2^\circ$--$3^\circ$ near-inline window. At $K{=}20$, the NE gain over uncoordinated transmission is 0.06\% nominally and 3.91\% in the near-inline case; all 50 near-inline runs converge, and the gain reaches 10.41\% at $K{=}50$.

I. INTRODUCTION

Co-frequency interference among independent NGSO operators motivates an operator-level power-control game with equilibrium guarantees. The paper formulates the equal-priority sharing problem, develops its theory and distributed solution, and evaluates a shared-resource setting.

I. INTRODUCTION

  • The paper formulates downlink power control between independent, equal-priority NGSO operators as a non-cooperative game with per-beam power vectors and operator-local constraints.The modeled snapshot has both operators reuse one resource block, with intra-operator inter-beam interference ideally suppressed.Source: §I · p. 1, §II · p. 1
  • A potential-like characterization includes an additive weak-coupling error, while pure-strategy Nash equilibrium existence is proved.A weighted spectral-norm sufficient condition addresses uniqueness and convergence under cumulative cross-constellation coupling.Source: §I · p. 1
  • Distributed water-filling uses local serving-channel estimates and aggregate interference measurements to compute best responses.The numerical evaluation considers convergence, scalability, geometric sensitivity, and efficiency–lower-tail trade-offs against multiple benchmarks.Source: §I · p. 1

II. SYSTEM MODEL

  • The modeled operators use different altitudes and activate spot beams that reuse one 250 MHz time–frequency resource-block snapshot.Intra-operator inter-beam interference is assumed ideally suppressed, so the model isolates inter-operator coexistence.Source: §II · p. 1

A. Signal Model

The signal model represents the received SINR using serving-channel gain, cross-interference, and thermal noise. The accompanying schematic distinguishes service links from cross-operator interference.

A. Signal Model

  • The received SINR for operator i’s beam k is specified by equation (1).The figure depicts solid service links and dashed cross-operator interference links; orbital tracks and beam counts are schematic.Source: A. Signal Model · p. 1, Figure 1 · p. 2
  • Serving-channel gain includes free-space path loss and antenna gains, while cross-interference gain describes interference from another operator’s beam to the user.Aggregate cross-interference is included in the received-SINR model.Source: A. Signal Model · p. 1
  • Thermal noise power is modeled as kBTB using a 300 K noise temperature and 250 MHz bandwidth.The model identifies aggregate cross-interference separately from thermal noise.Source: A. Signal Model · p. 1

B. Interference Channel Model

The interference-channel model accounts for antenna discrimination at both the receiving terminal and interfering satellite. It specifies reference patterns and parameters for those off-axis gains.

B. Interference Channel Model

  • The 0.6-m terminal’s receive gain toward an interfering satellite follows the ITU-R S.1428-1 reference pattern for q between 20 and 25.The pattern is defined piecewise by off-axis angle, with Gmax = 20 log10 q + 7.7.Source: B. Interference Channel Model · p. 2
  • The interfering satellite’s off-axis angle is measured from its beam’s scheduled-user boresight toward the victim user.With no measured pattern available, the model uses the LEO branch of ITU-R S.1528-1.Source: B. Interference Channel Model · p. 2
  • The satellite antenna pattern uses Gs,max = 30 dBi, a 2.77° beamwidth, and the conservative Annex-1 value LF = 5 dBi.The model specifies the satellite pattern in piecewise angular regions and combines gains at both link ends.Source: B. Interference Channel Model · p. 2

C. Utility and Strategy

Each operator maximizes weighted sum spectral efficiency over its strategy set. The objective and power limits do not provide fairness or rate guarantees.

C. Utility and Strategy

  • Each operator independently maximizes weighted sum spectral efficiency, also described as weighted sum-rate utility.The strategy is an operator-specific power vector.Source: C. Utility and Strategy · p. 2
  • When total power exceeds the sum of per-beam power caps, the budget is inactive and box upper bounds limit usable power.This condition is stated for the objective’s power constraints.Source: C. Utility and Strategy · p. 2
  • The aggregate spectral-efficiency objective imposes no proportional-fairness, minimum-rate, or outage guarantees.Source: C. Utility and Strategy · p. 2

III. EQUILIBRIUM ANALYSIS

The game is defined with two players, operators A and B.

III. EQUILIBRIUM ANALYSIS

A. Potential Structure

The section characterizes how cross-interference perturbs the operators’ separable utilities using a potential-like function. Its approximation error vanishes with weak coupling, but the result is asymptotic and need not preserve utility-change signs when changes are comparable to the error.

A. Potential Structure

  • At zero cross-interference, the potential exactly tracks unilateral utility changes and is an exact ordinal potential.Source: A. Potential Structure · pp. 2–3
  • For fixed beam counts, the additive potential-approximation error is O(η) as the weak-interference ratio tends to zero.The bound does not by itself preserve the sign of a utility change when that change is of the same order as the error.Source: A. Potential Structure · pp. 2–3
  • The weak-coupling potential result is asymptotic and does not characterize the experimental regime.The experiments instead assess convergence using a sufficient contraction diagnostic, direct iterations, and best-response residuals.Source: A. Potential Structure · p. 3

B. Existence

The game admits at least one pure-strategy Nash equilibrium under the stated strategy-set and utility conditions.

B. Existence

  • The game admits at least one pure-strategy Nash equilibrium.Source: B. Existence · p. 3
  • Existence follows because each strategy set is compact, convex, and non-empty, while each utility is continuous and concave in its own power vector.The proof applies Rosen’s concave-game existence result.Source: B. Existence · p. 3

C. Distributed Water-Filling Best Response

Each operator’s best response is implemented by distributed water-filling, and a weighted spectral-norm condition gives sufficient convergence and uniqueness guarantees.

C. Distributed Water-Filling Best Response

  • The water-filling best response solves a convex optimization problem for fixed opponent power, with active total power bounded by the per-beam caps.The multiplier is found by bisection with complexity O(K log(1/δ)) per operator per iteration.Source: C. Distributed Water-Filling Best… · p. 3
  • The algorithm requires only local serving-channel estimates and aggregate interference measurements; full cross-channel information is used only in offline simulation.Source: C. Distributed Water-Filling Best… · p. 3
  • If ρ(J2) < 1, the game has a unique Nash equilibrium and simultaneous and alternating water-filling iterations converge to it geometrically.For alternating updates, a full sweep contracts by at most ρ(J2)^2.Source: C. Distributed Water-Filling Best… · p. 3

IV. NUMERICAL RESULTS

The numerical evaluation compares nominal and near-inline NGSO geometry using spherical-Earth simulations. The NE gain over uncoordinated transmission is larger in near-inline conditions and increases with beams per operator.

Near-Inline Performance

  • 5th-percentile SINRs were −3.29 dB uncoordinated, −17.08 dB NE, −3.60 dB max-min, and −21.99 dB centralized; max-min forfeits 45.9% of NE utility.Sum-utility optimization sacrifices the pooled lower tail, while max-min preserves a tail similar to uncoordinated transmission.Source: Figure 4 · p. 4, §IV · pp. 4–5

Evaluation Setup

  • The 50 spherical-Earth realizations compare nominal geometry, with a 3° receive-angle floor representing satellite selection, against unavoided [2°, 3°] near-inline cases.The nominal scenario samples the stated elevation and azimuth ranges; the near-inline scenario conditions center-look separation uniformly on [2°, 3°].Source: §IV · p. 4

Nominal Reference; Near-Inline Performance

  • The NE gains 0.06% over uncoordinated transmission nominally and 3.91% in the near-inline case.The nominal NE reaches 95.7% of the multi-start benchmark; the near-inline NE reaches 56.5% of the centralized benchmark.Source: §IV · p. 4

Near-Inline Convergence

  • All 50 near-inline runs converged in 9–76 sweeps (mean 24.48), although the sufficient condition held for only 19/50 runs.The additive potential approximation is not asserted tight in this regime because mean η = 13.195.Source: §IV · p. 4

Scalability

  • 0.61% at K=10, 3.34% at K=20, and 10.41% at K=50 are the near-inline NE utility gains over uncoordinated transmission.Each Fig. 5 point averages 20 realizations.Source: Figure 5 · p. 5, §IV · p. 5

Geometry Sensitivity

  • At separations 2°, 3°, 5°, and 15°, NE gain falls from 10.75% to 0.75%, 0.02%, and effectively zero, respectively.Mean ρ(J2) falls from 1.816 to 0.613, 0.023, and 0.001 at those separations.Source: Figure 6 · p. 5, §IV · p. 5

V. CONCLUSION

The paper models equal-priority NGSO operators as a non-cooperative power-control game, proves pure-strategy Nash equilibrium existence, and derives distributed water-filling with a weighted spectral-norm convergence condition. Dual-ended off-axis discrimination yields gains that vary sharply with geometry, while centralized and lower-tail gaps expose an efficiency–fairness trade-off.

V. CONCLUSION

  • The paper models equal-priority NGSO operators as a non-cooperative power-control game, proves pure-strategy NE existence, and derives distributed water-filling with a weighted spectral-norm convergence condition.The convergence condition is weighted spectral-norm based.Source: §V · p. 5
  • 0.06% nominally, 3.91% in the 2°–3° near-inline window, and 10.41% at K=50 are the reported NE gains.The geometry sweep localizes useful adaptation; all values and comparisons are as stated in the conclusion.Source: §V · p. 5
  • Centralized and lower-tail gaps expose the efficiency–fairness cost of independent sum-utility maximization; minimum-rate-aware, pricing, and ephemeris-driven extensions remain future work.Source: §V · p. 5
Loading 2609.38935v1…