Dynamic Visibility-Aware Satellite Selection

• The paper presents a dynamic visibility-aware multi-orbit satellite selection framework that leverages Markov approximation, matching games, and dual decomposition to optimize LEO network sum-rate.
• It models time-varying user-satellite visibility constraints and addresses NP-hard challenges in resource allocation and user association with efficient algorithmic strategies.
• Simulations show a 7.85% improvement in average sum-rate over baselines, demonstrating robustness across varying network conditions and practical scalability.

A dynamic visibility aware multi-orbit satellite selection framework refers to a class of optimization and control solutions for multi-orbit Low Earth Orbit (LEO) satellite networks, in which the set of candidate serving satellites varies dynamically with time due to orbital motion, leading to phase-shifted ground tracks and nonstationary coverage patterns. Such frameworks are essential for maximizing network sum rate, ensuring per-satellite resource constraints, and provisioning robust connectivity in space-air-ground integrated networks. The framework proposed in "Visibility-aware Satellite Selection and Resource Allocation in Multi-Orbit LEO Networks" (Sun et al., 16 Nov 2025) models user visibility constraints, satellite selection, user association (UA), bandwidth allocation (BA), and power allocation (PA) as a joint NP-hard combinatorial optimization problem, addressed through a coupling of Markov approximation and matching game theory.

1. System Model and Notation

The considered system is a multi-orbit LEO user downlink network segmented in time slots $t \in \{1,\dots,T\}$ with the following components:

• Orbits and satellites: $\mathcal{O} = \{1, ..., O\}$ denotes orbital planes; for each $o$, $\mathcal{S}_o$ is the set of satellites, and $$\mathcal{S} = \bigcup_{o \in \mathcal{O}} \mathcal{S}_o$$ the complete satellite set.
• Users: $\mathcal{U} = \{1,\dots,J\}$ is the user set.
• Resources: Each satellite has total bandwidth $B$ and maximum power $P_s$.
• Channel and visibility: $g_{u,s}(t)$ is the channel gain between user $u$ and satellite $s$ at time $t$; $v_{u,s}(t) \in \{0,1\}$ indicates if $s$ is within $u$'s zenith angle cone.
• Decision variables: For each $t$,
  • UA: $x_{u,s}(t) \in \{0,1\}$ (1 if $u$ is associated to $s$),
  • BA: $b_{u,s}(t) \ge 0$,
  • PA: $p_{u,s}(t) \ge 0$.

Constraints enforce: (i) at most one association per user, (ii) only visible satellites can serve a user, (iii) per-satellite bandwidth/power limits, (iv) optional minimum rate guarantees (per time/user slot).

2. Joint Optimization Problem Formulation

The core problem is to maximize the time-averaged sum-rate under aforementioned constraints. The objective is:

$$\max_{x, b, p} \; \sum_{t=1}^T \sum_{u \in \mathcal{U}} \sum_{s \in \mathcal{S}} x_{u,s}(t)\,b_{u,s}(t) \log_2\left(1 + \frac{p_{u,s}(t)\,g_{u,s}(t)}{N_0\,b_{u,s}(t)}\right)$$

Subject to decision variable feasibility for UA, visibility, resource constraints, and (optionally) minimum per-user rates. The sum-rate maximization is non-convex and mixed-integer, with NP-hard complexity driven by the combinatorial user association and continuous (bandwidth, power) resource allocation.

3. Algorithmic Framework: Markov Approximation and Block Coordinate Descent

The Dynamic Visibility-aware Multi-Orbit Satellite Selection ("DV-MOSS" — Editor's term) framework decomposes the problem along two major axes:

3.1 Markov Approximation for Satellite Subset Selection

• State space: Each state $\boldsymbol{f} = (x, b, p)$ comprises feasible network allocations under a particular subset of active satellites $z$ (subject to a maximum constellation size).
• Transition dynamics: Moves $\boldsymbol{f} \to \boldsymbol{f}'$ are sampled with probability $$q(\boldsymbol{f} \to \boldsymbol{f}') = 1 / [1+\exp(\beta(E(\boldsymbol{f}') - E(\boldsymbol{f})))]$$ for energy function $E(\boldsymbol{f})$ (negative sum-rate objective), and inverse temperature $\beta$.
• Steady-state: The process converges to the Boltzmann distribution $$\pi(\boldsymbol{f}) \propto e^{-\beta E(\boldsymbol{f})}$$; as $\beta \to \infty$, global optima are sampled with higher probability. The theoretical guarantee is provided via detailed-balance.

3.2 Block Coordinate Descent for User/Resource Assignment

Given an active set $z$ of satellites, the framework alternates:

3.2.1 User Association and Bandwidth Allocation via Matching Games

• Two-sided matching: (1) Users and satellites for association, (2) associated users and subcarriers for bandwidth. User preferences (marginal rate improvement) and satellite/subcarrier preferences (SINR surplus, co-channel cost) drive stable matchings via deferred-acceptance.
• Stability and monotonicity: Theorem 1 guarantees monotonic increase in sum-rate and convergence to stable matching under this protocol.

3.2.2 Power Allocation via Dual Decomposition

• Optimization: For fixed $(x, b)$, power allocation per satellite is solved via dual decomposition, introducing Lagrange multipliers for power and minimum-rate constraints.
• Closed-form updates: KKT conditions yield water-filling-like updates:

$p_{u,s}^* = \frac{1 + \nu_u}{\lambda_s \ln 2}$

with dual multipliers updated via subgradient descent. Strong duality and convergence to the saddle-point are guaranteed (Theorem 2).

3.2.3 Algorithmic Structure

The overall framework iteratively samples satellite subsets via Markov dynamics (outer loop) and solves the resource assignment subproblem via matching + power allocation (inner block coordinate descent). Convergence is achieved when the exploration probability vanishes and allocations stabilize.

4. Theoretical Properties and Performance Characterization

• Optimality and mixing: The Markov approximation achieves detailed balance; as $\beta$ increases, the distribution converges on the globally optimal solution, within a log-sum-exp relaxation bound.
• Matching game properties: The matching subroutine guarantees monotonic improvement and stable user-satellite assignments.
• Dual decomposition: The power allocation subproblem enjoys strong duality, and subgradient-based updates converge efficiently to optimality.
• Simulation results: Against four baselines (closest-sat, random-sat, $\varepsilon$-Markov, fixed-UA), DV-MOSS achieves ~7.85% higher average sum-rate over the best baseline, with robustness to cone angle, user density, and shadowing regimes.

Summary of simulation parameters:

Parameter	Value
Orbits $O$	40
Satellites $|\mathcal{S}|$	$25 \times 40$
Users $|\mathcal{U}|$	30
Subcarriers per sat $K$	25
Bandwidth per sat $B$	10 MHz
Carrier $f_c$	6 GHz
Satellite altitude $h$	550 km
Power budget $P_s$	5 W
Constellation size $Z_{th}$	10
Shadowing $SF$	1–3 dB
Cone angle $\varphi$	$\le 7\pi/20$ rad

A plausible implication is that real-time adaptation to both visibility sets and resource states yields significant performance improvements over greedy or static satellite selection policies.

5. Key Features and Contributions

• Dynamic Visibility Modeling: Explicit incorporation of $v_{u,s}(t)$ to account for time-varying candidate sets, addressing the unique dynamics of phase-shifting multi-orbit constellations.
• Joint Approach: Direct coupling of satellite selection (Markov approximation), user association/bandwidth allocation (matching games), and power allocation (dual decomposition) in a single unified optimization.
• Provable Convergence: The method guarantees convergence for all major algorithmic components: Markov chain (detailed balance), matching assignments (stable matching), and PA (strong duality).
• Practical Gains: +7.85% sum-rate improvement, demonstrated robustness across varying network and environmental parameters, and the ability to adapt constellation size in real time.
• Implementation Scalability: Demonstrated feasibility on networks with $|O| = 40$ orbits, $|\mathcal{S}|=1000$ satellites, and dynamic multi-user traffic (Sun et al., 16 Nov 2025).

6. Context, Significance, and Research Trajectory

The dynamic visibility aware multi-orbit satellite selection framework advances the design of LEO satellite networks by bridging the gap between traditional single-layer selection methods and the requirements of modern mega-constellations exhibiting variable, phase-shifted coverage. A plausible implication is the enhanced viability of space-air-ground integrated networks, where real-time adaptability to fast-changing link topologies is essential for meeting performance and reliability objectives. The multi-level decomposition employed by DV-MOSS reflects a maturing trend in joint resource allocation for large-scale wireless systems, integrating stochastic search (Markov chain), combinatorial optimization (matching), and convex analysis (dual decomposition). Future research may extend these principles to incorporate additional real-world constraints such as inter-satellite link coordination, mobility prediction uncertainty, and network slicability for differentiated services (Sun et al., 16 Nov 2025).