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Inter-Plane Inter-Satellite Connectivity in Dense LEO Constellations

Israel Leyva-Mayorga, Beatriz Soret, Petar Popovski

arXiv:2005.07965v3cs.NI

TL;DR

Dense LEO constellations face limited satellite capabilities, high orbital velocities, and strongly Doppler-affected cross-seam inter-plane ISLs. The paper models ISL establishment as a weighted dynamic matching problem and proposes algorithms that improve achievable sum rates over benchmark and random allocation schemes.

  • Problem

    Limited satellite capabilities and high orbital velocities make cross-seam inter-plane ISLs highly dynamic and considerably affected by Doppler effects.

  • Method

    The paper formulates constellation connectivity as a weighted dynamic matching problem and proposes a framework and algorithms for maximizing inter-plane ISL rates.

  • Results

    The algorithm increases achievable sum rates by 42% with FDMA and 71% with CDMA, while its GRA algorithm achieves rates up to 71% greater than random allocation.

  • Takeaways & Limitations

    The framework maximizes rates in inter-plane ISLs of dense LEO constellations and provides a simple approach to constellation design.

Abstract

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With numerous ongoing deployments owned by private companies and startups, dense satellite constellations deployed in low Earth orbit (LEO) will play a major role in the near future of wireless communications. In addition, the 3rd Generation Partnership Project (3GPP) has ongoing efforts to integrate satellites into 5G and beyond-5G networks. Nevertheless, numerous challenges must be overcome to fully exploit the connectivity capabilities of satellite constellations. These challenges are mainly a consequence of the low capabilities of individual small satellites, along with their high orbital speeds and small coverage due to the low altitude of deployment. In particular, inter-plane inter-satellite links (ISLs), which connect satellites from different orbital planes, are greatly dynamic and may be considerably affected by the Doppler shift. In this paper, we present a framework and the corresponding algorithms for the dynamic establishment of the inter-plane ISLs in LEO constellations. Our results show that the proposed algorithms increase the sum of rates in the constellation 1) by up to 115% with respect to the state-of-the-art benchmark schemes in an interference-free environment and 2) by up to 71% when compared to random resource allocation in a worst-case scenario for interference.

I. INTRODUCTION

Dense LEO constellations require dynamic inter-plane ISLs because low altitude limits individual coverage while high orbital velocities continually change topology and complicate cross-plane communication. The paper formulates this establishment problem as weighted dynamic matching and proposes algorithms that improve aggregate rates.

  • Low altitude limits individual satellite coverage, motivating dense LEO deployments for global and continuous coverage.
  • High orbital velocities create frequent topology changes and complicate communication between satellites in different orbital planes.
  • Cross-seam inter-plane ISLs experience considerably large Doppler effects because satellites in opposite orbital directions have large relative velocities.
  • The paper formulates unicast inter-plane ISL establishment as a weighted dynamic matching problem that maximizes the selected sum of rates.
  • The framework supports constellations regardless of geometry or symmetry, including initial deployments with few satellites and substantial asymmetries.
  • 115% is the maximum increase in constellation sum rates reported for greedy matching algorithms relative to the benchmark algorithm.
  • 71% is the increase in achievable sum rates from the proposed resource allocation algorithm over random allocation with CDMA.
  • Less than 10 ms is the propagation delay for 80% of inter-plane ISLs when matching algorithms are combined with an appropriate constellation design.

II. SYSTEM MODEL AND PROBLEM FORMULATION

This section presents the system model and formulates the optimization problem, with relevant notation listed in Table I.

  • The section presents the system model and formulates the optimization problem.
  • Relevant notation introduced in the section is listed in Table I.

A. System model

The system models inter-plane ISLs in a dynamic Walker star LEO constellation as weighted satellite connections constrained by geometry, line of sight, Doppler shift, antenna direction, and interference.

  • Constellation and graph model: The constellation is modeled as a dynamic weighted multipartite graph whose vertices are satellites and whose edges are feasible inter-plane ISLs.Each edge weight represents the achievable data rate between its satellite endpoints.
  • Link characterization: Inter-plane rates are calculated from free-space path loss, antenna gain, SNR, and resource allocation under a zero-outage constraint.The model assumes free-space propagation, AWGN thermal noise, fixed transmit power, similar steerable directional antennas, and Pout = 0.
  • Link feasibility: Feasible inter-plane pairs are determined by line of sight and Doppler-shift constraints, while intra-plane pairs are excluded from the inter-plane edge set.The Earth blocks links whose distance exceeds the maximum slant range, and cross-seam links are excluded because of their high relative Doppler shift.
  • Interference model: Interference arises when multiple inter-plane ISLs share wireless resources, affecting their achievable data rates.The framework accounts for interference-free rates and for resource sharing when interference occurs.

B. Problem formulation

The paper formulates inter-plane ISL establishment as a weighted matching problem that maximizes constellation-wide rates while accounting for shared-resource interference and unknown instantaneous interference.

  • Objective: The objective is to maximize the sum of rates over inter-plane ISLs in the satellite constellation.The formulation builds on maximum weighted matching, where the weight of an edge set is the sum of its edge weights.
  • Resource allocation: Resource allocation is represented as a many-to-one maximum weighted matching with externalities in a bipartite graph.A resource can be shared by several satellite pairs, and each resource allocation can affect the weights of other edges through interference.
  • Interference uncertainty: The worst-case model assumes that neither the centralized entity nor satellites have instantaneous knowledge of interference.Rates are selected to ensure Pout = 0 for every permissible interference pattern and at all times.
  • Interference uncertainty: A permissible interference pattern allows at most one simultaneously transmitting satellite for each other satellite pair sharing the same resource.This condition is expressed through the allocation variables for the shared resource.
  • Optimization complexity: The optimal resource-allocation solution is the maximum matching A⋆, but finding it has tremendous complexity.The paper therefore presents a relatively low-asymptotic-complexity framework for finding a near-optimal solution.

III. PROPOSED FRAMEWORK

The proposed framework first selects feasible inter-plane satellite pairs and then allocates orthogonal wireless resources when interference affects the selected links.

  • Framework stages: The framework first solves inter-plane satellite matching over the feasible edge set E.This separates link selection from subsequent resource allocation.
  • Framework stages: When interference affects the selected ISLs, the framework performs orthogonal resource allocation after selecting the satellite pairs.The sequence is to select all satellite pairs first and allocate wireless resources afterward if needed.

A. Satellite matching

The satellite-matching phase selects a feasible weighted subgraph subject to transceiver and directional-neighborhood constraints, using interference-free edge weights before any resource allocation.

  • Matching formulation: Satellite matching with Q transceivers is formulated as a many-to-many maximum weighted matching.The selected subgraph must satisfy the satellite transceiver and directional-neighborhood conditions.
  • Matching constraints: The matching must satisfy ∆(G_M) ≤ Q, with Q = 1 yielding a one-to-one matching.A vertex cannot have more selected incident links than its available inter-plane transceivers.
  • Matching constraints: A satellite neighbor is permissible only when no other adjacent satellite uses the same direction d(u, v).This enforces directional compatibility among selected inter-plane links.
  • Matching weights: The matching objective uses edge weights based on SNR-derived rates in an interference-free environment.If selected pairs do not interfere, the matching directly solves ISL establishment for any positive number of resources K.
  • Resource allocation: Resource allocation remains a many-to-one matching with externalities, but its complexity is reduced because the satellite-pair set M is fixed.The fixed matching allows the allocation phase to operate on an already selected set of pairs.
  • Resource allocation: A perfect matching is guaranteed at the end of resource allocation because the formulation enforces |A| = |M|.The allocation graph therefore contains one allocated resource edge for every selected satellite pair.

IV. ALGORITHMS

The paper develops centralized greedy algorithms for dynamic satellite matching and resource allocation, while recognizing that centralized operation requires advance computation and communication across the constellation.

  • IV. ALGORITHMS: The centralized entity computes a new matching every T seconds and communicates the result to the constellation.The algorithms can be deployed at the ground or space segment and support arbitrary constellation geometry, P, and Q.
  • IV. ALGORITHMS: Centralized matching must be solved ahead of time because processing and communication overhead must be included in timely execution.The approach therefore cannot account for satellite buffer status and activation patterns in real time.
  • IV. ALGORITHMS: The activation pattern is assumed unknown to the centralized entity, while predicting satellite activity is outside the paper’s scope.The paper instead considers a general case without prior knowledge of satellite activation.
  • IV. ALGORITHMS: The proposed framework uses greedy algorithms to solve satellite matching first and resource allocation second.Greedy decisions generally offer relatively low complexity, but are not guaranteed to reach the optimal matching solution.
  • IV. ALGORITHMS: The greedy matching guarantee is characterized through a worst-case result for one-to-one maximum weighted matching.The theorem compares a greedy matching that selects the maximum-weight edge at each iteration with an optimal matching.

A. Satellite matching

The satellite-matching section introduces GIEM and its Markovian extension GMM, using ordered weighted edges to build feasible matchings while reducing recomputation across realizations.

  • A. Satellite matching: GIEM is a centralized greedy matching algorithm that orders feasible weighted edges and selects permissible high-weight pairs.The algorithm uses the number of transceivers Q and excludes edges below Rmin.
  • A. Satellite matching: GIEM has sorting-dominated complexity O(|L| log2(|L|)) and terminates after |L| + 1 executions.Its solution is unique when the same input produces a consistent sorted list.
  • A. Satellite matching: GIEM avoids unstable pairs by selecting the greatest-weight feasible edge while preserving permissible neighborhoods.An unstable pair is one preferred by both endpoints over their assigned partners.
  • A. Satellite matching: GMM extends GIEM by retaining feasible pairs from the previous realization before applying GIEM to the remaining edges.Pair persistence makes rate maximization secondary and reduces pair changes and associated handshakes.
  • A. Satellite matching: GMM reduces execution time relative to GIEM primarily by sorting a smaller edge list.The previous matching supplies the initial search set, reducing the remaining list size.

B. Greedy resource allocation (GRA)

The GRA algorithm allocates orthogonal resources greedily after matching, selecting at each iteration the resource that maximizes the interference-aware sum of rates.

  • B. Greedy resource allocation (GRA): GRA centrally allocates resources after satellite matching by repeatedly selecting the next pair and its best resource.The ordered matching list is processed one pair at a time.
  • B. Greedy resource allocation (GRA): The selected resource maximizes the sum of rates as a function of SINR while accounting for interference.The algorithm evaluates possible interfering-link combinations through exhaustive search.
  • B. Greedy resource allocation (GRA): The GRA algorithm has complexity O(KM^2).At most 4K(m −1) operations and K additions or comparisons are performed per iteration across M iterations.
  • B. Greedy resource allocation (GRA): GRA is unique with consistent sorting and achieves the optimal solution obtainable by any greedy algorithm.Exhaustive search selects the resource k⋆ that maximizes the sum of rates at each iteration.
  • B. Greedy resource allocation (GRA): GRA terminates after M +1 executions because one element is deleted from the ordered list at each iteration.The initial list contains M matched satellite pairs.

V. EXPERIMENT DESIGN AND PARAMETER SELECTION

The experiments evaluate matching and resource-allocation algorithms across limited and full inter-plane connectivity using representative Walker-star constellation settings and interference regimes.

  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: The constellation model uses a lowest orbital altitude of 600 km and 10 km separation between orbital planes.Plane altitude is defined as hp = h1 + 10(p −1) km.
  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: The evaluation studies limited and full inter-plane ISL connectivity for P ∈{5, 6, 7, 8}, with full connectivity guaranteed only for P ≥7.The study fixes f = 2.4 GHz, Rmin = 10 kbps, and MPL = L⋆(7, 40, f) dB.
  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: The communication range for the selected parameters is 3527 km, requiring EIRPG ≥3.74 W.This range is based on the adjacent-plane setting L⋆(7, 40).
  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: The interference evaluation compares an interference-free narrow-beam scenario with a worst-case isotropic-antenna scenario.These scenarios provide upper and lower performance bounds under the selected Rmin and EIRPG.
  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: Performance is simulated over 1000 matching periods with 30-second intervals, totaling 30000 seconds.The constellation is rotated before each matching and resource-allocation realization.
  • V. EXPERIMENT DESIGN AND PARAMETER SELECTION: GEO is the satellite-matching benchmark, matching neighboring-plane satellites within equal logical locations in O(PNp) complexity.GRA is compared with round-robin and random allocation after GIEM matching.

VI. RESULTS

The evaluation compares satellite-matching and resource-allocation algorithms for dynamic inter-plane ISLs, examining connectivity, rates, delay, and complexity. GIEM and GRA generally improve rate performance, while connectivity and complexity depend on orbital-plane configuration and algorithm choice.

  • Connectivity: Full inter-plane ISL connectivity is observed for P ≥7, although complete connectivity is not necessary to obtain benefits from inter-plane links.Packets can use intra-plane links toward the poles when a direct inter-plane link is unavailable.
  • Rates and delay: For GIEM with P = 7, almost 50% of rates are below 20 kbps, less than 20% exceed 100 kbps, and around 4% exceed 1 Mbps.More than 80% of established ISLs have propagation delay below 10 ms, with only slight differences among matching algorithms.
  • Complexity: The GEO matching algorithm has the lowest complexity, followed by GMM and GIEM, while resource allocation mainly determines the complexity of establishing ISLs.The greedy resource-allocation algorithm performs on the order of P −1 more operations than GIEM.

VII. CONCLUSIONS

The paper presents a framework for maximizing rates in inter-plane ISLs through constellation design, satellite matching, and resource allocation. GIEM generally improves sum rates over GMM and GEO, while GRA provides substantial allocation gains but has the highest complexity.

  • Framework and design: The framework maximizes rates in inter-plane ISLs and includes a simple approach to constellation design.The design calculates minimum transmission power and antenna gains to guarantee full inter-plane ISL connectivity.
  • Satellite matching: GIEM achieves greater sum rates than GMM and GEO, with the GIEM–GMM performance difference growing as constellation density increases.GMM maintains previous satellite pairs for as long as possible, whereas GIEM solves satellite matching from scratch at each realization.
  • Resource allocation: 71% greater sum rates are achieved by GRA than by random allocation.This comparison concerns the maximum sum of rates achieved with the proposed resource allocation algorithm.
  • Trade-offs: GRA has the greatest complexity among the satellite matching and resource allocation algorithms considered.Its use is therefore tied to settings where the centralized control entity has sufficient computing power and communication overhead is relatively low.
  • Trade-offs: GMM is an efficient alternative when ISL-establishment communication overhead is large or full inter-plane connectivity is not guaranteed.The paper recommends GIEM and GRA together in highly dense LEO constellations under specified operating conditions.
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