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Stochastic Geometry Analysis and Optimization for Multi-LEO Satellite-Assisted Communications

Xu Hu, Bin Lin, Ping Wang, Xiao Lu, Weihua Zhuang, Shuang Qi

arXiv:2610.31373v1eess.SP

TL;DR

Single-satellite LEO links face fading and satellite-failure risks, while multi-satellite performance under serial service is challenging to characterize. The paper develops stochastic-geometry analyses and approximations for this setting, then evaluates reliability, rate, delay, and satellite-count optimization. Numerical results validate the analysis and show that adding serving satellites improves success probability before gains stabilize.

  • Problem

    Characterizing multi-satellite LEO performance is challenging because realistic fading, uplink and downlink transmission, sequential service, and satellite failures must be considered together.

    Source: A. Related Works · p. 3

  • Method

    The paper develops a stochastic-geometry framework for serial multi-LEO service, derives asymptotic success expressions, proposes a cluster-based approximation, and optimizes the success-probability-to-delay ratio.

    Source: §V · p. 11

  • Results

    Numerical results validate the cluster-based approximation, and overall transmission success probability rises with serving-satellite count before gradually stabilizing.

    Source: B. Results and Discussion · p. 10

  • Takeaways & Limitations

    The optimal number of serving satellites varies with constellation altitude, with higher altitudes requiring more satellites to improve overall performance.

    Source: B. Results and Discussion · p. 11

    The analysis assumes negligible co-channel interference; extending it to severe interference is identified as future work for more interference-limited scenarios.

    Source: §V · p. 12

Abstract

from arXiv · show

Low Earth orbit (LEO) satellites are vital in providing seamless connectivities to ground users (GUs). However, complex satellite-to-ground propagation and non-negligible satellite failure probabilities make single-satellite transmission unreliable. To address this issue, we consider a multi-LEO-satellite communication network (mLEO-CN), where a typical GU is served by multiple LEO satellites. The satellite locations are modeled as a binomial point process on a sphere and ordered according to their distances to the GU. To improve communication reliability, a serial communication scheme is adopted, in which the subsequent satellite is activated only when all preceding satellites fail to satisfy the target signal-to-noise ratio threshold. By employing the tool of stochastic geometry, we derive key performance metrics, including overall transmission success probability, average transmission rate, and end-to-end delay under the serial communication scheme by incorporating both shadowed Rician (SR) fading and satellite failure probabilities. To improve analytical tractability, we derive asymptotic expressions for the end-to-end transmission success probability in two special cases by leveraging the asymptotic expression of the SR fading model and a Gamma-distribution, and further propose a cluster-based approximation approach for the overall transmission success probability. Building on these analyses, we formulate an optimization problem to maximize the ratio of the overall transmission success probability to the end-to-end delay, and obtain its solution using a one-dimensional search approach. Numerical results validate the analysis, reveal the impact of key parameters, and identify the optimal number of serving LEO satellites under different constellation altitudes.

I. INTRODUCTION

The paper addresses unreliable single-satellite links by analyzing a multi-satellite network that serves ground users sequentially. It develops a stochastic-geometry framework for performance analysis and satellite-count design, including satellite failures.

I. INTRODUCTION

  • The framework targets unreliable single-satellite service, where fading and satellite failures threaten link quality and continuity.The proposed multi-satellite network is intended to improve reliability and permit comparison with an individual-satellite network.Source: §I · p. 1

A. Related Works

  • Existing multi-satellite schemes often require simultaneous participation, coordination, or cluster formation; the paper identifies serial activation as a lower-complexity alternative and notes that its performance remains challenging to characterize.The stated challenges include sequential-analysis complexity, shadowed Rician channels, end-to-end uplink and downlink performance, and satellite failures.Source: A. Related Works · pp. 2–3

B. Contributions and Organization

  • The paper develops a tractable stochastic-geometry framework for serial service that accounts for channel fading and satellite failures and evaluates success probability, average transmission rate, and end-to-end delay.It also derives asymptotic success-probability expressions for two special cases.Source: B. Contributions and Organization · p. 3
  • The cluster-based approximation represents each satellite cluster by its central satellite to reduce the complexity of evaluating overall transmission success probability.The authors state that the approximation maintains high accuracy.Source: B. Contributions and Organization · p. 3
  • The numerical study examines constellation size, altitude, SNR threshold, and satellite failure probability as factors in end-to-end performance.The simulations are described as validating the analytical framework and the cluster-based approximation.Source: B. Contributions and Organization · p. 3

A. Network Model

  • The model places N satellites uniformly on a spherical surface and ranks them by distance from a typical ground user; the serving satellites are selected from those visible to that user.The network also models satellite failure probability and uses decode-and-forward transmission over orthogonal channels.Source: A. Network Model · p. 4

B. Communication Scheme

The communication scheme probes satellites in nearest-to-farthest order and uses a target SNR threshold to decide whether to proceed. The signal model assumes distance-dependent path loss, shadowed Rician fading, and negligible co-channel interference.

B. Communication Scheme

  • The ground user probes satellites sequentially, selecting one only when both satellite-side and user-side SNR exceed the target threshold; transmission stops if none qualifies.Each subsequent satellite is tried only after the preceding candidate fails the threshold test.Source: B. Communication Scheme · p. 4

1) Path-loss Model:

  • The path-loss model depends on GU-to-satellite distance; the excerpt identifies distance, light speed, carrier frequency, and path-loss exponent as model quantities.The supplied passage does not state the path-loss formula.Source: 1) Path-loss Model · p. 4

2) Fading model:

  • The satellite links use shadowed Rician fading, with parameters for line-of-sight power, Nakagami fading, and multipath power.The channel-gain CDF and PDF are described under this fading model.Source: 2) Fading model · p. 4

D. Signal Model

  • The uplink and downlink SNR models use independent channel gains across the K links, and the analysis assumes co-channel interference is negligible in a noise-dominated environment.The stated assumption suits relatively abundant spectrum and settings that prioritize reliability over aggressive frequency reuse.Source: D. Signal Model · pp. 4–5

E. Distance distribution

The analysis uses an ordered-distance distribution for satellites and defines end-to-end success by threshold satisfaction on both uplink and downlink. Under serial service, it accounts for satellite failures through the probability that all serving satellites fail.

E. Distance distribution

  • Lemma 1 gives the distance PDF for the kth satellite, over the range from constellation altitude to the maximum visible-satellite distance.The satellites are indexed from nearest to farthest from the ground user.Source: E. Distance distribution · p. 5

A. Exact Overall Transmission Success Probability

  • End-to-end transmission succeeds through a satellite when the received SNR exceeds the threshold on both the uplink and downlink.The per-satellite success probability is defined for a predefined threshold τ.Source: A. Exact Overall Transmission Success… · p. 5
  • Under serial service, overall success is expressed through the sequential satellite success probabilities and equivalently as one minus the probability that all serving satellites fail.Theorem 1 incorporates each satellite’s operational probability, 1−q_k, into this overall success probability.Source: A. Exact Overall Transmission Success… · p. 5

B. Asymptotic Overall Transmission Success Probability

Because the exact success probability is difficult to derive, the section presents two simplifications: a high-SNR asymptotic expression and a Gamma-distribution approximation.

B. Asymptotic Overall Transmission Success Probability

  • The exact overall success probability is difficult to derive because it involves the shadowed Rician fading CDF and the satellite-distance PDF.The section therefore derives asymptotic expressions for specific cases.Source: B. Asymptotic Overall Transmission… · p. 5
  • At high SNR, Corollary 1 approximates the end-to-end transmission success probability through the kth LEO satellite.This case uses the asymptotic expression of the shadowed Rician fading CDF.Source: B. Asymptotic Overall Transmission… · pp. 5–6
  • Approximating shadowed Rician fading with a Gamma distribution simplifies the end-to-end transmission success probability in Corollary 2.The Gamma model uses shape and scale parameters αs and βs.Source: B. Asymptotic Overall Transmission… · p. 6

C. Cluster-based Approximation Approach for Complexity Reduction

The section introduces a cluster-based approximation to reduce the complexity of overall success-probability calculations, then defines the serial scheme’s average transmission rate capacity.

C. Cluster-based Approximation Approach for Complexity Reduction

  • The cluster-based approximation estimates overall transmission success probability for K > 2 by grouping serving satellites and using a central satellite to represent each cluster.Proposition 1 gives the resulting approximation.Source: C. Cluster-based Approximation… · p. 6
  • Representing each cluster’s performance by its central satellite reduces the mathematical complexity of calculating overall transmission success probability.The original expression involves integrals and multiplications across satellites; the approximation replaces cluster members’ performance with a power of the central satellite’s performance.Source: C. Cluster-based Approximation… · p. 6

D. Average Transmission Rate Capacity

  • Average transmission rate capacity includes each subsequent satellite only when all preceding satellites fail to meet the SNR threshold.The end-to-end rate through a satellite depends on the minimum of the received uplink and downlink SNRs.Source: D. Average Transmission Rate Capacity · p. 7

E. End-to-End Delay

End-to-end delay includes transmission and handover delays, with a constant handover delay assumed per switch. The section then optimizes success probability relative to delay using a one-dimensional search over serving-satellite count.

E. End-to-End Delay

  • The delay model combines uplink and downlink transmission delays with a constant handover delay for each satellite switch.Switching from the first satellite to the kth adds (k −1) Tho; end-to-end delay averages total delays over satellites that successfully transmit the message.Source: E. End-to-End Delay · p. 7

F. Optimal Number of Serving LEO Satellites

  • The objective is to maximize overall transmission success probability divided by end-to-end delay, balancing reliability against added handover delay.The optimization selects the number of serving satellites K.Source: F. Optimal Number of Serving LEO… · p. 7
  • The objective is unimodal in the number of serving satellites, enabling a one-dimensional search that stops when the objective begins to decrease.The search is applied for a given constellation altitude because K is integer-valued and the objective is analytically intractable for conventional convex optimization.Source: F. Optimal Number of Serving LEO… · p. 8

IV. NUMERICAL RESULTS

The numerical results validate the analysis and show how satellite count, constellation conditions, and packet size shape reliability, rate, and delay. The preferred number of serving satellites depends on altitude and the balance between transmission and handover delay.

A. Parameter Settings

  • The simulations use N=400 satellites, a 1000 km constellation altitude, a -10 dB SNR threshold, and satellite failure probability q=0 or 0.1 unless otherwise specified.Source: A. Parameter Settings · p. 8

B. Results and Discussion

  • Analytical and simulated success probabilities agree; serial service with K=2 or K=3 improves reliability over K=1, especially when satellite failures occur.For K=1, success probability falls from 1 to 0.908 as the SNR threshold rises from -30 dB to -20 dB when q=0; with q=0.1, it falls from 0.8 to 0.726.Source: B. Results and Discussion · p. 9, Figure 2 · p. 9
  • Success probability declines with increasing altitude but rises as constellation size grows from 100 to 600.The paper attributes these trends to longer GU–satellite distances at higher altitudes and shorter serving-satellite distances in denser constellations; low-altitude results show a distance-distribution mismatch.Source: B. Results and Discussion · p. 9, Figure 3 · p. 9, Figure 4 · p. 10
  • The two asymptotic expressions agree with simulations in their examined regimes, but the Case 1 expression is limited to high SNR.Case 1 examines thresholds from -30 dB to -20 dB; Case 2 examines -7 dB to 3 dB and uses a Gamma approximation for the SR-fading CDF.Source: B. Results and Discussion · pp. 9–10, Figure 5 · p. 10
  • For K=3, the cluster-based approximation closely matches simulations, while success probability gains from adding serving satellites gradually stabilize.The approximation represents each cluster by its central satellite.Source: Figure 6 · p. 10, B. Results and Discussion · p. 10
  • Average transmission rate capacity decreases with altitude, increases as constellation size grows from 100 to 600, and improves for K=2 or K=3 versus K=1.Source: Figure 7 · p. 10, B. Results and Discussion · p. 10
  • End-to-end delay rises with altitude and serving-satellite count, but decreases slightly as constellation size grows from 100 to 500.The paper attributes the increases to longer transmission distance and added handover delay, and the size-related decrease to closer serving satellites; small constellations show delay deviations associated with distance-PDF mismatch.Source: B. Results and Discussion · pp. 10–11, Figure 9 · p. 11, Figure 10 · p. 11
  • At higher constellation altitudes, the optimal serving-satellite count increases; for large packets, adding satellites is recommended, while frequent switches can make serial service less attractive.The optimization maximizes the ratio of overall transmission success probability to end-to-end delay.Source: Figure 11 · p. 11, B. Results and Discussion · p. 11

V. CONCLUSION

The paper develops a stochastic-geometry framework for evaluating key performance metrics in a serial multi-LEO network. It derives tractability-oriented approximations and uses the analysis to optimize the number of serving satellites for the reliability–latency trade-off.

V. CONCLUSION

  • The framework evaluates overall transmission success probability, average transmission rate capacity, and end-to-end delay for a multi-LEO network with serial service.It also derives asymptotic success-probability expressions and a cluster-based approximation, then formulates an optimization to select the serving-satellite count.Source: §V · p. 11
  • The optimization selects the number of serving satellites that maximizes the trade-off between transmission reliability and latency.The paper identifies deriving more tractable distance PDFs under different spatial distributions and studying severe co-channel interference as follow-up directions.Source: §V · p. 12

APPENDIX

The appendix derives the end-to-end success-probability expressions by starting from the kth-satellite probability and applying channel-fading and distance-distribution results. It then obtains a high-SNR asymptotic expression through term-by-term evaluation.

A. Proof of Theorem 1

  • Theorem 1’s success-probability expression follows from uplink–downlink independence, the fading-gain CCDF, and the kth-satellite distance PDF.Substituting the shadowed Rician fading CDF yields expression (9).Source: A. Proof of Theorem 1 · p. 12

B. Proof of Corollary 1

  • Corollary 1’s asymptotic expression uses the high-SNR shadowed Rician CDF approximation and evaluates the resulting terms to obtain expression (11).The derivation calculates the first term using the PDF’s unit integral and follows a representative derivation for the remaining terms.Source: B. Proof of Corollary 1 · p. 12

C. Proof of Corollary 2

The proof derives expression (13) using an approximated shadowed Rician fading CDF and incomplete-gamma-function relationships.

C. Proof of Corollary 2

  • Expression (13) is obtained from the approximated shadowed Rician CDF, the relationship between upper and lower incomplete gamma functions, and an integer-shape gamma-function identity.The proof begins from the expression used for the kth-satellite success probability.Source: C. Proof of Corollary 2 · pp. 12–13

D. Proof of Theorem 2

The appendix derives rate capacity and delay expressions by averaging channel- and distance-dependent quantities. It uses the shadowed Rician fading and kth-satellite distance distributions in these derivations.

D. Proof of Theorem 2

  • Theorem 2’s rate-capacity expression follows by converting uplink and downlink SNR conditions to indicator functions, then averaging over fading and distance.The fading gain uses the shadowed Rician PDF, and the distance uses the PDF in Lemma 1.Source: D. Proof of Theorem 2 · p. 13

E. Proof of Theorem 3

  • Theorem 3’s delay derivation averages the uplink and downlink delay terms using the shadowed Rician and kth-satellite distance PDFs.The proof derives one delay component as an example and applies similar steps to the remaining component.Source: E. Proof of Theorem 3 · p. 13

F. Proof of Proposition 2

The proof establishes that F(K), the objective over the number of serving satellites, is unimodal across the discrete domain K ≥ 1. Its shape follows from comparing the objective with a non-increasing marginal gain-to-cost ratio.

F. Proof of Proposition 2

  • The proof studies the sign of F(K + 1) − F(K), which is determined by comparing M_K with F(K).This comparison characterizes the objective’s monotonic behavior.Source: F. Proof of Proposition 2 · p. 14
  • The marginal gain-to-cost ratio M_K is non-increasing because added satellites bring smaller performance gains and larger delay costs.The proof states that a_K+1 and b_K+1 decrease with K, while T_K+1 increases.Source: F. Proof of Proposition 2 · p. 14
  • F(K) is unimodal over the discrete domain K ≥ 1, so it can transition from increasing to decreasing at most once.The K = 1 expression is the boundary case of the general objective.Source: F. Proof of Proposition 2 · p. 14
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