PaperlayerView exact revision on arXiv

Source-linked AI summary

Spatial Distribution of Mean PAoI in LEO Satellite-enabled IoT Networks

Badiaa Gabr, Mustafa A. Kishk

arXiv:2610.23597v2cs.ITmath.IT

TL;DR

Rural IoT networks need fresh uplink updates where cellular coverage may be limited, and satellite motion makes large-scale performance analysis challenging. This paper uses stochastic geometry to derive the spatial distribution of conditional mean PAoI and studies satellite count, altitude, and beamwidth; its results show that useful beamwidth choices depend on deployment conditions.

  • Problem

    Rural IoT nodes may lack cellular coverage, while real-time applications depend on fresh data and satellite motion complicates large-scale network analysis.

    Source: §I · p. 1

  • Method

    Stochastic geometry is used to derive the uplink conditional-coverage-probability meta-distribution and spatial distribution of conditional mean PAoI in a LEOSAT-enabled IoT network.

    Source: §V · pp. 10–11

  • Results

    At 500 km altitude, narrower beamwidth does not reduce average conditional mean PAoI, whereas at 1500 km it can minimize it; denser constellations call for narrower beams to manage interference.

    Source: §IV · pp. 9–10

  • Takeaways & Limitations

    Beamwidth selection should be tailored to satellite altitude and constellation density to reduce conditional mean PAoI in the examined deployment scenarios.

    Source: §IV · p. 10, §V · p. 11

Abstract

from arXiv · show

Non-terrestrial networks (NTNs), such as Low Earth Orbit (LEO) satellites, can provide consistent coverage for Internet of Things (IoT) nodes, particularly in rural areas where cellular coverage is lacking. These IoT nodes are used to monitor and record their surroundings, supporting various applications, including real-time applications. In such cases, transmission of the most up-to-date data is essential, making the Age of Information (AoI) a critical metric to consider. The AoI measures the elapsed time from the data generation at the source node to its reception at the destination. Moreover, the Peak AoI (PAoI), which denotes the maximum AoI in the system before an update is received, provides valuable insights into the network's performance. Therefore, this encourages us to focus on the PAoI within a large-scale and fully loaded IoT network, which is covered by LEOSATs. Using tools from stochastic geometry, this paper derives the spatial distribution of the uplink mean PAoI in a LEOSAT-enabled IoT network. To achieve that purpose, the meta distribution of the uplink conditional coverage probability is derived. Numerical results articulate how varying the number of deployed LEOSATs and their altitudes affects the coverage probability meta distribution and the spatial distribution of the mean PAoI. Additionally, by examining various scenarios with different numbers of LEOSATs and altitudes, we can derive valuable insights to determine the optimal beamwidth for LEOSATs tailored to each specific scenario to minimize the conditional mean PAoI.

I. INTRODUCTION

Real-time IoT applications need fresh updates, while rural nodes may lack cellular coverage and satellite motion complicates large-scale performance analysis. This work uses stochastic geometry to derive the spatial distribution of uplink conditional mean PAoI and examine how deployment choices affect it.

I. INTRODUCTION

  • PAoI captures the maximum Age of Information before an update arrives, making it relevant to network reliability in real-time IoT applications.AoI measures time from data generation to reception.Source: §I · p. 1
  • Satellite motion randomizes distances to ground nodes, complicating analysis as transmissions scale, motivating tractable stochastic-geometry models.The work considers rural IoT settings where cellular coverage may be limited and LEOSATs provide connectivity.Source: §I · p. 1

A. Related Work

  • Unlike, which derives average PAoI, this work uses a meta-distribution to characterize spatial PAoI reliability across time deadlines.The distribution also provides time percentiles needed to achieve a specified network reliability.Source: A. Related Work · p. 3

B. Contributions

  • The contribution is a spherical stochastic-geometry model for uplink conditional mean PAoI under shadowed Rician fading, with coverage-probability moments and meta-distribution.The model considers a rural IoT network and satellite coverage for uplink updates.Source: B. Contributions · p. 3
  • The analysis examines how satellite count and altitude affect performance and identifies scenario-dependent beamwidth choices and an altitude that can minimize first-moment conditional mean PAoI.The beamwidth analysis concerns reducing conditional mean PAoI across deployment scenarios.Source: B. Contributions · p. 4

C. Paper Outline

The paper proceeds from its system model to the spatial distribution analysis, numerical results, and conclusion. Its system model focuses on uplink updates from an IoT node to an associated LEOSAT in an area without cellular infrastructure.

C. Paper Outline

  • Section II introduces the system model, Section III the spatial distribution of conditional mean PAoI, Section IV simulation results and insights, and Section V the conclusion.Table I summarizes the paper's notation.Source: C. Paper Outline · p. 4

II. SYSTEM MODEL

  • The modeled network relies on LEOSAT connectivity because its deployment region lacks cellular infrastructure.Source: §II · p. 4
  • The system-model scenario is uplink transmission of updates from an IoT node to its associated, or typical, LEOSAT.Source: §II · p. 4

A. Spatial Model

The spatial model places ground IoT nodes and LEOSATs in independent Poisson processes, with satellites on a sphere concentric with Earth. A beamwidth-defined footprint determines which nodes can connect and which may interfere.

A. Spatial Model

  • Ground IoT nodes follow a planar HPPP, while LEOSATs follow an independent HPPP on a concentric sphere of radius r_s = a_s + r_e.Satellite intensity is γ_s = N_s/(4πr_s^2), where N_s is the total satellite count.Source: A. Spatial Model · p. 4
  • The typical LEOSAT's beamwidth sets its footprint and maximum ground-node connection distance through the contact angle θ_c.The satellite is positioned at spherical coordinates (r_s, 0, 0).Source: A. Spatial Model · p. 4
  • Ground nodes are uniform within the footprint; nodes there served by another LEOSAT are potential interferers, with their impact depending on distance.The interferer distance r_i is measured to the typical LEOSAT and is bounded by the model's stated range.Source: A. Spatial Model · p. 4

B. Channel Model

The channel model accounts for line-of-sight and non-line-of-sight links between ground nodes and moving LEOSATs. It represents their fading with a Shadowed Rician distribution.

B. Channel Model

  • Orbital motion means ground-to-LEOSAT communication can involve either a direct LoS path or an NLoS path intermittently blocked by obstacles.Source: B. Channel Model · p. 4
  • The channel power gain from a ground node to its serving LEOSAT follows a Shadowed Rician distribution parameterized by Ω, b_0, and m.The text identifies Ω as average LoS power, 2b_0 as average NLoS multipath power, and m as the Nakagami parameter.Source: B. Channel Model · pp. 4–5

C. Queuing Model

The model treats each source as a zero-buffer M/G/1 queue with satellite availability represented by server vacations. Updates arriving while the server is busy or uncovered are dropped.

C. Queuing Model

  • Each source uses a zero-buffer M/G/1 queue with server vacations tied to LEOSAT availability, so updates are processed only when the server is empty.An update arriving during service is ignored until the in-service update is successfully transmitted.Source: C. Queuing Model · p. 5
  • Updates generated during an ongoing transmission or while the source is outside LEOSAT coverage are dropped until service finishes or coverage returns.The vacation probability Pv is the probability that no LEOSAT covers the intended source node.Source: C. Queuing Model · p. 5
  • The no-coverage probability depends on the LEOSAT footprint, whose spherical cap is described by its radius, length, and contact angle.Source: C. Queuing Model · p. 5

III. SPATIAL DISTRIBUTION OF THE CONDITIONAL MEAN PAOI ANALYSIS

The analysis formulates the spatial distribution of uplink mean PAoI by first defining conditional uplink coverage at a typical LEOSAT. The coverage event is that its SINR exceeds a threshold.

III. SPATIAL DISTRIBUTION OF THE CONDITIONAL MEAN PAOI ANALYSIS

  • The analysis targets the spatial distribution of uplink mean PAoI and defines conditional coverage as the typical LEOSAT's probability of meeting an SINR threshold.Source: §III · pp. 5–6
  • Fig. 2 marks update generation and reception with blue arrows, service starts with dashed blue arrows, and ignored arrivals with green arrows.Source: Figure 2 · p. 6
  • The SINR calculation includes the intended source's distance and channel gain, receiver noise, and aggregate uplink interference from potential interferers within the LEOSAT footprint.Interferer distance and channel power gain enter the aggregate interference.Source: §III · p. 6

A. Mean PAoI

The section defines PAoI under slotted, Poisson-arrival updates and a non-preemptive service process. It then relates the conditional mean PAoI to coverage probability and averages across network realizations.

A. Mean PAoI

  • AoI rises between updates and drops to the update's service time upon reception; the non-preemptive source ignores arrivals during service.Source: A. Mean PAoI · p. 6
  • Updates are generated at slot beginnings according to a Poisson process, giving exponential interarrival times; the minimum AoI is one slot and PAoI is the maximum before reception.Source: A. Mean PAoI · p. 6
  • Service and waiting times are modeled as geometric, with parameters based on conditional uplink coverage probability and the probability of being covered.The total update time runs from generation to successful reception.Source: A. Mean PAoI · p. 6
  • The uplink PAoI at the typical LEOSAT includes a propagation-time component that depends on the source-to-satellite distance.Source: A. Mean PAoI · p. 6
  • The conditional mean PAoI is defined for a given ground-node point-process realization, with expectation taken over channel fading.Source: A. Mean PAoI · p. 6
  • Stationarity and ergodicity make the spatial PAoI average an ensemble mean, while coverage varies across network realizations and its moments are needed to evaluate mean PAoI.Source: A. Mean PAoI · p. 6

B. b-th Moment of Conditional Coverage Probability

The section derives moments of conditional coverage and mean PAoI, then uses them to approximate the spatial distribution of conditional mean PAoI. The distribution is expressed as a CDF evaluated against a time deadline.

B. b-th Moment of Conditional Coverage Probability

IV. NUMERICAL RESULTS

The numerical results examine how satellite altitude, constellation size, and beamwidth relate to uplink coverage reliability and conditional mean PAoI. They also show that the beamwidth minimizing average conditional mean PAoI depends on altitude and constellation size.

IV. NUMERICAL RESULTS

  • Analytical and simulation meta-distributions show a minor gap, which the authors attribute to Gamma fading and Beta-distribution assumptions.The gap is reported as minimal at lower altitudes and small at higher altitudes; the analysis is validated using Monte Carlo simulations over 10^3 network realizations and 10^4 time iterations.Source: §IV · p. 8
  • Increasing satellite altitude or constellation size reduces the fraction of transmissions exceeding a reliability threshold τ, as shown for 900 and 1300 satellites.The paper attributes this trend to larger spherical caps containing more potential interferers.Source: §IV · p. 8, Figure 3 · p. 9
  • The probability of achieving a given conditional mean PAoI decreases as altitude rises from 500 to 1500 km or satellite count rises from 900 to 1300.The paper links these trends to more potential interferers, with higher altitude also increasing propagation delay.Source: §IV · p. 8, Figure 4 · p. 9
  • The spatial average of conditional mean PAoI generally increases with satellite count at higher altitudes, but 500 and 700 satellites can benefit from higher altitude.For 900 satellites, the minimum occurs at 700 km; at 500 km, increasing satellite count can reduce the average by shortening the wait to begin service.Source: §IV · pp. 8–9, Figure 5 · p. 10
  • At 1500 km, narrower beams minimize average conditional mean PAoI; preferred widths are 30°, 22.5°, and 18° for 1000, 1500, and 2000 satellites.At 500 satellites, the mean PAoI is constant from 30° beamwidth, making 45° the stated hardware-cost choice; at 500 km, narrower beams do not reduce the average.Source: §IV · pp. 9–10, Figure 7 · p. 10, Figure 6 · p. 10
  • At 1500 km, increasing the arrival rate from 0.1 to 0.9 reduces average conditional mean PAoI across the listed constellation sizes.The paper attributes this to the source continually transmitting more recent updates.Source: §IV · p. 10

V. CONCLUSION

Using stochastic geometry, the paper derives the spatial distribution and moments of conditional mean PAoI for updates sent from a ground IoT node to its serving LEOSAT. It examines how satellite count, altitude, and beamwidth relate to conditional coverage and mean PAoI.

V. CONCLUSION

  • The analysis derives the spatial distribution of conditional mean PAoI for updates transmitted from a ground IoT node to its serving LEOSAT.It also formulates the b-th moment of conditional mean PAoI.Source: §V · p. 10
  • Varying LEOSAT count and altitude affects the first moment of conditional mean PAoI, the conditional coverage probability meta-distribution, and the spatial distribution of conditional mean PAoI.Source: §V · p. 11
  • Evaluating scenarios with different LEOSAT counts and altitudes provides insights for tuning beamwidth to minimize conditional mean PAoI.Source: §V · p. 11

APPENDIX A PROOF OF THEOREM 1

The appendices outline derivations for the conditional coverage-probability moment and related spatial quantities. They invoke independence, a Gamma approximation, distributional expansions, and an interference-region split.

APPENDIX A PROOF OF THEOREM 1

  • The proof invokes independence of locations and channel gains and approximates channel power with a Gamma distribution when deriving the conditional coverage-probability moment.Source: Appendix A · p. 11
  • The derivation applies binomial and multinomial expansions, independence of source and interferer distances, and the Laplace-transform definition.Source: Appendix A · p. 11

APPENDIX B PROOF OF LEMMA 1

  • The Lemma 1 proof uses the Gamma-distribution moment-generating function and treats active IoT nodes within rmax as potential interferers.Source: Appendix B · p. 12

APPENDIX C PROOF OF THEOREM 2

The supplied proof passages describe expansions and expectations for Theorem 2, followed by a Beta-distribution approximation for Theorem 3.

APPENDIX C PROOF OF THEOREM 2

  • Theorem 2's proof applies a binomial-theorem expansion to expression (22).Source: Appendix C · p. 12
  • Theorem 2's proof uses linearity of expectation and evaluates an expectation across different network realizations for a given setup.Source: Appendix C · p. 12

APPENDIX D PROOF OF THEOREM 3

  • Theorem 3's proof applies a Beta-distribution approximation to the coverage-probability distribution using the regularized incomplete beta function.Source: Appendix D · p. 12
Loading 2610.23597v2…