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Wireless Broadcast Gossip for Decentralized Drone Swarms: Success Probability, Contraction, and Optimal Aloha

Ali Khalesi

Abstract

We study broadcast gossip for decentralized drone swarms over an interference-limited wireless medium. Modeling drone locations as a planar Poisson point process and medium access via slotted Aloha, we derive (i) a closed-form SIR success probability under Rayleigh fading, (ii) a mean-square contraction bound in which the consensus rate factorizes into an ideal mixing term and an explicit wireless thinning term, and (iii) a closed-form access probability that optimizes a sharp availability--reliability proxy. Simulations corroborate the predicted operating point by matching the fastest convergence region.

Wireless Broadcast Gossip for Decentralized Drone Swarms: Success Probability, Contraction, and Optimal Aloha

Abstract

We study broadcast gossip for decentralized drone swarms over an interference-limited wireless medium. Modeling drone locations as a planar Poisson point process and medium access via slotted Aloha, we derive (i) a closed-form SIR success probability under Rayleigh fading, (ii) a mean-square contraction bound in which the consensus rate factorizes into an ideal mixing term and an explicit wireless thinning term, and (iii) a closed-form access probability that optimizes a sharp availability--reliability proxy. Simulations corroborate the predicted operating point by matching the fastest convergence region.
Paper Structure (15 sections, 4 theorems, 25 equations, 2 figures)

This paper contains 15 sections, 4 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. Success probability under PPP+Aloha
  5. Wireless-thinned contraction bound
  6. Ideal mixing
  7. A per-slot successful-update lower bound
  8. Optimal Aloha probability
  9. Simulation Example
  10. Setup (PPP swarm + broadcast PHY)
  11. Metric and confidence intervals
  12. Result: predicted $p^\star$ matches the fastest mixing
  13. Practical takeaway for drone swarms
  14. Conclusion
  15. Future work.

Key Result

Theorem 1

Under the PPP+Aloha model with intensity $\lambda$, Aloha probability $p$, Rayleigh fading, and path-loss $\ell(r)=r^{-\alpha}$ with $\alpha>2$, the success probability of a transmission over distance $r$ is where

Figures (2)

  • Figure 1: System model (single slot) for wireless broadcast gossip in a drone swarm. Drone locations form a planar PPP of intensity $\lambda$. In each slot, each drone transmits with probability $p$ (slotted Aloha) and otherwise listens (half-duplex). A typical listener $i$ considers neighbors within range $R$ and selects one transmitting neighbor $j$. Decoding succeeds if $\mathrm{SIR}_{j\to i}(t)>\theta$ under Rayleigh fading and path-loss exponent $\alpha>2$. To preserve average consensus, successful exchanges in a slot are executed as a matching $\mathcal{M}(t)$ so that each node participates in at most one averaging update per slot, yielding a symmetric, doubly-stochastic update $x(t{+}1)=W(t)x(t)$.
  • Figure 2: Broadcast gossip in an interference-limited PPP swarm: achieved contraction $\epsilon(T)=V(T)/V(0)$ at horizon $T$ versus Aloha probability $p$ (geometric mean, $95\%$ CI). The dashed vertical line is the predicted $p^\star$ from \ref{['eq:pstar_closed']} using an empirical effective distance $r_{\mathrm{eff}}=\sqrt{\mathbb{E}[r^2]}$ computed from neighbor links; the dotted line shows the dense-neighborhood simplification \ref{['eq:pstar_simple_clean']}.

Theorems & Definitions (11)

  • Theorem 1: SIR success probability
  • proof
  • Remark 1
  • Lemma 1: Ideal one-step contraction
  • proof
  • Theorem 2: Consensus contraction under wireless thinning
  • proof
  • Remark 2: Interpretation
  • Proposition 1: Closed-form $p^\star$ for a broadcast availability--reliability proxy
  • proof
  • ...and 1 more