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Spatiotemporal Analysis of Parallelized Computing at the Extreme Edge

Yasser Nabil, Mahmoud Abdelhadi, Sameh Sorour, Hesham ElSawy, Sara A. Elsayed, Hossam S. Hassanein

TL;DR

This work develops a rigorous spatiotemporal framework for Extreme Edge Computing (EEC) in large-scale millimeter-wave networks by fusing stochastic geometry with an Absorbing Continuous-Time Markov Chain to capture the coupling and temporal overlap between communication and parallel computation across Extreme Edge Devices (EEDs). It provides closed-form or semi-closed-form expressions for the offloading success probability and characterizes the average task delay and task completion probability, revealing optimal task segmentation levels and the benefits of location-aware offloading under failures. The analysis accounts for spatial randomness, device reliability, and the interaction between D2D communication and distributed parallel processing, offering design insights for latency-reliability tradeoffs. It also demonstrates how MEC–EEC collaboration can mitigate congestion when EED availability is limited, informing practical deployment and cross-layer optimization in 6G-era edge networks.

Abstract

Extreme Edge Computing (EEC) pushes computing even closer to end users than traditional Multi-access Edge Computing (MEC), harnessing the idle resources of Extreme Edge Devices (EEDs) to enable low-latency, distributed processing. However, EEC faces key challenges, including spatial randomness in device distribution, limited EED computational power necessitating parallel task execution, vulnerability to failure, and temporal randomness due to variability in wireless communication and execution times. These challenges highlight the need for a rigorous analytical framework to evaluate EEC performance. We present the first spatiotemporal mathematical model for EEC over large-scale millimeter-wave networks. Utilizing stochastic geometry and an Absorbing Continuous-Time Markov Chain (ACTMC), the framework captures the complex interaction between communication and computation performance, including their temporal overlap during parallel execution. We evaluate two key metrics: average task response delay and task completion probability. Together, they provide a holistic view of latency and reliability. The analysis considers fundamental offloading strategies, including randomized and location-aware schemes, while accounting for EED failures. Results show that there exists an optimal task segmentation that minimizes delay. Under limited EED availability, we investigate a bias-based EEC and MEC collaboration that offloads excess demand to MEC resources, effectively reducing congestion and improving system responsiveness.

Spatiotemporal Analysis of Parallelized Computing at the Extreme Edge

TL;DR

This work develops a rigorous spatiotemporal framework for Extreme Edge Computing (EEC) in large-scale millimeter-wave networks by fusing stochastic geometry with an Absorbing Continuous-Time Markov Chain to capture the coupling and temporal overlap between communication and parallel computation across Extreme Edge Devices (EEDs). It provides closed-form or semi-closed-form expressions for the offloading success probability and characterizes the average task delay and task completion probability, revealing optimal task segmentation levels and the benefits of location-aware offloading under failures. The analysis accounts for spatial randomness, device reliability, and the interaction between D2D communication and distributed parallel processing, offering design insights for latency-reliability tradeoffs. It also demonstrates how MEC–EEC collaboration can mitigate congestion when EED availability is limited, informing practical deployment and cross-layer optimization in 6G-era edge networks.

Abstract

Extreme Edge Computing (EEC) pushes computing even closer to end users than traditional Multi-access Edge Computing (MEC), harnessing the idle resources of Extreme Edge Devices (EEDs) to enable low-latency, distributed processing. However, EEC faces key challenges, including spatial randomness in device distribution, limited EED computational power necessitating parallel task execution, vulnerability to failure, and temporal randomness due to variability in wireless communication and execution times. These challenges highlight the need for a rigorous analytical framework to evaluate EEC performance. We present the first spatiotemporal mathematical model for EEC over large-scale millimeter-wave networks. Utilizing stochastic geometry and an Absorbing Continuous-Time Markov Chain (ACTMC), the framework captures the complex interaction between communication and computation performance, including their temporal overlap during parallel execution. We evaluate two key metrics: average task response delay and task completion probability. Together, they provide a holistic view of latency and reliability. The analysis considers fundamental offloading strategies, including randomized and location-aware schemes, while accounting for EED failures. Results show that there exists an optimal task segmentation that minimizes delay. Under limited EED availability, we investigate a bias-based EEC and MEC collaboration that offloads excess demand to MEC resources, effectively reducing congestion and improving system responsiveness.

Paper Structure

This paper contains 7 sections, 1 theorem, 8 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Baseline Spatiotemporal Analysis
  4. Baseline System Model
  5. Successful Random EEDs Offloading Probability
  6. Average Task Response Delay Calculation
  7. ACTMC and EDTMC

Key Result

Theorem 1

The spatially averaged probability of successful segment offloading via mmWave D2D communication to a randomly selected LoS worker from $\Phi_w$ is given as follows: where $M_n(\xi) = - \frac{\eta_L n r_0^{\alpha_L} \xi}{C_L M_r M_w}$, while $W_n(\xi)$ and $Z_n(\xi)$ are given by (eqCoverageW) and (eqCoverageZ). where $\bar{a}_k = \frac{a_k}{M_r M_w}$, and $b_k$ along with $a_k$ for $1\leq k\leq

Figures (1)

  • Figure 1: The spatial system model: LoS workers (blue), NLoS workers (orange), and requesters (green). The typical requester offloads task segments to LoS workers. The typical worker receives the intended link from the typical requester, along with LoS and NLoS interference from other requesters.

Theorems & Definitions (1)

  • Theorem 1