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Lyapunov-based analysis of functional stability for edge computing systems
Oleksii S. Bychkov
TL;DR
Under z(t0) ≠0, (19) certifies residual or asymptotic tolerance within δᵢ, although transients may briefly violate the strong form. The paper maps quality degradation into an admissible region and applies Lyapunov, UUB, ISS, common-function, and finite-horizon analyses. The convergence rate k sets recovery-time bounds, disturbance tolerance, and finite-horizon requirements across the derived stability results.
Problem
Under z(t0) ≠0, (19) certifies residual or asymptotic tolerance within δᵢ, although transients may briefly violate the strong form.
Method
The paper maps quality degradation into an admissible region and applies Lyapunov, UUB, ISS, common-function, and finite-horizon analyses.
Results
The convergence rate k sets recovery-time bounds, disturbance tolerance, and finite-horizon requirements across the derived stability results.
Takeaways & Limitations
Functional stability becomes a constructive bounded-trajectory verification framework for edge services, including switched operation and mission-bounded workloads.
Takeaways & Limitations
Strong stability under arbitrary switching additionally requires per-mode invariance and safe post-switch states, especially after re-initialization.
Abstract
from arXiv · showhide
Functional stability has been proposed as a per-function reliability concept for edge computing, in which the unit of analysis is the individual service rather than the whole system and the binary working/failed evaluation is replaced by a continuous quality function with per-function thresholds. Verifying these properties for a concrete edge orchestrator, migration policy, or distributed inference service requires a constructive method. This article connects functional stability to the direct Lyapunov method. The strong form is characterized through positive invariance of the admissible region; the weak form is connected to uniform ultimate boundedness and input-to-state stability with analytical recovery-time bounds. Switched dynamics from service migration and node failover are treated via common Lyapunov functions. Finite-horizon variants address mission-bounded edge workloads.
2. Related work
Prior work provides Lyapunov tools for disturbed and switched nonlinear systems, while functional stability defines per-function quality requirements descriptively. The article closes the gap between these constructive stability methods and edge-specific functional stability.
- Lyapunov methods for nonlinear systems under disturbances: UUB bounds disturbed trajectories within a disturbance-dependent set, while ISS formalizes how state behavior depends on input magnitude.These tools were developed for equilibrium-centered stability analysis.
- Lyapunov methods for nonlinear systems under disturbances: Barrier Lyapunov functions keep trajectories inside prescribed regions by making the Lyapunov function diverge at the boundary.The prescribed region is conceptually analogous to the functional-stability admissible region Ωδ.
- Switched systems and common Lyapunov functions: Switched systems can become unstable under some switching signals even when every individual mode is asymptotically stable.A common Lyapunov function provides a sufficient condition for stability under arbitrary switching.
- Switched systems and common Lyapunov functions: Edge switching includes service migration, node failover, and adaptive inference-mode changes, but corresponding stability analysis has been largely absent.The article addresses these events using common Lyapunov functions.
- Functional stability in edge and fog computing: Functional stability shifts analysis from the whole system to individual functions and replaces binary failure labels with continuous quality evaluation.Its framework includes per-function parameters and aggregate metrics, but remains descriptive without constructive verification methods.
- Functional stability in edge and fog computing: The article closes the gap between descriptive functional stability and constructive Lyapunov methods for disturbed and switched edge systems.This connection supplies a formal verification basis for concrete edge dynamics and controllers.
3. Methods
The methods recast per-function quality degradation as bounded trajectories in an admissible region, then use Lyapunov conditions to verify strong, weak, switched, disturbed, and finite-horizon stability. Recovery bounds depend on convergence rate, while several conditions remain sufficient and conservative.
- Strong form: Strong functional stability is equivalent to positive invariance of the admissible region Ωδ for the degradation dynamics.Boundary derivative conditions provide constructive sufficient tests, including component-wise and Lyapunov-sublevel-set forms.
- Strong form: A strict decrease condition, V̇ ≤−kV, yields strong stability with margin and exponential convergence to zero at rate k.The associated inscribed Lyapunov sublevel set remains invariant.
- Weak form: UUB and ISS conditions establish weak functional stability with piecewise analytical recovery-time upper bounds.The non-trivial UUB recovery branch scales as k−1ln(⋅), and actual recovery can occur earlier than the bound.
- Switched dynamics: A common practical Lyapunov function preserves weak stability under arbitrary switching when every mode satisfies a suitable UUB inequality.This covers service migration and failover, whereas strong stability additionally requires per-mode invariance and safe post-switch states.
- Finite-horizon stability: Finite-horizon weak stability requires the observation horizon to exceed disturbance onset plus the recovery bound.Finite-horizon guarantees can differ from infinite-horizon guarantees because workload duration determines the relevant interval.
- Parameter relationships: The convergence rate k links recovery speed and disturbance tolerance, with the condition cδ>ρ/k coupling controller speed and disturbance intensity.Increasing k improves supported stability measures, while reducing ρ or enlarging δi provides alternative trade-offs.
4. Edge computing instantiation
The paper maps functional-stability concepts to concrete edge constructs and applies Lyapunov analysis to autoscaling, failover, and switching scenarios. The resulting bounds identify design levers for recovery-time guarantees and are compared with numerical simulations.
- Mapping abstract objects to edge constructs: Edge constructs are represented by resource states, containerized services, service-level quality functions, operating modes, switching events, horizons, and Lyapunov degradation energies.Quality functions encode indicators such as latency, accuracy, throughput, and frame rate, while modes include node assignments and inference paths.
- Analytical case studies: Analytical recovery bounds certify weak functional stability for autoscaled inference and failover services under explicit Lyapunov conditions.For autoscaling, the condition is c_L > γ(A_λ)/k; for failover, the corresponding design condition constrains the slowest operating mode.
- Containerized inference service: Doubling autoscaler responsiveness raises the effective Lyapunov rate and reduces both the residual degradation term and the recovery-time logarithm.Reducing arrival-rate prediction error likewise lowers the disturbance gain and moves operation toward the safe case.
- Edge node failover: A common Lyapunov function yields failover-robust weak stability when the inscribed-ellipsoid condition holds for both primary and standby modes.The slowest mode governs the design constraint; severe cold-cache penalties can require pre-warming or standby over-provisioning.
- Edge node failover: The Lyapunov analysis predicts failover recovery time analytically before deployment, enabling standby-resource sizing against a target recovery budget.This converts descriptive functional-stability scoring into a design-time calculation.
- Numerical simulation: Simulation recovery times were 3.1 s versus a 3.9 s bound for autoscaling and 4.8 s versus a 6.2 s bound for failover.The corresponding simulation-to-bound ratios were approximately 0.79 and 0.77, respectively, and parameter sweeps varied the identified design levers.
5. Discussion
The discussion presents Lyapunov analysis as a constructive way to verify and design functional stability, while identifying operational trade-offs, switching constraints, sampling effects, and scope limitations.
- 5.1. Constructive verification of functional stability: The Lyapunov approach provides analytical conditions, bounds, and design levers for verifying functional stability and sizing controllers, autoscalers, and replication policies.It complements descriptive measurement and simulation by predicting whether the target functional-stability score is attainable under a dynamics model.
- 5.2. Strong form, weak form and the role of k: Increasing the convergence rate k shortens recovery bounds and improves disturbance tolerance, while reducing persistent disturbance or relaxing the admissible degradation margin provides alternative trade-offs.The discussion links k to autoscaling latency, controller bandwidth, and orchestrator responsiveness, while ρ and δ_i define alternative design choices.
- 5.4. Switched dynamics and the conservatism of CLF: For switched dynamics, a common Lyapunov function certifies stability under arbitrary switching, whereas multiple Lyapunov functions with an average dwell-time condition can exploit mode structure and controlled switching.The MLF workflow designs per-mode certificates, estimates the switch-growth factor μ, and requires τ_a > ln μ/λ*.
- 5.5. Continuous-time analysis and discrete-time controllers: Discrete controller operation adds up to T_s to analytical recovery time because disturbances may occur immediately after a sampling instant.For a 15 s HPA period, this sampling term can dominate bounds of a few seconds; with T_s = 100 ms, it is negligible.
- 5.6. Limitations: The continuous-time analysis has three principal limitations: Lyapunov certificates are sufficient rather than necessary, disturbances are bounded but non-random, and simulations are illustrative rather than a completed live-cluster validation.A full validation campaign would require parameter sweeps, measurements on a live edge cluster, and integration with the outlined orchestration path.