End-to-End Delay Estimation Under Multi-Hop Recurrence in Dynamic Integrated Networks

The increasing scale of modern networks, particularly dynamic integrated networks spanning space, air, ground, and sea, introduces substantial challenges to end-to-end delay estimation due to strong node heterogeneity and volatile link conditions. These dynamics impair the accuracy and generalizability of traditional delay estimation methods, especially under diverse Quality of Service (QoS) requirements. To address this issue, we propose a stochastic particles and variational Bayesian estimation (SPAVE) scheme. The method first adopts stochastic network calculus (SNC) to model the arrival and service processes, and derives a recurrence relation for downstream delay as a function of upstream delay and local load. This recurrence relation is reformulated as a conditional distribution that characterizes the downstream delay at each hop, enabling hop-by-hop delay estimation through local parameter optimization and structured propagation across multi-hop nodes. By reducing the effective parameter dimensionality via stochastic particle sampling and rapidly refining the selected high probability particles through variational Bayesian inference, enabling SPAVE to adapt to dynamic integrated networks. Simulation results show that SPAVE achieves a mean absolute error below 10⁻³.