First-Passage Asymptotics Under Physical-Time and Operational-Time Resetting in a Two-State Semi-Markov Model

The question is whether the reset-time distribution alone determines long-time first-passage behavior when intrinsic waiting times are heavy-tailed. This paper studies complete resetting in a two-state semi-Markov chain motivated by spatial variable-order diffusion. For the spatial diffusion, the long-time result is conditional on an assumed expansion of the survival probability within one renewal cycle. Exact renewal equations connect long-time survival in the chain to the joint distribution of reset and intrinsic waiting times. For independent power-law resetting in the slow state, a critical reset-time tail exponent separates slow-state and fast-state dominance. For identical Mittag–Leffler reset-time distributions, independent physical-time resetting yields faster survival decay, whereas operational-time resetting through the same stable subordinator as the intrinsic waiting time preserves the slow-state exponent. For Poisson resetting in both states with a deterministic reset delay, an explicit mean first-passage formula reduces optimization over two reset rates to one dimension. A unique finite global minimizer exists under the stated assumptions. Monte Carlo simulations and numerical Laplace inversion for the chain support the analytical results. The findings identify dependence between reset and waiting times as essential to first-passage behavior in the two-state model.