Freshness-aware computation offloading has garnered great attention recently in the edge computing arena, with the aim of promptly obtaining up-to-date information and minimizing the transmission of outdated data. However, most of the existing work assumes that wireless channels are reliable and neglect the dynamics and stochasticity thereof. In addition, varying priorities of offloading tasks along with heterogeneous computing units also pose significant challenges in effective task scheduling and resource allocation. To address these challenges, we cast the freshness-aware task offloading problem as a multi-priority optimization problem, considering the unreliability of wireless channels, the heterogeneity of edge servers, and prioritized users. Based on the nonlinear fractional programming and ADMM-Consensus method, we propose a joint resource allocation and task offloading algorithm to solve the original problem iteratively. To improve communication efficiency, we further devise a distributed asynchronous variant for the proposed algorithm. We rigorously analyze the performance and convergence of the proposed algorithms and conduct extensive simulations to corroborate their efficacy and superiority over the existing baselines.

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