Power-mean homogenization of dependent PHR systems: Exact parallel-system thresholds under Archimedean copulas

Replacing heterogeneous dependent component models with tractable homogeneous surrogates while preserving system-level stochastic comparisons is a recurring problem in reliability analysis. Existing mean-based results typically fix the surrogate parameter and do not identify the generator-dependent power-mean boundary for universal validity. In this paper, we developed a power-mean homogenization framework for parallel and series systems with proportional hazard rate (PHR) margins under Archimedean dependence. For parallel systems whose generator was valid in every dimension, we established an exact necessary-and-sufficient criterion: For every system size, every positive PHR parameter vector, and every non-degenerate baseline, the heterogeneous system dominated its homogeneous power-mean surrogate in the usual stochastic order if and only if a generator-induced one-variable transform was convex. The criterion yielded an exact critical power index and a constructive two-component counterexample below the admissible region. A relative-curvature condition characterized a dimension-free sufficient Jensen mechanism. For the Gumbel, Clayton, Ali-Mikhail-Haq, and Joe families, the exact and curvature-based criteria produced the same admissible sets over the studied nonzero power-index range with absolute value at most one. Strictness and equality conditions and a second-order local expansion quantified the heterogeneity gain over the arithmetic-mean benchmark. Complementary stochastic comparisons were established for series systems under appropriate convexity assumptions. Under decreasing-failure-rate baselines and additional generator-shape conditions, both system architectures admitted dispersive-order extensions; these results were sufficient rather than exact. The framework provided a rigorous basis for selecting homogeneous PHR benchmarks and comparing dependent heterogeneous systems.