Imperfect delayed recovery \(\delta\)-shock model under a mean geometric process

Abstract

A biological reliability model is presented that integrates \(\delta\)-shock failure, deteriorating tolerance, delayed imperfect recovery, and mean-geometric-process recovery deterioration. The model identifies the optimal number of failure-recovery cycles before system renewal by minimizing the long-run average biological burden. A control-limit theorem is established to characterize the optimal replacement threshold, with numerical validation confirming the result. Sensitivity analysis quantifies how tolerance deterioration rate, recovery efficiency, and delay probability influence optimal renewal timing. This framework yields quantitative insight into life-history trade-offs in stress-prone biological systems and provides a rigorous mathematical basis for assessing recovery-replacement decisions under progressive deterioration.