A Mean Geometric Process -Shock Model for Optimal Replacement Policy in Deteriorating Systems

Abstract

This paper introduces a Mean Geometric Process  -shock (MGP- ) model for determining optimal replacement policies in deteriorating repairable systems. Extending Lam’s geometric process framework, the model replaces strict geometric scaling with a mean-based adjustment to capture smoother deterioration dynamics. System failures occur either due to the end of an operating period or upon the arrival of a fatal  -shock, modeled as a Poisson process. Using the renewal–reward theorem, the long-run average cost function   is derived and proven to be discretely convex, ensuring a unique optimal replacement number  . Comparative results show that the MGP-  model retains the analytical consistency of the GP-  model while providing smoother cost curves, improved numerical stability, and greater robustness to parameter uncertainty.