Optimal replacement policies for deteriorating systems under imperfect delayed repair via a quasi renewal–alpha process

Maintenance optimization of repairable deteriorating systems remains an important problem in reliability engineering, particularly when repairs are imperfect and maintenance activities are subject to delays. This paper proposes an imperfect delayed repair maintenance model based on a Quasi Renewal–Alpha Process. An N-failure replacement policy is considered, and renewal–reward theory is used to derive expressions for the expected renewal cycle length and the long-run average cost rate. Analytical conditions are established for the existence and uniqueness of the optimal replacement number. Numerical examples are presented to illustrate the analytical results, and an empirical comparison with the classical Quasi Renewal Process is conducted using a publicly available failure-time dataset. Based on the Akaike Information Criterion (AIC), the Quasi Renewal–Alpha Process provides a better fit than the classical Quasi Renewal Process for this dataset.