Extended repair model for a deteriorating system with its repairman having vacations

A new repairable system model with a single component and a repairman is proposed. It is assumed that the system has two types of failure and is replaced at the occurrence of the Nth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\hbox {th}$$\end{document} type I failure, or the first type II failure (catastrophic failure), whichever occurs first. The system is repaired after type I failure. The repairman has multiple vacations when the system is working. And the system is detected by repairman after his vacation. The system will work again when it repaired. Successive working time and the successive repair time are extended geometric process. The explicit expression of the long-run average cost rate function of the system is derived. Numerical cases are provided to verify the effectiveness of the theoretical approaches. Finally, sensitive analysis of parameters is carried out.