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Journal of Function Spaces
Volume 2015, Article ID 671756, 18 pages
Research Article

A Simple Repairable System with Warning Device

1Key Laboratory of Highway Construction and Maintenance Technology in Loess Region, Shanxi Transportation Research Institute, Taiyuan 030006, China
2Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Received 3 December 2014; Revised 23 April 2015; Accepted 28 April 2015

Academic Editor: Jaeyoung Chung

Copyright © 2015 Xiao Zhang and Lina Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper considers a simple repairable system with a warning device which can signal an alarm when the system is not in good condition and a repairman who can have delayed-multiple vacations. By using Markov renewal process theory and the probability analysis method, the system is first described into a group of integrodifferential equations. Then the unique existence and asymptotic stability, especially the exponential stability of the system dynamic solution, are studied by using the strongly continuous semigroup theory or semigroup theory and the spectrum theory. The reliability indices and some applications (such as the comparisons of some indexes and profit of systems with and without warning device) as well as numerical examples are presented at the end of the paper.