A Simple Repairable System with Warning Device

Lina Guo,Xiao Zhang

doi:10.1155/2015/671756

Abstract

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 orC0semigroup 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.

Highlights

A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method rather than the replacement of the entire system
From the perspective of rational use of human resources, the introduction of repairman’s vacation makes modeling of the repairable system more realistic and flexible. This is due to the fact that, in practice, the vast majority of small- and medium-sized enterprisers (SMEs) cannot afford to hire a full-time repairman
We propose a simple repairable system with a warning device which can send alarms when the system

Summary

Introduction

A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method rather than the replacement of the entire system. The available references concerning repairman vacation in repairable systems can be classified into two categories: one focuses on system indices and the other deals with optimization problems. Hu et al [7] studied the steady-state availability and the mean uptime of a seriesparallel repairable system, which consists of one master control unit and two slave units and a single repairman runs single vacation via the method of supplementary variable and the vector Markov process theory. We are devoted to studying the asymptotic behavior of the system by strongly continuous semigroup theory and make comparisons of reliability indexes (such as reliability, availability, and the probability of the repairman’ vacation) and profit of the two systems with and without warning device.

System Formulation

Stability of System Solution

Reliability Indices

Applications and Numerical Examples

Conclusion