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Abstract
This paper considers a two-unit complex system, in which one of the components has priority with preemptive repeat repair disciplines, described by partial differential equations with integral boundary conditions. First, we prove that the system has a unique nonnegative time-dependent solution by using the strong continuous semigroup theory of linear operators. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution by means of the spectral properties of the corresponding operator. We also provide the asymptotic behavior of some time-dependent reliability indices, a numerical illustration showing the effects of various parameters on the system, and the validity of the theoretical analysis.
Highlights
In many real-life situations, system reliability plays a very important role
In 2001, Gupur [13] firstly did dynamic analysis for reliability models, which was established by the supplementary variable technique, by means of the C0-semigroup theory
7 Conclusions This paper investigates the two-unit complex system, in which one of the components has priority with preemptive repeat repair disciplines
Summary
In many real-life situations, system reliability plays a very important role. With an increasing complexity of industrial systems, the reliability-related problems are quite challenging due to the diversity of factors that can lead to failures in industrial systems. This technique leads to mathematical models described by partial differential equations with integral boundary conditions, and there are significant difficulties to obtain exact solutions Due to this fact, some papers related to reliability models assume that time-dependent solutions converge to their steady-state solutions, but they do not answer whether this assumption holds. In 2001, Gupur [13] firstly did dynamic analysis for reliability models, which was established by the supplementary variable technique, by means of the C0-semigroup theory. After that, he and his coauthors studied several reliability models. We discuss the asymptotic behavior of the time-dependent reliability indices, such as time-dependent availability, failure frequency, renewal frequency, and reliability of the system, and illustrate, with numerical examples, the effect of changes in the system parameters on those indices in a particular case
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