Abstract
This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k-out-of-n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system.
Highlights
Introduction and MotivationThe paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of k-out-of-n systems
For a stochastic process X = { X (t) : t ∈ R} with its flow of σ-algebras FtX, it is supposed that a sequence of points of time — regeneration times
I where: n o πi (t, Γ) = P X (1) Sk + t ∈ Γ, t < Tk+1 | X (1) Sk + 0 = i is the process state probabilities during the embedded regeneration period of the second level and embedded Markov renewal matrix (EMRM) H (1) (t) satisfies Equation (7). This procedure can be developed for the construction of decomposable semi-regenerative process (DSRP) with several levels of decomposition
Summary
The paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of k-out-of-n systems. ~ (2) (t, Γ), Hij (du)π j (t − u, Γ) ≡ H i where: πi (t, Γ) = P X (1) Sk + t ∈ Γ, t < Tk+1 | X (1) Sk + 0 = i is the process state probabilities during the embedded regeneration period of the second level and EMRM H (1) (t) satisfies Equation (7) This procedure can be developed for the construction of DSRP with several levels of decomposition. To study the k-out-of-n system, we used an approach based on the methods of DSRP that allows extending the previous results and finding closed form representations for time-dependent and stationary characteristics of the system In the future, these results can be used for the sensitivity analyses of the system characteristics to the shape of components’ repair time distributions.
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