Reliability and Optimization for k-out-of-n: G Mixed Standby Retrial System with Dependency and J-Vacation

Shao, Qi; Hu, Linmin; Xu, Fan

doi:10.1007/s11009-024-10078-x

Reliability and Optimization for k-out-of-n: G Mixed Standby Retrial System with Dependency and J-Vacation

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Published: 28 February 2024

Volume 26, article number 8, (2024)
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Qi Shao¹,
Linmin Hu¹ &
Fan Xu¹

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Abstract

Based on the design and potential application of wind-solar storage intelligent power generation systems in engineering practice, this paper develops a novel reliability model of k-out-of-n: G mixed standby retrial system with failure dependency and J-vacation policy. The working components in the system have redundant dependencies. When any component of the system fails and the repairman is working or on vacation, the failed component goes into the retrial space. If the retrial space has no failed components, the idle repairman goes on vacation, which may last for up to J consecutive vacations, until at a minimum one failed component appears in the retrial space on a vacation return. Firstly, the performance indexes of the system under steady state are analyzed based on the Markov process theory. Secondly, an algorithm for modelling the failure process of the proposed model is developed through a Monte Carlo method, and numerical solutions for the reliability function and mean time to first failure (MTTFF) are presented. Then, some numerical examples are provided to demonstrate the influence of different parameters on the system reliability indexes. Finally, a system cost optimization model based on availability control is developed, and the optimal component configuration schemes for systems with no vacations and different maximum numbers of vacations J are compared and analyzed by genetic algorithm (GA).

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Cost analysis of a repairable warm standby k-out-of-n: G and 2-out-of-4: G systems in series configuration under catastrophic failure using copula repair

Article 02 January 2021

Maintenance policy determination for a complex system consisting of series and cold standby system with multiple levels of maintenance action

Article 09 January 2015

Bi-objective Reliability Optimization of Switch-Mode k-out-of-n Series–Parallel Systems with Active and Cold Standby Components Having Failure Rates Dependent on the Number of Components

Article 19 June 2017

Data Availability

No data was used for the research described in the article.

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Funding

This work was supported by the National Natural Science Foundation of China [grant number 72071175], the Project of Hebei Key Laboratory of Software Engineering [grant number22567637H], and the Basic Innovative Research and Cultivation Project of Yanshan University [grant number 2023LGZD003].

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School of Science, Yanshan University, Qinhuangdao, 066004, Hebei, China
Qi Shao, Linmin Hu & Fan Xu

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Qi Shao
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Qi Shao, Linmin Hu and Fan Xu. The first draft of the manuscript was written by Qi Shao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Linmin Hu.

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Appendices

Appendix 1 Sub-blocks of the Transition Rate Matrix Q

The sub-blocks $\boldsymbol{Q}_{i,i}$($i=0, 1, \cdots , L$), $\boldsymbol{Q}_{i,i-1}$($i=1, 2, \cdots , L$) and $\boldsymbol{Q}_{i,i+1}$($i=0, 1, \cdots , L-1$) are provided as follows:

$$\begin{aligned} \boldsymbol{Q}_{0,0}= \left( \begin{array}{ccccccccc}{-\lambda }_{0}&{}{\lambda }_{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}-({\lambda }_{1}+{\mu })&{}{\mu }&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}-({\lambda }_{0}+{\theta })&{}{\theta }&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}-({\lambda }_{0}+{\theta })&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {\theta }&{}{0}&{}{0}&{}{0}&{}\cdots &{}-({\lambda }_{0}+{\theta }) \end{array}\right) _{(J+2)\times (J+2)}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{Q}_{i,i-1}= \left( \begin{array}{ccccccccc}{0}&{}{\eta }_{i}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0} \end{array}\right) _{(J+2)\times (J+2)},{i=1, 2, \cdots , L-1}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{Q}_{i,i}= \left( \begin{array}{ccccccccc}-({\lambda }_{i}+{\eta }_{i})&{}{\lambda }_{i}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\mu }&{}-({\lambda }_{i+1}+{\mu })&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{0}&{}-({\lambda }_{i}+{\theta })&{}{\theta }&{}\cdots &{}{0}\\ {\theta }&{}{0}&{}{0}&{}-({\lambda }_{i}+{\theta })&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {\theta }&{}{0}&{}{0}&{}{0}&{}\cdots &{}-({\lambda }_{i}+{\theta }) \end{array}\right) _{(J+2)\times (J+2)}, \end{aligned}$$

$i=1, 2, \cdots , L-1$,

$$\begin{aligned} \boldsymbol{Q}_{i-1,i}= \left( \begin{array}{ccccccccc}{0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{\lambda }_{i}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{\lambda }_{i-1}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{\lambda }_{i-1}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{\lambda }_{i-1} \end{array}\right) _{(J+2)\times (J+2)},{i=1, 2, \cdots , L-1}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{Q}_{L-1,L}= \left( \begin{array}{ccccccccc}{0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{\lambda }_{L-1}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{\lambda }_{L-1}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{\lambda }_{L-1} \end{array}\right) _{(J+2)\times (J+1)}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{Q}_{L,L-1}= \left( \begin{array}{ccccccccc}{0}&{}{\eta }_{L}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0} \end{array}\right) _{(J+1)\times (J+2)}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{Q}_{L,L}= \left( \begin{array}{ccccccccc}{-\eta }_{L}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{-\theta }&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{0}&{}{-\theta }&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{0}&{}{0}&{}{-\theta }&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {\theta }&{}{0}&{}{0}&{}{0}&{}\cdots &{}{-\theta } \end{array}\right) _{(J+1)\times (J+1)}. \end{aligned}$$

Appendix 2 Sub-blocks of the Matrix $D_{1}$

The sub-blocks $\boldsymbol{A}_{L-1,L-1}$, $\boldsymbol{A}_{L-1,L-2}$ and $\boldsymbol{A}_{L-2,L-1}$ are provided as follows:

$$\begin{aligned} \boldsymbol{A}_{L-1,L-2}= \left( \begin{array}{ccccccccc}{0}&{}{\eta }_{L-1}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0} \end{array}\right) _{(J+1)\times (J+2)}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{A}_{L-2,L-1}= \left( \begin{array}{ccccccccc}{0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{\lambda }_{L-2}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {0}&{}{0}&{}{\lambda }_{L-2}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {0}&{}{0}&{}{0}&{}{0}&{}\cdots &{}{\lambda }_{L-2} \end{array}\right) _{(J+2)\times (J+1)}, \end{aligned}$$

$$\begin{aligned} \boldsymbol{A}_{L-1,L-1}= \left( \begin{array}{ccccccccc}-(\lambda _{L-1}+\eta _{L-1})&{}{0}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{-(\lambda _{L-1}+\theta )}&{}{0}&{}{0}&{}\cdots &{}{0}\\ {\theta }&{}{0}&{}{-(\lambda _{L-1}+\theta )}&{}{0}&{}\cdots &{}{0}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ {\theta }&{}{0}&{}{0}&{}{0}&{}\cdots &{}{-(\lambda _{L-1}+\theta )} \end{array}\right) _{(J+1)\times (J+1)}. \end{aligned}$$

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Shao, Q., Hu, L. & Xu, F. Reliability and Optimization for k-out-of-n: G Mixed Standby Retrial System with Dependency and J-Vacation. Methodol Comput Appl Probab 26, 8 (2024). https://doi.org/10.1007/s11009-024-10078-x

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Received: 02 February 2023
Revised: 24 July 2023
Accepted: 31 January 2024
Published: 28 February 2024
DOI: https://doi.org/10.1007/s11009-024-10078-x

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Reliability and Optimization for k-out-of-n: G Mixed Standby Retrial System with Dependency and J-Vacation

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Cost analysis of a repairable warm standby k-out-of-n: G and 2-out-of-4: G systems in series configuration under catastrophic failure using copula repair

Maintenance policy determination for a complex system consisting of series and cold standby system with multiple levels of maintenance action

Bi-objective Reliability Optimization of Switch-Mode k-out-of-n Series–Parallel Systems with Active and Cold Standby Components Having Failure Rates Dependent on the Number of Components

Data Availability

References

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Appendices

Appendix 1 Sub-blocks of the Transition Rate Matrix Q

Appendix 2 Sub-blocks of the Matrix \(D_{1}\)

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Reliability and Optimization for k-out-of-n: G Mixed Standby Retrial System with Dependency and J-Vacation

Abstract

Access this article

Similar content being viewed by others

Cost analysis of a repairable warm standby k-out-of-n: G and 2-out-of-4: G systems in series configuration under catastrophic failure using copula repair

Maintenance policy determination for a complex system consisting of series and cold standby system with multiple levels of maintenance action

Bi-objective Reliability Optimization of Switch-Mode k-out-of-n Series–Parallel Systems with Active and Cold Standby Components Having Failure Rates Dependent on the Number of Components

Data Availability

References

Funding

Author information

Authors and Affiliations

Contributions

Corresponding author

Ethics declarations

Competing Interest

Additional information

Publisher's Note

Appendices

Appendix 1 Sub-blocks of the Transition Rate Matrix Q

Appendix 2 Sub-blocks of the Matrix \(D_{1}\)

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