On reliability assessment of weighted k-out-of-n systems with multiple types of components

ABSTRACT

This paper studies the reliability function of a weighted $\mathit{k}$-out-of-$\mathit{n}$ system with multiple types of components. A weighted system with $\mathit{M}$ different types of components is considered and two real-life scenarios for the operation of the system are defined. In the first one, it is assumed that the system operates if the accumulated weight of all active components is more than a predetermined threshold $\mathit{k}$, otherwise the system fails. In the second scenario, the system is assumed to operate if the total weight of the operating components of type $\mathit{i}$ is more than or equal to the corresponding threshold $k_i$, for at least $\mathit{m}$ types of components, $\mathit{m}\le \mathit{M}$. Our approaches in assessing the reliability function of the system lifetime rely on the notion of survival signature corresponding to multi-type weighted $\mathit{k}$-out-of-$\mathit{n}$ system. The formulas for the proposed survival signature are presented and the corresponding computational algorithms are also given. To examine the theoretical results, the reliability function and other aging characteristics of a wind farm (system) consisting of three plants located in three different regions are assessed numerically and graphically. Finally, an allocation problem to determine the best choice for the distribution of each type in the weighted $\mathit{k}$-out-of-$\mathit{n}$ systems is studied.

KEYWORDS:

• Weighted $\mathit{k}$-out-of-$\mathit{n}$ system
• reliability
• failure model
• survival signature
• allocation

Notations

$\mathit{M}$	=	Number of types of components
$\mathit{n}$	=	Number of system components
$n_i$	=	Number of components of type $\mathit{i}$
$w_j^{(i)}$	=	Weight of the $\mathit{j}$th component of type $\mathit{i}$
$\mathit{w}$	=	Weights vector of the system
$\mathit{k}$	=	System threshold (Minimum required weight/capacity for the operating system)
${\epsilon }_j^{(i)}(\mathit{t})$	=	State of $\mathit{j}$th component of type $\mathit{i}$ at time $\mathit{t}$
$k_i$	=	Threshold for type $\mathit{i}$ of components
$C_i(\mathit{t})$	=	Number of operating components of type $\mathit{i}$ at a time instant $\mathit{t}$
$F_i(\mathit{t})$	=	Cumulative distribution function of the components of type $\mathit{i}$
${\bar{F}}_{\mathit{i}}(\mathit{t})$	=	Reliability function of the components of type $\mathit{i}$
$T_s$	=	System lifetime
$\mathit{R}(\mathit{t})$	=	System reliability function
$\mathit{h}(\mathit{t})$	=	System hazard rate
${\varphi }_1(\mathit{\epsilon }(\mathit{t}))$	=	Structure function of the system at time $\mathit{t}$ in the first model of failure
${\mathrm{\Phi }}_{\mathrm{I}}(l_1,l_2,\dots ,l_M)$	=	Survival signature of the system in the first model of failure
${\varphi }_2(\mathit{\epsilon }(\mathit{t}))$	=	System structure function at time $\mathit{t}$ in the second model of failure
${\mathrm{\Phi }}_{\mathrm{II}}(l_1,l_2,\dots ,l_M)$	=	Survival signature of the system in the second model of failure
${\mathrm{\Phi }}^{(\mathrm{i})}$	=	Survival signature of the type $\mathit{i}$
${\mathrm{\Phi }}_{\mathrm{j}}^{(\mathrm{i})}$	=	Probability that the $\mathit{i}$th type operates with exactly $\mathit{j}$ components
${\mathit{v}}_{l_i}^{(\mathit{i})}$	=	State vector of the components of type $\mathit{i}$ with exactly $l_i$ operating components
$V_{l_i}^{(\mathit{i})}$	=	Set of all state vectors ${\mathit{v}}_{l_i}^{(\mathit{i})}$ of type $\mathit{i}$
$V_{l_1,l_2,\dots ,l_M}$	=	Set of all state vectors of the system with exactly $l_i$ operating components of type $\mathit{i}$

Acknowledgements

We would like to express our sincere thanks to an Associate Editor and two anonymous referees for their constructive comments and suggestions which led to the improvements of the paper. Asadi’s research was carried out in IPM Isfahan branch and was in part supported by a grant from IPM (No. 1400620212).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/16843703.2023.2238339.

Additional information

Funding

The work was supported by the Institute for Research in Fundamental Sciences [1400620212].

Notes on contributors

Kinan Hamdan

Kinan Hamdan is an assistant professor of Statistics at the Department of Statistics, Faculty of Science, University of Albaath, Syria. He received BS in Statistics from Albaath University, Syria, and MS in Statistics from Isfahan University of Technology, Iran and a PhD in Statistics from the University of Isfahan, Iran. His research interests include reliability theory, reliability modeling of systems and networks, ordered random variables, and maintenance strategies.

Majid Asadi

Majid Asadi is a Professor of Statistics at the Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Iran and a Senior Associate Researcher at the School of Mathematics, IPM. He is an elected member of the International Statistical Institute. He received BS and MS in Statistics from Ferdowsi University of Mashhad, Iran and a PhD in Statistics from the University of Sheffield, UK. His research interests include reliability theory, reliability modeling of systems and networks, information-theoretic probability modeling, and ordered random variables.

Mahdi Tavangar

Mahdi Tavangar received BS, MS and Ph.D. in Statistics from the University of Isfahan, Iran, in 2009. He is presently an Associate Professor in the Department of Statistics at the University of Isfahan, Iran. He has published over 30 research papers in international journals. His research interests include reliability theory, ordered random variables, maintenance models, and characterization of probability distributions.