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Reliability of a coherent system equipped with two cold standby components

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Abstract

In this paper, we focus on a particular type of coherent system which may fail either on the failure of its first component or on the failure of its second component. We investigate the renewal of such a coherent system using two cold standby components. We obtain the reliability function of the considered coherent system which is equipped with two cold standby components. We study the problem to optimize the reliability of the system. Some stochastic ordering results are also presented. Examples are provided to illustrate the theoretical results presented in this study. Our results subsume some of the earlier results in the literature.

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Acknowledgements

The authors would like to thank the editor, the associate editor and the anonymous referees for the useful suggestions which helped improve the presentation of the paper to a great extent. The first author would also like to thank Indian Institute of Technology Kharagpur for the financial assistance.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Achintya Roy & Nitin Gupta

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  1. Achintya Roy
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  2. Nitin Gupta
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Correspondence to Achintya Roy.

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Appendix

Appendix

Proof of Theorem 2

The reliability function of a coherent system with two cold standby components can be written as

$$\begin{aligned}&P(T_{R}(Z_{1},Z_{2})>t)\nonumber \\&\quad =P(T_{R}(Z_{1},Z_{2})>t,T=X_{1:n})\nonumber \\&\quad \quad +P(T_{R}(Z_{1},Z_{2})>t,T=X_{2:n}) +P(T_{R}(Z_{1},Z_{2})>t,T>X_{2:n})\nonumber \\&\quad =p_{1}P(T+\phi (X^{'}_{1},\ldots ,X^{'}_{u-1},Z_{1},X^{'}_{u+1},\ldots ,X^{'}_{n})>t|T=X_{1:n})\nonumber \\&\quad \quad +p_{2}P(T + \phi (X^{''}_{1},\ldots ,X^{''}_{v-1},Z_{1},X^{''}_{v+1},\ldots ,X^{''}_{w(v)-1},Z_{2},\nonumber \\&\quad X^{''}_{w(v)+1},\ldots ,X^{''}_{n})>t|T=X_{2:n})\nonumber \\&\quad \quad +\sum _{i=3}^{n} p_{i} P(X_{i:n}>t). \end{aligned}$$
(19)

Using Eq. (4) of Eryilmaz (2014b), we can write

$$\begin{aligned}&P(T+\phi (X^{ '}_{1},\ldots ,X^{'}_{u-1},Z_{1},X^{'}_{u+1},\ldots ,X^{'}_{n})>t|T=X_{1:n})= P(X_{1:n}>t) \nonumber \\&\quad \quad + \sum _{i=1}^{n} P(u=i) \int _{0}^{t} P\{ \phi (X^{'}_{1},\ldots ,X^{'}_{i-1},Z_{1},X^{'}_{i+1},\ldots ,X^{'}_{n})\nonumber \\&\quad >t-x|X_{1:n}=x\} f_{X_{1:n}}(x) dx. \end{aligned}$$
(20)

Next, consider the conditional probability in (19),

$$\begin{aligned}&P(T+\phi (X^{''}_{1},\ldots ,X^{''}_{v-1},Z_{1},X^{''}_{v+1},\ldots ,X^{''}_{w(v)-1},Z_{2},X^{''}_{w(v)+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad>t|T=X_{2:n})=\frac{1}{P(T=X_{2:n})} \nonumber \\&\quad \times P\{T+\phi (X^{''}_{1}\ldots ,X^{''}_{v-1},Z_{1},X^{''}_{v+1},\ldots ,X^{''}_{w(v)-1},Z_{2},X^{''}_{w(v)+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad>t,T=X_{2:n}\}=\frac{1}{P(T=X_{2:n})} \nonumber \\&\quad \times \sum _{i=1}^{n} P\{T+\phi (X^{''}_{1}\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{w(i)-1},Z_{2},X^{''}_{w(i)+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad>t,X_{1:n}=X_{i},T=X_{2:n}\} =\frac{1}{P(T=X_{2:n})} \nonumber \\&\quad \times \sum _{i=1}^{n}\sum _{j\ne i} ^{n} P\{\phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2},X^{''}_{j+1},\ldots ,X^{''}_{n}) +T\nonumber \\&\qquad>t,X_{1:n}=X_{i},X_{2:n}=X_{j},T=X_{2:n}\} =\frac{1}{P(T=X_{2:n})} \nonumber \\&\quad \times \sum _{i=1}^{n}\sum _{j\ne i} ^{n} P\{\phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2},X^{''}_{j+1},\ldots ,X^{''}_{n}) +X_{j}\nonumber \\&\qquad>t,X_{1:n}=X_{i},X_{2:n}=X_{j},T=X_{2:n}\} \nonumber \\&\qquad =\sum _{i=1}^{n}\sum _{j\ne i} ^{n} \frac{P(X_{1:n}=X_{i},X_{2:n}=X_{j},T=X_{2:n})}{P(T=X_{2:n})}\nonumber \\&\quad \times P\{ \phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2},X^{''}_{j+1},\ldots ,X^{''}_{n}) +X_{j}\nonumber \\&\qquad>t|X_{1:n}=X_{i},X_{2:n}=X_{j},T=X_{2:n}\} =\sum _{i=1}^{n}\sum _{j\ne i} ^{n}P(v=i) P(w(i)=j) \nonumber \\&\quad \times \int _{0}^{\infty } P\{ \phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2}, X^{''}_{j+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad>t-x|X_{2:n}=x\} f_{X_{2:n}}(x) dx = \sum _{i=1}^{n}\sum _{j\ne i}^{n} P(v=i) P(w(i)=j)\nonumber \\&\qquad \int _{t}^{\infty } f_{X_{2:n}}(x) dx + \sum _{i=1}^{n}\sum _{j\ne i}^{n} P(v=i) P(w(i)=j) \nonumber \\&\quad \times \int _{0}^{t} P\{ \phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2},X^{''}_{j+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad>t-x|X_{2:n}=x\} f_{X_{2:n}}(x) dx = P(X_{2:n}>t) + \sum _{i=1}^{n}\sum _{j\ne i}^{n} P(v=i)P(w(i)=j)\nonumber \\&\quad \times \int _{0}^{t} P\{ \phi (X^{''}_{1},\ldots ,X^{''}_{i-1},Z_{1},X^{''}_{i+1},\ldots ,X^{''}_{j-1},Z_{2},X^{''}_{j+1},\ldots ,X^{''}_{n})\nonumber \\&\qquad >t-x|X_{2:n}=x\} f_{X_{2:n}}(x) dx. \end{aligned}$$
(21)

Therefore, using Eqs. (20) and (21) in (19), we have Eq. (4). \(\square \)

Proof of equation (13)

By the definition of five-component bridge system,

$$\begin{aligned}&P\{ \phi (Z_{1},Z_{2},X^{''}_{3},X^{''}_{4},X^{''}_{5})>t-x|X_{2:5}=x\} \\&\quad =P(min(Z_{1},X^{''}_{4})>t-x|X_{2:5}=x)\\&\qquad +P(min(Z_{2},X^{''}_{5})>t-x|X_{2:5}=x)\\&\qquad +P(min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x|X_{2:5}=x)\\&\qquad +P(min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x)\\&\qquad -P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\qquad -P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x|X_{2:5}=x\}\\&\qquad -P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\qquad -P\{min(Z_{2},X^{''}_{5})>t-x,min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x|X_{2:5}=x\}\\&\qquad -P\{min(Z_{2},X^{''}_{5})>t-x,min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\} \\&\qquad -P\{min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x,min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\qquad +P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{5})>t-x,\\&\quad \quad \qquad min(Z_{2}, X^{''}_{3},X^{''}_{4})>t-x|X_{2:5}=x\}\\&\qquad +P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{5})>t-x,\\&\quad \quad \qquad min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\ \end{aligned}$$
$$\begin{aligned}&\qquad +P\{min(Z_{2},X^{''}_{5})>t-x,min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x,\\&\quad \quad \qquad min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\qquad +P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x,\\&\quad \quad \qquad min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\qquad -P\{min(Z_{1},X^{''}_{4})>t-x,min(Z_{2},X^{''}_{5})>t-x,\\&\quad \quad \qquad min(Z_{2},X^{''}_{3},X^{''}_{4})>t-x, min(Z_{1},X^{''}_{3},X^{''}_{5})>t-x|X_{2:5}=x\}\\&\quad =P(Z_{1}>t-x)P(X^{''}_{4}>t-x|X_{2:5}=x)\\&\quad \quad +P(Z_{2}>t-x)P(X^{''}_{5}>t-x|X_{2:5}=x)\\&\qquad +P(Z_{1}>t-x)P(X^{''}_{3}>t-x|X_{2:5}=x)\\&\quad \quad \quad P(X^{''}_{5}>t-x|X_{2:5}=x)+P(Z_{2}>t-x) \\ \end{aligned}$$
$$\begin{aligned}&\quad P(X^{''}_{3}>t-x|X_{2:5}=x)P(X^{''}_{4}>t-x|X_{2:5}=x)\\&\quad \quad -P(Z_{1}>t-x)P(Z_{2}>t-x) \\&\quad P(X^{''}_{3}>t-x|X_{2:5}=x)P(X^{''}_{5}>t-x|X_{2:5}=x)\\&\quad \quad -P(Z_{1}>t-x)P(Z_{2}>t-x) \\&\quad P(X^{''}_{3}>t-x|X_{2:5}=x) P(X^{''}_{4}>t-x|X_{2:5}=x)\\&\quad \quad -P(Z_{1}>t-x)P(Z_{2}>t-x) \\&\quad P(X^{''}_{4}>t-x|X_{2:5}=x)P(X^{''}_{5}>t-x|X_{2:5}=x)\\&\quad \quad -P(Z_{1}>t-x)P(X^{''}_{3}>t-x|X_{2:5}=x) \\&\quad P(X^{''}_{4}>t-x|X_{2:5}=x)P(X^{''}_{5}>t-x|X_{2:5}=x)\\&\quad \quad - P(Z_{2}>t-x)P(X^{''}_{3}>t-x|X_{2:5}=x) \\&\quad P(X^{''}_{4}>t-x|X_{2:5}=x)P(X^{''}_{5}>t-x|X_{2:5}=x) \\&\quad \quad + 2P(Z_{1}>t-x)P(Z_{2}>t-x) \\&\quad P(X^{''}_{3}>t-x|X_{2:5}=x) P(X^{''}_{4}>t-x|X_{2:5}=x)P(X^{''}_{5}>t-x|X_{2:5}=x) \\&\quad ={\overline{G}}(t-x)\frac{{\overline{F}}(t)}{{\overline{F}}(x)}+{\overline{H}}(t-x)\frac{{\overline{F}}(t)}{{\overline{F}}(x)}+{\overline{G}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{2}+{\overline{H}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{2}\\&\qquad -{\overline{H}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{3} -{\overline{G}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{3}-3{\overline{H}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{2}{\overline{G}}(t-x) \\&\qquad +2{\overline{H}}(t-x)\left( \frac{{\overline{F}}(t)}{{\overline{F}}(x)}\right) ^{3}{\overline{G}}(t-x). \end{aligned}$$

\(\square \)

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Roy, A., Gupta, N. Reliability of a coherent system equipped with two cold standby components. Metrika 83, 677–697 (2020). https://doi.org/10.1007/s00184-019-00752-3

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