Reliability Analysis of Weighted- k-out-of- n: G System Consisting of Two Different Types of Nonidentical Components Each with its Own Positive Integer-Valued Weight

Eisa Mahmoudi; RahmatSadat Meshkat

doi:10.2991/jsta.d.200917.002

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Volume 19, Issue 3, September 2020, Pages 408 - 414

Reliability Analysis of Weighted- k-out-of- n: G System Consisting of Two Different Types of Nonidentical Components Each with its Own Positive Integer-Valued Weight

Authors

Eisa Mahmoudi^*^,, RahmatSadat Meshkat

Department of Statistics, Yazd University, 89175-741, Yazd, Iran

^*Corresponding author. Email: emahmoudi@yazd.ac.ir

Corresponding Author

Eisa Mahmoudi

Received 5 May 2019, Accepted 14 June 2020, Available Online 30 September 2020.

DOI: 10.2991/jsta.d.200917.002 How to use a DOI?
Keywords: Component importance; Mean time to failure; Reliability; Weighted- k-out-of- n:G system
Abstract: This paper introduces a special case of weighted- k-out-of- n:G system formed from two types of nonidentical components with different weights. This system consists of n nonidentical components each with its own positive integer-valued weight which are categorized into two groups with respect to their duties and services. In fact, we have a system consisting n components such that n1 of them each with its own weight ωi and reliability p1i and n2 of them each with its own weight ωi∗ and reliability p2i. If the total weights of the functioning components exceeds a prespecified threshold k, the system is supposed to work. The reliability of system is obtained based on the total weight of all working components in both group. The survival function and mean time to failure are presented. Also, the component importance of this system are studied.
Copyright: © 2020 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In most real-life systems, the total contribution of the components plays an important role and must be above a specified performance level. In many situations, the components contribute differently to the capacity of the system. The systems with weighted components with unequal weights are introduced by Wu and Chen [1] to deal with this situation which has been studied in the literature. A system including n components with their different positive integer weights is known as weighted- k-out-of- n:G system when it works if and only if the total weight of working components is above a given threshold k.

Chen and Yang [2] extended the one-stage weighted- k-out-of- n model to the two-stage weighted- k-out-of- n model with components in common. Samaniego and Shaked [3] presented a review on weighted- k-out-of- n systems. Li and Zuo [4] provided two models of multi-state weighted- k-out-of- n system models and presented the recursive algorithms for their reliability evaluation. Navarro et al. [5] extended the signature-based representations of the reliability functions of coherent systems to systems with heterogeneous components. Eryilmaz [6] introduced a k-out-of- n system with random weights for components and investigated the reliability properties of the system. Eryilmaz and Bozbulut [7] studied a multi-state weighted- k-out-of- n:G system model in a dynamic setup and yielded an algorithmic approach for its dynamic reliability analysis. In order to measure the components importance in k-out-of- n system with random weights, Rahmani et al. [8] defined the weighted importance (WI) measure that depends only on the distribution of component weights and also, Meshkat and Mahmoudi [9] generalized this measure for two component i and j and the relation of these measures is investigated with Birnbaum reliability importance measure.

Eryilmaz and Sarikaya [10] studied the special case of weighted- k-out-of- n: G system containing two types of components, each group having different weights and reliabilities such that one group have the common weight ω and reliability p1, while the other have common weight ω∗ and reliability p2. They also obtained the nonrecursive equations for the system reliability, survival function and mean time to failure (MTTF). Recently, Eryilmaz [11] introduced the (k1,k2,⋯,km)-out-of- n system including ni components of type i for i=1,⋯,m and n=∑i=1mni. The corresponding system is assumed to work if at least k1 components of type 1, k2 components of type 2, …, km components of type m function. Its reliability and the setup of weighted- (k1,k2,⋯,km)-out-of- n system is also defined and studied.

In this study, we consider a general setup of weighted- k-out-of- n:G system in which one group consists of n1 components each with its own positive integer-valued weight ωi and reliability p1i, while the other group consists of n2 components each with its own positive integer-valued weight ωi∗ and reliability p2i, (n=n1+n2). This system operates if the total weight of all working components is at least k. Therefore, its reliability is obtained based on the total weight of all working components in both group. This setup of weighted- k-out-of- n:G system containing two types of nonidentical components might be useful in practice. As an illustration, assume that a production line (system) is constituted by n devices (components) in two groups with n1 and n2 devices (n=n1+n2), respectively, which each group has their own duties and services and different capacities (weights). For an instance, a production line in tile factory consists of two groups of devices, one group including some devices to combine the materials and bake the tiles and the other group including some devices to implement the layout and color on the tiles and glaze them. Finally to have a complete tile, some devices in the line should be operate such that a certain capacity of total capacities is provided. Therefore, we have a system consisting n components such that n1 of them each with its own weight ωi and reliability p1i and n2 of them each with its own weight ωi∗ and reliability p2i. To operate the system, a certain capacity of components denoted by threshold k, should be provided.

The remainder of the paper is arranged as follows: in Section 2, for mentioned weighted- k-out-of- n:G system, the description of system and its reliability evaluation is provided. The survival function and MTTF of this system are studied in Section 3. In Section 4, the Birnbaum reliability importance is investigated. Finally, the concluding remarks are given in Section 5.

2. THE SYSTEM MODEL

In this section, first some notations and the description of modelling main assumptions are provided for this general setup of weighted- k-out-of- n:G system.

2.1. Notations

n	Number of components in a system
Xi	State of i-th component of the system:
	Xi=1 if component is functioning; Xi=0 if it is failed
k	Minimum total weight(capacity) of all working components to operate system
C	The set of all components
C1	First group of components
C2	Second group of components
n1	Number of components in C1
n2	Number of components in C2
ωi	Weight of i-th component in C1
ωi∗	Weight of i-th component in C2
p1i	Reliability of i-th component in C1
p2i	Reliability of i-th component in C2

2.2. System Description

Consider a system including n independent and nonidentical components which are placed in two groups with respect to their duties and services. Each component has its own positive integer-valued weight. Let the state of i-th component Xi be an independent binary random variable such that p1i=P(Xi=1) if i∈C1 with corresponding weight ωi and p2i=P(Xi=1) if i∈C2 with corresponding weight ωi∗. According to the above representation, clearly C=C1∪C2, C1∩C2=∅, |C1|=n1 and |C2|=n2. Assume that ϕ(X) denotes the structure function of the system where X=(X1,⋯,Xn) is the state vector of components. As mentioned, in this general setup of weighted- k-out-of- n:G system, if the total weight of all functioning components is above predefined threshold k, the system is supposed to work. So, the structure function of this system is defined by

ϕ(X)=1if∑i∈C1ωiXi+∑i∈C2ωi∗Xi≥k,0o.w.

Evidently, the reliability of system defined as probability that system is working, can be given by the probability

R=P(ϕ(X)=1)=P∑i∈C1ωiXi+∑i∈C2ωi∗Xi≥k.(1)

2.3. Reliability Evaluation

Now, taking the independent and nonidentical components into account with the reliability of i-th component p1i if i∈C1 with corresponding weight ωi and p2i if i∈C2 with corresponding weight ωi∗, the reliability of system defined by (1) can be formulated as follows:

R=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈E⁡∏m=1sp1lm∏m=s+1n1(1−p1lm)∏m=1up2lm∏m=u+1n2(1−p2lm),(2)

where s=∑i∈C1Xi, u=∑i∈C2Xi, summation Es(Eu) extends over all combinations l1,⋯,ls(l1,⋯,lu) of {1,⋯,n1}({1,⋯,n2}) and

E=(Es,Eu):∑m=1sωlmXlm+∑m=1uωlm∗Xlm≥k.

In the following, illustrative results are presented to observe the value of R for a weighted- k-out-of-5:G system with respect to the different values of k, n1 and the component weights and reliabilities.

Example 2.1.

Consider a weighted- k-out-of- 5:G system with two especial case n1=2 and n1=3. First case n1=2, suppose that ω1=3,ω2=1,ω1∗=1,ω2∗=2 and ω3∗=3 with corresponding component reliabilities as

Then for k=7,

R=p11(1−p12)(p21(1−p22)p23+(1−p21)p22p23+p21p22p23)+(1−p11)p12p21p22p23 +p11p12((1−p21)(1−p22)p23+p21p22(1−p23)+p21(1−p22)p23+(1−p21)p22p23+p21p22p23),

and for k=9,

R=p11(1−p12)p21p22p23)+p11p12((1−p21)p22p23+p21p22p23).

Now case n1=3, let ω1=3, ω2=1, ω3=2, ω1∗=1 and ω2∗=2 with corresponding component reliabilities as

The calculation of R for k=7 and k=9 is similar.

In Table 1, the reliability of the weighted- k-out-of- 5:G system is computed with respect to the different values of k, n1 and the weight of components. As observed, the system reliability is sensitive to the values of n1 which determines the number of components in each group and also, the values of threshold k plays a significant role in system reliability.

n	n1	k	R
5	2	7	0.9893
		9	0.8910
	3	7	0.9210
		9	0.6697

Table 1

Reliability of the weighted-k-out-of-5:G system.

Proposition 2.1.

As an especial case, consider a weighted- k-out-of- n:G system including independent and identical components in each of two groups with the reliability of i-th component p1 if i∈C1 and p2 if i∈C2, but with different corresponding weight ωi and ωi∗ respectively in each group. Then, the reliability given by (2) can be rewritten as

R=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈Ep1s(1−p1)n1−sp2u(1−p2)n2−u,

where the notations are mentioned for Equation (2).

Remark 2.1

If all the weights ωi and ωi∗ are respectively equal to ω and ω∗ with identical components in each of two groups, the especial case of weighted- k-out-of- n:G system containing two types of components presented by Eryilmaz and Sarikaya [10] is concluded.

3. SURVIVAL ANALYSIS

Let T1,⋯,Tn represent the lifetimes of n independent and nonidentical components. As representative of the performance of the weighted- k-out-of- n:G system, the total weight of the system at time t(≥0) can be described by

Wn(t)=∑i∈C1ωiI(Ti>t)+∑i∈C2ωi∗I(Ti>t),

where I(Ti>t) is indicator function (which is 1 if Ti>t and 0 if Ti≤t). Suppose that the survival function of i-th component is F̄i(t)=P(Ti>t),∀i∈C1 and Gi(t)=P(Ti>t),∀i∈C2, then the mean weight of the system at time t is obtained as

E[Wn(t)]=∑i∈C1ωiP(Ti>t)+∑i∈C2ωi∗P(Ti>t)=∑i=1n1ωiF¯i(t)+∑i=1n2ωi∗Gi(t).

Now, considering T as the lifetime of the system, then it is defined as

T=inf{t:Wn(t)<k}.

So, by substituting p1lm and p2lm in Equation (2) by F̄lm(t) and Glm(t), respectively, the survival function defined as probability that system is working at time t for weighted- k-out-of- n:G system with two type nonidentical components can be established as follows:

S(t)=P(T>t)=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈E∏m=1sF¯lm(t)∏m=s+1n1Flm(t)∏m=1uGlm(t)∏m=u+1n2Glm(t),(3)

where the notations are mentioned for Equation (2) in previous section.

Proposition 3.1.

For the especial case mentioned in Proposition 2.1, substituting F(t) and G(t), respectively, the survival function (3) at time t can be rewritten as

S(t)=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈EF¯s(t)Fn1−s(t)Gu(t)Gn2−u(t),

where the notations are mentioned for Equation (2) in previous section.

As one of the most important reliability characteristics, the MTTF of a system is defined as expected value of the lifetime of the system

MTTF=E(T)=∫0∞P(T>t)dt.

Hence, the MTTF of our proposed system is computed by

MTTF=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈E∫0∞∏m=1sF¯lm(t)∏m=s+1n1Flm(t)∏m=1uGlm(t)∏m=u+1n2Glm(t)dt.(4)

Example 3.1.

Considering the weighted- k-out-of- 5:G system mentioned in Example 2.1, let the survival function of i-th component for each classes is Fi(t)=e−λit and Gi(t)=e−μit, respectively. For first case n1=2, suppose that ω1=3,ω2=1,ω1∗=1,ω2∗=2 and ω3∗=3 with corresponding component survival function λ=(0.2,0.3) and μ=(0.6,0.1,0.4). For second case n2=3, suppose that ω1=3,ω2=1,ω3=1,ω1∗=1 and ω2∗=2 with corresponding component survival function λ=(0.2,0.6,0.3) and μ=(0.1,0.4).

From Figure 1, the graphs of survival function with respect to different values of k indicate that survival of system depends on the combination of the survival function and the weight of components in each of classes. Obviously, survival function is sensitive to the values of n1 which determines the number of components in each group and also, the values of threshold k plays a significant role in system reliability.

As well, MTTF is

MTTF=∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈E∫0∞∏m=1se−λlmt∏m=s+1n1(1−e−λlmt)∏m=1ue−μlmt∏m=u+1n2(1−e−μlmt)dt, =∑s=0n1∑u=0n2∑Es∑Eu(Es,Eu)∈E∫0∞e−t∑m=1sλlm+∑m=1uμlm∏m=s+1n1(1−e−λlmt)∏m=u+1n2(1−e−μlmt)dt.

In Table 2, the MTTF of the weighted- k-out-of- 5:G system is computed with respect to the different values of k, n1 and the weight of components. As observed, MTTF is sensitive to the values of n1 which determines the number of components in each group and also, the values of threshold k plays a significant role in system reliability.

n	n1	k	MTTF
5	2	7	1.8597
		9	1.1442
	3	7	1.4637
		9	0.6250

Table 2

MTTF) of the weighted-k-out-of-5:G system.

4. COMPONENT IMPORTANCE

Considering the important role of ranking the components according to their importance measure in system reliability, the Birnbaum reliability importance is obtained in a weighted- k-out-of- n:G system with two type nonidentical components. Birnbaum [12] introduced the importance of the i-th component in a coherent system. If the event S shows that the system works, then Birnbaum reliability importance of the i-th component is defined by

Ii=P(S|Xi=1)−P(S|Xi=0).

In this setup of weighted- k-out-of- n:G system, the Birnbaum reliability importance of the i-th component is rewritten as follows:

For i∈C1,

P(S|Xi=1)=∑s=1n1∑u=0n2∑Es\i∑Eu(Es\i,Eu)∈E∏m=1s−1p1lm∏m=s+1n1(1−p1lm)∏m=1up2lm∏m=u+1n2(1−p2lm),(5)

and

P(S|Xi=0)=∑s=0n1−1∑u=0n2∑Es\i∑Eu(Es\i,Eu)∈E∏m=1sp1lm∏m=s+1n1−1(1−p1lm)∏m=1up2lm∏m=u+1n2(1−p2lm),(6)

where summation Es\i extends over all combinations l1,⋯,ls of {1,⋯,i−1,i+1,⋯,n1}, denoted by {1,⋯,n1}\{i}.

For i∈C2,

P(S|Xi=1)=∑s=0n1∑u=1n2∑Es∑Eu\i(Es,Eu\i)∈E∏m=1sp1lm∏m=s+1n1(1−p1lm)∏m=1u−1p2lm∏m=u+1n2(1−p2lm),(7)

and

P(S|Xi=0)=∑s=0n1∑u=0n2−1∑Es∑Eu\i(Es,Eu\i)∈E∏m=1sp1lm∏m=s+1n1(1−p1lm)∏m=1up2lm∏m=u+1n2−1(1−p2lm),(8)

where summation Eu∖i extends over all combinations l1,⋯,lu of {1,⋯,n2}\{i}. Let I1(I2) denote the importance of the components in C1(C2) that equals the difference between Equations (5) and (6) (Equations (7) and (8)). Next example presents the value of Birnbaum reliability importance for a weighted- k-out-of- 5:G system with respect to the values of k, n1, and the component weights and reliabilities.

Example 4.1.

Consider a weighted- k-out-of- 5:G system with the especial case n1=3 and k=5. Suppose that ω1=3, ω2=1, ω3=2, ω1∗=1 and ω2∗=2 with corresponding component reliabilities p1=(p11,p12,p13) and p2=(p21,p22).

For instance, if i=3∈C1,

P(S|Xi=1)=(1−p11)(1−p12)p21p22+p11(1−p12)((1−p21)(1−p22)+p21(1−p22)+(1−p21)p22+p21p22) +(1−p11)p12((1−p21)p22+p21p22)+p11p12((1−p21)(1−p22)+p21(1−p22)+(1−p21)p22+p21p22),

and

P(S|Xi=0)=p11(1−p12)((1−p21)p22+p21p22) +p11p12(p21(1−p22)+(1−p21)p22+p21p22).

In Table 3, the Birnbaum reliability importance of the weighted- k-out-of- 5:G system is computed with respect to the different vectors of reliability of the components in both group C1 and C2. As observed, the value of I1 and I2 depends on the combination of the weight and the reliability of the components.

P1	P2	i	ωi	ωi∗	I1	I2
(0.95,0.97,0.85)	(0.90,0.95)	1	3	1	0.1940	0.0102
		2	1	2	0.0105	0.0605
		3	2	–	0.0534	–
(0.80,0.90,0.95)	(0.90,0.85)	1	3	1	0.1992	0.0472
		2	1	2	0.0516	0.1957
		3	2	–	0.1911	–
(0.80,0.90,0.85)	(0.95,0.97)	1	3	1	0.1790	0.0243
		2	1	2	0.0117	0.1866
		3	2	–	0.1965	–
(0.70,0.75,0.65)	(0.85,0.65)	1	3	1	0.5490	0.1472
		2	1	2	0.0919	0.2765
		3	2	–	0.2765	–

Table 3

Birnbaum reliability importance of the weighted-k-out-of-5:G system for n1=3 and k=5 with respect to reliability of the components.

5. CONCLUSION

In many situations, the components contribute differently to the capacity of the system which the weighted- k-out-of- n:G system are used to deal with. A system including n components with their different positive integer weights that it works if and only if the total weight of working components is above a given threshold k. In most of the studies in the literature, the weighted- k-out-of- n:G systems have been studied with identical components.

In this paper, we introduce a special case of weighted- k-out-of- n:G system formed from two types of nonidentical components with different weights in which one group consists of n1 components each with its own positive integer-valued weight ωi and reliability p1i, while the other group consists of n2 components each with its own positive integer-valued weight ωj∗ and reliability p2j, (n=n1+n2). The components are categorized into two groups with respect to their duties and services with different capacity. If the total weights of the functioning components exceeds a prespecified threshold k, the system is supposed to work. The reliability and component importance of this system are studied. The survival function and MTTF are presented. This setup of weighted- k-out-of- n:G system containing two types of components might be useful in practice. As result, the system reliability, survival function, MTTF and component importance of this system are sensitive to the values of n1 which determines the number of components in each group and also, the values of threshold k plays a significant role in system reliability.

CONFLICTS OF INTEREST

There is no conflict of interest in this article. Mrs Meshkat is the phd student under supervision of Prof. Eisa Mahmoudi and this work is related to her thesis.

AUTHORS' CONTRIBUTIONS

RahmatSadat Meshkat wrote the initial draft of the paper and did the analysis. Eisa Mahmoudi helped in analyzing the finding and supervised overall work.

Funding Statement

The work is sponsored by the Yazd University, Iran. We have received no Funding for this paper.

ACKNOWLEDGMENTS

The authors would like to sincerely thank the Editor-in-Chief, Associate Editor, and referees for carefully reading the paper and for their useful comments. The authors are also indebted to Yazd University for supporting this research.

REFERENCES

1.J.S. Wu and R.J. Chen, IEEE Trans. Reliab., Vol. 43, 1994, pp. 327-328.

2.Y. Chen and Q. Yang, IEEE Trans. Reliab., Vol. 54, 2005, pp. 431-440.

3.F.J. Samaniego and M. Shaked, Stat. Probab. Lett., Vol. 78, 2008, pp. 815-823.

4.W. Li and M.J. Zuo, Reliab. Eng. Syst. Saf., Vol. 93, 2008, pp. 160-167.

5.J. Navarro, F.J. Samaniego, and N. Balakrishnan, J. Appl. Probab., Vol. 48, 2011, pp. 856-867.

6.S. Eryilmaz, Reliab. Eng. Syst. Saf., Vol. 109, 2013, pp. 41-44.

7.S. Eryilmaz and A.R. Bozbulut, Comput. Ind. Eng., Vol. 72, 2014, pp. 255-260.

8.R.A. Rahmani, M. Izadi, and B.E. Khaledi, J. Comput. Appl. Math., Vol. 296, 2016, pp. 1-9.

9.R.S. Meshkat and E. Mahmoudi, Comput. Appl. Math., Vol. 326, 2017, pp. 273-283.

10.S. Eryilmaz and K. Sarikaya, Proc. Inst. Mech. Eng. O J. Risk Reliab., Vol. 228, 2014, pp. 265-271.

11.S. Eryilmaz, J. Comput. Appl. Math., Vol. 346, 2019, pp. 591-598.

12.Z.W. Birnbaum, P.R. Krishnaiah (editor), Multivariate Analysis II, Academic Press, New York, 1969, pp. 581-592.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 3
Pages: 408 - 414
Publication Date: 2020/09/30
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.200917.002 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Eisa Mahmoudi
AU  - RahmatSadat Meshkat
PY  - 2020
DA  - 2020/09/30
TI  - Reliability Analysis of Weighted- k-out-of- n: G System Consisting of Two Different Types of Nonidentical Components Each with its Own Positive Integer-Valued Weight
JO  - Journal of Statistical Theory and Applications
SP  - 408
EP  - 414
VL  - 19
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200917.002
DO  - 10.2991/jsta.d.200917.002
ID  - Mahmoudi2020
ER  -

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