Reliability equivalence of a series–parallel system

Applied Mathematics and Computation 154 (2004) 257–277 www.elsevier.com/locate/amc

Reliability equivalence of a series–parallel system Ammar M. Sarhan a,b,*, A.S. Al-Ruzaiza a, I.A. Alwasel a, Awad I. El-Gohary a,b a

b

Department of Statistics and O.R., College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Abstract In this paper we study equivalence of different designs of a four independent and identical components series–parallel system. The failure rates of the system components are assumed to be constant. Then two types of reliability equivalence factors of the system are obtained. Numerical studies are given in order to explain how one can utilize the theoretical results obtained. Ó 2003 Elsevier Inc. All rights reserved.

1. Introduction In reliability analysis, sometimes different system designs should be comparable based on a reliability characteristic such as the reliability function or mean time to failure in case of no repairs. Recently, the concept of comparing different designs is considered in the literature, e.g. [4–7], to derive the reliability equivalence factors of some certain systems. R
ade [4,5] derived the reliability equivalence factors for two component parallel and two component series systems with independent and identical components. Sarhan [6] applied the concept of comparing different designs to derive the reliability equivalence

*

Corresponding author. E-mail addresses: [email protected], [email protected] (A.M. Sarhan).

0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00709-4

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factors of a series system consists of n independent and non-identical components. He used the reliability function as a characteristic measure to compare different system designs. In [7], Sarhan applied such concept to derive two types of reliability equivalence factors of a basic series/parallel system consists of three independent and non-identical components. He used both the reliability function and mean time to failure as characteristic measures to compare different system designs in obtaining these factors. Generally, there are two main methods to improve a system design. The first is a reduction method and the second is a redundancy method. It is assumed in the reduction method that the system design can be improved by reducing the failure rates of a set of its components by a factor q, 0 < q < 1. The redundancy method is divided into some other types. From these types hot, cold and cold with imperfect switch redundancy. Sarhan [6] called these types respectively as hot, cold and cold with imperfect switch duplication methods. Using redundancy method may not be the optimal solution in system in which the minimum size and weight are overriding considerations: for example, in satellites or other space applications, in well-logging equipment, and in pacemakers and similar biomedical applications, Lewis [3]. In such applications space or weight limitations may indicate an increase in component reliability rather than redundancy. Then more emphasis must be placed on robust design, manufacturing quality control, and on controlling the operating environment. Therefore, the concept of reliability equivalence takes place. In such concept, the design of the system which is improved according to reduction method should be equivalent to the design of the system improved according to one of the reset of the redundancy methods. The following definition gives the general definition of the reliability equivalence factor. Definition 1 [7]. A reliability equivalence factor is a factor by which a characteristic of components of a system design has to be multiplied in order to reach equality of a characteristic of this design with a different design. The objective of this paper is to deduce the reliability equivalence factors of a series–parallel system. In obtaining these factors, the reliability function and mean time to failure of the system as performance measures to compare different system designs of original system and another improved systems. Section 2 introduces the description of the system studied here. The structural importance and joint structural importance of the system are also calculated in Section 2. The reliability functions and mean time to failures of the designs of improved systems are presented in Section 3. Also, theoretical studies are established in this section to compare different methods used to improve the system design. In Section 4 we obtain two types of reliability equivalence factors of the system. Section 5 gives the a-fractiles of the original

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259

design and improved designs. Numerical results and conclusion are listed in Section 6. 2. Series–parallel system The system studied here consists of four independent and identical components, indexed 1 to 4. Components 1 and 2 connected in series to form module 1. Also components 3 and 4 connected in series to form module 2. Module 1 is connected in parallel with module 2. Fig. 1 presents the structural diagram of the system. It is assumed, in the current work, that the lifetime of each component is exponential with failure rate k, k > 0. The structure function of the system can be obtained to be UðxÞ ¼ x1 x2 þ x3 x4
x1 x2 x3 x4 ;

ð1Þ

where x ¼ ðx1 ; x2 ; x3 ; x4 Þ is a binary random vector with random components xi , i ¼ 1, 2, 3, 4, defined by
1 if component i is functioning; xi ¼ 0 if component i has failed: The following definition gives the structural importance of system components. Definition 2 [2]. The structural importance of component i in a coherent system of n components is 1 X IU ðiÞ ¼ n
1 ½Uð1i ; xÞ
Uð0i ; xÞ ð2Þ 2 fxjx ¼1g i

for i ¼ 1; 2; . . . ; n. Then to calculate the structural importance of component 1, IU ð1Þ, the summation will run over all state vectors ð1; 1; 0; 0Þ, ð1; 1; 0; 1Þ, ð1; 1; 1; 0Þ, ð1; 0; 0; 0Þ, ð1; 0; 1; 0Þ, ð1; 0; 0; 1Þ, ð1; 0; 1; 1Þ and ð1; 1; 1; 1Þ, since all theses vectors have x1 ¼ 1. Of these eight vectors, Uð1i ; xÞ
Uð0i ; xÞ ¼ 1 for the first three vectors listed. Therefore, 3 IU ð1Þ ¼ : 8 By symmetry, IU ðiÞ ¼ 38, i ¼ 2; 3; 4. This means that all components have the same importance for the system. This implies that, if there is a possibility for the designer to improve one component, he can choose any one of the system components. The following definition gives the joint structural importance of two components of a system.

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Fig. 1. Series–parallel system diagram.

Definition 3 [8]. The joint structural importance of components i, j in a coherent system of n components is JSIU ði; jÞ ¼

1 2n
2

X

½Uð1i ; 1j ; xÞ þ Uð0i ; 0j ; xÞ

fxjxi ¼1;xj ¼1g


Uð1i ; 0j ; xÞ
Uð0i ; 1j ; xÞ

ð3Þ

for i; j ¼ 1; 2; . . . ; n. One of the components of i; j becomes more (less) important when the other is functioning if JSIU ði; jÞ > 0 ðJSIU ði; jÞ < 0Þ, see Sarhan and Abouammoh [8]. For the given system, we have 3 JSIU ði; jÞ ¼ ; 4

ði; jÞ ¼ ð1; 2Þ or ð3; 4Þ

and 1 JSIU ði; jÞ ¼
; 4

ði; jÞ ¼ ð1; 3Þ or ð1; 4Þ or ð2; 3Þ or ð2; 4Þ:

Therefore, one of the components 1, 2 (or 3, 4) becomes more important when the other is functioning. While one of the components 1, 3 (or 1, 4 or 2, 3 or 2, 4) becomes less important when the other is functioning. Let Rs ðtÞ be the reliability function of the system. One can obtain Rs ðtÞ as follows. The reliability function of a coherent system is given by Rs ðtÞ ¼ P ½UðxðtÞÞ ¼ 1 ¼ E½UðxðtÞÞ: Using both (1) and the assumption that the systemÕs components are independent, we get Rs ðtÞ ¼ Eðx1 ðtÞÞEðx2 ðtÞÞ þ Eðx3 ðtÞÞEðx4 ðtÞÞ
Eðx1 ðtÞÞEðx2 ðtÞÞEðx3 ðtÞÞEðx4 ðtÞÞ: But Eðxi ðtÞÞ ¼ P ðxi ðtÞ ¼ 1Þ ¼ Ri ðtÞ, where Ri ðtÞ is the reliability function of component i, i ¼ 1, 2, 3, 4. Then

A.M. Sarhan et al. / Appl. Math. Comput. 154 (2004) 257–277

Rs ðtÞ ¼ R1 ðtÞR2 ðtÞ þ R3 ðtÞR4 ðtÞ
R1 ðtÞR2 ðtÞR3 ðtÞR4 ðtÞ:

261

ð4Þ

Since the system components are identical and each has a constant failure rate k, then Ri ðtÞ ¼ e
kt , i ¼ 1, 2, 3, 4. Therefore, the system reliability function becomes Rs ðtÞ ¼ f2
e
2kt ge
2kt : The system mean time to failure, say mttf, is Z 1 3 Rs ðtÞ dt ¼ : mttf ¼ 4k 0

ð5Þ

ð6Þ

3. Designs of improved systems Different improved designs of the system can be obtained by improving some of its components according to one of the following methods: 1. Reduction method; 2. Hot duplication method; 3. Cold duplication method. In the following subsections, we discuss how one can utilize these methods to improve the system design. 3.1. Reduction method In the reduction method, it is assumed that the system can be improved by reducing the failure rates of the set A  f1; 2; 3; 4g of system components by a factor q, 0 < q < 1. Let RA;q ðtÞ be the reliability function of the design obtained by improving the set A components according to reduction method. Let mttf A;q be the mean time to failure of the system that has the reliability function RA;q ðtÞ. The reliability function of component i after reducing its failure rate k by a factor q is Ri;q ðtÞ ¼ e
qkt . In what follows, we deduce the function RA;q ðtÞ and mttf A;q for all possible sets A. For A such that jAj ¼ 1, that is A ¼ fig, i ¼ 1, 2, 3, 4:   RA;q ðtÞ ¼ e
kt e
qkt þ e
kt
e
ðqþ2Þkt

ð7Þ

and mttf A;q ¼

1 1 1 þ
: 2k ð1 þ qÞk ð3 þ qÞk

ð8Þ

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For A ¼ f1; 3g or f1; 4g or f2; 4g or f2; 3g:   RA;q ðtÞ ¼ e
ðqþ1Þkt 2
e
ðqþ1Þkt

ð9Þ

and mttf A;q ¼

3 : 2ð1 þ qÞk

ð10Þ

For A ¼ f1; 2g or f3; 4g: RA;q ðtÞ ¼ e
2qkt þ e
2kt
e
2ðqþ1Þkt

ð11Þ

1 1 1 þ
: 2k 2qk 2ð1 þ qÞk

ð12Þ

and mttf A;q ¼

For A such that jAj ¼ 3: RA;q ðtÞ ¼ e
2qkt þ e
ðqþ1Þkt
e
ð3qþ1Þkt

ð13Þ

and mttf A;q ¼

1 1 1 þ
: 2qk ð1 þ qÞk ð3 þ qÞk

For A such that jAj ¼ 4:   RA;q ðtÞ ¼ e
2qkt 2
e
2qkt

ð14Þ

ð15Þ

and mttf A;q ¼

3 : 4qk

ð16Þ

3.2. Hot duplication method In the hot duplication method, it is assumed that each component of the set B is improved by assuming a hot duplication of another identical one. Let RH i ðtÞ denote the reliability function of the component i if it is improved according to
kt
kt hot duplication method. That is, RH Þe . Let RH i ðtÞ ¼ ð2
e B ðtÞ be the reliability function of the design obtained by improving the components belonging to the set B according to hot duplication method. In what follows we give RH B ðtÞ for all possible sets B. For B such that jBj ¼ 1, that is B ¼ fig, i ¼ 1, 2, 3, 4:  
2kt RH 1 þ ½2
e
kt ½1
e
2kt  : B ðtÞ ¼ e For B ¼ f1; 3g or f1; 4g or f2; 3g or f2; 4g:

ð17Þ

A.M. Sarhan et al. / Appl. Math. Comput. 154 (2004) 257–277

RH B ðtÞ ¼ e


2kt

2
e


kt

2
ð2
e
kt Þe


2kt

:

For B ¼ f1; 2g or f3; 4g: n o
2kt
kt 2
2kt RH ðtÞ ¼ e 1 þ ð2
e Þ ð1
e Þ : B

263

ð18Þ

ð19Þ

For B such that jBj ¼ 3:  
2kt RH ð2
e
kt Þ 1 þ ð2
e
kt Þ½1
ð2
e
kt Þe
2kt  : B ðtÞ ¼ e

ð20Þ

For B such that jBj ¼ 4:

n o
2kt
kt 2
kt 2
2kt RH ðtÞ ¼ e ð2
e Þ 2
ð2
e Þ e : B

ð21Þ

The mean time to failures of the improved systems designed that obtained by improving the set B components according to hot duplication method, say H mttf H B , can be deduced by using the above formulae of RB ðtÞ. Table 1 gives H mttf B for all possible sets B  f1; 2; 3; 4g. 3.3. Cold duplication method In cold duplication method, we assume that each component of the set B is connected with an identical component via a perfect switch. Let RCi ðtÞ denote the reliability function of the component i when it is improved according to cold duplication method. That is, RCi ðtÞ ¼ ð1 þ ktÞe
kt , see Billinton and Allan [1]. Let RCB ðtÞ be the reliability function of the design obtained by improving the components belong to the set B according to cold duplication method. In what follows we give RCB ðtÞ for all possible sets B. For B such that jBj ¼ 1, that is B ¼ fig, i ¼ 1, 2, 3, 4:   RCB ðtÞ ¼ e
2kt 1 þ ð1 þ ktÞ½1
e
2kt  :

ð22Þ

For B ¼ f1; 3g or f1; 4g or f2; 3g or f2; 4g:   RCB ðtÞ ¼ ð1 þ ktÞe
2kt 2
ð1 þ ktÞe
2kt :

ð23Þ

For B ¼ f1; 2g or f3; 4g: n o RCB ðtÞ ¼ e
2kt 1 þ ð1 þ ktÞ2 ½1
e
2kt  :

ð24Þ

For B such that jBj ¼ 3:   RCB ðtÞ ¼ ð1 þ ktÞe
2kt 1 þ ð1 þ ktÞ½1
ð1 þ ktÞe
2kt  :

ð25Þ

For B such that jBj ¼ 4:

n o 2 2 RCB ðtÞ ¼ ð1 þ ktÞ e
2kt 2
ð1 þ ktÞ :

ð26Þ

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The mean time to failure of the improved systems obtained by improving the set B components according to cold duplication method, say mttf CB , can be deduced by using the above formulae of RCB ðtÞ. For all possible sets B  f1; 2; 3; 4g mttf CB are obtained and given in Table 1. Theorem 1. For a given subset B  f1; 2; 3; 4g of the system’s components, the following is satisfied mttf CB > mttf H B:

ð27Þ

Proof. The proof of this theorem can be reached easily by comparing the results booked in Table 1. h Theorem 1 summarizes that a design of the system obtained by improving a set B of its components according to cold redundancy is better than that design obtained by improving the same set B according to hot redundancy, in the sense of having a higher mean time to failure. Theorem 2. For a series–parallel system with independent and identical components, the following statements are fulfilled: mttf DB:jBj¼4 > mttf DB:jBj¼3 > mttf Df1;2g > mttf Df1;3g > mttf DB:jBj¼1 ;

D ¼ H; C: ð28Þ

Proof. Based on the results shown in Table 1, one can reach the proof.

h

From Theorem 2, one can say that 1. Duplication of all system components, that is B : jBj ¼ 4, yields a design of an improved system with the highest mean time to failure. 2. For B1  B2 , a design obtained by improving the set B2 components has a higher mean time to failure than the design obtained by improving the set B1 components. Table 1 C mttf H B and mttf B B

k  mttf H B

k  mttf CB

jBj ¼ 1 f1; 3g f1; 2g jBj ¼ 3 jBj ¼ 4

0.867 0.967 1.050 1.126 1.251

0.938 1.094 1.344 1.445 1.695

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265

3. For B : jBj ¼ 2, duplicating two components with a positive joint structure importance (such as components 1, 2) gives a design with a higher mean time to failure than that of a design obtained by duplicating two components with a negative joint structure importance (such as components 1, 3). Fig. 2 shows the reliability functions of the original system and of the design obtained using hot and cold duplication methods against time t for all possible sets B  f1; 2; 3; 4g. Fig. 2(a) gives the case when jBj ¼ 1. Fig. 2(b) gives the

Fig. 2

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A.M. Sarhan et al. / Appl. Math. Comput. 154 (2004) 257–277

case when B ¼ f1; 3g. Fig. 2(c) gives the case when B ¼ f1; 2g. Fig. 2(d) gives the case when jBj ¼ 3. Fig. 2(e) gives the case when jBj ¼ 4.

4. Reliability equivalent factors The reliability equivalence factor of a system, say REF, is generally defined as that factor q, 0 < q < 1, by which the failure rates of some of the systemÕs components should be reduced so that one could obtain equality of some characteristic of the original system with that of a better system [7]. He introduced two different REFs: (i) survival reliability equivalence factor, say SREF; and (ii) mean reliability equivalence factor, say MREF. In what follows, we derive SREF and MREF of the underlying system. Definition 4. The SREF (say qDA;B ðaÞ, D ¼ HðCÞ for hot (cold)) is defined as that factor q by which the failure rate of the set A components should be reduced so that one could obtain a design of the system components with a reliability function equals the reliability function of a design obtained from the original system by assuming hot (cold) duplications of the set B components. Definition 4 implies that the reliability function is used as a characteristic of comparison between the two improved designs of the system obtained by using reduction method and hot (cold) duplication method. Therefore the SREF q ¼ qDA;B ðaÞ can be obtained by solving the following system of equations: RA;q ðtÞ ¼ a;

RDB ðtÞ ¼ a;

D ¼ H; C:

ð29Þ

For notation, we call q ¼ qDA;B ðaÞ as hot SREF, shortly HSREF, when D ¼ H and cold SREF, shortly CSREF, when D ¼ C. We can obtain different SREF by choosing different sets A and B. In the following we present different SREF of the system studied here. If A ¼ fig, i ¼ 1, 2, 3, 4, then Eq. (7) and the first equation in (29) give e
ðqþ1Þkt
e
ðqþ3Þkt þ e
2kt ¼ a:

ð30Þ

Let x ¼ e
kt , then (30) becomes xqþ1
xqþ3 þ x2 ¼ a: Using Eq. (31), one can get   1 a
x2 ln qDA;B ðaÞ ¼ q ¼ ; ln x xð1
x2 Þ

ð31Þ

0 0:5. 5.2. three components according to hot duplication method at level a ¼ 0:9. 5.3. three components according to cold duplication method at a level a > 0:3. 5.4. four components according to hot duplication method at a level a > 0:6. 5.3. four components according to cold duplication method at a level a > 0:1. 6. Reducing the failure rates of two components with positive joint structural importance gives a design for which the reliability function is larger than that of the design obtained by improving any set of components according to hot or cold duplication method, see Tables 3 and 4. 7. One can read the reset of the results shown in Tables 2–8. Table 9 gives all possible mean reliability equivalence factors of the system. Based on the data booked in Tables 1 and 9, one can conclude that: 1. Hot duplication of one component increases the system mean time to failure from 0:75=k to 0:867=k, Table 1. The same increase in mttf can be obtained by one of the following, Table 9: 1.1. reducing the failure rate of one component by the factor nH A;B ¼ 0:540, 1.2. reducing the failure rates of components 1, 3 by the factor nH A;f1;3g ¼ 0:730, 1.3. reducing the failure rates of components 1, 2 by the factor nH A;f1;2g ¼ 0:770, Table 9 The mean reliability equivalence factors nDA;B A

B jBj ¼ 1

jAj ¼ 1 f1; 3g f1; 2g jAj ¼ 3 jAj ¼ 4

f1; 3g

f1; 2g

jBj ¼ 3

jBj ¼ 4

H

C

H

C

H

C

H

C

H

0.540 0.730 0.770 0.824 0.865

0.360 0.599 0.680 0.742 0.800

0.298 0.551 0.649 0.713 0.776

0.090 0.371 0.545 0.605 0.686

0.153 0.429 0.577 0.639 0.741

)0.16 0.116 0.418 0.463 0.558

0.048 0.332 0.524 0.583 0.666

)0.24 )0.09 )0.37 0.038 0.199 )0.12 0.383 0.457 0.318 0.423 0.508 0.347 0.519 0.600 0.443

C

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A.M. Sarhan et al. / Appl. Math. Comput. 154 (2004) 257–277

1.4. reducing the failure rates of any three components by the factor nH A;B ¼ 0:824, 1.5. reducing the failure rates of all systemÕs components by the factor nH A;B ¼ 0:865. 2. Cold duplication of one component increases the system mean time to failure from 0:75=k to 0:938=k, Table 1. The same increase in mttf can be obtained by one of the following, Table 7: 2.1. reducing the failure rate of one component by the factor nCA;B ¼ 0:360, 2.2. reducing the failure rates of components 1, 3 by the factor nCA;f1;3g ¼ 0:599, 2.3. reducing the failure rates of components 1, 2 by the factor nCA;f1;2g ¼ 0:680, 2.4. reducing the failure rates of any three components by the factor nCA;B ¼ 0:742, 2.5. reducing the failure rates of all systemÕs components by the factor nCA;B ¼ 0:800. 3. Hot duplications of components 1,2 increase the system mean time to failure from 0:75=k to 1:050=k, Table 1. The same phenomena can be done by one of the following, Table 2: 3.1. reducing the failure rate of one component by the factor nH A;B ¼ 0:153, 3.2. reducing the failure rates of components 1, 3 by the factor nH A;f1;3g ¼ 0:429, 3.3. reducing the failure rates of components 1, 2 by the factor nH A;f1;2g ¼ 0:577, 3.4. reducing the failure rates of any three components by the factor nH A;B ¼ 0:639, 3.5. reducing the failure rates of all systemÕs components by the factor nH A;B ¼ 0:741. 4. Cold duplications of components increase the system mean time to failure from 0:75=k to 1:344=k, Table 1. Since nCfig;f1;2g ¼
0:160, then it is not possible to reduce the failure rate of one component in order to get an improved design with mttf fig;q ¼ 1:344=k. On the other hand, the design with mttf CB ¼ 1:344=k can be obtained by doing one of the following: 4.1. reducing the failure rates of components 1, 3 by the factor nCA;f1;3g ¼ 0:116, 4.2. reducing the failure rates of components 1, 2 by the factor nCA;f1;2g ¼ 0:418, 4.3. reducing the failure rates of any three components by the factor nCA;B ¼ 0:463, 4.4. reducing the failure rates of all systemÕs components by the factor nCA;B ¼ 0:558. 5. In the same manner one can interpret the rest of the data shown in Table 9.

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