Научная статья на тему 'Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems'

Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems Текст научной статьи по специальности «Математика»

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two-state system / consecutive “ k out of n : F” system / reliability / algorithm

Аннотация научной статьи по математике, автор научной работы — Guze Sambor

An approach to reliability analysis of two-state systems is introduced and basic reliability characteristics for such systems are defined. Further, a two-state consecutive “ k out of n : F” system composed of two-state components is defined and the recurrent formulae for its reliability function are proposed. The algorithm for numerical approach to reliability evaluation is given. Moreover, the application of the proposed reliability characteristics and formulae to reliability evaluation of the system of pump stations composed of two-state components is illustrated.

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Текст научной работы на тему «Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems»

-¥< g(a)< aç í x - ar ö Fè J <a< 1 0

aç £ g (a ) < b œ x - ^ö œ x - arö FV oM o^ x - oF-1 (a)- aç 2(bç- aç)

£ g (a)< cç œ x - crö œ x - ö FV „T^H o?ö x -oF-1 (a) + cç- 2bç 2 ( cç- b)

g (a ) ^ cç 0 <a<F^x cçJ 1

Table 3. Integration range with respect to a

degenerates to a crisp number, and is maximum when the fuzzy variable is an equi-possible one, i.e., all values have the same possibility. In order to address such a requirement, Li and Liu [6] provided a new definition based on credibility measure.

Definition 6.1.(Fuzzy Entropy) Let x be a continuous fuzzy variable defined on a credibility space (©,2®, Cr ), then the fuzzy entropy, H[x], is defined by

where Z = g (a) = x - cO-1 (a) .

Then we obtain the average chance measure for the

event {X (w ,9) < x}

F

%

s

( x v / x -oF 1(a) - a

ch{X(w,0) £ x}= J 1 V.

œ x-^ö 2bz - a )

o

F

| x-bç ! f| x-cç

è s J x-oF-1(a) + cç - 2b è s

+ J -y ' -Z-da + J 1x (47)

of

2(cç - bJ

which leads to the average chance distribution

Y(x ) =

x - a.

ç

œ œ x - a ö

2(bç - aJ

F

V V ° 0

-F

œ x - bz öö

v s 00

x + cç - 2bç œ œ x - bz ö

2(cç - bJ

F

V v s 0

-F

œx -cz öö

+ F

V s 0

o

2(b - aç)

V s 00

x-aç o

Juf(u)du

x

o

2(cç - bJ

Juf(u)du

z x-cz

o

(48)

6. Fuzzy repair effect estimation under fuzzy maximum entropy principle

Entropy is a measure of uncertainty. The entropy of De Luca and Termini [1] characterizes uncertainty resulting primarily from the linguistic vagueness rather than resulting from information deficiency, and vanishes when the fuzzy variable takes all the values with membership degree 1. However, we hope that the degree of uncertainty is 0 when the fuzzy variable

H [X ] = J S (Cr ({0 : X (0 ) = u}))du

where

S (t) = -t ln t - (1 -1)ln(1 -1)

(49)

(50)

For convenience, we name S (t) as entropy density at point t.

The maximum entropy principle provides a route such that it is possible to select the parameter(s) l that maximizes the value of entropy function and satisfies certain given constraints for specifying a membership function with a given form. However, what we aim at is not obtaining parameters from the theoretical entropy function rather we must determine the parameters based on observations of the fuzzy variable, say, x . In other words, we need to develop a criterion to obtain data-assimilated membership function. Therefore, we suggest an empirical fuzzy entropy function for parameter searching since the optimal value of the data-dependent object function has to reflect the constraints specified by observational data implicitly. The data assimilated object function is the average of entropy densities evaluated at {z1,z2,L ,zn}respectively, i.e.,

J [- A, L2 ] =1 S S (Cr {Z (0 ) = z, ; (S )}) (51)

n i=1

where a finite interval [-Lj, L2], L2 > L1 > 0 is defined for the domain of the entropy. Note that with the finiteness of empirical entropy, J[-L1, L2 ] ® H[Z; l ] asymptotically with parameter constrained by the data structure and Z e [-L1, L2], L2 > L1 > 0 which guarantees the theoretical entropy H [Z] exists and finite in general. Then, we can estimate the parameter (az, , cz) of

the membership of fuzzy composite error in terms of maximum entropy principle. Furthermore, we can isolate a few repair as bad-as-old regime and thus

x-

o

+

o

repair effect is zero for estimation parameter o for specifying e i, the translation error because under triangular membership assumption, the empirical membership can be defined and satisfies the asymptotical requirements.

7. Conclusion

In this paper, we argue that a differential equation motivated regression model will result in a regression model with random fuzzy error terms and thus complete our mission for solidifying a rigorous mathematical foundation for the grey modelling on system repair effects proposed by Guo [3], [4]. The maximum entropy principle facilitates a way for fuzzy parameter estimation. However, the average change distribution is also providing a way for parameter data-assimilation.

Acknowledgements

This work was supported partially by South Africa Natural Research Foundation Grant FA2006042800024. We are grateful to Professor B.D. Liu for his constant encouragements of applying his fuzzy credibility measure theory. The error frequency in Figure 1 is provided by my MSc student, Mr Li Xiang.

References

[1] De Luca, A. & Termini, S. (1972). A Definition of Non-probabilistic Entropy in the Setting of Fuzzy Sets Theory. Information and Control 20, 301-312.

[2] Deng, J. L. (1985). Grey Systems (Social L Economical). The Publishing House of Defence Industry, Beijing (in Chinese).

[3] Guo, R. (2005a). Repairable System Modelling Via Grey Differential Equations. Journal of Grey System, 8 (1), 69-91.

[4] Guo, R. (2005b). A Repairable System Modelling by Combining Grey System Theory with Interval-Valued Fuzzy Set Theory. International Journal of Reliability, Quality and Safety Engineering, 12 (3), 241-266.

[5] Liu, B. (2004). Uncertainty Theory: An Introduction to Its Axiomatic Foundations. Springer-Verlag Heidelberg, Berlin.

[6] Li, P. & Liu, B. Entropy of Credibility Distributions for Fuzzy Variables. IEEEE Transactions on Fuzzy Systems,( to be published).

Guze Sambor

Maritime University, Gdynia, Poland

Numerical approach to reliability evaluation of two-state consecutive "k out of n: F" systems

Keywords

two-state system, consecutive " k out of « : F" system, reliability, algorithm Abstract

An approach to reliability analysis of two-state systems is introduced and basic reliability characteristics for such systems are defined. Further, a two-state consecutive " k out of n: F" system composed of two-state components is defined and the recurrent formulae for its reliability function are proposed. The algorithm for numerical approach to reliability evaluation is given. Moreover, the application of the proposed reliability characteristics and formulae to reliability evaluation of the system of pump stations composed of two-state components is illustrated.

1. Introduction

The assumption that the systems are composed of two-state components gives the possibility for basic analysis and diagnosis of their reliability. This assumption allows us to distinguish two states of system reliability. The system works when its reliability state is equal to 1 and is failed when its reliability state is equal to 0. In the stationary case the system reliability is the independent of time probability that the system is in the reliability state 1. The main results determining the stationary reliability and the algorithms for numerical approach to this reliability evaluation for consecutive "k out of n: F" systems are given for instance in [1], [5]-[6], An exemplary technical consecutive "k out of n: F" system can be found in [3]. There is considered the ordered sequence of n relay stations /',,, /',,, • ,En,, which have to reroute a signal from a source station E(l to a target station En+j. A range of each station is equal to k. It means, when the station En i = 0,1,...,«, is operating, it sends a signal directly to a station EM,...,Emm(j+t . The failed station does not send any

signal. The probability of efficiency of the stations E(l and En+1 is equal to 1. The signal from E(l to En+1

cannot be sent, if at least k consecutive stations out of /',,, /',,,• , En, are damaged.

The paper is devoted to extension of these stationary results to the non-stationary case and applying them in transmitting then for two-state consecutive "" k out of n: F" systems with dependent of time reliability functions of system components ([3]). Then, the reliability function, the lifetime mean value and the lifetime standard deviation are basic characteristics of the system.

2. Reliability of two-state consecutive "k out of n: F" systems

In the non-stationary case of two-state reliability analysis of consecutive "k out of n: F" systems we assume that ([3]):

- «is the number of system components,

- E-, i = 1,2are components of a system,

- Tj are independent random variables representing the lifetimes of components En i = 1.2.......

- Ri(t) = P(Ti >t),t g<0,cc\ is a reliability function of component E1, / = 1,2

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- ^(0 = 1-R, (0 = P{T, < t),t g< 0,oo), is the

distribution function (unreliability function) of component Ei, i = 1,2,...,n.

Definition 1. A two-state system is called a two-state consecutive " k out of n: F" system if it is failed if and only if at least its k neighbouring components out of n its components arranged in a sequence of E1, E2, ..., En, are failed.

The following auxiliary theorem is proved in [3], [6].

Lemma 1. The stationary reliability of the two-state consecutive " k out of n: F" system composed of components with independent failures is given by the following recurrent formula

Rkn = <

1

n

1 -Ek

j=i

Pn Rk ,n-1

k-1

+ ^ pn-i Rk, n—i-1 ¿=1

for n < k, for n = k,

(1)

• n

j=n-i+1

for n > k,

where

- pt is a stationary reliability coefficient of component Ei, i = 1,2,...,n,

- qi is a stationary unreliability coefficient of component Ei, i = 1,2,...,n,

- Rkn is the stationary reliability of consecutive "k out of n: F" system.

After assumption that:

- Tkn is a random variable representing the lifetime of a consecutive "k out of n: F" system,

- Rkn(t) = P(Tkn > t), t î< 0, ¥), is the reliability function of consecutive "k out of n: F" system,

- Fk n (t) = 1 — Rkn (t) = P(Tkn £ t), t î< 0, ¥), is the distribution function of consecutive "k out of n: F" system,

we can formulate the following result.

Lemma 2. The reliability function of the two-state

consecutive " k out of n : F" system composed of

components with independent failures is given by the following recurrent formula

Rk.n (t ) =

1

1—Ef (t )

j=1

Rn (t)Rk,n—1(t)

k—1

for n < k, for n = k,

(2)

+ ^ Rn_t (t ) R i =1

• nn Fj (t)

k,n—i—1

(t)

for n > k,

j=n—i+1

for t e< 0, ¥).

Motivation. When we assume in formula (1) that

pt (t) = R (t), q (t) = F (t) for t e< 0,¥), i = 1,2,..., n,

we get formula (2).

From the above theorem, as a particular case for the system composed of components with identical reliability functions, we immediately get the following corollary.

Corollary 1. If components of the two-state consecutive "k out of n : F" system are independent and have identical reliability functions, i.e.

R (t) = R(t), Ft (t) = F(t) for t e< 0,¥), i = 1,2,..., n,

then the reliability function of this system is given by

Rk.n (t ) =

1

1 — [ F (t)]n

R(t ) Rk ,n—1 (t )

k—1

+ R(t ) Z F1 (t )

i=1

• R

for n < k, for n = k,

(3)

k, n—i

-1(t )

for n > k,

for t e< 0, œ).

In further considerations we will used the following reliability characteristics:

- the mean value of the system lifetime,

¥

E[TKn ] = J Rk,n(t)dt, (4)

- the second order ordinary moment of the system lifetime,

E[Tk2n ] = 2J t Rk, n(t)dt,

(5)

- the standard deviation of the system lifetime,

s=V D[Tkn ],

where

D[T,n)] = E[Tk2n ] - (E[Tkn ])2.

(6)

(7)

3. Algorithm for reliability evaluation of a two-state consecutive „k out of n: F" system

For numerical approach to evaluation of the reliability characteristics, given by (3)-(6), we use the trapezium rule of numerical integration. In particular situation, for 10 = 0, step h, we have

E[Tkn ] = J Rk,n(t)dt

h E Rk,n (t 0 + ih) + Rkn (t 0 + (i +1) • h)l (8)

2 i=0

E[T2n ] = 2J t Rk, n(t)dt

n-1 t

= h£{(t 0 + ih) • Rk ,n (t 0 + ih)

i=0

+ (t0 + (i +1) • h) • Rk n (t0 + (i +1) • h)}.

(9)

2. If k > n then Rk,n (t) = 1

3. else if k = n Rk,n (t) = 1 -[F(t)]n

4. else

5. for i = 0 to t do

6. {

7. for j = 1 to k - 1 do

8. temp := temp + [F (i)]1 • Rk ,n-1 -1 (i);

9. Rk ,n (i) = R(i) • Rk ,n-:(i) + temp;

10. }

where

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Necessary in (7)-(8) values of function Rknn(t) are calculated from (2) using the following algorithm.

- k is a length of the sequence of consecutive components,

- n is a number of all components in sequence,

- t is an end of the time interval,

- F(t) is a distribution function of components,

- R(t) is a reliability function of components.

Example implementation of Algorithm 1 and formulas (3), (8)-(9) in the D programming language is given in Appendix.

4. Application

From Corollary 1, in a particular case, substituting k = 3 in (3), we get:

- for n = 1

R31 (t) = 1 for t e< 0,¥), (10)

- for n = 2

R3 2(t) = 1, for t e< 0,¥), (11)

- for n = 3

R3 3 (t) = 1 - F3(t) for t e< 0,¥), (12)

- for n > 4

RXn (t) = R(t) RXn-1 (t) + R(t)F(t) RXn-2 (t) + R(t)[F(t)]2 R3 n-3 (t) for t e< 0, ¥), (13)

U.i

Algorithm 1.

1. Given: t, k, n, F(t), R(t);

Example 1. Let us consider the pump stations system with n = 20 pump stationsEj,E2,...,E20. We assume that this system fails when at least 3 consecutive pump stations are down. Thus, the

considered pump stations system is a two-state consecutive "3 out of 20: F" system, and according to (9)-(12), its the reliability function is given by

R320(t)=R(t) Rug(t)

+ R(t)F(t)Rus(t)

+ R(t)[F(t)]2RU7(t) (14)

for t e< 0, oo).

In the particular case when the lifetimes '/,', of the pump stations /v, 7 = 1,2,* ,20 have exponential distributions of the form

F(t) = l-e-°mt for f > 0,

i.e. if the reliability functions of the pump stations /v, / = 1,2,* ,20 are given by

R(t) = e-°mt for t > 0,

considering (9)-(12), (13) we get the following recurrent formula for the reliability Rr ,n (!) of pump stations system

Ru(t) = 1 for fe<0,oo), (15)

R32(t) = \ for fe<0,oo), (16)

R33(t) = l- [l-e-0'01']3 for fe<0,oo), (17)

R3 Jt)=e-omt R3 n^)

+ e""u[]-e""u]Ryn2U)

+ e-001' [1 -e-001']2 R3^3(t) for t e< 0,oo), (18) 77 = 4,5,...,20.

The values of reliability function of the system of pump stations given by (14), calculated by the computer programme based on the formulae (10)-(18) and Algorithm 1, are presented in Table 1 and illustrated in Figure 1.

Table 1. The values of the two-state reliability function of the pump stations system for X = 0.01

t R 2 (>) 21 RX20(t)

0.0 1.0000 0.0000

5.0 0.9980 9.9800

10.0 0.9859 19.7189

15.0 0.9583 28.7499

20.0 0.9137 36.5474

25.0 0.8535 42.6743

30.0 0.7811 46.8657

35.0 0.7008 49.0561

40.0 0.6170 49.3614

45.0 0.5337 48.0347

50.0 0.4541 45.4117

55.0 0.3805 41.8584

60.0 0.3144 37.7282

65.0 0.2564 33.3331

70.0 0.2066 28.9274

75.0 0.1647 24.7024

80.0 0.1299 20.7893

85.0 0.1016 17.2662

90.0 0.0787 14.1688

95.0 0.0605 11.5004

100.0 0.0462 9.2416

105.0 0.0350 7.3588

110.0 0.0264 5.8107

115.0 0.0198 4.5531

120.0 0.0148 3.5426

125.0 0.0109 2.7385

130.0 0.0081 2.1044

135.0 0.0060 1.6082

140.0 0.0044 1.2229

145.0 0.0032 0.9255

150.0 0.0023 0.6974

155.0 0.0017 0.5235

160.0 0.0012 0.3916

165.0 0.0009 0.2918

170.0 0.0006 0.2168

175.0 0.0004 0.1607

180.0 0.0003 0.1188

185.0 0.0002 0.0876

190.0 0.0002 0.0644

195.0 0.0001 0.0473

200.0 0.0000 0.0347

205.0 0.0000 0.0253

210.0 0.0000 0.0185

215.0 0.0000 0.0135

220.0 0.0000 0.0098

225.0 0.0000 0.0072

230.0 0.0000 0.0052

235.0 0.0000 0.0038

240.0 0.0000 0.0027

245.0 0.0000 0.0020

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