CC BY
10
250.0 0.0000 0.0014
255.0 0.0000 0.0010
260.0 0.0000 0.0007
265.0 0.0000 0.0005
270.0 0.0000 0.0004
275.0 0.0000 0.0003
280.0 0.0000 0.0002
285.0 0.0000 0.0001
R3,20(t) 1
0.9 -0.8 0.7 -0.6 -0.5 0.4 0.3 0.2 H 0.1 0
0
50
100
150
200
Figure 1. The graph of the pump stations system reliability function
Using the values given in the Table 1, the formulae (4)-(9) and numerical integration we find:
- the mean value of the pump stations system lifetime
E[T3,20] = J RX20(t)dt @ 50.8639,
two-state consecutive "k out of n: F" system has been shown as well. The formulae and algorithm for two-state reliability function of a homogeneous two-state consecutive " k out of n: F" system have been applied to reliability evaluation of the pump stations system. The considered pump stations system was a two-state consecutive "3 out of 20: F" system composed of components with exponential reliability functions. On the basis of the recurrent formula and the algorithm for two-state pump stations system reliability function the approximate values have been calculated and presented in table and illustrated graphically. On the basis of these values the mean value and standard deviation of the pump stations system lifetime have been estimated. The input structural and reliability data of the considered pump stations system have been assumed arbitrarily and therefore the obtained its reliability characteristics evaluations should be only treated as an illustration of the possibilities of the proposed methods and solutions.
The proposed methods and solutions and the software are general and they may be applied to any two-state consecutive "k out of n: F" systems.
Appendix
We present the D programming language code for formulas (3), (8)-(9) and Algorithm 1.
import std.stdio; import std.stream; import std.math; import std.string;
const real LAMBDA1 = 0.01;
- the second order ordinary moment of the pump stations system lifetime
real Ft(real t) {
return (1-exp(-(LAMBDA1)*t));
}
£[T22q(1)] = 2JtR3,20(t)dt @ 3246.69,
- the standard deviation of the pump stations system lifetime
s = VD[T3 20 ] = V659.558 @25.6819.
5. Conclusion
Two recurrent formulae for two-state system reliability functions, a general one for non-homogeneous and its simplified form for homogeneous two-state consecutive " k out of n : F" systems have been proposed. The algorithm for reliability evaluation of
real Rt(real t) { return exp(-(LAMBDA1)*t);
}
real SigmaFi(real ii, real k, real t, real n) { real result = 0; for(real i = ii; i < k; i++) {
result += pow(Ft(t),i)*Rkn(t,k,n-i-1);
}
return result;
}
real Rkn(real t, real k, real n) { if (n < k) return 1;
t
if (n == k)
return 1 - pow(Ft(t),n); return Rt(t)*(Rkn(t, k, n-1) + SigmaFi(1, k, t, n));
}
real trapeziumT(real k, real n, uint p, real t){ real integ = 0; real step = 0;
step=t/p;
for(real i = 0; i < p; i = i + step){ integ += (((Rkn(i, k, n) + Rkn(i + step, k, n))*step)/2);
}
return integ;
}
real trapezium2T(real k, real n, uint p, real t){ real integ = 0; real step = 0;
step=t/p;
for(real i=0; i < p; i = i + step){ integ += (((i*Rkn(i, k, n) + (i + step)*Rkn(i + step, k, n))*step));
}
return integ;
}
int main(char[] [] args) { real integral = 0; real integral1 = 0; real dif = 0; real sq = 0;
if (args.length < 3) {
writefln("Usage:\n args[0] t k n\n"); return 0;
}
for(real i = 0; i < atoi(args[1]); i = i + 5){ writefln("%s\t%4s\t%4s\t%s", i, Rkn(i,
atoi(args[2]), atoi(args[3])), 2*i*Rkn(i, atoi(args[2]),atoi(args[3])), 1 - Rkn(i, atoi(args[2]), atoi(args[3])) );
}
integral=trapeziumT(atoi(args[2]), atoi(args[3]), atoi(args[4]) );
integral1=trapezium2T(atoi(args[2]), atoi(args[3]), atoi(args[4]) );
diff=(integral1)-pow(integral,2); sq=sqrt(diff);
writefln("The mean value of the system lifetime"); writefln("%s", integral );
writefln("The second order ordinary moment of the
system lifetime"); writefln("%s", integral1 ); writefln("%s", diff);
writefln("The standard deviation of the system
lifetime"); writefln("%s",sq);
return 0;
}
References
[1] Antonopoulou, J. M. & Papstavridis, S. (1987). Fast recursive algorithm to evaluate the reliability of a circular consecutive-k-out-of-n: F system. IEEE Transactions on Reliability, R-36, 1, 83 - 84.
[2] Barlow, R. E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York.
[3] Guze, S. (2007). Wyznaczanie niezawodnosci dwustanowych systemow progowych typu „kolejnych k z n: F". Materialy XXXV Szkoly Niezawodnosci, Szczyrk.
[4] Kossow, A. & Preuss, W. (1995). Reliability of linear consecutively connected systems with multistate components. IEEE Transactions on Reliability, 44, 3, 518-522.
[5] Malinowski, J. & Preuss, W. (1995). A recursive algorithm evaluating the exact reliability of a consecutive k-out-of-n: F system. Microelectronics and Reliability, 35, 12, 1461-1465.
[6] Malinowski, J. (2005). Algorytmy wyznaczania niezawodnosci systemow sieciowych o wybranych typach struktur. Wydawnictwo WIT, ISBN 8388311-80-8, Warszawa.
[7] Shanthikumar, J. G. (1987). Reliability of systems with consecutive minimal cut-sets. IEEE Transactions on Reliability, 26, 5, 546-550.
Guze Sambor Kolowrocki Krzysztof
Maritime University, Gdynia, Poland
Reliability analysis of multi-state ageing consecutive „k out of n: F" systems
Keywords
multi-state system, ageing system, consecutive " k out of n : F" system, reliability Abstract
A multi-state approach to reliability analysis of systems composed of ageing components is introduced and basic reliability characteristics for such systems are defined. Further, a multi-state consecutive " k out of n : F" system composed of ageing components is defined and the recurrent formulae for its reliability function are proposed. Moreover, the application of the proposed reliability characteristics and formulae to reliability evaluation of the steel cover composed of ageing sheets is illustrated.
1. Introduction
Taking into account the importance of the safety and operating process effectiveness of technical systems it seems reasonable to expand the two-state approach to multi-state approach in their reliability analysis. The assumption that the systems are composed of multistate components with reliability states degrading in time [4]-[5], [10] gives the possibility for more precise analysis and diagnosis of their reliability and operational processes' effectiveness. This assumption allows us to distinguish a system reliability critical state to exceed which is either dangerous for the environment or does not assure the necessary level of its operational process effectiveness. Then, an important system safety characteristic is the time to the moment of exceeding the system reliability critical state and its distribution, which is called the system risk function. This distribution is strictly related to the system multi-state reliability function that is a basic characteristic of the multi-state system. The main results determining the multi-state reliability functions and the risk functions of typical series, parallel, seriesparallel, parallel-series, series-"k out of n" and "k out of n"- series systems with ageing components are given in [4]-[5]. The paper is devoted to transmitting these results on the multi-state ageing consecutive " k out of n : F" systems [1], [2]-[3], [6], [7]-[8], [9].
2. Multi-state system with ageing components
In the multi-state reliability analysis to define systems with degrading components we assume that [4]-[5], [10]:
- , i = 1,2,...,n, are components of a system,
- all components and a system under consideration have the reliability state set {0,1,...,z}, z > 1,
- the state indexes are ordered, the state 0 is the worst and the state z is the best,
- Ti (u), i = 1,2,...,n, are independent random variables representing the lifetimes of components Ei in the state subset {u,u+1,...,z}, while they were in the state z at the moment t = 0,
- Ti (u), is a random variable representing the lifetime of a system in the state subset {u,u+1,...,z} while it was in the state z at the moment t = 0,
- the system state degrades with time t without repair,
- ei (t) is a component Ei state at the moment t, t > 0,
- s(t) is a system state at the moment t, t > 0.
The above assumptions mean that the reliability states of the system with degrading components may be changed in time only from better to worse. The way in which the components and the system reliability states change is illustrated in Figure 1.
transitions
Em
worst state best state
Figure 1. Illustration of reliability states changing in system with ageing components
The basis of our further consideration is a system component reliability function defined as follows.
Definition 1. A vector
R(t ) = [Ri(t,0),Ri(t,1),...,Ri(t,z)], t > 0,
where
R(t,u) = P(e(t) > u | e,-(0) = z) = P(T(u) > t)
for t > 0, u = 0,1,...,z, i = 1,2,...,n, is the probability that the component Ei is in the reliability state subset {u, u +1,..., z} at the moment t, t > 0, while it was in the reliability state z at the moment t = 0, is called the multi-state reliability function of a component Ei.
Similarly, we can define a multi-state system reliability function.
Definition 2. A vector
Rn (t,•) = [1Rn (t,0)Rn (t,1),...Rn (t,z)], t > 0,
where
R n (t,u) = P(s(t) > u | s(0) = z) = P(T(u) > t),
for t > 0, u = 0,1,...,z, is the probability that the system is in the reliability state subset {u,u +1,..., z} at the moment t, t > 0, while it was in the reliability state z at the moment t = 0, is called the multi-state reliability function of a system.
Under this definition we have
Rn(t,0) > Rn(t,1) > . . . > Rn(t,z), t > 0,
and if
p(t) = [p(t,0),p(t,1),...,p(t,z)], t > 0, where
p(t,u) = P(s(t) = u | s(0) = z),
for t > 0, u = 0,1,...,z, is the probability that the system is in the state u at the moment t, t > 0, while it was in the state z at the moment t = 0, then
Rn(t,0) = 1, Rn(t,z) = p(t,z), t > 0, (1)
and
p(t,u) = Rn(t,u) - Rn(t,u+1), u = 0,1,...,z-1, t > 0. (2) Moreover, if
Rn(t,u) =1 for t < 0, u = 1,2,..., z, then
¥
M(u) = E[T(u)]= J Rn(t,u)dt, u = 1,2,...,z, (3)
0
is the mean lifetime of the system in the state subset {u, u + 1,..., z},
s (u) = JD[T(u)] =JN(u) - [M(u)]2, (4)
u = 1,2,...,z,
where
¥
N(u) = 2JtRn(t,u)dt, u = 1,2,...,z, (5)
0
is the standard deviation of the system lifetime in the state subset {u, u +1,..., z} and moreover
_ ¥
M (u) = Jp(t,u)dt, u = 1,2,...,z, (6)
0
is the mean lifetime of the system in the state u while the integrals (3), (4) and (5) are convergent. Additionally, according to (1), (2), (3) and (6), we get the following relationships
M(u) = M(u) -M(u+1), u = 1,2,...,z-1, (7)
M (z) = M(z).
Close to the multi-state system reliability function its basic characteristic is the system risk function defined as follows.
Definition 3. A probability
r(t) = P(s(t) < r | s(0) = z) = P(T(r) £ t), t > 0,
that the system is in the subset of states worse than the critical state r, r e{1,...,z} while it was in the reliability state z at the moment t = 0 is called a risk function of the multi-state system.
Considering Definition 3 and Definition 2, we have r(t) = 1 - Rn (t, r), t > 0, (8)
and if t is the moment when the system risk function exceeds a permitted level 5, then
t = r^(5),
(9)
where r -1(t) , if it exists, is the inverse function of the risk function r(t).
3. Reliability of a multi-state ageing consecutive „k out of n: F" system
Definition 4. A multi-state system is called an ageing consecutive " k out of n: F" system if it is out of the reliability state subset {u,u+1,...,z} if and only if at least its k neighbouring components out of n its components arranged in a sequence of E1, E2, ..., En, are out of this reliability state subset.
In our further analysis, we denote by sk n (t) the reliability state of the ageing consecutive " k out of n: F" system at the moment t, t e< 0, »), and by Tk n (u)
the lifetime of this system in the reliability subset {u,u+1,...,z}. Moreover, we denote by
Rk,n (t,u) = P(Skn (t) > u | s(0) = z) = P(Tk,n (u) > t)
for t > 0, u = 0,1,...,z, the probability that the ageing consecutive "k out of n: F" system is in the reliability state subset {u,u +1,..., z} at the moment t, t > 0, while it was in the reliability state z at the
moment t = 0 and by
Fkn (t, u) = 1 - Rk n (t,u) = P(Tk,n (u) £ t)
for t > 0, u = 0,1,...,z, the distribution function of
the lifetime Tk,n (u) of this system in the reliability
state subset {u,u+1,...,z} while it was in the state z at the moment t = 0.
Theorem 1. The reliability function of the ageing consecutive "k out of n: F" system composed of
components with independent failures is given by the following recurrent formula
Rk,n(t,•)= [1,Rk,n(t,1), Rk,n(t,2), ...,Rk,n(t,z)],
where
1 for n < k,
1 -n F} (t, u) for n = k,
Rn (t, u)Rk,n-1 (t, u) k-1
+ Z Rn-, (t, u)Rk,n-i-1 (t, u) i=1
Rk,n (t, u) =
(10)
n Fj (t, u)
for n > k,
j=n-i+1
for t e< 0,¥ >, u = 1,2,...,z.
Motivation. Since for each fixed u, u = 1,2,..., z, the assumptions of this theorem as the same as the assumptions of Theorem 2 proved in [2] and the formula (10) is equivalent with the formula (12) from [2], then after considering Definition 4, we conclude that this theorem is valid.
From the above theorem, as a particular case for the system composed of components with identical reliability, we immediately get the following corollary.
Corollary 1. If components of the ageing consecutive " k out of n : F" system are independent and have identical reliability functions, i.e.
Rt (t,u) = R(t,u), Ft (t,u) = F(t,u) for t e< 0,«), u = 1,2,...,z, i = 1,2,...,n,
then the reliability function of this system is given by
Rk,n(t,•)= [1,Rk,n(t,1), Rk,n(t,2), ...,Rk,n(t,z)], where
Rk ,n (t, u) =
1
1 - [ F (t, u)]n
R(t, u) Rk ,n-1 (t, u) k-1
+ R(t, u) z F' (t, u)
i=1
• Rk,n-'-l(t, u)
for n < k, for n = k,
(11)
for n > k,
for t e< 0,¥), u = 1,2,...,z.
From Corollary 1, in a particular case, substituting k = 2 in (11), we get:
- for n = 1
R^(t, •)= [1, R2,1 (t,1), R2,1(t,2), ..., R2,1 (t, z)], (12) where
R21 (t,u) = 1 for t e< 0,¥), u = 1,2,...,z, (13)
- for n = 2
R„(t, •)= [1, R2,2(t,1), R2,2(t ,2), ..., R2a(t, z)], (14) where
R2 2 (t,u) = 1 - F2(t,u) for t e< 0,¥), (15)
u = 1,2,..., z,
- for n > 3
R2,n (t, •)= [1, R2,n (t,1), R2,n (t ,2), ..., R2,n (t, z)], (16) where
RXn(t,u) = R(t,u) RXn-X(t,u)
+ R(t,u)F(t,u) R2 n-2 (t,u) for t e< 0,«), (17) u = 1,2,... , z.
4. Application
Example 1. Let us consider the steel cover composed of n = 24 arranged identical sheets Ej,E2,...,E24. We assume that z = 4, i.e. the cover and the sheets it is composed of may be in the one of the reliability states from the set {0,1,2,3,4}. The cover is out of the reliability state subset {u,u +1,...,4} if at least k = 2 of its neighbouring sheets is out of this reliability state subset. If the considered steel cover critical reliability state is r = 2 , then this steel cover is failed if at least 2 neighbouring sheets from 24 sheets are out of the reliability state subset {2,3,4}. Thus, the considered steel cover is a five-state ageing consecutive "2 out of 24: F" system, and according to (16)-(17), its the reliability function is given by
R2,24 (t, •) =
[1, R2,24(t,1), R2,24(t,2), R2,24(t,3), R^M)], (18) where
R2 24 (t, u) = R(t,u) R2 23 (t,u)
+ R(t,u)F(t,u) R2,22 (t,u) for t e< 0,«), (19)
u = 1,2,3,4.
In the particular case when the lifetimes Ti (u) ,
u = 1,2,3,4, of the sheets Et, i = 1,2,3,4,5, in the
reliability state subsets have Weibull distributions of the form
F(t, u) = 1 - e(u)t2 for t > 0, u = 1,2,3,4, where
l(1) = 0.01, l(2) = 0.02, l(3) = 0.05, l(4) = 0.10,
i.e. if the reliability function of the sheets Ei , i = 1,2,3,4,5, is given by
R(t, •) = [1,R(t,1), R(t,2), R(t,3), R(t,4)], t e<0,¥),
where
R(t,1) = e-001i2, R(t,2) = e-0 02i2, R(t,3) = e-0 05i2,
R(t,4) = e-0-1012 for t > 0,
considering (12)-(19), we get the following recurrent formula for the cover reliability
R2,24 ^ •) =
[1, R2,24 (t,1), R2,24 (t,2), R2,24 (t,3), R^24 (t,4) ], (20)
where
- R2 24 (t,1) is determined by the formulae R21 (t,1) = 1 for t e< 0,¥), (21)
R2 2 (t,1) = 1 - [1 - e-°-0U ]2 for t e< 0,¥), (22)
2
R2,n (t,1) = e-001t R2,n-1(t,1)
2 2
+ e~0'01t [1 - e~001i ] R2,n-2(t,1) for t e< 0,»), (23) n = 3,4,...,24,
- R2 24 (t,2) is determined by the formulae
R21 (t,2) = 1 for t e< 0,¥), (24)
R2 2 (t,2) = 1 - [1 - e-0'02t2 ]2 for t e< 0,»), (25)
R2, (t,2) = e -0 02t2 RXn- (t,2)
22
+ e~0 02t [1 - e-omt ] R2n-2(t, 2) for t e< 0,»), (26) n = 3,4,...,24,
- R2 24 (t,3) is determined by the formulae
R21 (t,3) = 1 for t e< 0,¥), (27)
R22 (t,3) = 1 - [1 - e-0-05t ]2 for t e< 0,»), (28)
2
R2,n(t,3) = e-0 05t R2,n-!(t,3) 22
+ e-omt [1 -e-°'05t ] R2 n-2(t,3) for t e< 0,»), (29) n = 3,4,...,24,
- R2 24 (t,4) is determined by the formulae
R21 (t,4) = 1 for t e< 0,¥), (30)
R2,2 (t,4) = 1 - [1 - e_0'10t ]2 for t e< 0,»), (31)
2
R2, (t, 4) = e -010t R2n- (t,4)
22
+ e"0'10t [1 - e-°'10t ] R2 n-2(t,4) for t e< 0,»), (32) n = 3,4,...,24.
The values of the particular vector components of the multi-state reliability function of the steel cover given by (20), calculated by the computer programme based on the formulae (21)-(32), are presented in the Tables 1-4 and illustrated in Figure 1. As earlier we have assumed that r = 2 is the cover critical reliability state, then according to (8) and (26) its risk function is given by
2
r(t) = 1 - R2 24 (t, 2) = 1 - e -omt R2 23 (t, 2)
22
- e-omt [1 - e-a02t ] R2 22 (t, 2) for t e< 0,»). (33)
The values of the steel cover risk function are given in Table 5 and illustrated in Figure 2.
Table 1. The values of the steel cover multi-state reliability function vector component u = 1
t R2,24 (t,1) 2t R2 24(t,1)
0.0 1.0000 0.0000
1.0 0.9978 1.9955
2.0 0.9664 3.8657
3.0 0.8531 5.1183
4.0 0.6362 5.0889
5.0 0.3750 3.7499
6.0 0.1664 1.9957
7.0 0.0538 0.7534
8.0 0.0125 0.2001
9.0 0.0021 0.0374
10.0 0.0002 0.0049
Table 2. The values of the steel cover multi-state reliability function vector component u = 2
t R2,24 (t,2) 2t R2 24(t,2)
0.0 1.0000 0.0000
0.5 0.9994 0.9994
1.0 0.9912 1.9824
1.5 0.9580 2.8742
2.0 0.8802 3.5207
2.5 0.7479 3.7398
3.0 0.5731 3.4388
3.5 0.3876 2.7131
4.0 0.2275 1.8200
4.5 0.1145 1.0307
5.0 0.0491 0.4905
5.5 0.0178 0.1958
6.0 0.0055 0.0655
6.5 0.0014 0.0184
7.0 0.0003 0.0044
Table 3 The values of the steel cover multi-state reliability function vector component u = 3
t R2,24 (t,3) 2t R2 24 (t,3)
0.0 1,0000 0,0000
0.2 0.9999 0.3999
0.4 0.9986 0.7988
0.6 0.9928 1.1914
0.8 0.9781 1.5649
1.0 0.9489 1.8978
1.2 0.9005 2.1613
1.4 0.8302 2.3246
1.6 0.7385 2.3632
1.8 0.6299 2.2675
2.0 0.5122 2.0489
2.2 0.3953 1.7392
2.4 0.2883 1.3837
2.6 0.1980 1.0298
2.8 0.1278 0.7158
3.0 0.0774 0.4642
3.2 0.0438 0.2806
3.4 0.0233 0.1581
3.6 0.0115 0.0830
3.8 0.0053 0.0406
4.0 0.0023 0.0185
Table 4. The values of the steel cover multi-state reliability function vector component u = 4
R2,24(t,u) 1
0.8 0.6 0.4 0.2 0
10
t
t R2,24 (t,4) 2t R2 24(t,4)
0.0 1.0000 0.0000
0.1 0.9999 0.0399
0.2 0.9996 0.1599
0.3 0.9982 0.3593
0.4 0.9943 0.6364
0.5 0.9864 0.9864
0.6 0.9725 1.4004
0.7 0.9508 1.8636
0.8 0.9195 2.3540
0.9 0.8775 2.8433
1.0 0.8244 3.2975
1.1 0.7605 1.6731
1.2 0.6875 1.6499
1.3 0.6076 1.5799
1.4 0.5242 1.4677
1.5 0.4406 1.3217
1.6 0.3602 1.1528
1.7 0.2862 0.9731
1.8 0.2207 0.7944
1.9 0.1650 0.6269
2.0 0.1195 0.4779
2.1 0.0838 0.3519
2.2 0.0569 0.2502
2.3 0.0373 0.1718
2.4 0.0237 0.1138
2.5 0.0146 0.0728
2.6 0.0086 0.0450
2.7 0.0050 0.0268
2.8 0.0028 0.0154
2.9 0.0015 0.0086
3.0 0.0008 0.0046
Figure 1. The graphs of the steel cover multi-state reliability function vector components
Table 5. The values of the steel cover multi-state reliability function vector component u = 2 and its risk function
t R2,24(t,2) r(t) = 1- R2^(t ,2)
0.0 1.0000 0.0000
0.5 0.9994 0.0006
1.0 0.9912 0.0088
1.5 0.9581 0.0419
2.0 0.8802 0.1198
2.5 0.7480 0.2520
3.0 0.5731 0.4269
3.5 0.3876 0.6124
4.0 0.2275 0.7725
4.5 0.1145 0.8855
5.0 0.0490 0.9510
5.5 0.0178 0.9822
6.0 0.0055 0.9945
6.5 0.0014 0.9986
7.0 0.0003 0.9997
