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9
MODELLING RELIABILITY OF COMPLEX SYSTEMS
Krzysztof Kolowrocki, Joanna Soszynska-Budny.
Gdynia Maritime University, Gdynia, Poland katmatkk@am. gdynia.pl, joannas@am.gdynia.pl
ABSTRACT
Modelling and prediction of the operation and reliability of technical systems related to their operation processes are presented. The emphasis is on multistate systems composed of ageing components and changing their reliability structures and their components reliability parameters during their operation processes that are called the complex systems. The integrated general model of complex systems' reliability, linking their reliability models and their operation processes models and considering variable at different operation states their reliability structures and their components reliability parameters is constructed. This theoretical tool is applied to modelling and prediction of the operation processes and reliability characteristics of the multistate non-homogeneous system composed of a series-parallel and a series-"m out of l" subsystems linked in series, changing its reliability structure and its components reliability parameters at variable operation conditions.
1 INTRODUCTION
Most real technical systems are very complex and it is difficult to analyze their reliability. Large numbers of components and subsystems and their operating complexity cause that the identification, evaluation and prediction of their reliability are complicated. The complexity of the systems' operation processes and their influence on changing in time the systems' structures and their components' reliability parameters are very often met in real practice. Thus, the practical importance of an approach linking the system reliability models and the system operation processes models into an integrated general model in reliability assessment of real technical systems is evident.
The convenient tools for analyzing these problems are semi-Markov modelling the systems' operation processes (Ferreir, Pacheco, 2007; Glynn, Hass, 2006; Habibullah et al. 2009; Kolowrocki, Soszynska, 2009; Mercier, 2008; Soszynska et al. 2010; Grabski, 2002; Kolowrocki, Soszynska-Budny, 2011; Limnios, Oprisan, 2001; Kolowrocki 2008) multistate approach to the systems' reliability evaluation (Kolowrocki, Soszynska, 2009; Xue, 1985; Xue, Yang 1995b; Kolowrocki, 2008). The common usage of the multistate systems' reliability models and the semi-Markov model for the systems' operation processes in order to construct the joint general system reliability model related to its operation process (Kolowrocki, 2006; Kolowrocki, 2007a; Kolowrocki 2007b; Kolowrocki, Soszynska, 2006; Kolowrocki, Soszynska, 2010, Soszynska, 2007a; Soszynska, 2007b; Kolowrocki, Soszynska-Budny, 2011; Soszynska 2007c; Kolowrocki et all 2008) and to apply it to the reliability analysis of complex technical systems is this paper main idea.
2 COMPLEX SYSTEM OPERATION PROCESS MODELLING
We assume that the system during its operation process is taking v,vg N, different operation states Zj, z 2,..., zv. Further, we define the system operation process Z (t), t e< 0,+«), with discrete operation states from the set (zj, z 2,..., zv}. Moreover, we assume that the system operation
process Z(t) is a semi-Markov process (Kolowrocki, Soszynska, 2009; Kolowrocki, Soszynska, 2010; Grabski, 2002; Soszynska, 2007b) with the conditional sojourn times 6bl at the operation
states zb when its next operation state is zl, b, l = 1,2,...,v, b ^ l. Under these assumptions, the system operation process may be described by:
- the vector of the initial probabilities pb (0) = P(Z(0) = zb), b = 1,2,...,v, of the system operation process Z(t) staying at particular operation states at the moment t = 0
[pb (0)Lv = [Pi(0), P 2 (0),..., pv (0)];
(1)
- the matrix of probabilities pbl, b, l = 1,2,...,v, b ^ l, of the system operation process Z(t) transitions between the operation states zb and zl
[pbl ]vxv
Pli Pl2 - Piv
P 21 P 22 . . . P 2v
Pvl Pv2 - Pvv
(2)
where by a formal agreement Pbb = 0 for b = 1,2,..., v;
- the matrix of conditional distribution functions Hbl (t) = P(6bl < t), b, l = 1,2,...,v, b ^ l, of the system operation process Z(t) conditional sojourn times 6bl at the operation states
[ Hbl (t )U =
H11 (t ) H i2 (t )... H iv (t )■
H 21 (t) H 22 (t)... H 2v (t)
Hvi (t) Hv2 (t)... Hvv (t)
(3)
where by formal agreement Hbb (t) = 0 for b = 1,2,..., v.
We introduce the matrix of the conditional density functions of the system operation process Z(t) conditional sojourn times 6bl at the operation states corresponding to the conditional distribution functions Hbl (t)
h (t )U =
hii(t ) hi2(t )... hiv (t )" h21 (t) h 22
(t)... h
2v
(t)
hvi(t ) hvl(t )... hvv (t )
(4)
where
hbl (t) = d[Hbl (t)] for b, l = 1,2,..., V, b * l, dt
and by formal agreement hbb (t) = 0 for b = 1,2,..., v.
As the mean values E[6bl ] of the conditional sojourn times dbl are given by
X X
mbl = E[6bl ] = J tdHu (t) = J thbl (t)dt, b, l = 1,2,..., v, b * l, (5)
0 0
then from the formula for total probability, it follows that the unconditional distribution functions of the sojourn times 6b, b = 1,2,...,v, of the system operation process Z(t) at the operation states zb, b = 1,2,..., v, are given by (Grabski, 2002; Kolowrocki, Soszynska-Budny, 2011; Soszynska, 2007b; Limnios, Oprisan, 2001)
Hb (t) = £pwHw (t), b = 1,2,..., v. (6)
l=1
Hence, the mean values E[db ] of the system operation process Z (t) unconditional sojourn times db, b = 1,2,..., v, at the operation states are given by
mb = E[6b ] = £pblmbl, b = 1,2,..., v, (7)
l=1
where mbl are defined by the formula (5).
The limit values of the system operation process Z(t) transient probabilities at the particular operation states
pb (t) = P(Z(t) = zb) , t e< 0,+x), b = 1,2,..., v,
are given by (Grabski, 2002; Kolowrocki, Soszynska-Budny, 2011; Soszynska, 2007b; Limnios, Oprisan, 2001)
nhmh
pb = lim Pb (t)= , b = 1,2,..., V, (8)
l=1
where mb, b = 1,2,...,v, are given by (7), while the steady probabilities nb of the vector [nb ]1xv satisfy the system of equations
K] = K][Pbi]
v (9)
£ = 1.
l=1
In the case of a periodic system operation process, the limit transient probabilities pb, b = 1,2,...,v, at the operation states defined by (8), are the long term proportions of the system operation process Z (t) sojourn times at the particular operation states zb, b = 1,2,..., v.
Other interesting characteristics of the system operation process Z (t) possible to obtain are its total sojourn times &b at the particular operation states zb, b = 1,2,...,v, during the fixed system opetation time. It is well known (Grabski, 2002; Kolowrocki, Soszynska-Budny, 2011; Soszynska, 2007b; Limnios, Oprisan, 2001) that the system operation process total sojourn times &b at the particular operation states zb, for sufficiently large operation time 6, have approximately normal distributions with the expected value given by
€ = E[&b ] = Pb6, b = 1,2,...,v, (10)
where pb are given by (8).
Example
We consider a series system S composed of the subsystems S1 and S2, with the scheme showed in Figure 1.
Fig. 1. The scheme of the exemplary system S reliability structure
We assume that the subsystem S1 is a series-parallel system with the scheme given in Figure 2 and the subsystem S2 illustrated in Figure 3 is either a series-parallel system or a series-"2 out of 4" system.
Fig. 2. The scheme of the subsystem S1 reliability structure
The subsystems S1 and S 2 are forming a general series reliability structure of the system presented in Figure 1. However, this system reliability structure and its subsystems and components reliability depend on its changing in time operation states (Kolowrocki, Soszynska, 2009; Soszynska, 2007b). Under the assumption that the system operation conditions are changing in time, we arbitrarily fix the number of the system operation process states v = 4 and we distinguish the following as its operation states:
• an operation state z1 - the system is composed of the subsystem S1 with the scheme showed in Figure 2 that is a series-parallel system,
• an operation state z2 - the system is composed of the subsystem S2 with the scheme showed in Figure 3 that is a series-parallel system,
• an operation state z3 - the system is a series system with the scheme showed in Figure 1 composed of the subsystems S1 and S2 that are series-parallel systems with the schemes respectively given in Figure 2 and Figure 3,
• an operation state z4 - the system is a series system with the scheme showed in Figure 1 composed of the subsystem S1 and S 2, while the subsystem S1 is a series-parallel system with the scheme given in Figure 2 and the subsystem S2 is a series-"2 out of 4" system with the scheme given in Figure 3.
The influence of the above system operation states changing on the changes of the exemplary system reliability structure is indicated in these operation states above definitions and illustrated in Figures 1-3. Its influence on the system components reliability will be defined in this example continuation in Section 3.
We arbitrarily assume that the probabilities pbl of the exemplary system operation process transitions from operation state zb into the operation state zl are given in the matrix below
[ Pbl ] =
0 0.25 0.30 0.45
0.20 0 0.25 0.55
0.15 0.20 0 0.65
0.40 0.25 0.35 0
(11)
We also arbitrarily fix the conditional mean values mbl = E[6bl ], b, l = 1,2,3,4, of the exemplary system sojourn times at the particular operation states as follows:
m12 = 190, m13 = 480, m14 = 200, m21 = 100, m23 = 80, m24 = 60, m31 = 870, m32 = 480, m34 = 300,
m41 = 320, m42 = 510, m43 = 440. (12)
This way, the exemplary system operation process is defined and we may find its main characteristics. Namely, applying (7), (11) and (12), the unconditional mean sojourn times at the particular operation states are given by:
ml = E[0J = p12 m12 + p13 m13 + p14 m14 = 0.25-190 + 0.30 • 480 + 0.45 • 200 = 281.5, (13)
m2 = E[02] = p 21 m21 + p 23 m23 + p 24 m24 = 0.20 -100 + 0.25 • 80 + 0.55 • 60 = 73.00, (14)
m3 = E[d 3] = p31m31 + p32m32 + p34m34 = 0.15 • 870 + 0.20 • 480 + 0.65 • 300 = 421.5, (15)
m4 = E[64] = p 41 m41 + p 42 m42 + p 43 m43 = 0.40 • 320 + 0.25 • 510 + 0.35 • 440 = 409.5. (16) Further, according to (9), the system of equations
f[K1,K2,K3,K4] = [K1,K2,K3,K4][p« Lx4
7 + k2 + K3 + K4 = 1, after considering (11), takes the form
K = 0.20K 2 + 0.15K3 + 0.40K 4 K2 = 0.25Kx + 0.20K3 + 0.25K 4 K3 = 0.30Kx + 0.25K 2 + 0.35K 4 K4 = 0.45Kx + 0.55K 2 + 0.65K3
Kx + K2 + K3 + K 4 = 1.
The approximate solutions of the above system of equations are:
K = 0.216, k2 = 0.191, K3 = 0.237, K4 = 0.356. (17)
After considering the result (17) and (13)-(16), we have
£ Km = 0.216 • 281.5 + 0.191-73.0 + 0.237 • 421.5 + 0.356 • 409.5 = 320.4245,
i=1
and according to (8), the limit values of the exemplary system operation process transient probabilities pb (t) at the operation states zb are given by
0.216 • 281.5 0.191 • 73.0
p, =-= 0.190, p 2 =-= 0.043,
320.4245 320.4245
0.237 • 421.5 0.356 • 409.5
p 3 =-= 0.312, p 4 =-= 0.455. (18)
320.4245 320.4245
Hence, the expected values of the total sojourn times &b, b = 1,2,3,4, of the exemplary system operation process at the particular operation states zb, b = 1,2,3,4, during the fixed operation time d = 1 year = 365 days, after applying (9.10), amount:
Mx = E[ &x] = 0.190 4 = 0.190 year = 69.3 days, m2 = E[#2] = 0.043 4 = 0.043 year = 15.7 days, m3 = E[#3] = 0.312 4 = 0.312 year = 113.9 days,
m4 = E[#4] = 0.455 = 0.455 year = 166.1 days. (19)
3 COMPLEX SYSTEM RELIABILITY MODELLING
We assume that the changes of the operation states of the system operation process Z(t) have an influence on the system multistate components E, i = 1,2,...,n, reliability and the system reliability
structure as well. Consequently, we denote the system multistate component Et, i = 1,2,...,n, conditional lifetime in the reliability state subset {u, u +1,..., z} while the system is at the operation state zb, b = 1,2,...,v, by Ti(b)(u) and its conditional reliability function by the vector
R(t, -)](b)= [1, [Ri(t,1)](b),..., [R,.(t,z)](b)], (20)
with the coordinates defined by
[R, (t, u)](b) = P(T,(b)(u) > t|Z(t) = zb) (21)
for t g< 0, x), u = 1,2,..., z, b = 1,2,..., v.
The reliability function [R, (t, u)](b) is the conditional probability that the component Et lifetime Ti(b)(u) in the reliability state subset {u,u +1,...,z} is greater than t, while the system operation process Z(t) is at the operation state zb.
Similarly, we denote the system conditional lifetime in the reliability state subset {u,u +1,...,z} while the system is at the operation state zb, b = 1,2,..., v, by T(b)(u) and the conditional reliability function of the system by the vector
[R(t,-)](b) = [1, [R(t,1)](b),..., [R(t,z)](b)], (22)
with the coordinates defined by
[R(t, u)](b) = P(T(b) (u) > t|Z(t) = zb) (23)
for t g< 0, x), u = 1,2,...,z, b = 1,2,...,v.
The reliability function [ R(t, u)](b) is the conditional probability that the system lifetime T(b) (u) in the reliability state subset {u, u +1,..., z} is greater than t, while the system operation process Z(t) is at the operation state zb.
Further, we denote the system unconditional lifetime in the reliability state subset {u, u +1,..., z} by T (u) and the unconditional reliability function of the system by the vector
R(t,-) = [1, R(t ,1),..., R(t, z)], (24)
with the coordinates defined by
R(t,u) = P(T(u) > t) for t g< 0, x), u = 1,2,...,z.
In the case when the system operation time d is large enough, the coordinates of the unconditional reliability function of the system defined by (24) are given by
R(t,u) =£pb[R(t,u)](b) for t > 0, u = 1,2,...,z, (25)
6=1
where [R(t,u)](b), u = 1,2,...,z, b = 1,2,...,v, are the coordinates of the system conditional reliability functions defined by (23) andpb, b = 1,2,..., v, are the system operation process limit transient probabilities given by (9).
Thus, the mean value /u(u) = E[T(u)] of the system unconditional lifetime T(u) in the reliability state subset {u, u +1,..., z} is given by (Kolowrocki, Soszynska-Budny, 2011; Soszynska, 2007b),
M(u) = £ pbMb (u), u = 1,2,..., z, (26)
b=1 b b
where Mb (u) = E[T(b)(u)] are the mean values of the system conditional lifetimes T (b)(u) in the reliability state subset {u, u +1,..., z} at the operation state zb, b = 1,2,..., v, given by
w
Mb (u) = J[R(t, u)](b)dt, u = 1,2,..., z, (27)
0
[R(t,u)](b), u = 1,2,...,z, b = 1,2,...,v, are defined by (23) and pb are given by (9). Since the
relationships between the system unconditional lifetimes T (u) in the particular reliability states and the system unconditional lifetimes T (u) in the reliability state subsets can be expressed by
T (u) = T (u) - T (u +1), u = 0,1,..., z -1, T (z) = T (z), (28)
then we get the following formulae for the mean values of the unconditional lifetimes of the system in particular reliability states
M (u) = M (u) - M (u +1), u = 0,1,..., z -1, M (z) = M (z), (29)
where M(u), u = 0,1,..., z, are given by (27).
Moreover, if s(t) is the system reliability state at he moment t, t e< 0, w), and r, r e {1,2,..., z},is the system critical reliability state, then the system risk function
r(t) = P(s(t) < r | s(0) = z) = P(T(r) < t), t e< 0, w),
defined as the probability that the system is in the subset of states worse than the critical state r, r e{1,...,z} while it was in the state z at the moment t = 0 is given by (Kolowrocki, Soszynska-Budny, 2011)
r(t) = 1 - R(t, r), t e< 0, w), (30)
where R(t, r) is the coordinate of the system unconditional reliability function given by (25) for u = r and if t is the moment when the system risk function exceeds a permitted level 5, then
T= r-1 (5), (31)
where r -1 (t), if it exists, is the inverse function of the risk function r(t) given by (30).
Further, we assume that the system components Et, i = 1,2,...,n, at the system operation states zb,
b = 1,2,...,v, have the exponential reliability functions, i.e. their coordinates are given by
[R (t, u)](b) = P(T(b) (u) > t|Z(t) = Zb) = exp[-[Ai (u)f) t] (32)
for t g< 0, w), u = 1,2,...,z, b = 1,2,...,v.
The reason for this strong assumption on the system components is that the exponential distribution has "no memory" (Kolowrocki, Soszynska-Budny, 2011). Both of them, the assumption about the exponential reliability functions of the system components and this property, justify the following form of the formula (25) (Kolowrocki, Soszynska-Budny, 2011)
R(t, u) =Z pb [ R(t, u)]
(b)
= 1 Pb[ R(exp[-[^(u)](b) t],exp[-[A!(u )](b) t ],...,exp[-[A„ (u)](b) t])](b) (33)
b=1
for t > 0, u = 1,2,..., z.
The application of the above formula and the results given in Chapter 3 of (Kolowrocki, Soszynska-Budny, 2011) yield the following results formulated in the form of the following proposition.
Proposition 1
If components of the multi-state system at the operation state zb, b = 1,2,...,v, have the exponential reliability functions given by
[R(t,-)](b) = [1,[Rj(t,1)](b),...,[R(t,z)](b)], t g (-»,»), b = 1,2,...,v, where
[R.(t, u)](b) = exp[-[Ay.(u)](b)t] for t > 0, [Xy. (u)](b) > 0, i = 1,2,...,k, j = 1,2,...,/, u = 1,2,...,z, b = 1,2,...,v,
then its multistate unconditional reliability function is given by the vector:
i) for a series-parallel system with the structure shape parameters k(b), /i(b), i = 1,2,...,k(b-1, at the operation state zb, b = 1,2,..., v,
R(t,-)= [1, R(t,1),..., R(t, z)], (34)
where
v
R(t, u) =ZPbRkib).l(b)/(b) /(b) (t,u) u = 1,2,..., z, (35)
b=1
k (b)
k(b ) l(b)
R (b) / (b) (b) l(b) (t, u) = 1 -n [1 -n [Rj (t, u)](b)] k 'l1 ,l2 ,..,lk(b) i=1 j=1
= 1 -n[1 -exp[-E(u)](b)t]], t > 0, u = 1,2,...,z, b = 1,2,..., v; (36)
i=1 j=1 ^
ii) for a series-"w out of k" system with the structure shape parameters w(b), k(b), l^-1, i = 1,2,..., k(b), at the operation state zb, b = 1,2,...,v,
R(t,0 = [1, R(t,1),..., R(t, z)], (37)
where
v (b)
R(t,u) =EPbR"(b),(b) ,(b) (t,u), u = 1,2,...,z, (38)
b=1 k ;l1 ,l2 ,...,lk(b)
(b) 1 k(b) l(b) (b) l(b) ( 1 R^,(b),(b) ,(b) (t,u) = 1 - X n[ n[Rj (t,u)](b)t]]" [1 -nRj(t, u)](b)]]1-"
k ;l1 2 ,...,% (b) 1 "2,...,"k =0 i=1 j=1 j=1 k (b)
"1 + "2 +... + "k <wv ' -1
1 k(b) l(b)
= 1 - X n [ n exp[-[Aj (u)](b) t]f
1,"2,..., rk =0 i=1 j=1 "1 + "2 +...+"k <w(b) -1
f)
• [1 -nexp[-[(u)](b)t]]1-i, t > 0, u = 1,2,...,z, b = 1,2,...,v. (39)
j=1 j
• Example (continuation)
• In Section 2, it is fixed that the exemplary system reliability structure and its subsystems and components reliability depend on its changing in time operation states. Considering the assumptions and agreements of these sections, we assume that its subsystems Sv, v = 1,2, are composed of
four-state, i.e. z = 3, components Efv, v = 1,2, having the conditional reliability functions given
by the vector
[ Rv)(t,- )](b )= [1, [ Rv)(t, 1)](b),[ Rv)(t ,2)](b),[ R(v)(t,3)](b)], b = 1,2,3,4, with the exponential co-ordinates
[ Rv)(t ,1)](b) = exp[-[tf >(1)](b)],
[ Rv)(t,2)](b) = exp[-[tf )(2)](b)], [ Rv)(t, 3)](b) = exp[-[tf )(3)](b)],
different at various operation states zb, b = 1,2,3,4, and with the intensities of departure from the reliability state subsets {1,2,3}, {2,3}, {3}, respectively
[tf}(1)](b), [tfv (2)](b), [tf}(3)](b), b = 1,2,3,4.
The influence of the system operation states changing on the changes of the system reliability structure and its components reliability functions is as follows.
At the system operation state z1, the system is composed of the series-parallel subsystem S1 with the structure showed in Figure 2, containing two identical series subsystems (k(1) = 2), each composed of sixty components (l(1) = 60, 12(1) = 60 ) with the exponential reliability functions. In both series subsystems of the subsystem S1 there are respectively:
- the components Ej1-1, i = 1,2, j = 1,2,...,40, with the conditional reliability function coordinates
[R,(1)(t,1)](1)= exp[-0.0008t], [Rj)(t,2)](1)= exp[-0.0009t], [R,(1)(t,3)](1)= exp[-0.0010t], i = 1,2, j = 1,2,...,40;
- the components Ej1-1, i = 1,2, j = 41,42,...,60, with the conditional reliability function coordinates
[Rf(t,1)](1)= exp[-0.0011t], [Rj15(t,2)](1)= exp[-0.0012t], [R,(1)(t,3)](1)= exp[-0.0013t], i = 1,2, j = 41,42,...,60.
Thus, at the operational state z1, the system is identical with the subsystem S1 that is a four-state series-parallel system with its structure shape parameters , l1(1) = 60, 12(1) = 60 , and according to the formulae appearing after Definition 3.11 in (Kolowrocki, Soszynska-Budny, 2011) and (34)-(36), its conditional reliability function is given by
[R(t, -)](1) = [1, [R(t,1)](1), [R(t,2)](1), [R(t,3)](1) ], t > 0, (40)
where
[ R(t ,1)](1) = R2;60,60(t,1) = 1 -n[1 -fi[Rj1)(t,1)](1)]
i=1 j=1
2 60 = 1 -n[1 - exp[-I[ j1)](1) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0008 • 40 + 0.0011- 20]t ]]2 = 1 - [1 - exp[-0.054t ]]2
= 2 exp[-0.054t ] - exp[-0.108t ], (41)
[ R(t ,2)](1) = R 2;60,60(t,2) = 1 -ri[1 -fi[ Rj (t ,2)](1) ]
i=1 j=1 2 60 = 1 -n[1 - exp[-]^[A(j}(2)](1) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0009 • 40 + 0.0012 • 20]t ]]2 = 1 - [1 - exp[-0.060t ]]2
= 2 exp[-0.060t ] - exp[-0.120t ], (42)
[ R(t ,3)](1) = R2;60,60 (t,3) = 1 -n[1 -fi[Rj1)(t,3)](1)]
i=1 j=1
2 60 = 1 -n[1 - exp[-]C[^(^^}(3)](1) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0010 • 40 + 0.0013 • 20]t ]]2 = 1 - [1 - exp[-0.066t ]]2 = 2 exp[-0.066t ] - exp[-0.132t ].
(43)
The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets {1,2,3}, {2,3}, {3} at the operation state zx, calculated from the results given by (40)-(43), according to (27), respectively are:
M 1(1) = J [R(t,1)](1) dt = 2/0.054 - 1/0.108 s 27.78, (44)
0
M1 (2) = J [R(t,2)](1)dt = 2/0.060 - 1/0.120 = 25.00, (45)
0
M 1(3) = J [R(t,3)](1)dt = 2/0.066 - 1/0.132 s22.73.
(46) 0
At the system operation state z 2, the system is composed of the series-parallel subsystem S 2 with the structure showed in Figure 3, containing four identical series subsystems (k(2) = 4), each composed of eighty components (/1(2) = 80, 12(2) = 80, 13(2) = 80, 142) = 80 ) with the exponential reliability functions. In all series subsystems of the subsystem S 2 there are respectively:
- the components Ej2), i = 1,2,3,4, j = 1,2,...,40, with the conditional reliability function coordinates
[Rj2)(t,1)](2)= exp[-0.0014t], [Rj2)(t,2)](2)= exp[-0.0015t], [Rj2)(t,3)](2) = exp[-0.0016t], i = 1,2,3,4, j = 1,2,...,40;
- the components E<j2), i = 1,2,3,4, j = 21,22,...,40,with the conditional reliability function coordinates
[Rj2)(t ,1)](2)=exp[-0.0018t], [Rj2)(t ,2)](2)=exp[-0.0020t], [R^ (t,3)](2) =exp[-0.0022t], i = 1,2,3,4, j = 41,42,...,80.
Thus, at the operation state z2 , the system is identical with the subsystem S2 that is a four-state series-parallel system with its structure shape parameters k(2) = 4), l1(2) = 80, 12(2) = 80, 13(2) = 80, 142) = 80, and according to the formulae appearing after Definition 3.11 in (Kolowrocki, Soszynska-Budny, 2011) and (34)-(36), its conditional reliability function is given by
[R(t, • )](2) = [1, [R(t,1)](2), [R(t,2)](2), [R(t,3)](2)], t > 0, (47)
where
[R(t,1)](2) = *4;80, 80,80,80(t,1) = 1 - 5 [1 "5 [Rf (M)]®]
¿=1 j=1
4 80 = 1 "II[1 "exp[-£[Ajj (1)](2)t]]
¿=1 j=1
= 1 " [1 " exp["[0.0014 • 40 + 0.0018 • 40]t ]4 = 1 " [1 " exp["0.128t ]]4
= 4exp[-0.128t] - 6exp[-0.256t] + 4exp[-0.384t] -exp[-0.512t], (48)
[R(t,2)](2) = R4;80,80,80,80 (t,2) = 1 - II [1 -fi [^ (t,2)](2)]
¿=1 j=1
4 80
= 1 -II[1 - exp[-Z[42)(2)](2) t ]]
¿=1 j=1
= 1 - [1 - exp[-[0.0015 • 40 + 0.0020 • 40]t ]4 = 1 - [1 - exp[-0.140t ]]4
= 4 exp[-0.140t] - 6 exp[-0.280t] + 4 exp[-0.420t] -exp[-0.560t], (49)
[ R(t ,3)](2) = R4;80,80,80,80 (t,3) = 1 -IP -n[Rf(t,3)](2)]
i=1 j=1
4 80
= 1 -II[1 -exp[-Z^ (3)](2)t]]
¿=1 j=1
= 1 - [1 - exp[-[0.0016 • 40 + 0.0022 • 40]t ]4 = 1 - [1 - exp[-0.152t ]]4
= 4exp[-0.152t] - 6exp[-0.304t] + 4exp[-0.456t] -exp[-0.608t]. (50)
The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets {1,2,3}, {2,3}, {3} at the operation state zx, calculated from the results given by (47)-(50), according to (27), respectively are:
M2(1) = J [R(t,1)](2) dt = 4/0.128 - 6/0.256 + 4/0.384 - 1/0.512 s 16.27, (51)
0
M2 (2) = J [R(t,2)](2)dt = 4/0.140 - 6/0.280 + 4/0.420 - 1/0.560 s 14.88, (52)
0
M2(3) = J [R(t,3)](2)dt = 4/0.152 - 6/0.304 + 4/0.456 - 1/0.608 s 13.71. (53)
0
At the system operation state z3, the system is a series system with the structure showed in Figure 1, composed of two series-parallel subsystems S1 and S2 illustrated respectively in Figure 2 and Figure 3.
The subsystem S1 with the structure showed in Figure 2, consists of two identical series subsystems (k(3) = 2), each composed of sixty components (/1(3) = 60, 123) = 60 ) with the exponential reliability functions. In both series subsystems of the subsystem S1 there are respectively: - the components E.p, 1 = 1,2, j = 1,2,...,40, with the conditional reliability function co-ordinates
[ R,(1)(i,1)]<3) = exp[-0.0009t],[ R,(1)(t ,2)](3)= exp[-0.0010t], [R,(1)(t,3)](3) = exp[-0.0011t], i = 1,2, j = 1,2,...,40;
- the components Ej\ i = 1,2, j = 41,42,...,60, with the conditional reliability function coordinates
[R,(1)(t,1)](3)= exp[-0.0012t], [R^t,2)](3)= exp[-0.0014t], [R-j>(t,3)](3)= exp[-0.0016t], i = 1,2, j = 41,42,...,60.
Thus, at the operation state z3, the subsystem S1 is a four-state series-parallel system with its structure shape parameters k(3) = 2 , /1(3) = 60, 123) = 60 , and according to the formulae appearing after Definition 3.11 in [18] and (34)-(36), its conditional reliability function is given by
[Rm(t,-)](3) = [1, [Rm(t,1)](3), [Rm(t,2)](3), [R(1)(t,3)](3)], t > 0,
(54) where
[ R(1) (t, 1)](3) = R2;60,60 (t,1) = 1 -ri[1 -fi [ R0) (t ,1)](3) ]
i=1 j=1
2 60 = 1 -n[1 - exp[-I^œF t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0009 • 40 + 0.0012 • 20]t ]2 = 1 - [1 - exp[-0.060t ]]2
= 2 exp[-0.060t ] - exp[-0.120t ], (55)
[ R ® (t, 2)](3) = R 2;60,60(t,2) = 1 -fi [1 -fi [Rj) (t,2)](3) ]
i=1 j=1
2 630
= 1 -n[1 - exp[ -I[A( (2)](3) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0010 • 40 + 0.0014 • 20]t ]2 = 1 - [1 - exp[-0.068t ]]2
= 2 exp[-0.068t ] - exp[-0.136t ], (56)
[ R(t,3)](1) = R2;60,60 (t,3) = 1 -fi[1 -fi[Rj1)(t,3)](1)]
i=1 j=1
2 60 = 1 -n[1 - exp[ (3)] t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0011- 40 + 0.0016 ^20]t ]2 = 1 - [1 - exp[-0.076t ]]2
= 2exp[-0.076t ] - exp[-0.152t ]. (57)
The subsystem S2 with the structure showed in Figure 3, consists of four identical series subsystems (k(3) = 4), each composed of eighty components (l1(3) = 80, 123) = 80, 13(3) = 80,
14(3) = 80 ) with the exponential reliability functions given below. In all series subsystems of the subsystem S 2 there are respectively:
- the components E(2), i = 1,2,3,4, j = 1,2,...,40, with the conditional reliability function coordinates
[Rj2)(t,1)](3)=exp[-0.0010t], [R(2)(t,2)](3)= exp[-0.0011t], [R(2)(t,3)](3) = exp[-0.0012t], i = 1,2,3,4, j = 1,2,...,40;
- the components E{j), i = 1,2,3,4, j = 41,42,...,80, with the conditional reliability function coordinates
[Rj2)(t ,1)](3)=exp[-0.0014t], [R(2)(t ,2)](3)= exp[-0.0016t], [R(2)(t,3)](3) = exp[-0.0018t], i = 1,2,3,4, j = 41,42,...,80.
Thus, at the operation state z3, the subsystem S2 is a four-state series-parallel system with its structure shape parameters k(3) = 4 , l1(3) = 80, 123) = 80, 13(3) = 80, 14(3) = 80 , and according to the formulae appearing after Definition 3.11 in (Kolowrocki, Soszynska-Budny, 2011) and (34)-(36), its conditional reliability function is given by
[R(2)(t, -)](3) = [1, [R(2)(t,1)](3), [R(2)(t,2)](3), [R(2) (t,3)](3)], t > 0,
(58) where
[ R(2)(t ,1)](3) = R4;80,80,80,80(t,1) = 1 "Il[1 -fi[Rj2)(t,1)](3)]
i=1 j=1
4 80 m
= 1 -n[1 - exp[-Z[42)(1)](3) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.0010 • 40 + 0.0014 • 40]t ]4 = 1 - [1 - exp[-0.096t ]]4
= 4exp[-0.096t]-6exp[-0.192t] + 4exp[-0.288t] -exp[-0.384t],
(59)
[R(2)(t,2)](3) = R4;80,80,80,80 (t,2) = 1 -Il[1 -fi[Rj2)(t,2)](3)]
i=1 j=1
4 80
= 1 -n[1 - exp[-]T[A(/2)(2)](3) t ]]
i=1 j=1
= 1 - [1 - exp[-[0.001140 + 0.0016 • 40]t ]4 = 1 - [1 - exp[-0.108t ]]4
= 4exp[-0.108t] - 6exp[-0.216t] + 4exp[-0.324t] - exp[-0.432t], (60)
[R <2)(t ,3)](3) = R4;80,80,80,80 (t,3) = 1 -n[1 -fi[Ri2)(t ,3)](3)]
i=1 j=1
4 80
= 1 -II[1 -exp[-]^[A(j }(3)](3)t]]
i=1 j=1
= 1 - [1 - exp[-[0.0012 • 40 + 0.0018 • 40]t ]4 = 1 - [1 - exp[-0.120t ]]4
= 4exp[-0.120t] - 6exp[-0.240t] + 4exp[-0.360t] - exp[-0.480t]. (61)
Considering that the system at the operation state z3 is a four-state series system composed of
subsystems S1 and S2, after applying the formulae appearing after Definition 3.4 in (Kolowrocki, Soszynska-Budny, 2011) and (54)-(57) and (58)-(61), its conditional reliability function is given by
[R(t,• )](3) = [1, [R(t,1)](3), [R(t,2)](3), [R(t,3)](3) ], t > 0, (62)
where
[R(t,1)](3) = R2(t,1) = [R(1)(t,1)](3) [R(2)(t,1)](3)
= 8exp[-0.156t ] - 12exp[-0.252t ] + 8exp[-0.348t ]
- 2 exp[-0.424t] - 4 exp[-0.216t] + 6 exp[-0.312t]
- 4 exp[-0.408t ] + exp[-0.504t ],
(63)
[R(t, 2)](3) = R2(t,2) = [R(1)(t,2)](3) [R(2) (t,2)](3)
= 8exp[-0.176t ] - 12exp[-0.284t] + 8exp[-0.392t]
- 2 exp[-0.500t] - 4 exp[-0.236t] + 6 exp[-0.344t]
- 4 exp[-0.452t ] + exp[-0.560t ], (64)
[R(t,3)](3) = R2(t,3) = [R(1)(t,3)](3) [R(2)(t,3)](3)
= 8exp[-0.196t ] - 12exp[-0.316t] + 8exp[-0.436t]
- 2 exp[-0.556t] - 4 exp[-0.256t] + 6 exp[-0.376t]
- 4exp[-0.496t ] + exp[-0.616t ].
(65)
The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets {1,2,3}, {2,3}, {3} at the operation state z3, calculated from the results given by (62)-(65), according to (27), respectively are:
M3(1) = J [R(t,1)](3)dt = 8/0.156 - 12/0.252 + 8/0.348 - 2/0.424 - 4/0.216
0
+ 6/0.312 - 4/0.408 + 1/0.504 = 14.82, (66)
M3 (2) = J [R(t,2)](3)dt = 8/0.176 - 12/0.284 + 8/0.392 - 2/0.500 - 4/0.236
0
+ 6/0.344 - 4/0.452 + 1/0.560 = 13.04, (67)
M3(3) = J [R(t,3)](3)dt = 8/0.196 - 12/0.316 + 8/0.436 - 2/0.556 - 4/0.256
0
+ 6/0.376 - 4/0.496 + 1/0.616 = 11.48. (68)
At the system operation state z 4, the system is a series system with the scheme showed in Figure 1, composed of the subsystem S1 and S 2 illustrated respectively in Figure 2 and Figure 3, whereas the subsystem S1 is a series-parallel system and the subsystem S 2 is a series-"2 out of 4" system. The subsystem S1 consists of two identical series subsystems (k(4) = 2), each composed of sixty components (/1(4) = 60,124) = 60 ) with the exponential reliability functions the same as at the operation state zv Thus, according to (54)-(57), the subsystem St conditional reliability function at the operation state z4 , is given by
[R(1)(t, • )](4) = [1, [R(1)(t,1)](4), [R(1)(t,2)](4), [R(1)(t,3)](4)], t > 0, (69)
where
[ R(1) (t,1)](4) = 2exp[-0.054t ] - exp[-0.108t], (70)
[ R (1)(t ,2)](4) = 2exp[-0.060t ] - exp[-0.120t ],
(71)
[ R(t ,3)](4) = 2exp[-0.066t] - exp[-0.132t ]. (72)
The subsystem S 2 consists of four identical series subsystems ( k(4) = 4 ), each composed of eighty components ( l1(4) = 80, 12(4) = 80, 13(4) = 80, 144) = 80 ) with the exponential reliability functions the same as at the operation state z 2 and is a series-"2 out of 4" system ( m = 2 ). Thus, at the operation state z4, the subsystem S2 is a four-state series-"2 out of 4" system, with its structure shape parameters ( k(4) = 4), each composed of eighty components l1(4) = 80, 12(4) = 80, 13(4) = 80, 144) = 80 , and according to the formulae appearing after Definition 8.1 in (Kolowrocki, Soszynska-Budny, 2011) and (37)-(39), its conditional reliability function is given by
[R(2)(t,• )](4) = [1, [R(2)(t,1)](4), [R(2)(t,2)](4), [R(2)(t,3)](4)], t > 0,
(73) where
[ R(2)(t, 1)](4) = Rlmm(t ,1) = 1 - I fi[ fi[ Ri2)(t ,1)](4)]ri [1 - fi[ Ri2)(t ,1)](4)]1-ri
1,r2,r3,r4 =0 i =1 J =1 J =1
r\ + r2 + r3 + r4 <1
1 4 80 80
= 1 - I n exp[-r II[^ (1)]t][1 - exp[-I[42)(1)](4)t]]1-ri
n,r2,r3,r4 =0 i =1 j =1 J=1
r[ + r2 + r3 + r4 <1
= 1 - ^ fi exp[-r [0.0014^40 + 0.0018 ^40]t ]
1 ,r2 ,^3 , r4 =0 i=1 + r2 + r3 + r4 <1
1-ri
• [1 - exp[-[0.0014 • 40 +1.0018^ 40]t ]]
1 4 1
= 1 - I n exp[-r 0.128t ][1 - exp[-0.128t]]1
1 ,r2 ,^3 ,r4 =0 i=1 + r2 + r3 + r4 <1
1
= 1 - I
i = 0
r 4 ^
exp[ -i • 0.128t ][1 - exp[ -0.128t ]]4
= 1 - exp[ -0 • 0.1281] [1 - exp[-0.128t]4 - 4 exp[ -1 • 0.1281] [1 - exp[-0.128t]3 = 6exp[-0.256t ] - 8 exp[-0.384t ] + 3exp[-0.512t ], (74)
[R<2)(t, 2)](4) = R42;80,80,80,80 (t,2) = 1 - i f[fjt,2)](4)]n [1 - fi[R2)(t,2)](4)]1-1
1,r2,r3,r4 =0 i=1 J =1 J =1
r\ + r2 + 13 + r4 <1
1 4 80 80
= 1 - i ff exp[-r i[A^ (2)](4)t][1 - exp[-i[42)(2)](4)t]]1-1
1,r2,r3,r4 =0 i =1 J=1 J=1
r[ + r2 + r3 + r4 <1
= 1 - i ffexp[-r [0.0015 • 40 + 0.0020 • 40]t ]
1 ,r2 ,r3 , r4 =0 i=1 r +12 +13 +14 <1
• [1 - exp[-[0.0015 • 40 +1.0020 • 40]t ]]1-ri
14
= 1 - i n exp[-r 0.140t ][1 - exp[-0.140t]]1-1
1,12,13,14 =0 i=1 1 +12 +13 +14 <1
1
= 1 - I
i = 0
r 4 ^
v1 /
exp[ -i • 0.1401 ][1 - exp[ -0.1401 ]]4-1
= 1 - exp[ -0 • 0.1401] [1 - exp[-0.140t]4 - 4 exp[ -1 • 0.1401] [1 - exp[-0.140t]3 = 6 exp[-0.280t] - 8 exp[-0.420t] + 3 exp[-0.560t], (75)
[R <2)(t, 3)](4) = R42;80,80,80,80 (t ,3) = 1 - i ff[ n[Ri2)(t ,3)](4)r [1 - n[Ri2)(t,3)](4)]1-"
1,12,13,14 =0 i =1 j =1 j=1
1 +12 +13 +14 <1
1 4 80 80
= 1 - i ff exp[-1 i [42)(3)](4) t ][1 - exp[-i[42)(3)](4) t ]]1-*
1,12,13,14 =0 i =1 J =1 J =1
1 +12 +13 +14 <1
= 1 - i ff exp[-1 [0.0016 • 40 + 0.0022 • 40]t ]
1,12 ,13 ,14 =0 i=1 1 +12 +13 +14 <1
• [1 - exp[-[0.0016 • 40 +1.0022 • 40]t ]]1"i
14
= 1 - i f exp[-10.152t ][1 - exp[-0.152t ]]1-1
1,12,13,14 =0 i=1 1 +12 +13 +14 <1
1 r 4 ^
= 1 - i exp[ -i • 0.1521 ][1 - exp[ -0.1521 ]]4-1
i=0 vi
= 1 - exp[ -0 • 0.1521] [1 - exp[-0.152t]4 - 4 exp[ -1 • 0.1521] [1 - exp[-0.152t]3 = 6 exp[-0.304t ] - 8 exp[-0.456t ] + 3 exp - 0.608t ]. (76)
Considering that the system at the operation state z4 is a four-state series system composed of subsystems S1 and S2, after applying the formulae appearing after Definition 3.4 in (Kolowrocki, Soszynska-Budny, 2011) and (69)-(72) and (73)-(76), its conditional reliability function is given by
[R(t, -)](4) = [1, [R(t,1)](4), [R(t,2)](4), [R(t,3)](4)], t > 0, (77)
where
[R(t,1)](4) = R2(t,1) = [R(1)(t,1)](4) [R(2)(t,1)](4)
= 12exp[-0.310t]-6exp[-0.364t] - 16exp[-0.438t]
+ 8 exp[-0.492t] + 6 exp[-0.566t] - 3 exp[-0.620t], (78)
[R(t, 2)](4) = R2(t,2) = [R(1)(t,2)](4) [R(2) (t,2)](4)
= 12exp[-0.340t]-6exp[-0.400t] - 16exp[-0.480t]
+ 8 exp[-0.540t] + 6 exp[-0.620t] - 3 exp[-0.680t], (79)
[R(t,3)](4) = R2 (t,3) = [R(1)(t,3)](4) [R(2)(t,3)](4)
= 12exp[-0.370t]-6exp[-0.436t] - 16exp[-0.522t]
+ 8 exp[-0.588t] + 6 exp[-0.674t] - 3 exp[-0.740t]. (80)
The mean values of the system sojourn times T(u) in the reliability state subsets after applying the formula (77)-(80) and (27), are:
M4(1) = J [R(t,1)](4) dt = 12/0.310 - 6/0.364 - 16/0.438 + 8/0.492+ 6/0.566 - 3/0.620
0
= 7.72, (81)
M4 (2) = J [R(t,2)](4)dt = 12/0.340 - 6/0.400 - 16/0.480 + 8/0.540 + 6/0.620 - 3/0.680
0
= 7.04, (82)
M4 (3) = J [R(t,3)](4)dt = 12/0.370 - 6/0.436 - 16/0.522 + 8/0.588 + 6/0.674 - 3/0.740
0
= 6.47. (83)
In the case when the system operation time is large enough its unconditional four-state reliability function is given by the vector
R(t, • ) = [1, R(t, 1), R(t, 2), R(t, 3)], t > 0, (84)
where according to (25) and considering the exemplary system operation process transient probabilities at the operation states determined by (18), the vector co-ordinates are given respectively by
R(t,1) = pj R(t ,1)](1) + p 2[R(t,1)](2) + p 3[R(t,1)](3) + p4[ R(t ,1)](4)
= 0.190 •[ R(t ,1)](1) + 0.043 •[ R(t,1)](2) + 0.312 •[ R(t,1)](3) + 0.455 •[ R(t,1)](4) (85)
for t > 0,
R(t,2) = Pj [ R(t,2)](1) + p 2[ R(t,2)](2) + p3[ R(t ,2)](3) + p 4[R(t,2)](4)
= 0.190 •[ R(t,2)](1) + 0.043 •[ R(t ,2)](2) + 0.312 •[ R(t ,2)](3) + 0.455 •[ R(t,2)](4) (86)
for t > 0,
R(t,3) = pj R(t,3)](1) + p 2[ R(t,3)](2) + p3[ R(t ,3)](3) + p4[ R(t,3)](4)
= 0.190- [ R(t ,3)](1) + 0.043 • [ R(t,3)](2) + 0.312 • [ R(t ,3)](3) + 0.455 • [ R(t,3)](4) (87)
for t > 0,
where coordinates [R(t,1)](1), [R(t,1)](2),[R(t,1)](3), [R(t,1)](4)are given by (41), (48), (62), (76), [R(t,2)](1),[R(t,2)](2),[R(t,2)](3), [R(t,2)](4)are given by (42), (49), (63), (77) and [R(t,3)](1), [R(t,3)](2), [R(t,3)](3), [R(t,3)](4) are given by (43), (50), (64), (80). The graph of the four-state exemplary system reliability function is illustrated in Figure 4.
Fig. 4. The graph of the exemplary system reliability function R(t, •) coordinates
The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets {1,2,3}, {2,3}, {3}, calculated from the results given by (84)-(87), according to (27) and considering (18), (44)-(46), (51)-(53), (66)-(68) and (81)-(83), respectively are:
M (1) = p1 M1 (1) + p2 M2 (1) + P3M3 (1) + p4M4 (1)
= 0.190• 27.78 + 0.043-16.27 + 0.312-14.82 + 0.455• 7.72 s 14.11,
(88)
M (2) = P1M1 (2) + p2 M2(2) + P3M3 (2) + p4 M4(2)
= 0.190• 25.00 + 0.043•14.88 + 0.31243.04 + 0.455• 7.04 s 12.66, (89)
M (3) = p, M1 (3) + p2M2 (3) + p3M3 (3) + p4M4 (3)
= 0.190• 22.73 + 0.04343.71 + 0.312•11.48 + 0.455• 6.47 s 11.43. (90)
Farther, considering (29) and (88), (89) and (90), the mean values of the system unconditional lifetimes in the particular reliability states 1, 2, 3, respectively are:
M(1) = M(1) -M(2) = 1.45, M(2) = M(2) -M(3) = 1.23, M(3) = M(3) = 11.43. (91)
Since the critical reliability state is 1 = 2, then the system risk function, according to (30), is given by
r(t) = 1 - R(t,2)for t > 0, (92)
where R(t,2) is given by (86).
Hence, by (31), the moment when the system risk function exceeds a permitted level, for instance 5 = 0.05, is
t = r_1(5 s 2.255. (93)
The graph of the risk function r(t) of the exemplary four-state system operating at the variable conditions is given in Figure 5.
Fig. 5. The graph of the exemplary system risk function r(t)
4 CONCLUSION
The integrated general model of complex systems' reliability, linking their reliability models and their operation processes models and considering variable at different operation states their reliability structures and their components reliability parameters is constructed and applied to the reliability evaluation of the exemplary system composed of a series-parallel and a series-"™ out of l" subsystems linked in series. The predicted reliability characteristics of the exemplary system operating at the variable conditions are different from those determined for this system operating at constant conditions. This fact justifies the sensibility of considering real systems at the variable operation conditions that is appearing out in a natural way from practice. This approach, upon the good accuracy of the systems' operation processes and the systems' components reliability parameters identification, makes their reliability prediction more precise.
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