CC BY
21
Soszynska Joanna
Maritime University, Gdynia, Poland
Systems reliability analysis in variable operation conditions
Keywords
reliability function, semi-markov process, large multi-state system Abstract
The semi-markov model of the system operation process is proposed and its selected parameters are defined. There are found reliability and risk characteristics of the multi-state series- "m out of k " system. Next, the joint model of the semi-markov system operation process and the considered multi-state system reliability and risk is constructed. The asymptotic approach to reliability and risk evaluation of this system in its operation process is proposed as well.
1. Introduction
Many technical systems belong to the class of complex systems as a result of the large number of components they are built of and complicated operating processes. This complexity very often causes evaluation of systems reliability to become difficult. As a rule these are series systems composed of large number of components. Sometimes the series systems have either components or subsystems reserved and then they become parallel-series or series-parallel reliability structures. We meet these systems, for instance, in piping transportation of water, gas, oil and various chemical substances or in transport using belt conveyers and elevators.
Taking into account the importance of safety and operating process effectiveness of such 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 state degrading in time without repair gives the possibility for more precise analysis of their reliability, safety 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 reliability 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 complexity of the systems' operation processes and their influence on changing in time the systems' structures and their components' reliability characteristics is often very difficult to fix and to analyse. A convenient tool for solving this problem is semi-markov modelling of the systems operation processes which is proposed in the paper. In this model, the variability of system components reliability characteristics is pointed by introducing the components' conditional reliability functions determined by the system operation states. Therefore, the common usage of the multi-state system's limit reliability functions in their reliability evaluation and the semi-markov model for system's operation process modelling in order to construct the joint general system reliability model related to its operation process is proposed. On the basis of that joint model, in the case, when components have exponential reliability functions, unconditional multi-state limit reliability functions of the series- m out kn system are determined.
2. System operation process
We assume that the system during its operation process has v different operation states. Thus, we can define Z(t), t e< 0,+¥>, as the process with discrete states from the set
Z = {zl , z2 , . . ., Zv }.
In practice a convenient assumption is that Z(t) is a semi-markov process [1] with its conditional sojourn times 0 bl at the operation state zb when its next operation state is zl, b, l = 1,2,..., v, b * l. In this case this process may be described by:
- the vector of probabilities of the initial operation states [pb (0)]i„ ,
- the matrix of the probabilities of its transitions
between the states [pbl ]VxV ,
- the matrix of the conditional distribution functions
[ H bl (t )]vxV of the sojourn times 0 bl , b *l.
If the sojourn times 0bl, b, l = 1,2,...,v, b * l, have Weibull distributions with parameters abl, bbl, i.e., if for b,l = 1,2,...,v, b * l,
£[(96 )2] = Jt2dHb (t)
= ZPbl J12ablbbl exp[-abltbbl ]tbbl-1 dt
l=1 0
v 2
= Z Pbl a bl%l r (1 + — ), b = l,2,...,v.
l=1 b bl
Limit values of the transient probabilities
pb (t) = P(Z(t) = ), t > 0, b = 1,2,...,v, at the operation states zb are given by
Hu (t) = P(9bl < t) =1 - exp[-autbbl ], t > 0, then their mean values are determined by
Mbl = E[9u ] =abibbl r(1 + ^),
b,l = 1,2,...,v, b * l.
1
bbl
(1)
The unconditional distribution functions of the process Z(t) sojourn times 0b at the operation states zb, b = 1,2,...,v, are given by
Hb (t) = Z Pbi [1 - exp[-a wtbbl t]],
l=1
= 1 -ZPbl exp[-abltbbl ], t > 0,
l=1
b = 1,2,..., v, and, considering (1), their mean values are
Mb = £[9b ] = Z PUMU l=1
= Z Pbl a bl *bl r (1 + —), b = 1,2,...,v,
l=1 b»
and variances are
Db = D[9b ] = £[(9b )2] - (Mb )2 where, according to (2),
(2)
(3)
(4)
Pb =lim Pb (t) =pbMb / ZplMl, b = 1,2,...,v, (5)
l=1
where Mb are given by (3) and the probabilities p b of the vector [p b ]1xv satisfy the system of equations
[p b ] = [p b ][ Pbl ]
Z P l = 1.
l=1
3. Multi-state series- "m out of kn" system
In the multi-state reliability analysis to define systems with degrading components we assume that all components and a system under consideration have the reliability state set {0,1,...,z}, z > 1, the reliability states are ordered, the state 0 is the worst and the state z is the best and the component and the system reliability states degrade with time t without repair. The above assumptions mean that the states of the system with degrading components may be changed in time only from better to worse ones. The way in which the components and system states change is illustrated in Figure 1.
transitions
worst state
best state
Figure 1. Illustration of states changing in system with ageing components
2
v
One of basic multi-state reliability structures with components degrading in time are series- "m out of kn" systems.
To define them, we additionally assume that Ej, i = j = 1,2,...,li, kn, li, I2,...,lkn, n e N, are
components of a system, Tij(u), i = 1,2,...,kn, j = 1,2,...,l, kn, l1, l2,...,lkn, n eN, are independent random
variables representing the lifetimes of components Eij in the state subset {u,u +1,..., z}, while they were in the state z at the moment t = 0, ejt) are components Ej states at the moment t, t e< 0, ¥), T(u) is a random variable representing the lifetime of a system in the reliability state subset {u,u+1,...,z} while it was in the reliability state z at the moment t = 0 and s(t) is the system reliability state at the moment t, t e< 0, ¥).
Definition 1. A vector
Rjt,•) = [Rj(t,0), Ry(t,1),..., Ry(t,z)], t e< 0,»),
where
Ry(t,u) = Pj > u | ey(0) = z) = P(Tj(u) > t)
for t e< 0,¥), u = 0,1,...,z, i = 1,2,...,kn, j = 1,2,...,li, is the probability that the component Eij is in the reliability state subset {u, u +1,..., z} at the moment t, t e< 0, ¥), while it was in the reliability state z at the moment t = 0, is called the multi-state reliability function of a component Eij.
Definition 2. A vector
rm (t,•)=[1,rm (t,0),rm (t,1),...,rm (t,z)],
knln L ' knln " knln ' knln v ' ,J'
where
R km,} (t,u) = P(s(t) > u | s(0) = z) = P(T(u) > t)
knln
for t e< 0,<x>), 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 e< 0, ¥), while it was in the reliability state z at the moment t = 0, is called the multi-state reliability function of a system.
It is clear that from Definition 1 and Definition 2, for u = 0, we have Ry(t,0) = 1 and R[mJn (t,0) = 1.
Definition 3. A multi-state system is called series- "m out of kn" if its lifetime T(u) in the state subset {u, u +1,..., z} is given by
T(u) = T(kn-m+1) (u) , u = 1,2,...,Z,
where T(kn_m+1) (u) is m-th maximal statistics in the random variables set
T(u) = min {Tj(u)}, i =1,2,...,kn, u = 1,2,...,z.
1£ j£li
Definition 4. A multi-state series- "m out of kn" system is called regular if l1 = l2 = . . . = lkn = ln, ln eN.
Definition 5. A multi-state series- "m out of kn" system is called homogeneous if its component lifetimes Tij (u) have an identical distribution function,
i. e.
F(t,u) = P(T;- (u) < t), t e< 0,¥), u = 1,2,...,z,
i = 1,2,...,kn, j = 1,2,...,l,
i.e. if its components Eij have the same reliability function, i.e.
R(t,u) = 1 _ F(t,u), t e< 0,¥), u = 1,2,...,z.
From the above definitions it follows that the reliability function of the homogeneous and regular series- "m out of kn" system is given by [3]
< (t.)=[1 (t,1), < (t,2),..., RS}n (t, z)] ,(6) where
R(km) (t, u)
nn
(7)
= 1 _ I?(kn)[Rln (t,u)] [1 _ Rln (t,u)]kn_'
i=0
for t e<0,¥), u = 1,2,...,z, or by
— (m) —(m) —(m) —(m)
Rknin (t,0 = [ 1, Rknin (t,1), Rknin (t,2),...,Rknln (t,z)], (8)
where
Rkr,i (t, u) = £ (kn )[1 _ Rln (t, u)]! [Rln (t, u )]kn _ (9)
i=0
for t e<0,¥), u = 1,2,...,z, m = kn _ m,
where kn is the number of series subsystems in the "m out of kn" system and ln is the number of components of the series subsystems.
Under these definitions, if R^/1 (t,u) = 1 for t < 0,
knln
u = 1,2,...,z, or (t,u) = 1 for t < 0, u = 1,2,...,z, then
M(u) = fRkm) (t,u)dt, u = 1,2,..., z,
Kfil fi
(10)
or
M(u) = fR(,m,} (t,u)dt, u = 1,2,..., z,
Kfil fi
(11)
is the mean lifetime of the multi-state non-homogeneous regular series "m out of kn" system in the reliability state subset {u,u +1,..., z},and the variance is given by
D[T(u)] = 2 ftRkm) (t,u)dt - E2[T(u)],
J knln
(12)
or by
D[T(u)] = 2 ftRm (t,u)dt- E2[T(u)].
J knln
(13)
and if t is the moment when the system risk function exceeds a permitted level 5, then
t= r-x(5),
(16)
where r l(t), if it exists, is the inverse function of the risk function r(t).
4. Multi-state series- "m out of kn" system in its operation process
We assume that the changes of the process Z(t) states have an influence on the system components Eij
reliability and the system reliability structure as well. Thus, we denote the conditional reliability function of the system component Eij while the system is at the
operational state zb, b = 1,2,...,v, by
[Rj)(t, -)](b)= [1, [Rj)(t, 1)](b),..., [R(',j)(t,z)](b)], where for t e< 0,¥), u = 1,2,..., z, b = 1,2,...,v, [R(i,»(t,u)](b) = P(T;ib)(u) > t|Z(t) = zb)
and the conditional reliability function of the system while the system is at the operational state zb , b = 1,2,...,v, by
The mean lifetime M(u), u = 1,2,..., z, of this system in the particular states can be determined from the following relationships
M(u) = M(u) -M(u +1), u = 1,2,..., z -1,
(14)
M (z) = M (z). Definition 6. A probability r(t) = P(s(t) < r | s(0) = z) = P(T(r) < t), t e< 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 homogeneous regular series "m out of kn " system.
Considering Definition 6 and Definition 2, we have
r(t) =1 - Rkml (t,r), t e< 0,¥),
(15)
[RSn(t,-)](b)= [1, [RSn(t,1)](b),..., [RU(t,z)](
for t e< 0,¥), u = 1,2,..., z, b = 1,2,...,v, where according to (7), we have
[R<m\ (t, u)](b) = P(T(b)(u) > t|Z(t) = zb)
#) _
(b).
1 - zfr)[[R(t,u)](b)]1
[1 - [[R(t,u)](b)]ln ]kn-i for t e< 0,¥),
u = 1,2,..., z, b = 1,2,...,v, or by
[R$n(t,-)](b)= [1, [R$n(t,1)](b),..., [R(l(t,z)](b)
for t e< 0,¥), u = 1,2,..., z, b = 1,2,...,v, where according to (9), we have
[R« (t, u)](b) = P(T(b)(u) > t|Z(t) = zb)
¥
¥
¥
¥
n
= £ (kn )[1 _ [[ R(t, u)](b)]lni
i=0
• [[[R(t,u)](b)]ln ]kn_ for t e< 0,¥), u = 1,2,..., z, b = 1,2,...,v.
The reliability function [ R(i, j )(t, u)](b) is the conditional probability that the component Eij lifetime
T(b)(u) in the reliability state subset {u, u +1,..., z} is
not less than t, while the process Z(t) is at the operation state zb . Similarly, the reliability function
[R(m), (t, u)](b) or [R(m) (t, u)](b) is the conditional probability that the system
lifetime T (b)(u) in the reliability state subset {u, u +1,..., z} is not less than t, while the process Z(t) is at the operation state zb. In the case when the system operation time 0 is large enough, the unconditional reliability function of the system
R(min (t,) = [1, R(Cln №..., Rlmi (t,z)],
where
R^ (t,u) = P(T(u) > t) for u = 1,2,...,z,
or
RSt = [1, Rici (MX,..., RiSn (t, z)],
where
R^n (t,u) = P(T(u) > t) for u = 1,2,...,z,
and T(u) is the unconditional lifetime of the system in the reliability state subset {u,u + 1,..., z}, is given by
Rl (t,u) @£Pb[On (t,u)](b), (17)
b=1
or
rs (t, u) @£ Pb [*sn (t, u )](b) (18)
b=1
for t > 0 and the mean values and variances of the system lifetimes in the reliability state subset {u, u +1,..., z} are
M(u) @ £ pbMb (u) for u = 1,2,..., z, (19)
b=1
where
¥
Mb (u) = J [R«n ](b)(t, u)dt, (20)
0
or
¥ _ _
Mb (u) = J [R£)n ](b)(t, u)dt, (21)
0
and
¥
D[T(b)(u)] = 2Jt[R(t, u)](b) dt_E2[T(b)(u)], (22)
knln
0
or
¥ _ _
D[T(b) (u)] = 2Jt[R (t, u)](b) dt_E2 [T(b) (u)] (23)
knln
0
for b = 1,2,...,v, t > 0, and pb are given by (4). The mean values of the system lifetimes in the particular reliability states u, by (14), are
M(u) = M(u) _M(u +1), u = 1,2,..., z _ 1,
M (z) = M (z). (24)
5. Large multi-state series- "m out of kn" system in its operation process
Definition 7. A reliability function
^ (t, •) = [1,K (t,1),...,^ (t,z)], t e (_¥,¥), where
H (t, u) = ]£pb^ (b)(t, u),
b=1
is called a limit reliability function of a multi-state homogeneous regular series- "m out of kn" system in its operation process with reliability function
fif' (t, •) = [1, R^) (t,1),..., Rm (t,z)],
knln knln knln
or
RS)(t, ■ ) = [1, RS(t,1),.., R£)(t,z)],
where Rkm/ (t,u), R(m,) (t,u), u = 1,2,...,z, are given
knln knln
by (17) and (18) if there exist normalising constants
a^)(u) > 0, bnb)(u) e (-¥,¥), b = 1,2,...,v,
,(b)
u = 1,2,..., z, such that for t e C
Â(b)(u)
u = 1,2,...,z, b = 1,2,...,v,
limR") ab)(u)t + bf(u),u)](b) = Â (b)(t,u),
or
lim[Rf) (anb)(u)t + bf(u),u)](b) = Â (b)(t, u).
n®¥ n' n
Hence, the following approximate formulae are valid
(25)
v t - b (b)
R & (t, u) (b)(u),
b=1
u = 1,2,... , z,
or
— - v t - b (b)
RCl(t,u) @Xpb (b)(u),
b=1
(26)
u = 1,2,..., z.
The following auxiliary theorem is proved in [7]. Lemma 1. If
(i) lim kn = ¥ , m = constant
n®¥
(m/i ® 0 and kn ® ¥),
/ kn
(ii) H (m)(t, u)
v m-1 ,,, [' (b)(tu)V
= 1 - I Pb I exp[-Fb (t, u)] i'—MiL
b=1 i=0 7!
is a non-degenerate reliability function,
(iii) Rk:)ln (t, •) = [1 <1 (t,1),...,RSn (^ z)],
t e (-¥, ¥), where
[ Rkm,i (t, u)](b)
= 1 - m^{kin )[R(b) (t, u)]n [1 - [R(b) (t, u)]ln ]kn-i ,
i=0
te(-¥,¥), u = 1,2,...,z ,
is its reliability function at the operational state zb , then
H (m)(t, •) = [1, H (m)(t,1),...,H (m)(t,z)], t e (-¥, ¥),
is the multi-state limit reliability function of that system if and only if [7]
lim kn [R(b) (anb) (u)t + bnb) (u),u)]ln
= F(b)(t, u), t e C
V
u = 1,2,...,z, b = 1,2,...,v.
(27)
Proposition 1. If components of the multi-state homogeneous, regular series- "m out of kn" system at the operational state zb
(i) have exponential reliability functions, R(b)(t, u) = 1 for t < 0,
R(b)(t,u) = exp[-1(b)(u)t] for t > 0, (28)
u = 1,2,...,z, b = 1,2,...,v,
(ii) m = constant, kn = n, ln > 0,
1 , (b) 1
(iii) anb)(u) =
b (b) =-
1(b)(u)l/ n 1(b)(u)l, u = 1,2,...,z, b = 1,2,...,v,
log n,
then
(29)
 3(m) (t, ■) = [1,  3(m) (t,1),..., 3(m) (t, z)], t e (-¥, ¥),
Where
Â3(m)(t,u) = 1 - ipbmîexp[-expt-t)]^-^ (30)
b=1 i=0 i!
Rmi (t ) Pb [ r , (t )]
b=1
is the reliability function of a homogeneous regular multi-state series- "m out of kn" system, where
for t e (-¥,¥), u = 1,2,...,z, is the multi-state limit reliability function of that system , i.e. for n large enough we have
v m-1 t - b(b)(u)
RSn (t, u) @ 1 -.SPb Sexp[- exp(--(b^Ai)]
b=1 i=0
an (u)
[ .t - bnb)(u)]
exP[-i (bw, ]
an )(u)
i!
v m-1 /»,-,
@ 1 - S Pb S exp[- exp(-1b\u)/nt - log n)]
b=1 i=0
(31)
exp[_i 1(b)( u )lnt _ i log n] i!
for t e (_¥,¥), u = 1,2,...,z. Proof. For n large enough we have
anb)(u)t + b'b(u) = t(+log n > 0 for t e (_¥,¥)
1( )(u)ln u = 1,2,...,z, b = 1,2,...,v.
Therefore, according to (28) for n large enough, we obtain
R (b)(anb)(u)t + bb)(u), u) = exp[ _1(b)(u)(anb)(u)t + b(nb)(u))]
= exp[ —t—log n] for t e (_¥,¥), u = 1,2,...,z,
ln
b = 1,2,..., v. Hence, considering (27), it appears that
[V(t,u)](b) = lim kn[R(b)(a(b)(u)t + b(nb)(u))]ln
1- r, - t - log nn r n = lim n exp[/n-] = exp[-t]
n®¥ /
for t e (_¥,¥), u = 1,2,...,z, b = 1,2,...,v, which means that according to Lemma 1 the limit reliability function of that system is given by (29)-(30).
The next auxiliary theorem is proved in [7].
Lemma 2. If m
(i)--, 0 <h < 1 for n ®¥ ,
k„
m r 1 \ kn jK
1
v(b)(t ,u )
(ii) Â >(t, u) = 1 —I=S Pb I exp[- — ]dx,
V2p b=1 2
is a non-degenerate reliability function, where v(b) (t, u ) is a non-increasing function
(m) Rimi (t, •)=[1, Rkm± (t,1),..., Rkm± (t, z)]
(m)
,(m)
t e (-¥, ¥), where
Rkl (t,u) @ SPb[Rkmi (t,u)](b), t e (-¥,¥X
is the reliability function of a homogeneous regular multi-state series- "m out of kn" system, where
[ d (t, u)](b)
= 1 _ )[R(b) (t, u)]ln' [1 _ [R(b) (t, u)]ln ]kn_' ,
i=0
t e(-¥,¥), u = 1,2,...,z , b = 1,2,...,v,
is its reliability function at the operational state zb , then
9~ (h)(t, •) = [1, 9~(h)(t,1),...,3~(h)(t,z)], t e (_¥, ¥)
is the multi-state limit reliability function of that system if and only if [7]
lim ^kn+î[R/n (anb)(u)t + bnb)(u),u)](b) -h] n®¥ ^(1 -h)
= v(b)(t,u) for t e C (b)_, u = 1,2,...,z, (32)
b = 1,2,..., v.
v(bW
Proposition 2. If components of the multi-state homogeneous, regular series- "m out of kn " system at the operational state zb
(i) have exponential reliability functions, R(b)(t, u) = 1 for t < 0,
R(b)(t,u) = exp[-1(b)(u)t] for t > 0, (33)
u = 1,2,...,v, b = 1,2,...,v, m
(ii )--> h ,0 < h < 1 for n ® ¥, kn = n, /n > 0,
k„
(iii) anb)(u) = /^V, bf)(u) =
1(b)(u)/n '
2
u = 1,2,...,z, b = 1,2,...,v,
then
= lim JK+~1[[R {b)(a{b)(u )t + b"b)(u), u)]ln -h ] n®¥ ^/h(1 -h)
â 7h)(t,■) = [1, â7h)(t,1),...,Â7h)(t,z)], t e (-¥, ¥),
where
(34)
= lim
Vn+Y(exp[-ln (^t - log^)]-h) _lWh n ln
Vh(1 -h)
-, 1 v t -
Â7h)(t,u) = 1 Pb Je 2 dx
V 2p b=1 ¥
for t e (-¥,¥), u = 1,2,..., z,
(35)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
t-bnb)(u)
,, 1 v an>b(u) -L
RSn (t,u) @ 1 Pb J e 2 dx
V2p b=1 ¥
@ 1 -
V2P
Vh" (1(b)(u )lnt +log n
■S Pb
b=1
J
e 2 dx
for t e (-¥,¥), u = 1,2,...,z.
Proof. Since, for sufficiently large n, we have
a(-b)(u)t + b(b)(u) =
(36)
1 (^lÎ t - log h ) > 0
i(b)(u)ln Vh"
for t e (-¥,¥), u = 1,2,...,z, b = 1,2,...,v,
+1 (exp[-^1=h t + log h ] -h )
= lim
a/H"
Vh(1 -h)
V!-
= lim
Vn+1 (h (exp[ -^J- t ] -1))
Vh n_
Vh(1 -h)
= lim
vn+ï (h (1 t+o^11 )-1))
ylh" Vh n
Vh(1 -h)
vn+1 (-
= lim
Vh (1 -h )
t +h^ 1 ))
Vh"
Vh(1 -h)
= -1 for t e (-¥, œ), b = 1,2,..., v,
which means that according Lemma 2 the limit reliability function of that system is given by (34)-(35).
then according to (33) for sufficiently large n, we obtain
The next auxiliary theorem is proved in [7].
R(b) (a"b) (u)t + b"b)(u), u):
= exp[-1(b) (u)(a"b) (u)t + b"b)(u))] 1 /1 -h
= exp[--(—-=— t - log h )] for t e (-¥, ¥),
ln Vh"
u = 1,2,...,z, b = 1,2,...,v.
Hence, considering (32), it appears that
v (b)(t, u)
Lemma 3. If
m
(i) kn ® ¥ ,--> 1, (kn - m) ® m = constant
k„
for n ® ¥ ,
(ii) Â(m)(t,u) =S Pb S exp[-r (t,u)]
b=1 i=0 i!
is a non-degenerate reliability function,
(iii) (t, ■)=[1, ri (t,1),..., rsl (t, z)],
v m
77(b),
— (b)
[V l)(t, u)]
2
1
2
x
¥
t e (-¥, ¥), where
— (m ) v —(m )
Rknin (t, u) @ S Pb [Rk„In (t, u)](b), t e (-¥, ¥),
b=1
is the reliability function of a homogeneous regular multi-state series- "m out of kn" system, where
— (m )
[Rknln (t)]
(b)
(kn-0
= V (kn )[1 - [R(b) (t )]/n ] [ R(b) (t )]/n
i=0
te(-¥,¥), u = 1,2,...,z , b = 1,2,..., v,
is its reliability function at the operational state zb , then
H)(t, •) = [1, H}(t,1),...,H(m}(t, z)], t e (_¥, ¥),
is the multi-state limit reliability function of that system if and only if [7]
lim kn/nF(b}(anb)(u)t + bnb)(u),u)
(b).
is the multi-state limit reliability function of that system , i.e. for n large enough we have
Rm (t, u)
1,
t < 0,
v m
t - bb)(u) S Pb S exp[--
b=1 i=0 an (u)
[ t - bnb)(u)f
[ a(b) ] un_
i!
t > 0,
1, t < 0,
v m
S Pb S exp[t1(b)(u)/nkn
b=1 i=0
[t1(b)(u )/nkn ]'
t > 0.
Proof. Since
(41)
= V (b)(t, u) for t e C_,b)
V 7 V(b)(u)
u = 1,2,...,z, b = 1,2,...,v.
(37) af{u)t + bb)(u) =
Proposition 3. If components of the multi-state homogeneous, regular series- "m out of kn" system at the operational state zb
(i) have exponential reliability functions, R(b)(t, u) = 1 for t < 0,
R(b)(t,u) = exp[_1(b)(u)t] for t > 0, (38)
u = 1,2,...,z, b = 1,2,...,v,
(ii) kn ® ¥ , lim kn _ m = m = constant,
(iii) anb)(u) = ■
1
1(b)(u) /nk,
-, bnb)(u) = 0,
u = 1,2,...,z, b = 1,2,...,v,
then
Â2(m) (t, •) = [1, Â2(m) (t,1),...,Â2(m)(t,z)], t e (-¥, ¥),
where
 2(m )(t, u) =
1, t < 0,
v m t'
S Pb S exp[-t]-, t > 0,
b=1 i=0 i!
(39)
(40)
1(b\u)/nkr
< 0 for t < 0,
u = 1,2,...,z, b = 1,2,...,v, and
anb) (u)t + b{b) (u) = ■
> 0 for t > 0,
1(b\u)lnk, u = 1,2,...,z, b = 1,2,...,v,
therefore, according to (38), we obtain
F(b)(a(b)(u)t + bb)(u),u) = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
F (b)(anb)(u)t + bnb)(u), u) = 1 _ exp[--t—] for t > 0, u = 1,2,...,z,
knln
b = 1,2,...,v. Hence, considering (37), it appears that
V (b)(t, u)
t
t
= lim knlnF^K)(af)(u)t + b"b)(u),u) = 0 for t < 0,
n®¥
u = 1,2,...,z, b = 1,2,...,v,
and
V(b) (t,u) = lim knlnF(b) (a"b) (u)t + b"b) (u),u)
= lim k"ln (1 - exp[- — ])
n®¥ k l
= lim knln (1 -1+ -L- - o(-i- )) n®¥ k l k l
n n n n
= t for t > 0, u = 1,2,...,z, b = 1,2,...,v,
which means that according Lemma 3 the limit reliability function of that system is given by (39)-(40).
The next auxiliary theorem is proved in [7]. Lemma 4. If
(i) lim kn = k, k > 0, 0 < m < k, lim ln = ¥,
n®¥ n®¥
(ii) H (t,u) = I pbH (b)(t, u) is a non-degenerate
b=1
reliability function,
(iii)R-krl (t, •) = [1, Rkmi (t,1),...,Rk! (t,z)],
nn> m
t e (-¥, ¥), where
Rkmi « @ sPb [ Rsn (t )]
b=1
is the reliability function of a homogeneous regular multi-state series- "m out of kn" system, where
[ Rkmi (t, u)](b)
= 1 - "î(kn )[R(b) (t, u)]n [1 - [R(b) (t, u)]ln ]
ln lkn i
i=0
te(-¥,¥), u = 1,2,...,z, b = 1,2,...,v,
is its reliability function at the operational state zb , then
H (t, •) = [1, H (t,1),...,H (t,z)], t e (-¥,¥),
lim[Rib)(af )(u)t + b"b)(u), u)]ln =Â 0b)(t, u) (42)
n®¥
for t e C
(b) , u = 1,2,...,z, b = 1,2,...,v, Âq (u)
where  0(b)(t, u), u = 1,2,...,z, is a non-degenerate reliability function and
 (t, u)
vm
= 1 -SPb S
b=1 i=0
Y k \
è10
[Â 0(b)(t, u)]1 [1 -Â 0(b)(t, u)]k-i (43)
(b).
nk-i
for t e(-¥,¥), u = 1,2,...,z.
Proposition 4. If components of the multi-state homogeneous, regular series- "m out of kn" system at the operational state zb
(i) have exponential reliability functions, R(b)(t, u) = 1 for t < 0,
R(b) (t, u) = exp[-1(b) (u)t] for t > 0, (44)
u = 1,2,...,z, b = 1,2,...,v,
(ii) kn ® k, k > 0, ln ® ¥, m = const, 1
(iii) a"b) (u) = ■
b"b)(u) = 0,
1(b)(u)l u = 1,2,... , z, b = 1,2,...,v, then
 9(m) (t, ■) = [1,  9(m) (t,1),..., 9m) (t, z)], t e(-¥,¥),
where
â 9(m)(t, u)
1, t < 0,
v m-1^ k \
1 -SPb S . [exp[-t]]!
b=1 i=0 è i
■ [1 - exp[-t]]k-i, t > 0,
(45)
(46)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
 9(m) (t, u)
is the multi-state limit reliability function of that system if and only if [7]
1, t < 0,
V m-1кö t - bib)(u) t
1 -SPb S . [exp[--b^r1]]'
b=1 i=0è i 0
аГ(и)
• [1 - exp[-^]]к
an (u)
v m-
iœ к ö
1 -sPb S . [exp[-t1(b)(u)/„]]
b=1 г=0^ i 0
t > 0,
t < 0,
• [1 - exp[-t1(b)(u)/„ ]]к
t > 0.
Proof. Since
anb) (u)t + b(nb\u ) =
t
1(b)(u)lr
< 0 for t < 0,
u = 1,2,...,z, b = 1,2,...,v, and
a(-b)(u)t + bb)(u ) = ■
t
> 0 for t > 0,
1(b)(u)lr u = 1,2,...,z, b = 1,2,...,v,
therefore, according to (44), we obtain
= lim[R ib)(af)(u)t + bnb)(u), u]ln
lim[exp[ - — ]]ln
П®¥ l
= exp[-t] for t > 0, u = 1,2,...,z, b = 1,2,...,v.
which, by Lemma 4, completes the proof.
6. Conclusion
(47) The purpose of this paper is to give the method of reliability analysis of selected multi-state systems in variable operation conditions. As an example a multistate series-"« out of k" systems are analyzed. Their exact and limit reliability functions, in constant and in varying operation conditions, are determined. The paper proposes an approach to the solution of practically very important problem of linking the systems' reliability and their operation processes. To involve the interactions between the systems' operation processes and their varying in time reliability structures a semi-markov model of the systems' operation processes and the multi-state system reliability functions are applied. This approach gives practically important in everyday usage tool for reliability evaluation of the large systems with changing their reliability structures and components reliability characteristic during their operation processes. The results can be applied to the reliability evaluation of real technical systems.
[R(b)(anb)(u)t + bnb)(u),u)]ln = 1 for t < 0,
u = 1,2,...,z, b = 1,2,...,v, and
[R{b)(a{b)(u)t + bnb)(u),u)] = exp[--t-] for t > 0
ln
u = 1,2,...,z, b = 1,2,...,v. Hence, according (42)-(43), it appears that
 0b)(t, u) = lim[ R (b)(a(lb)(u )t + bf^u), u]ln = for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
 0b)(t,u )
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1
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