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ASYMPTOTIC APPROACH TO RELIABILITY EVALUATION OF LARGE "M OUT OF L"- SERIES SYSTEM IN VARIABLE OPERATION CONDITIONS
Joanna Soszynska
Gdynia Maritime University, Gdynia, Poland e-mail: [email protected]
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 "m out of l "-series 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 (Kolowrocki 2004). The assumption that the systems are composed of multi-state 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 (Grabski 2002, Kolowrocki & Soszynska 2005, Soszynska 2006 a, b, Soszynska 2007 a, b, c) 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 "m out /„"-series system are determined.
2 SYSTEM OPERATION PROESS
We assume that the system during its operation is operating in v, v e N, different operation states. After this assumption we can define the system operation process Z(t), t e< 0,+ro>, with discrete states from the set of states
Z = , z2 ' . . .' Zv
In practice a convenient assumption is that Z(t) is a semi-markov process (Grabski 2002, Kolowrocki & Soszynska 2005, Soszynska 2006 a, b, Soszynska 2006 a, b, c) with its conditional sojourn times 9bl at the operation state zb when its next operation state is zl, b, / = 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)]1xv,
- the matrix of the probabilities of its transitions between the states [pbl ]vxv,
- the matrix of the conditional distribution functions [Hbl (t)]vxv of the sojourn times 9bl, b *l.
If the sojourn times 9bl, b, l = 1,2,..., v, b * l, have Weibull distributions with parameters abl, /3bl, i.e., if for b,l = 1,2,...,v, b * l,
Hbl (t) = P(0bl< t) = 1 - exp[-awt ^ ], t > 0,
then their mean values are determined by
1
Mbl = E[dbl ] = ajblr(1+ J-), b,l = U...,v, b *l. (1)
Pbl
The unconditional distribution functions of the process Z(t) sojourn times db at the operation states zb, b = 1,2,..., v, are given by
Hb (t) = ipbl [1 - exp[-awtPbl t]], = 1 -V-Zpbl exp[-awtPbl ], t > 0, b = 1,2,..., v, (2)
l=1 l=1
and, considering (1), their mean values are
1
v v 1
Mb = E[6b ] = X pblMbl =E pblajbl r(1 + —), b = 1,2,...,v, (3)
l=1 l=1 Pbl
and variances are
Db = D[0b ] = E[(0b )2] - (Mb)2, (4)
where, according to (2),
œ v œ v ----2
E№)2] = J12dHb(t) = Xpbl J12ablßbl exp[-awtßbl ]tßbldt = Xpblablßbl r(1 + —), b = 1,2,...,v.
0 l=1 0 l=1 Pbl
Limit values of the transient probabilities
pb (t) = P(Z(t) = Zb), t > 0, b = 1,2,...,v,
at the operation states zb are given by
,=lim pb (t) = nbMb / JJnlMl, b = 1,2,...,v,
l=1
(5)
where Mb are given by (3) and the probabilities nb of the vector [nb ]1xv satisfy the system of equations
[nb ] = K ][pbl]
1 v
Xn = 1.
J=1
3 MULTI STATE " M OUT OF L"- SERIES 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. One of multi-state reliability structures with components degrading in time (Kolowrocki 2004, Kolowrocki et. al 2005) are "m out of ln"- series systems.
transitions
Figure 1. Illustration of states changing in system with ageing components.
To define them, we additionally assume that Eij, i = 1,2,...,kn, j = 1,2,...,l-, kn, l1, l2,...,lkn, n e N, are components of a system, T1](u), i = 1,2,...,kn, j = 1,2,...,I-, kn, I1, I2,...,lk , n e N, 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, eij(t) are components Eij states at the moment t, t e<0,<x>), 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,w).
Definition 1. A vector
Rj(t,•) = [R(t,0), Rj(t,1),..., Rj(t,z)], te<0,w),
where
Rij(t,u) = P(etJ(t) > u | ej(0) = z) = P(TIJ(u) > t)
for t e<0,w), u = 0,1,...,z, i = 1,2,...,kn, j = 1,2,...,l;-, is the probability that the component Ej is in the reliability state subset {u, u +1,..., z} at the moment t, t e< 0,w), while it was in the reliability state z at the moment t = 0, is called the multi-state reliability function of a component Ej.
Definition 2. A vector
<1 (t,) = [1 <1 (t,0X<1 (M),..(t,z)],
where
(t,u) = P(s(t) > u | s(0) = z) = P(T(u) > t)
for te<0,w), 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, w), 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 Rj(t,0) = 1 and R;m) (t,0) = 1.
Definition 3. A multi-state system is called "m out of ln"- series if its lifetime T(u) in the state subset {u, u +1,..., z} is given by
T(u) = min T(h +1)(uX mi < lt, , u = z,
1<i<kn 1 1
where T{i. _m.+1)(u) is mrth maximal statistics in the random variables set
Ti(u), Tl2(u),...,Tlh (u), i = 1,2,..., kn, u = 1,2,..., z.
Definition 4. A multi-state "m out of ln"- series system is called regular if l1 = l2 = . . . = lkn = ln
and m1 = m2 =...= mk = m, ln , me N, m < ln.
Kn
Definition 5. A multi-state "m out of ln"- series system is called homogeneous if its component lifetimes TJJ (u) have an identical distribution function, i.e.
F(t,u) = P(T1} (u) < t), t e< 0,o>), u = 1,2,..., z, i = 1,2,...,kn, j = 1,2,...,, i.e. if its components Ej have the same reliability function, i.e.
R(t,u) = 1 - F(t,u), te<0,<x>), u = 1,2,...,z.
From the above definitions it follows that the reliability function of the homogeneous and regular "m out of ln"- series system is given by (Kolowrocki 2004, Kolowrocki et al 2005)
Äf' (t,-) = [1,Rm (t,1),...,Rkm) (t,z)],
kn ,'n V '' L ' kn Jn V ' '' ' kn ,1n V ' n '
(6)
where
Rm~(t,«)=[i-Hi W,«)]' [i-R(t,u)]n 1 ]kn ,te<°,œ), u=1,2,...,z, (7)
i=°
or by
—(m) —(m) —(m)
Rknn (t,-) = [1, R kn in (t,1),..., R kn in (t, z)], (8)
where
(9)
—(m) ln —m/., \ j , ,
Rk„n (t,u) = [ Z ((n )[1 — R(t,u)]1 [R(t,u)]'n—1 ]kn, te<°,œ), u = 1,2,...,z,
i=°
where kn is the number of "m out of ln" subsystems connected series and ln is the number of components of the "m out of ln" subsystems.
Under these definitions, if R^ (t,u) =1 for t < 0, u = 1,2,..., z,or Rkmn>Jn (t,u)= 1 for t < 0, u = 1,2,..., z, then
M(u) = J R« (t,u)dt, u = 1,2,..., z, (1°)
or
x —(m)
M(u) = J tfknin(t,u)rft, u = 1,2,..., z, (11)
0
is the mean lifetime of the multi-state non-homogeneous regular "m out of ln"- series system in the reliability state subset {u,u +1,...,z}, and the variance is given by
D[T (u )] = 2 J t R™ (t, u)dt — E 2[T (u)], (12)
0
TO
TO
or by
D[T(u)] = 2 Jt RZn (t,u)dt -E2[T(u)]. (13)
0
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, M (z) = M (z). (14)
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 "m out of ln"- series system.
Considering Definition 6 and Definition 2, we have
r(t) = 1 - RfX (t,r), t e< 0, (15)
and if ris the moment when the system risk function exceeds a permitted level 8, then
r= r -x(8), (16)
where r -1 (t), if it exists, is the inverse function of the risk function r(t).
4 MULTI STATE " M OUT OF L"- SERIES SYSTEM IN ITS OPERATION PROCESS
We assume that the changes of the process Z(t) states have an influence on the system components Etj reliability and the system reliability structure as well. Thus, we denote the
conditional reliability function of the system component Etj while the system is at the operational state z6, b = 1,2,...,v, by
[R(i,j)(t, -)](b)= [1, [R(i,j)(t, 1)](b),..., [R(i,j)(t,z)](b)], where for t e< 0,«), u = 1,2,...,z, b = 1,2,...,v,
[R(1,J)(t,u)](b) = P(T(b)(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
[Rti(t,-)](fe)= [1, [Rg„(t,1)](fe),..., [Rgn(t,z)](b)] for t E< 0,«), u = 1,2,...,z, b = 1,2,...,v,
where according to (7), we have
[Rg~(t,u)](b) = P(T(b)(u) > t|Z(t) = zb) = [1 - zf? )[[R(t,u)](b)]' [1 -[R(t,u)](b)]ln-i]kn
i=0
for t e< 0,«), u = 1,2,..., z, b = 1,2,...,v,
or by
[R{Cn (t,-)](b)= [1, [R-ZÏ (t,1)](b),..., [Rk3„ (t, z)](b) for t e< 0, «), u = 1,2,..., z, b = 1,2,...,v, where according to (9), we have
[RCn (t, u)](b) = P(T (b)(u) > t|Z(t) = zb) = ["£ (n )[1 - [ R(t, u)](b)]! [[ R(t, u)](b)]ln-i ]kn
i=0
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^"] (t,u)](b) or [Rmln (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 is large enough, the unconditional reliability function of the system
Rk! (t ,•) = [1, Rkl (t,1),..., Rkl (t, z)],
where
Rkmln (t,u) = P(T(u) > t) for u = 1,2,..., z,
or
( m)
Rkn ,ln (t,0 = [1, Rkn ,ln (t,1),..., Rkn ,ln (t, z)],
where
R km i (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
Rf(t,u) = ipb (t,u)](b), b=1
(17) or
[R'Ci (t,u)](b) = ¿Pb [Rimln (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
V
M (u) = £ pbMb (u) for u = 1,2,..., z,
b=1
(19) where
TO -
Mb(u) = J [Rgn (t, u)](b) dt, (20)
0
or
Mb (u) = J [R'd (t, u)](b) dt, (21)
0
and
to -
D[T (b)(u)] = 2 J t [Rkm) (t, u )](b) dt — E 2[T (b)(u)],(22)
J kn ,ln
0
or
D[T(b)(u)] = 2 Jt [R^ (t,u)](b) dt - E2[T(b)(u)] (23)
0
for b = 1,2,..., v, t > 0, and pb are given by (5).
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 " M OUT OF L"- SERIES SYSTEM IN ITS OPERATION PROCESS
Definition 7. A reliability function
X(t, •) = [1,#(t,1),...,#(t,z)], t e (-«,
where
X(t, u) = Z Pb [Ä(t, u)]
(b)
b=1
is called a limit reliability function of a multi-state homogeneous regular "m out of ln"- series system in its operation process with reliability function
Rkmi (t ,•)=[1, Rkmi (t ,1),..., Rkmi (t, z)],
or
—(m) / x ri —(m) / x —(m) , ,
Rkn,ln (t,V [1, Rkn,ln
(t,1) ,... , Rkn
n
where R^] (t,u), R^ (t,u), u = 1,2,...,z, are given by (17) and (18) if there exist normalising constants
,(b)
(u) > 0, bnb)(u) e (-«,«), b = 1,2,...,v, u = 1,2,...,
such that for t e C[)](b), u = 1,2,...,z, b = 1,2,...,v,
lim[ Rkmn)n (anb)(u )t + bnb)(u), u)](b) = [#(t, u)](b),
n^w n' n
or
lim[Rk(l (anb)(u)t + bnb)(u), u)](b) = [Ä(t, u)](b).
z
Hence, the following approximate formulae are valid
or
- v t - h(b)(u)
Rkmi (t, u ) = Z Pb [#( "^TV", u )](b), u = 1,2,..., z, (25)
n n b=1 an '(u)
t - b(b) (u)
Rl (t,u) = ZPb[*( -(bn//,u)](b), u = 1,2,...,z. (26)
b=i an)(u)
The following auxiliary theorem is proved in (Kolowrocki et al 2005). Lemma 1. If
(i) kn ^ k = const, ln = n , m/ ^ 0, m = const, as n ^ o ,
\ / n n/n
(ii) *(m)(t, u)= VPb [1 — "l"V (t, u)] ] exp[—[V (t, u)](b)]]k is
b=1 1=0 i!
a non-degenerate reliability function,
(b)n i
(iii) R'Cl (t,■) = P, Kl (t,!), Ki (t,z)], te (-«V»), is the reliability function of a homogeneous regular multi-state "m out of ln "- series system, in variable operation conditions, where
Rm (t,u) = zPb [R-Ti (t,u)](b), t g (-to,to)
b=1
where
[R^ (t,u)](b) = [1 — Îfr )[[R(t,u)](b)]1 [1 — R(t,u)](b)]ln —1 ]kn, t g (-to,to), u = 1,2,...,z, b = 1,2,...,v,(27)
n' n 1=0
is its reliability function at the operational state zb then
#(m)(t,-) = [1,^(m)(t,1), ...,%{m)(t, z)] , t G (-TO,to), is the multi-state limit reliability function of that system if and only if (Kolowrocki et al 2005)
limn[[R(anb)(u)t + bf(u\u)](b)] = [V(t, u)](b), t g C,b), u = 1,2,..., z, b = 1,2,..., v. (28)
[V (u)r
Proposition 1. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1 for t < 0, [tf(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,..., z, b = 1,2,..., v,
(29)
(ii) kn ^ k = const, ln = n , m/ ^ 0, m = const, as n ^ «,
(iii) anb)(u )= bnb)(u)= -±-log n , u = 1,2,..., z, b = 1,2,..., v,
— J(u) — '(u)
(30)
then
#3(m)(t,-) = [1,#3(m)(t,1), ...,#3(m)(t, z)] :
(31)
where
t g (-TO,to),
#3(m)(t,u) = £pb [1 -Iexp[ lt] exp[-exp[-t]]]k for t e (-w, w) u = 1,2,...,z, (32)
b=1 i=0 i!
is the multi-state limit reliability function of that system , i.e. for n large enough we have
RiiC, u) S £p, [1 -'¿exp[-itl")(") +i 'og "] exp[exp[-tXb)(«) - log n]]]k (33)
for t e (-«,«), u = 1,2,..., z. Proof. Since
t + log n
z,
a„(u)t + bnb)(u) = B-->w as n ^w for t e (-w, w), b = 1,2,..., v, u = 1,2,...,
X )(u)
then, according to (29) for n large enough, we obtain
[R(a(nb) (u)t + b^ (u), u)](b) = exp[-Xb) (uXa^ (u)t + bf) (u))] = exp[-t - logn] for t e (-w, w), u = 1,2,..., z, b = 1,2,..., v. Hence, considering (28), it appears that
[V(t, u)](b) = lim n[R(anb)(u )t + bfV), u)](b) = lim n exp[-t - log n] = exp[-t]
for t e (-w, w), 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 (31)-(32). □
The next auxiliary theorem is proved in (Kolowrocki et al 2005). Lemma 2. If
(i) kn ^ k = const, ln = n , mn ^ n, 0 <n< 1, as n ^ w,
1 -[v(t,u )](b) -fi
(ii) #(n)(t,u)= £ pb [ - _ j e 2 dx]k is a non-degenerate reliability function, where
b=1 V2n -w
[v(t, u)](b) is a non-increasing function
(iii) rJ,]" (t,0 = [1, rJ,]" (t,1),..., R^ (t, z)], te(-w,w), is the reliability function of a homogeneous regular multi-state "m out of ln"- series system, in variable operation conditions, where
Rk;i (t, u)=i Pb [ Rk;i (t, u )](b), t e (-«,«),
b=1
and
[Rkl (t,u)](b) = [1 — Z^ )[[R(t,u)](b)]1 [1 — [R(t,u)](b)]ln—1 ]kn t g (-to,to),b = 1,2,...,v, u = 1,2,...,z,(12)
is its reliability function at the operational state zb, then
#(n)(t,0 = [1,#(n)(t,1), ...,#(n)(t, z)], t G (-O,o),
is the multi-state limit reliability function of that system if and only if
(n + 1)[R(anb)(u)t + bnb)(u),u)](b) -m )](b) for , C 12 . 12
lim---.--[v(t,u)]( ) for tG C (b), u = 1,2,...,z, b = 1,2,...,v.
m(n - m +1) [v(u)]( )
n +1
(35)
Proposition 2. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1 for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,..., z, b = 1,2,..., v, (36)
(ii) kn ^ k = const, ln = n , m/n ^ n, 0 <n< 1, as n ^ o ,
(iii) ^(u ) = -1— n—mH, (37)
V 7 n W —(b)(u)i(n + 1)m V 7
bnb)(u) = —Fb^Tiog—, u = 1,2,...,z, b = 1,2,...,v, (38)
—(b) (u) m
then
«/"(t,-) = [1,Ä1("(t,1), ...,m(n)(t,z)], t g (-to,to), (39)
where
1 t „ x2
m^tt,u) = ZPb[1 —t= J exp[—— ] dx] k for t g (-to,to),u = 1,2,...,z, (40)
b=1
V2n—TO 2
is the multi-state limit reliability function of that system , i.e. for n large enough we have
tib)(a)-log—
t —b(n>(u ) In—m+1
1 a(nb)(u ) x 2 v r 1 \(n+1)
2
Rjin (t,u) = Z Pb [1 — ^ J exp[ —] dx] k = Z Pb [1 —f= I exP[ ~] dx] k (41)
n n b=1 yj2n —to 2 b=1 V2n —to 2
for t g (-to,to), u = 1,2,..., z.
m
Proof. For n large enough we have
af\u)t + +^(b77-7log—>0 for t e (■",")> " = i'2'-'z, h = v
A (u)\(n + 1)m X-D) (u) m Therefore, according to (37)-(38) for n large enough we obtain
[ R(a(b) (u)t + b(nb) (u), u)](b) = exp[-Ab) (u)(a(b) (u)t + b(nb) (u))]
n - m + 1 n + ln n - m +1 ,1s-, m = exp[-t I-----log-] = [1 -tl--— + o(-=)]-
(n + 1)m m \ (n + 1)m Jn n +1
for t e (-œ, œ), u = 1,2,..., z, b = 1,2,..., v. Hence, considering (35), it appears that
Mt. u)](b) = lim (n + QW^frV + b?(u ),u)]lbl - m = lim[-t + Q(-hJ(n + 1)(n - m +1) ] = -t
m(n - m +1) Jn V m
u)]v^ - m
m(n - m +1) ^ 4n V m
n +1
for t e (-<x>, <x>), u = 1,2,..., z, b = 1,2,..., v,
which means that according to Lemma 2 the limit reliability function of that system is given by (39)-(40). □
The next auxiliary theorem is proved in (Kolowrocki et al 2005). Lemma 3. If
(i) kn ^ k = const, ln = n , m/n ^ 1, (n - m) ^ m = const, as n ^ «,
(ii) m(m)(t,u) = Xpb[2u)]()] exp[-[V(t,u)](b)]]k is a non-degenerate reliability function,
b=1 i=o i!
(iii)R™n (t,-) = [1, R^i (t,1),..., RQn (t,z)], t e (-»,œ), is the reliability function of a homogeneous regular multi-state "m out of ln "- series system, in variable operation conditions, where
R™ (t, u) =X pb [RS„ (t, u)](b), t e (-œ, œ),
b=1
and
[RZ)in (t,u)](b) = [ "£" ( )[1 - [R(t,u)](b)]! [1 - [R(t,u)](b)]ln-i ]kn , te (-œ,œ), u = 1,2,..., z, b = 1,2,..., v, (42)
is its reliability function at the operational state zb, then
m(m)(t,-) = [1,m{m)(t,1), ...,m{m)(t,z)], t g (-to,to)
"(m )
(m ) 1
is the multi-state limit reliability function of that system if and only if
lim n[F(anb)(u)t + bf)(u), u)](b) = [V (t, u)](b)for t g C_ (b), u = 1,2,..., z, b = 1,2,..., v.
(b)i
[V (u )]'
i(b) '
(43)
Proposition 3. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1 for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,..., z, b = 1,2,..., v, (44)
(ii) kn ^ k = const, ln = n , m/ ^ 1, (n - m) ^ m = const, as n ^ o
(iii) anb)(u)= —-b— , bnb)(u)= 0, u = 1,2,...,z, b = 1,2,...,! n— ;(u )
(45)
then
m2(m )(t,-) = [1,m2(m )(t,1), ...,m2(m )(t, z)], t g (-to, to),
(46)
where
mm )(t, u) =
for t < 0,
v |- m t ik
Z Pb [Z— exp[—t ]]k for t > 0, u = 1,2,..., z,
yb _ , b=1 1=0 1!
(47)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
R(km] (t,u)
kn ,ln
1 for t < 0,
(n—(b)(u)t )1
"Ü for t > 0,
v r m
ZPb [Z-
b=1 1=0
exp[—n—(b)(u)t ]]k
(48)
u = 1,2,..., z.
Proof. Since
at\u)t + bnb)(u) = ■
n—b) (u)
< 0 for t < 0, u = 1,2,..., z, b = 1,2,...,v,
a(b)(u)t + bnb)(u) = ■
n—b)(u)
> 0 for t > 0, u = 1,2,..., z, b = 1,2,...,v,
then according to (44) we obtain
[F(anb)(u)t + bb)(u),u)](b) = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
1
t
t
and
[ F (a(b) (u )t + b(nb) (u), u)](b) = 1 - expM(b) (u)(a(b) (u)t + b™ (u))]
= 1 - exp[--] for t > 0, u = 1,2,...,z, b = 1,2,...,v. n
Hence, considering (43), it appears that
[V(t,u)](b) = limn[F(a(b)(u)t + bnb)(u),u)](b) = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
n^ro
and
[V(t, u)](b) = lim n[F(a(b)(u )t + b(b)(u), u )](b) = lim n[1 - exp[- -]] = lim n[- + o(1)] = t
n^ro n^ro n n^ro n n
for t > 0, u = 1,2,..., z, b = 1,2,...,v,
which means that according to Lemma 3 the limit reliability function of that system is given by (46)-(47). □
The next auxiliary theorem is proved in (Kolowrocki et al 2005). Lemma 4. If
(i) kn = n , ln ^ l = const, m < ln, as n ^ ro
(ii) 9t(t, u) =XPb exp[-[V(t,u)](b)] is a non-degenerate reliability function,
b=1
(iii) (t,0 = [1, R1 (M), •••, RI (t, z)], t e (-œ, œ), is the reliability function of a
homogeneous regular multi-state "m out of ln "- series system, in variable operation conditions, where
(t,u) = iPb Kl(t,u)](b), t e (-«,«),
b=1
and
[R™ (t, u)](b) = [1 - xfa )[[R(t, u)](b)]i [1 - [R(t, u)](b)]ln-i ]kn t e (-ro,ro), u = 1,2,..., z, b = 1,2,..., v,
n i=0
(49)
is its reliability function at the operational state zb , then
X(t,•) = [1,«(t,1), ...,X(t,z)], t e (-ro,ro),
is the multi-state limit reliability function of that system if and only if
lim knZl'r )[[R(a (b)(u)t + bnb\u), u)](b)Y [[F(a'P(u)t + ^(u), u)](b)]ln—1 = [V(t, u)](b) (50)
n ^TO 1=0
for t g ni(b), u = 1,2,..., z, b = 1,2,...,v.
[V (u )]'
Proposition 4. If components of the multi-state homogeneous, regular "m out of ln "-series system
at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[——(b)(u)t] for t > 0, u = 1,2,..., z, b = 1,2,..., v,
(51)
(ii) kn = n, ln i l, l g (0,to), m = const, m < ln,
(iii) anb)(u) = (52)
b(b \u) = 0, u = 1,2,...,z, b = 1,2,...,v,
—b\u)[n(ml—n1)]1/(ln—m+1) '
(53)
then
m2(t,-) = [1,m2(t,1), ...,m2(t,z)], t g (-to,to),
(54)
where
(55)
m(t,u)
1 for t < 0, u = 1,2,..., z,
Mi(t,u) = ZPb exp[—tl—m+1] for t > 0, u = 1,2,...,z,
b=1
(56)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
Rm (t,u)
nn
1 for t < 0, u = 1,2,..., z,
(57)
( l ^
L M
Rkm), (t,u) = ZPb exp[—[t—(b)(u)[n n ]1/(ln—m+1] for t > 0, u = 1,2,...
kn ,ln
b=1
vm — 1/
(58) Proof. Since
anb)(u )t + b(nb )(u) =
—(b)(u)[n( ml—n1)]1/(ln—m+1)
< 0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
and
1
z
t
^(u)t + bnb)(u) = —(b)(u )[n(j, )]1/(ln —m+1) > 0 for t > 0, u = 1,2,..., z, b = 1,2,..., v, then, according to (51), we obtain
[R(anb)(u )t + bnb)(u), u)](b) = 1
and
(b) (b) (b)
and
[ F (a„'(u )t + bf' (u ), u )]( = 0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
[R(anb) (u)t + b(b) (u), u )](b) = exp[——(b) (u)(a(nb") (u)t + ^ (u ), u)]
= exp[--] = 1 — o(1).
p[ [n ( ml—1)]1/(ln — m + 1)]
t
[ F (anb) (u)t + bnb) (u), u )](b) = 1 — exp[——(b) (u)(af) (u )t + b^ (u), u )] = 1 — exp[————-—]
[n( ml—")]1/(ln—m+1)'
1
— o^-^ for t > °, u = 1,2_ z, b = 1,2_ v.
[n( ln )]1/(ln —m+1) n1/(ln—m+1)
Then for each 1 = 0,1,..., m — 1 we have
[[R(anb)(u)t + bb)(u), u)](b)]1 = 1
and
-h(b)i,,\ ,,yi(b)1ln—1 -
[[F(anb)(u)t + b{b)(u),u)](b)]ln—1 = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
[[R(anb)(u)t + b(b)(u),u)](b)]1 = [1 — o(1)]1 — 1as n —^ to for t > 0, u = 1,2,...,z, b = 1,2,...,v,
[[F{a(-b\u)t + ^(u)u)](b)]ln—1 = [^ —m+1) — o("„7/(1„1—m+T^)]ln—1
[1 — o( , ,,:,/(,—m+1))]ln —1 for t > °, u = 1,2,...,z, b = 1,2,...,v.
[n( m—1 )]1(ln_
tln 1
[n( \ )](ln^1)/(ln —'m+1) [n( — )](ln —1)/(ln —m+1) From last equation we obtain
[[F(anb)(u)t + b(nb)(u),u)](b)]ln—1 = o(1) for 1 = 0,1,...,m — 2, t > 0, u = 1,2,...,z, b = 1,2,...,v,
t
tln-;+1
[[F( a (u)t + bnb)(u),u)](b)]ln-i =-— [1 - o(1)] for i = ; -1, t > 0, u = 1,2,...,z, b = 1,2,...,v.
n( ;'-:)
Hence, considering (50), it appears that
[V(t,u)](b) = lim k^(n )[[R(a{b)(u)t + ^(u),u)](b)]!
niœ i=0
and
[[F(a(b)(u)t + bnb)(u),u)](b)]ln-i = lim n ■ 0 = 0 for t < 0, u = 1,2,...,z, b = 1,2,..., v,
[V(t,u)](b) = limk^ )[[R(af)(u)t + b(nb\u),u)](b)]!
niœ i=0
An-;+1
[[F(anb) (u)t + bnb) (u), u)](b) ]ln-i = lim n(£ ) [1 - o(1)];-1 [1 - o(1)] = t
n
l-;+1
n( ^
for t > 0, u = 1,2,...,z, b = 1,2,..., v,
which means that according to Lemma 4 the limit reliability function of that system is given by (54)-(56). □
Proposition 5. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,...,z, b = 1,2,...,v,
(59)
(ii) kn = n, c << ln, c log n - ln >> s, c > 0, s > 0, m = constant ( m/ ^ 0, as n ^ œ ) or ^ ^ n ,
0 <n < 1, as n iœ.
(iii) anb)(u) =--
[[n(;-1)]ln-;+1 - 1]2(b)(u)(/n - ; +1)
(60)
bnb)(u) = lQg[1 -[n(;l-n1)] ln-;+1], u = 1,2,...,z, b = 1,2,...,v
(61)
then
(62) where
Ä3(t,^) = [1,#3(t,1),...,Ä3(t, z)]:
t e (-œ,œ),
1
i'
%3(t,u) = I pb exp[-exp[t]] for t e (-œ,œ), u = 1,2,..., z,
b=1
(63)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
( I V
n nl.
RkM)i (t,u) = I Pb exp[- exp[t[[n n ]ln-;+1 - 1] A(b)(u)(ln - ; +1)]
b=1
v; -1/
( i \ -— ( i ^ — + log[1 - [n n ] ln-;+'][[n n ]ln-;+1 - 1](ln - ; +1)] for t e (-œ,œ), u = 1,2,..., z.
v; - 1/
v; - 1/
(64)
Proof. Since
and
?nb)(u)t + b(nb) (u) > 0 for t e (-œ,œ), u = 1,2,..., z, b = 1,2,...,i
a{b)(u)t + bnb)(u) ^ 0 as n ^ro for t e (-ro,ro), u = 1,2,...,z, b = 1,2,...,v, then, according to (59) for n large enough, we obtain
[R(anb) (u)t + bnb) (u), u)](b) = exp[-2(b) (u)(aib) (u)t + b(b) (u))] for t e (-ro, ro),
u = 1,2,..., z, b = 1,2,...,v,
and
[F(anb) (u)t + bnb) (u),u)](b) = 1 - exp[-2(b) (u)(anb) (u)t + b'f (u))] for t e (-œ,œ), u = 1,2,..., z, b = 1,2,..., v. Moreover for n large enough, we obtain
[R(anb) (u)t + b'b (u ), u)](b) = exp[-2(b) (u)(anb) (u)t + bb) (u ))]
= 1 - o(-
anb)(u )t + b(b) (u)
)for t e (-œ,œ), u = 1,2,..., z, b = 1,2,..., v,
and considering
a(b) (u) i 0 as n i œ for t e (-œ, œ),
we obtain
[F (anb) (u )t + b(b (u), u)](b) = 1 - exp[-2(b) (u)(anb) (u )t + bb) (u))]
1
= 1 — [1 — — (u )anb) (u )t + o(-L^] exp[——(b) (u bb) (u)]
an)(u)
= 1 — exp[——-b\u)bib)(u )] + o(-1-) + —(b)(u)anb)(u )exp[——b)(u)b(b\u )]t
anb)(u)
for t g (—to, to), u = 1,2,..., z, b = 1,2,..., v. Hence, for each 1 = 0,1,..., m — 1 we have
[[R(a"Wt + b"(u)'u)f)]' =[1 - o( a?'(u» + bf'(u)* '
as n ^ < for t g (-<,<), u = 1,2,..., z, b = 1,2,..., v,
and
[[F (anb)(u)t + bnb)(u), u)](b)]ln -i = [1 - exp[-2(b)(u )b{b)(u)] + o(ab1—)
an (u)
+ l(b) (u )anb) (u) exp[-l(b) (u )b(nb) (u)]t ]ln -i = [1 - exp[-l(b) (u)b(nb) (u)] + o(-b^-)]ln -i
an )(u)
[1 , ^(b)(u)a(b)(u)exp[-A(b)(u)bnb)(u)] , -
[1 +-1-t ]
1 - exp[-l(b) (u)bnb) (u)] + o(^—)
an )(u)
1 ln-i
= [n(J-:) + o(-(b—)] ln-m+: [1 +-t-1-]'n-i for t G (-O,<), u = 1,2,...,z, b = 1,2,...,v.
an (u) (ln - m +1)[1 + o^-^b—)]
an)(u)
From last equation we obtain
[[F(anb)(u)t + bb)(u),u)](b)]ln-i = o(1) for i = 0,1,...,m - 2, t g (-o,<), u = 1,2,...,z, b = 1,2,...,v,
[[F(anb)(u)t + bb)(u),u)](b)]ln-i = n[n(m'-,)]-1 [1 + t n]n-M+ [1 -o(1)]
(ln - m +1)
for i = m -1, t g (-<, <), u = 1,2,..., z, b = 1,2,..., v. Hence, considering (50), it appears that
[V(t,u)](b) = limkn^ )[[R(a(b)(u)t + bnb)(u),u)](b)]1 [[F(a<-b)(u)t + b(nb\u),u)](b)V
„—to 1=0
= limn(J-) [[F(a^(u)t + bf\u),u)](b)]ln-;+1
= lim n(;l-n1) [n(;l-n1)]-1 [1 + --<—^z]ln= exp[t] for t e (-œ, œ), u = 1,2,..., z, b = 1,2,..., v,
(ln - ; +1)
which means that according to Lemma 4 the limit reliability function of that system is given by
(62)-(64). □
Proposition 6. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,...,z, b = 1,2,...,v,
(65)
(ii) kn = n, ln - c log n ~ s, c > 0, s e (-œ,œ), m = constant ( m/ — 0, as n — œ ) or
/ ln
—> n , 0 <n< 1, as n — œ,
(iii) anb)(u ) =
[[n(;l-n:)]ln-;+1 - 1]^(b)(u)(ln - ; + 1)
bnb)(u) = -^j lQg[1 - [n(;l-")^ln-;+1], u = 1,2,...,z, b = 1,2,...,v,
(66) (67)
then
(68) where
Ä3(t,0 = [1,Ä3(t,1),...,Ä3(t, z)]:
t e (-œ,œ),
&3(t,u) = I pb exp[-exp[t]] for t e (-œ,œ),
b=1
(69)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
( n 1
b=1
v; -1/
( ln 1
R; (t, u) = I Pb exp[-exp[t[[n ln ]ln-;+1 -1] A(b (u)(ln - ; + 1) + log[1 - [n *n ] ln-;+1 ]
v; -1/
( I 1 —
[[n n ]ln-;+! - 1](ln - ; +1)] for t e (-œ,œ), u = 1,2,..., z.
v; - 1J
(70)
Proof. Since
,(b)
(u)t + b'lb)(u) > 0 for t e (-œ,œ), b = 1,2,...,v, u = 1,2,...,z,
1
n
and
a(b)(u) ^ 0 as n , b = 1,2,...,v, u = 1,2,...,z, then, according to (65) for n large enough, we obtain
[R(anb)(u)t + bnb)(u),u)](b) = exp[-2(b)(u)(anb)(u)t + ^(u))] for t g (-<,<), b = 1,2,...,v, u = 1,2,...,z, and
[F(anb) (u)t + b™ (u), u)](b) = 1 - exp[-2(b) (u)(anb) (u)t + b^ (u))] for t g (-<, <), u = 1,2,.., z, b = 1,2,.., v. Moreover for n large enough, we obtain
[R(anb) (u)t + bnb) (u), u)](b) = exp[-2(b) (u)(anb) (u)t + bb) (u))]
= [1 -A(b)(u)a(nb)(u)t + o(—1—)] exp[-2(b) (u)b(b) (u)] for t g (-<,<), u = 1,2,..., z, b = 1,2,..., v,
a'„)(u)
and
[ F ab) (u )t + bnb) (u), u )](b) = 1 - exp[-2(b) (u )(anb) (u)t + b^ (u))] = 1 - [1 - 2(b) (u)anb) (u)t + o(-L-)] exp[-2(b) (u)^ (u)]
an (u)
= 1 - exp[-2(b)(u)bnb)(u)] + o(-L, +2(b)(u)anb)(u)exp[-2(b)(u)bnb)(u)]t
an )(u)
for t g (-<, <), u = 1,2,..., z, b = 1,2,..., v.
Hence, for each i = 0,1,...,m -1 for n large enough, we have
[[R(anb) (u)t + bnb) (u),u)](b)r = [1 - 2(b) (u)anb) (u)t + o^^-)]1 exp[-U(b) (u)b^ (u)] ^ 1
a()(u)
as n ^ < for t g (-<,<), u = 1,2,..., z, b = 1,2,..., v,
and
[[F (a(b)(u)t + b(nb)(u\u)](b)]ln -i = [1 - exp[-2(b)(u )b(-b)(u)] + o(-b-)
an (u)
+ —b (u)a(b) (u) exp[——b (u)b(b) (u)]t ]
n
= [1 -exp[-,»(u)b,?>M] + « ')]n-i [, + W'W'M-W'V.y t]n-i
an (u) 1 -expM^MbfM] + 0(^^-0
an (u)
1 ln-i
= [n(m-) + «(-(byr-r)] ln-m+' [1 +-t-1-]ln-i
an (u) (ln - m +1)[1 + o(-1b—)]
an (u)
for t e (-ro,ro), u = 1,2,..., z, b = 1,2,..., v. From last equation we obtain
[[ F (u)(anb) (u)t + b(nb) (u), u)](b)]ln= o(1) for i = 0,1,...,m - 2, t e (-ro,ro), u = 1,2,...,z, b = 1,2,..., v, and
[[F(a(b)(u)t + b(nb)(u\u)](b)]ln- = n[n(m^)]-1 [1 + t n]ln-m+' [1 -o(1)]
(ln - m +1)
for i = m -1, t e (-ro,ro), u = 1,2,..., z, b = 1,2,..., v. Hence, considering (50), it appears that
m-i/ _
n
[V(t,u)](b) = limknlX'r )[[R(anb)(u)t + bnb)(u),u)](b)]i
n
niœ i=0
[[F(anb)(u)t + b(b)(u),u)](b)ln-i = limn(;ln1) [[F(anb)(u)t + bnb)(u),u)](b)]^
, u )r j'n i = lim n( ;l-n ) [[F(a (b)(u)t + b (b)(u) u)](b)]ln-;+!
= l—m n(m'n1) [n(m^)]-1 [1 + t n]ln-m+1 = exp[t] for t e (-œ,œ), u = 1,2,...,z, b = 1,2,...,v,
n—œ (ln - m +1)
which means that according to Lemma 4 the limit reliability function of that system is given by (68)-(69). □
Proposition 7. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions,
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,...,z, b = 1,2,...,v, (71)
(ii) kn = n, ln - c log n >> s, c > 0, s > 0, m = constant ( mA — 0, as n — œ ) or
/ ln
—n, 0 <n< 1, as n — œ ,
(iii) anb)(u )= -1- , (72)
l(b)(u)log n( ml-n1)
b-"(u >= log<
ln - m +1
—-;-, u = 1,2,..., z, b = 1,2,...,V,
(73)
log n( n )
m -1
then
(74) where
Ä3(i,-) = [1,Ä3(i,1),...,Ä3(i, z)],
t G (-œ,œ),
Ä3(i,u) = Épb exp[-exp[t]] for t g (-œ,œ), u = 1,2,...,z,
b=1
(75)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
Rm (t,u) = ÉPb exp[-exp[l(b)(u)tlogn
b=1
r i \ i - m+1 ( i ^ - iog[in m \ !°g n| i"1|]]]
ln V m -
vm -1/
log n
vm -1/
(76)
for t G (-œ,œ), u = 1,2,...,z. Proof. Since
a(nb")(u)t + bnb)(u) ^ +œ as n ^œ for t g (-œ,œ), b = 1,2,...,V, u = 1,2,...,;
and
a<b)(u) ^ 0 as n ^ œ,
then, according to (71) for n large enough, we obtain
[Ria^iu )t + b(nh\u), u )](b) = exp[-2(b) (u )(af)(u )t + b(nh\u))\ for t e (-w, w), u = 1,2,..., z, b = 1,2,..., v, and
[F(aib)(u )t + bnb)(u), u)](b) = 1 - exp[-2(b)(u)(anb)(u)t + bnb)(u))]for t e (-w, w), u = 1,2,..., z, b = 1,2,..., v. Moreover for n large enough, we obtain
[R(anb) (u)t + bnb) (u), u)](b) = exp[-2(b) (u)(anb) (u)t + b'b (u))]
= [1 -A(b)(u)af)(u)t + o(—b-—)]exp[-2(b)(u)bnb)] for t e (-w,w), u = 1,2,...,z, b = 1,2,...,v,
an)(u)
and
[F (a^ (u)t + bnb) (u), u )](b) = 1 - exp[-2(b) (u)^ (u)t + bf» (u))] = 1 - [1 - 2(b) (u)a<b) (u)t + o(-^)] exp[-2(b) (u^^ (u)]
an)(u)
= 1 - exp[-2(b)(u)bnb)(u)] + o(-L, +2(b)(u)anb)(u)exp[-2(b)(u)bnb)(u)]t
an )(u)
for t e (-ro, ro), u = 1,2,..., z, b = 1,2,..., v.
Hence, for each i = 0,1,...,m -1 for n large enough, we have
[[R(anb)(u)t + bnb) (u),u)](b)]i = [1 - 2(b) (u)a(b) (u)t + o(^-)]! exp[-U(b) (u)bnb) (u)] ^ 1
an)(u)
and
as n i œ for t e (-œ, œ), u = 1,2,..., z, b = 1,2,..., v,
[[F(anb)(u)t + bnb)(u), u)](b)]ln = [1 - exp[-2(b)(u )bnb)(u)] + o(-b-)
an (u)
+ A(b) (u)anb) (u) exp[-l(b) {u)b(nb) (u)]t ]ln-
= [1 -exp[-,»(„b^u)] + « '>]n- [1 + ^»»pM'^^(-M t]n-i
a' (u) 1 - exp[-^")(u)b:*)(B)] + o(ab1—)
an (u)
1 ln-i
= [n( ml-n1) + o^-^b^-)] ln-m+1 [1 +-t-1-]ln-i
an (u) (ln - m +1)[1 + o^^—)]
an (u)
for t e (-ro,ro), u = 1,2,...,z, b = 1,2,...,v. From last equation we obtain
[[F(anb)(u)t + bnb)(u),u)](b)]ln- = o(1) for i = 0,1,...,m - 2, t e (-ro,ro), u = 1,2,...,z, b = 1,2,...,
[[F(anb)(u)t + bnb)(u), u)](b)]ln - = n[n( m^)]-1 [1 + t n ]ln-m+1 [1 - o(1)]
(ln - m +1)
for i = m -1, t e (-ro, ro), u = 1,2,..., z, b = 1,2,..., v. Hence, considering (50), it appears that
v
[V(t, u)](b) = lim kn Éln )[[R(anb) (u)t + ^b) (u),u)](b)]
niœ i=o
[[F(a(b)(u)t + bnb)(u),u)](b)ln-i = limn(Jln_l) [[F(a^(u)t + bnb)(u),u)](b)]ln-mt1
niœ
= lim n(J-,) [n(jn,)]-1 [1 + t ]n-m+l = exp[t] for t e (-w,w), u = 1,2,...,z, b = 1,2,...,v.
niw (l n - m +1)
which means that according to Lemma 4 the limit reliability function of that system is given by (74)-(75). □
The next auxiliary theorem is proved in (Kolowrocki 2005). Lemma 5. If
(i) kn = n , ln i l = const, m < ln, as n i w
(ii) ¡H(t) = VPb exp[-[V(t)](b)] is a non-degenerate reliability function,
b=1
(iii) Rn}n (t,-) = [1, Rn}n (t,1),..., R^ (t, z)], t G (-œ,œ), is the reliability function of a
homogeneous regular multi-state "m out of ln "- series system, in variable operation conditions, where
Rk(l (t, u) = É Pb [RZI (t, u)](b), t G (-œ, œ),
n n b=1
and
[ R (Cn (t, u )](b) = ["£" (ln )[1 - [ R(t, u)](b)ï [[ R(t, u)](b)]ln-i ]kn , tG(-œ,œ), u = 1,2,..., z, b = 1,2,..., v,(77)
is its reliability function at the operational state zb, then
#(t,•) = [1,#(t,1), ...,#(t, z)], t e (-w,w), is the multi-state limit reliability function of that system if and only if
lim kn [1 -if ( )[[F(anb)(u )t + bf\u), u)]^]' [[R(a(-b)(u)t + b<-b)(u), u)](b)]ln ] = [V(t, u)](b) (78)
n
niœ i=0
for t G C_ (b), u = 1,2,..., z, b = 1,2,..., v.
[V (u )](b) , , ,
Proposition 8. If components of the multi-state homogeneous, regular "m out of ln "-series system
at the operational state zb
(i) have exponential reliability functions
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[-l(b)(u)t]
for
t > 0, u = 1,2,..., z, b = 1,2,...,v,
(79)
(ii) kn = n, c << ln, c log n - ln >> 5, c > 0, 5 > 0, (ln-;) = ; = const, ( i 1 as n i œ ),
(iii) anb)(u) =
1
^(b)(u)[n( ;+!)]
ln \ i1/(;+1)
bnb)(u) = 0, u = 1,2,...,z, b = 1,2,...,
(80)
then
#2(t,0 = [1,«2(t,1), ...,»2(t,z)], t e (-œ,œ),
(81)
where
m2(t,u) = 1 for t < 0,
(82)
Ä2(t,w) = Ipb exp[-t;+!]for t > 0, u = 1,2,...,z,
b=1
(83)
is the multi-state limit reliability function of that system , i.e. for n large enough we have
R(;) (t, u)= 1 for t < 0,
kn ,ln
(84)
Rn)n (t,u) = I Pb exp[-[tl(b)(u)n
b=1
( l 1 L M
1/ ;+1
v; + 1/
] ; +1 ]
(85)
Proof. Since
for t > 0, u = 1,2,...,
an\u)t + bnb)(u ) =
^(b)(u)[n( ;lni)]1/(; +1)
--0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
ln Yi1/(;+1) ' ' ' ' ' ' ' ' '
and
anb)(u )t + b(nb)(u) =
^(b)(u)[n( ;+1 )]1/(;+1)
> 0 for t > 0, u = 1,2,..., z, b = 1,2,..., v,
then, according to (79), we obtain
[R(anb)(u)t + b(b)(u),u)](b) = 1 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
[F(anb)(u)t + bnb)(u),u)](b) = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
[R(anb) (u)t + bnb) (u ), u)](b) = exp[-2(b) (u)(anb) (u)t + b'f (u ))]
v
z
t
t
= exp[--, t _ ] = 1 - o(—-—1 _ ) for t > 0, u = 1,2,..., z, b = 1,2,..., v,
[n( m+1)]1/(m+1)] ([n(m;+l )]1/(m+1)) , ,,,, , ,
[ F ab (u )t + bnb) (u), u )](b) = 1 - exp[-2(b) (u )(anb) (u)t + bnb) (u))] = 1 - exp[--—t—=—] =-—t—=--o(--1—=—) for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[n(m+1)]1/(m+1)] [n(m+1)]1/(m+1) ([n(m+1)]1/(m+1)) , ,,,, ,,,
Next, for each i = m +1,m + 2,...,ln we have
[[R(a(b)(u)t + bf )(u),u)](b)]ln-i = 1 for t < 0, u = 1,2,...,z, b = 1,2,...,v, [[F(anb)(u)t + bb)(u),u)](b)]! = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
[[R(a{b\u)t + ^(u),u)](b)]ln- = [1 -o([ (n 1 1) )]ln- ^ 1 as n ^ro
[n( m+1)]
for t > 0, u = 1,2,..., z, b = 1,2,..., v,
[[F (anb)(u)t + bb)(u), u)](b)]! = [-—^--o(-, 1 _ )]'
[[ (n () n (),)] ] [[n(m+1)]1/(m+1) ([n(m;+l )]1/(m+1))]
ti i
[1 - o(1)]! for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[n( m+1)]! /(m+1)
From last equation we obtain
[[F(anb)(u)t + bnb)(u),u)](b)]! = o( 1 ) for i = m + 2,m + 3,...,ln, t > 0, u = 1,2,...,z, b = 1,2,...,
n( m+1)
u.m +1
[[F(a(b) (u)t + bnb) (u),u)](b)]' =-— [1 - o(1)] for i = m +1, t > 0, u = 1,2,..., z, b = 1,2,..., v.
n( m+1)
Since
1 - £ ((n )[[F(anb)(u)t + bnb)(u),u)](b)]' [[R(anb)(«)t + b'b(u), w)](b)]ln-i ] i=0
= 1 - £ (n )[[F(anb) (u)t + bb) (u),u)](b)]! [[R(a'nb) (u)t + bnb) («), w)](b) ]ln-i ]
i=0
+ £ (n )[[F(anb) (u)t + bnb) (u),u)](b)r [[R(anb) (u)t + bnb) (u), w)](b) ]ln-i ]
i=m +1
v
= 1 - [[F (a^ (u)t + bf (u), u)](b) + [ R(anb) (u)t + bf (u), u)](b)]ln + I (n )[[F(a<b) (u)t + bf (u), u)](b) ] [[R(a(^) (u)t + b(b) (u), u)](b) ]l"-i ]
i=m+1
= I (in )[[F(a<b) (u)t + bf (u),u)](b)]i [[R(anb) (u)t + b™ (u), M)](b) ]ln-i ]
i=m +1
u = 1,2,...,z, b = 1,2,...,v,
then, considering (78), it appears that
[V(t,u)](b) = limkn[1 - I( )[F^a^^t + bf(u),u)]i [[R(a(b\u)t + b(nb\u),u)](b)]ln-i]
¿=0
= limkn I()[[F(b)(anb)(u)t + bnb)(u),u)](b)]i [[R(a?\u)t + b(nb\u),u)](b)]ln-i]
niw i=m+1
= lim n • 0 = 0 for t < 0, u = 1,2,..., z, b = 1,2,...,v,
and
[V(t,u)](b) = limkn[1 - É((n )[F(b)ab)(u)t + bnb)(u),u)]'' [[R(anb)(u)t + b(nb\u),u)](b)]ln]
niœ '=o
limkn É(ln )[[F(anb)(u)t + bb)(u),u)](b)]' [[R(anb)(u)t + b(-b)(u),u)](b)]ln]
n
niœ ' =m+1
^.m+1
n( m+1)
= lim n(m+J-t—rz [1 - o(1)] = tm+1 for t > 0, u = 1,2,...,z, b = 1,2,..., v.
which means that according to Lemma 5 the limit reliability function of that system is given by (81)-(83). □
Proposition. 9 If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions
[tf(t,u)](b) = 1for t < 0, [tf (t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,...,z, b = 1,2,...,v,
(86)
(ii) kn = n, ln - c log n ~ s, c > 0, s G (-œ,œ), (n - m) = m = const, ( ^ ^ 1 as n ^ œ )
(iii) anb)(u}= )]1/(m+1) , bnb)(u) = 0, u = 1,2,...,z, b = 1,2,...,v, (87)
then
Ä2(t,-) = [1,#2(t,1), ...,#2(t,z)], t e (-œ,œ),
(88)
where
9k(t,u) = 1 for t < 0,
9ti(t,u) = Ipb exp[-t;+1]for t > 0, u = 1,2,...,z,
b=1
is the multi-state limit reliability function of that system , i.e. for n large enough we have
R(;) (t,u)= 1 for t < 0,
Rf) (t,u) =Ipb exp[-[tib)(u)n
b=1
( ln 1
n
v ; + 1y
1/;+1
];+1] for t > 0, u = 1,2,..., z.
Proof. Since
a^(u)t + bnb)(u ) =
^(b)(u)[n( 5l+1)]1/(;+1)
< 0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
and
a^(u)t + bnb)(u ) =
^(b)(u)[n( ;+1 )]1
--— > 0 for t > 0, u = 1,2,..., z, b = 1,2,...,v,
ln \ l!/(; +1) 5 5 5 5
then, according to (86), we obtain
[ R(af')(u)t + b(b)(u ), u )](b) = 1 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
[F(a{b)(u)t + ô+ô)(u),u)](b) = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
\(b)(„\ „Ylb)
and
[R(anb) (u)t + b(b (u ), u )](b) = exp[-2(b) (u )(anb) (u)t + b^ (u ), u )]
= exp[-
i+ )]1/(;+1)] 1 o('
[n( ;+1 )]
[n( ;l+1 )]1/(;+1)
—■) for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[ F (a(b (u)t + ^ (u), u )](b) = 1 - exp[-2(b) (u )(anb) (u)t + b^ (u ))]
= 1 - exp[-
-=—] =-=--o(- _
[n( )]1/(; +1) [n( ;l+1)]1/(; +1) [n( ;l+1)]1/( ; +1)
(89)
(90)
(91)
(92)
for t > 0, u = 1,2,..., z, b = 1,2,..., v.
t
t
1
Next, for each i = m +1,m + 2,...,ln we have
[[R(ajb)(u)t + bf(u),u)]^]1"-i = 1 for t < 0, u = 1,2,...,z, b = 1,2,...,v, [[F(anb)(u)t + b^u),u)]^]' = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
[[R(a<b)(u)t + b(b)(u), u = [1 - o([ ( ln J^.,)]1"-i i 1 as n iw
[n( m +1 )]
for t > 0, u = 1,2,..., z, b = 1,2,..., v, [[F(a^ (u)t + hnh) (u),u)](h)]i = [-—^--o(-, 1 „ )]
[[ (n () ^ (), )] ] [[n(fl+l)]1/(m+-) >(LXTm+-))]
ti i = [ni^m+-) [1 -0(1)]i for t> 0, u =-,2,. .,z, b =-,2,. .,v.
From last equation we obtain
[[F(ajb(u)t + b^u),u)](h)]i = o( -- ) for i = m + 2,m + 3,...,ln, t > 0, u = 1,2,...,z, b = 1,2,...,
n( m+-)
v m +1
[[F(anb)(u)t + b^tM), u)](h)]i =-— [1 - o(1)] for i = m +1, t > 0, u = 1,2,..., z, b = 1,2,..., v.
n( m+i)
Since
1 - I ( )[[F(a(„b) (u)t + bf (u), u)](h)]i [[R(af (u)t + bf (u), u)]^]1"-i i=0
= - -I( )[[F(anb)(u)t + b^u),u)](b)]! [[R(a<b)(u)t + b^^),u)](b)]l"-i
i=0
+ I ( )[[F(a<b) (u)t + bnb) (u),u)](b)r [[R(a<b) (u)t + bf (u),uM^]1"-i
i=m+1
= 1 - [[F (a<b) (u)t + (u), u)](b) + [ R(a<b) (u)t + (u), u)]^]'» + I ( )[[F (a<b) (u)t + bb) (u), u)](b)]! [[R(a<b) (u)t + (u), u -i
i=m+1
= I ( )[[F(a<b) (u)t + b^ (u), u)]b)] [[R(a<b) (u)t + bf (u), u)\(b) ]ln-i
V
u = 1,2,..., z, b = 1,2,...,v,then, considering (78), it appears that
[V(t, u)](b) = lim kn [1 - £ ( )[№nb)(u)t + bf(u), u)](b)] [[R(a(-b)(u)t + bf^u), u )](b)]1n - ] lim kn £ (i« )[[F(a«b) (u)t + b«b) (u), u)](b)]! [[R(a«b) (u)t + b«b) (u), u)](b)]1"]
n
= lim n • 0 = 0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
niœ
and
[F (t, u)](b) = lim kn [1 - £ ((n )[[F(af(H)i + u)](b)] [[R(anb)(u)t + u)](b)]ln-i ]
niœ j-0
= limkn £(n)[F(anb)(u)t + bnb)(u),u)] [[R(a(b)(u)t + ^(u),u)](b)]ln-i]
u.m+1
= lim n(m+1)-— [1 - o(1)] = tm+1 for t > 0, u = 1,2,..., z, b = 1,2,..., v,
niœ n( m+1)
which means that according to Lemma 5 the limit reliability function of that system is given by (88)-(90). □
Proposition 10. If components of the multi-state homogeneous, regular "m out of ln "-series system at the operational state zb
(i) have exponential reliability functions
[R(t,u)](b) = 1for t < 0, [R(t,u)](b) = exp[-l(b)(u)t] for t > 0, u = 1,2,...,z, b = 1,2,...,v,
(93)
(ii) kn = n, ln - c log n >> s, c > 0, s > 0, (n - m) = m = const, ( m< i 1 as n i œ )
(iii) °;')<B )= ¿»CM • bi"(")- "=''b=V (94)
then
#2(t,0 = [1,^2<t,1), . .,^2<t,z)], t e (95)
where
&2(t,u) = 1 for t < 0, (96)
%i(t,u) = £ pb exp[-tm+1 ]for t > 0, (97)
b=1
is the multi-state limit reliability function of that system , i.e. for n large enough we have
R(min (t,u) = 1 for t < 0, (98)
=— V ( i V m+1 _
R^ (t,u) = ÉPb exp[-[tl(b)(u)n _^ ]m+1] for t > 0, u = 1,2,...,z.
b=1
(99)
vm +1/
Proof. Since
^(u)t + bnb) (u) = Â(b)(u)[n(ài )]1/(m+!) < 0 for t < u = 1,2,...,z, b =1,2,...,V,
and
anb)(u)t + bf)(u) = —-n > 0 for t > 0, u = 1,2,...,z, b = 1,2,...,v,
^ (u)[n( fl+l)]1/(m+-)
then, according to (93), we obtain
[R(anb)(u)t + bnb)(u),u)](b) = 1 for t < 0, u = 1,2,...,z, b = 1,2,...,v, [F(anb)(u)t + bnb)(u),u)](b) = 0 for t < 0, u = 1,2,..., z, b = 1,2,..., v,
and
[R(anb) (u)t + bf (u), u )](b) = exp[-2(b) (u)(a(^) (u)t + b(nb (u), u)\ = exp[---—] = 1 - o(-, for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[n( s/+i)\1/(m+-) [n( m:-)]1^+1)7 '
[ F (af) (u )t + bnb) (u), u)]^) = 1 - exp[-2(b) (u^ (u )t + b^ (u), u)]
= 1 - exp[--—t—-—] =-—t—---o(-—1—-—) for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[n(f+l)]1/(m+1)] [n(f+l)]1/(m+-) ([n(f+"l)]1/(m+-)) , , ,, , , -
Next, for each i = m +1,m + 2,...,ln we have
[[R(anb)(u)t + h"h)(u),u)](h)]l"-i = 1 for t < 0, u = 1,2,...,z, b = 1,2,...,v, [[F(aib)(u)t + b^u),u)]^ = 0 for t < 0, u = 1,2,...,z, b = 1,2,...,v,
and
[[R(a{b)(u)t + bnb)(u),u)](b)]ln-i = [1 -o( n ) ^-' i 1 as n i<x>
[n( m +1 )]
for t > 0, u = 1,2,..., z, b = 1,2,..., V,
[[F (a{b)(u)t + Mb)(u), u)](b)]' = [-—^--o(-, 1 _ )|'
[[ (n () n (),)] ] [[n(m+1 )]1/(m+1) ([n(mn+1 )]1/(m+1))] t'
=----- [1 - o(1)]' for t > 0, u = 1,2,..., z, b = 1,2,..., v.
[n(m+1)]' +1) L WJ '
From last equation we obtain
[[F(a(b)(u)t + bnb)(u),u)](b)]' = o(^) for i = m + 2,m + 3,...,ln, t > 0, u = 1,2,...,z, b = 1,2,...,v.
n( £+1)
,m+1
[[F(anb) (u)t + bnb) (u),u)](b)]' =-— [1 - o(1)] for i = m +1, t > 0, u = 1,2,..., z, b = 1,2,..., v.
n( ml+1)
Since
1 - £ ((n )[[F(anb)(u)t + bnb)(u),u)](b)]' [[R(anb) (u)t + b(b)(u),u)](b)]ln-' ] i=0
= 1 -£(ln)[[F(anb)(u)t + bnb)(u),u)](b)](b)]' [[R(anb)(u)t + bb)(u),u)](b)]ln-']
i=0
+ £ (in )[[F(anb) (u)t + bnb) (u),u)](b)]' [[R(a(b) (u)t + bnb) (u),u)](b)]ln-' ] +1
= 1 - [[F (anb) (u)t + bnb) (u), u)](b) + [ R(anb) (u)t + ^ (u), u)](b)]ln £ (ln )[[F(af) (u)t + bnb) (u), u)](b) ]' [[R(a™ (u)t + bf (u),u)](b)]ln-' ]
=m+1
= £(ln )[[F(anb)(u)t + bb)(u),u)](b)]' [[R(anb)(u)t + bnb)(u),u)](b)]ln-'], u = 1,2,...,z, b = 1,2,...,v,
i=m+1
then, considering (78), it appears that
[V(t,u)](b) = limkn[1 -£((n )[[F(a(b)(u)t + bnb)(u),u)](b)]' [[R(a(b)(u)t + b(nb)(u),u)](b)]ln-']
ni<» ' = 0
= limkn £('n)[[F(a(b)(u)t + bnb)(u),u)](b)]' [[R(a(b)(u)t + b(nb)(u),u)](b)]ln-']
ni i=m+1
i=m +1
+
i=m+1
= lim n • 0 = 0 for t < 0, u = 1,2,...,z, b = 1,2,..., v,
and
[V(t,u)](b) = limkn[1 - £((")[[F(a"b)(u)t + b^u),u)]^ [[R(anh)(u)t + bnh)(u),u)]^]'"-i] i=0
= limk" £((")[[F(b)(a"b)(u)t + b"b)(u),u)](b)]! [[R(anh)(u)t + b"b\u),u)]1^"-i] i=m+i
v m+1
= lim n(m+1)-— [1 - o(1)\ = tm+- for t > 0, u = 1,2,...,z, b = 1,2,..., v,
niro n( l" A
"vm+i >
which means that according to Lemma 5 the limit reliability function of that system is given by (95)-(97). □
6 CONCLUSION
The purpose of this paper is to give the method of reliability analysis of multi-state "m out of l"- series systems in variable operation conditions. 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.
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International Journal of Automation and Computing Vol. 3, No 2: 199-206.
