CC BY
7
Kwiatuszewska-Sarnecka Bozena
Gdynia Maritime University, Poland
On asymptotic approach to reliability improvement of multi-state systems with components quantitative and qualitative redundancy: series and parallel systems
Keywords
reliability improvement, limit reliability functions Abstract
The paper is composed of two parts, in this part after introducing the multi-state and the asymptotic approaches to system reliability evaluation the multi-state homogeneous series and parallel systems with reserve components are defined and their multi-state limit reliability functions are determined. In order to improve of the reliability of these systems the following methods are used: (i) a warm duplication of components, (ii) a cold duplication of components, (iii) a mixed duplication of components, (iv) improving the reliability of components by reducing their failure rate. Next, the effects of the systems' reliability different improvements are compared.
1. Introduction
Most real systems are very complex and it is difficult to analyze and to improve their reliability. Large numbers of components and subsystems and their operating complexity cause that the evaluation of their reliability is complicated. As a rule these are series systems, parallel systems or "m out of n" systems composed of a large number of components. One of the important techniques for reliability evaluation of large systems is the asymptotic approach. The mathematical methods are based on the limit theorems of order statistics distributions considered in a wide literature. These theorems generated investigations on limit reliability functions for systems with two-state components. Next, more general systems with multistate components began to be considered. The asymptotic approach is also very useful in reliability improvement of large multi-state systems because of simplifying the calculation.
2. Multi-state and asymptotic approach
In multi-state reliability analysis presented in this paper it is supposed that:
- Ei, i = 1,2,...,n, are components of a system,
- all components and a system under consideration have the state set {0,1,...,z},
- the state indices are ordered, the state 0 is the worst and the state z is the best,
- Ti(u), i = 1,2,...,n, are independent random variables representing the lifetimes of the components Ei in the state subset {u,u+1,...,z} while they were in the state z at the moment t = 0,
- T(u) is a random variable representing the lifetime of a system in the state subset {u,u+1,...,z} while it was in the state z at the moment t = 0,
- the system state degrades with time t without repair,
- ei(t) is a component Ei state at the time t, t > 0,
- s(t) is a system state at the moment t, t > 0.
Definition 2.1. A vector
Rt (t,-) = [Rt (t,0), Rt (t,1),...,Rt (t, z)],
t e(-<x>,<x>), i = 1,2,...,n,
where
Rt (t, u) = P(et (t) > u | ei (0) = z) = P(Tt (u) > t), t e(-<x>,<x>), u = 0,1,...,z,
is the probability that the component Ei is in the state subset {u,u+1,...,z} at the time t, t e(-<x>,<x>) while it was in the state z at the moment t = 0, is called the multistate reliability function of a component Ei.
Definition 2.2. A vector
Rn (t,-) = [ Rn (t,0), Rn (t,l),..., Rn (t, z)],
t e(-<x>,<x>),
where
Rn (t, u) = P(s(t) > u | 5(0) = z) = P(T(u) > t), t e(-<x>,<x>), u = 0,1,...,z,
is the probability that the system is in the state subset {u,u+1,...,z} at the moment t, t e(-<x>,<x>) while it was in the state z at the moment t = 0, is called the multi-state reliability function of a system.
In the asymptotic approach to system reliability analysis we are interested in limit distributions of a standardized random variable
(T(u) - bn(u))/an(u), u = 1,2,...,z,
where T(u) is the lifetime of the system in the state subset {u,u+1,...,z} and an (u) > 0, bn (u) e (-<, <), u = 1,2,...,z, are some suitably chosen numbers, called normalizing constants.
Since
P((T(u) - bn (u))/ an (u) > t)
= P(T(u) > an (u)t + bn (u)) = Rn (an (u)t + bn (u), u), u = 1,2,...,z, where
Rn(t,-) = [Rn(t,0),Rn(t,1),...,Rn(t,z)], t e(-¥,<),
is the multi-state reliability function of the system, then we assume the following definition.
Definition 2.3. A vector
^ (t,-) = [1,^ (t,1),...,^ (t, z)], t e(-<,<),
is called the limit multi-state reliability function of the system if there exist normalizing constants an(u) > 0, bn(u) e(-< ,< ) such that
lim Rn (an (u)t + bn (u), u) = ^ (t, u), t eC<R(u), u = 1,2,...,z,
where C^(u) is the set of continuity points of ^(t,u).
The knowledge of the system limit reliability function allow us, for sufficiently large n, to apply the following approximate formula
Rn (t,-) = * ((t - bn (u))/ a„ (u),-), t e(-¥,¥). (1)
3. System reliability improvement
3.1. Reliability improvement of a multi-state series system
Definition 3.1. A multi-state system is called a series system if its lifetime T(u) in the state subset {u,u+1,...,z} is given by
T (u) = min{Ti (u)}, u = 1,2,...,z.
E1 E2 ---- En-1 En
Figure 1. The scheme of a homogeneous series system
Definition 3.2. A multi-state series system is called homogeneous if its component lifetimes Ti(u) in the state subsets {u,u+1,...,z} have an identical distribution function
Fi(t,u) = F(t,u), u = 1,2,...,z, t e(-<,<), i = 1,2,...,n,
The reliability function of the homogeneous multi-state series system is given by
Rn (t,-) = [1, Rn (t,1),..., Rn (t, z)],
where
Rn(t,u) = [R(t,u)]n, t e(-<,<), u = 1,2,...,z.
Definition 3.3. A multi-state series system is called a system with a hot reserve of its components if its lifetime T(1)(u) in the state subset {u,u+1,...,z} is given by
T(1)(u) = min{max{T..(u)}}, u = 1,2,...,z,
1<i<n 1<. <2 1
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components.
The reliability function of the homogeneous multistate series system with a hot reserve of its components is given by
IRn<x'(t,-) = [1, IRn(1)(t,1),..., IRn(1) (t,z)],
where
IRn(1)(t,u) = [1 -(F(t,u))2]n,t e(-<,<). (2)
Lemma 3.1. If
(i) I—(1)(t,u) = exp[- V(t,u) ], u = 1,2,...,z, is non-degenerate reliability function,
(ii) lRn(1)(t,u), t e (-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multistate series system whit a hot reserve of its components defined by (2),
(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z, then
lim lRn<x>(an(u)t + bn(u)) = I—(1)(t,u), t e C - , u = 1,2,...,z,
if and only if
lim n[F(an (u)t + bn (u))]2 = V(t,u), t e C- , u = 1,2,...,z.
Proposition 3.1. If components of the homogeneous multi-state series system with a hot reserve of its components have multi-state exponential reliability functions
and an(u) = —1—, bn(u) =0, u = 1,2,...,z,
l (u)
then
I—(1) (t,u) = 1, t < 0,
I—(1) (t,u) = exp[-t2], t > 0, u = 1,2,...,z,
is its limit reliability function.
The proof of Proposition 3.1 is given in [9].
Corollary 3.1. The reliability function of exponential series system whit a hot reserve of its components is given by
lRn <x>(t, u) = 1, t < 0,
IRn(1)(t,u) @ exp[-12 (u)nt2], t > 0 , u = 1,2,...,z. (3)
Definition 3.4. A multi-state series system is called a system with a cold reserve of its components if its
lifetime T(2)(u)
in the state subset {u,u+1,...,z} is given
by
T(2)(u) = min{£Tj (u)}, u = 1,2,...,z,
1<i<n j-1 J
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components. The reliability function of the homogeneous multi-state series system with cold reserve of its components is given by
IRn(2)(t,-) = [1, IRn(2) (t,1),..., IRn(2)(t,z)], where
IRn(2)(t,u) = [1 -F(t,u)*F(t,u)]n, t e (-<,<), (4) u = 1,2,...,z.
Lemma 3.2. If
(i) I—(2)(t, u) = exp[- V(t, u)] u = 1,2,...,z, is non-degenerate reliability function,
(ii) IRn(2)(t,u), t e (-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multistate series system whit a cold reserve of its components defined by (4),
(iii) an (u) > 0, bn (u)e (-<,<), then
lim IRn<2)(an(u)t + bn(u)) = I—(2)(t,u), t e CI—, u = 1,2,...,z,
if and only if
lim n[F(an (u)t + bn (u)) * F(an (u)t + bn (u))]
= V(t,u), t e CV , u = 1,2,...,z.
Proposition 3.2. If components of the homogeneous multi-state series system with a cold reserve of its components have multi-state exponential reliability functions
and an (u) =-—, bn (u) = 0, u = 1,2,...,z,
l (u^ n
then
I—(2)(t, u) = 1, t < 0, I—(2)(t,u)= exp[-t2], t > 0, u = 1,2,...,z, is its limit reliability function. The proof of Proposition 3.2 is given in [9].
Corollary 3.2. The reliability function of exponential series system whit a cold reserve of its components is given by
IRn(2>(t, u) = 1, t < 0,
IRn(2) (t, u) @ exp[-12 (u)nt1 / 2], t > 0 u =1,2,...,z.
(5)
Definition 3.5. A multi-state series system is called a system with a mixed reserve of its components if its lifetime T3)(u) in the state subset {u,u+1,...,z} is given by
T (3)(u)
2
= min{ min {max{Ti1 (u)}}, min { 2 T. (u)}},
1<i<51« 1< 1 <2 S1H+1<i<n 1=1
u = 1,2,...,z,
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components and s1, s2, where s1+ s2 = 1 are fractions of the components with hot and cold reserve, respectively.
The reliability function of the homogeneous multi-state series system with a mixed reserve of its components is given by
IRn (3)(t ,• ) = [1, IRn ^(t,1),..., IRn w(t, z )],
(3)
(3).
where
IRn (3) (t, u) = [1 - (F (t, u))2]S1n [1 - F (t, u) * F (t, u)]S2n , (6) t e (-<,<), u = 1,2,...,z.
Lemma 3.3. If
(i) (3)(t,u) = exp[ -V(t,u)], u = 1,2,...,z, is non-degenerate reliability function,
(ii) IRn(3) (t,u), t e(-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multistate series system whit a mixed reserve of its components defined by (6),
(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z,
then
lim IRn{i> (an (u)t + bn (u)) = IÂ (3) (t, u), t e CIÂ , u = 1,2,...,z,
if and only if
lim 2ns 1 [F(an (u)t + bn (u))]
+ ns2 [F(an (u)t + bn (u)) * F(a„ (u)t + bn (u))] = V(t, u), t e CV , u = 1,2,...,z.
Proposition 3.3. If components of the homogeneous multi-state series system with a mixed reserve of its components have multi-state exponential reliability functions
and an(u)=
then
1
l (u)^
, bn (u) = 0, u = 1,2,...,z,
IÂ (t, u) = 1 for t < 0,
IÂ(t,u) = exp[-(2s:+s2) t2/2] , t > 0, u = 1,2,...,z, is its limit reliability function.
Corollary 3.3. The reliability function of exponential series system whit mixed reserve of its components is given by
IRn <y>(t, u) = 1, t < 0,
IRn <y>(t, u) @ exp[-12 (u)n(2sj + s 2)t2 /2], t > 0, (7) u = 1,2,...,z.
Proposition 3.4. If components of the homogeneous multi-state series system have improved component reliability functions i.e. its components failure rates l(u) is reduced by a factor p(u), p(u)e <0,1>, u = 1,2,...,z, i.e.
R (t,u) = 1, t < 0,
R(t,u) = exp[-1 (u) p (u )t ], t > 0, l (u )> 0, u = 1,2,.,z,
and an(u) = then
1
1 (u ) p (u)n
bn(u) = 0, u = 1,2,...,z,
I^ (4)(t, u) = 1, t < 0, I^(4)(t,u) = exp[-t], t > 0, u = 1,2,...,z, is its limit reliability function.
Corollary 3.4. The reliability function of exponential series system whit improved reliability functions of its components is given by
IRn (4>(t, u) = 1, t < 0,
IRn<4)(t,u) = exp[-1(u)np(u)t], t > 0, u = 1,2,...,z. (8)
3.2. Reliability improvement of a multi-state parallel system
Definition 3.6. A multi-state system is called a parallel system if its lifetime T(u) in the state subset {u,u+1,...,z} is given by
T(u) = max{T (и)}, u = 1,2,...,z.
1<i<n
IRn(1)(t,. ) = [1, IRnw(t,1),..., IRnW(t,z)],
n(1)(t,1),..., IRn(1)'
Figure 2. The scheme of a homogeneous parallel system
Definition 3.7. A multi-state parallel system is called homogeneous if its component lifetimes Ti(u) in the state subsets {u,u+1,...,z} have an identical distribution function
Fi(t,u) = F(t,u), u = 1,2,...,z, t e(-<,<), i = 1,2,...,n.
The reliability function of the homogeneous multi-state parallel system is given by
Rn(t,-) = [1, Rn(t,1),..., Rn(t, z)],
where
Rn(t,u) = 1 -[F(t,u)]n, t e(-<,<), u = 1,2,...,z.
Definition 3.8. A multi-state parallel system is called a system with a hot reserve of its components if its lifetime T(1)(u) in the state subset {u,u+1,...,z} is given by
T(1)(u) = max{max{Ti;.(u)}}, u = 1,2,...,z,
1<i<n 1< j<2 J
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components.
The reliability function of the homogeneous multi-state parallel system with a hot reserve of its components is given by
where
IRnw (t, u) = 1 - [(F(t, u))2]n, t e(-œ,œ), u = 1,2,...,z.
(9)
Lemma 3.4. If
(i) I— (1) (t,u) = exp[- V(t,u) ], u = 1,2,...,z, is non-degenerate reliability function,
(ii) IRn(1)(t,u), t e (-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multistate parallel system whit a hot reserve of its components defined by (9),
(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z,
then
lim IRnw (an (u)t + bn (u)) = IÂ (t, u), t e CIÂ :
(1)
n
u = 1,2,...,z,
if and only if
lim 2n[R(an (u)t + bn (u))] =V(t,u), t e CV, u = 1,2,...,z.
Proposition 3.5. If components of the homogeneous multi-state parallel system with a hot reserve of its components have multi-state exponential reliability functions
and an(u) = , bn(u) = 2 , u = 1,2,...,z,
l(u) l(u)
then
/ (1)(t) = 1 - exp[-exp[-t]], t e (-œ,œ), u = 1,2,...,z, is its limit reliability function. Proof: Since for all fixed u, we have
, / x t + log2n
an (u)t + bn (u) =--as n ® <x>
l (u)
for t e (-¥,œ), u = 1,2,...,z.
Therefore
F(t, u) = lim 2nR(an (u)t + bn (u))
= lim 2n exp[-l(u)(an (u)t + bn (u))]
= lim 2n exp[-t - log 2n]
= exp[-t], t e (-¥, <),
which by Lemma 3.4 completes the proof. Corollary 3.5. The reliability function of exponential parallel system whit a hot reserve of its components is given by
IRn(1) (t, u) @ 1 - exp[- exp[-1(u)t + log 2n],
t e (-¥,¥), u = 1,2,...,z.
(10)
Definition 3.9. A multi-state parallel system is called a system with a cold reserve of its components if its lifetime T(2)(u) in the state subset {u,u+1,...,z} is given by
T(2)(u) = max{2Tij (u)}, u = 1,2,...,z,
1<i<n j=1
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components.
The reliability function of the homogeneous multi-state parallel system with a cold reserve of its components is given by
IRn(2)(t,• ) = [1, IRn{2> (t,1),...,IRn{2>(t,z)],
(2)
(2).
where
IRn(2) (t, u) = 1 - [F(t, u) * F(t, u)]n, t e (-œ,œ), u = 1,2,...,z.
(11)
Lemma 3.5. If
(i) rn (2)(t,u) = exp[-V(t,u)], u = 1,2,...,z, is non-degenerate reliability function,
(ii) IRn(2)(t,u), t e (-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multistate parallel system whit a cold reserve of its components defined by (11),
(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z,
then
lim IRn(2) (an (u)t + bn (u)) = m (2) (t, u), t e C H , u = 1,2,...,z,
if and only if
lim n[1 - F(an (u)t + bn (u)) * F(an (u)t + bn (u))]
= V(t,u), t e CV, u = 1,2,...,z.
Proposition 3.6. If components of the homogeneous multi-state parallel system with a cold reserve of its components have multi-state exponential reliability functions
1 exp[1b„(u)] , , and an (u) =-, -= n, u = 1,2,...,z,
1 (u) 1 (u)bn (u )
then
I^(2) (t,u) = 1 - exp[-exp[-t]], t e (-<,<), u = 1,2,...,z,
is its limit reliability function.
Proof: Since for all fixed u, we have
an (u )t + bn (u) ®¥ as n , t e (-<, <), u = 1,2,...,z.
Therefore
V(t,u)
= lim n[1 - F(a„ (u)t + bn (u)) * F(a„ (u)t + bn (u))] = lim n[(1 +1(u)(an (u)t + bn (u))) -
exp[-1 (u)(an (u)t + bn (u))]
= lim n[(1 +1 +1 (u)bn (u)) exp[-(t +1 (u)bn (u))]
= lim n[_1±t_+ 1 (u)b"(u) ]exp[-t]
exp[1 (u)bn (u)] exp[1 (u)bn (u)]
= exp[-t] for t e (-<, <),
which by Lemma 3.5 completes the proof.
Corollary 3.6. The reliability function of exponential parallel system whit a cold reserve of its components is given by
IRn(2)(t,u) @ 1 -exp[-exp[-1(u)t + 1(u)bn(u)], (12) t e (-<,<), u = 1,2,...,z.
Definition 3.10. A multi-state parallel system is called a system with a mixed reserve of its components if its lifetime T(3)(u) in the state subset {u,u+1,...,z} is given by
T(3) (u) = max{ max {maxT (u)}},
1<i<S1n 1< j<2 J
max { £ Tj (u)}},
S]n+1<i<n j=\
u = 1,2,...
where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components and si, s2, where si+ s2 = 1 are fractions of the components with hot and cold reserve, respectively.
The reliability function of the homogeneous multi-state parallel system with mixed reserve of its components is given by
IRn{i>(t,-) = [1, IRnPJ(t,1),...,IRnw(t,z)], where
IRn (3) (t, u) = 1 - [(F (t, u))2]s1n [F (t, u) * F (t, u)]s2n , (13) t e (-<,<), u = 1,2,...,z.
Lemma 3.6. If
(i) I— (3)(t,u) = exp[-V(t,u)], u = 1,2,...,z, is non-degenerate reliability function,
(ii) IRn(3) (t,u), t e(-<,<), u = 1,2,...,z, is the reliability function of non-degenerate multi-state parallel system with a mixed reserve of its components defined by (13),
(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z, then
lim IRn<3)(an (u)t + bn (u)) = IÂ (3)(t, u) ,te C Â ,
= тя? (3),
u = 1,...,z, if and only if
lim 2ns1 [R(an (u)t + bn (u))]
+ ns2[1 - F (an (u)t + bn (u)) * F (an (u)t + bn (u))] = V(t,u), t e CV, u = 1,2,...,z.
Proposition 3.7. If components of the homogeneous multi-state parallel system with a mixed reserve of its components have multi-state exponential reliability functions and
an (u) =
then
1
exp[1 (u )bn (u)]
1(u)bn (u) -1 1(u)bn (u)
I—(3) (t,u) =1 - exp[-exp[-t]], te (-<,<), u = 1,...,z, is its limit reliability function.
Proof: Since for all fixed u, we have
an (u)t + bn (u) ® < as n ® <, t e (-<, <),
and
1
l (u)bn (u ) -1 Therefore
->0 as n , t e (-¥, œ).
F(t, u) = lim n[2^j exp[-1 (u)(an (u)t + bn (u))]
+ 52 [(1 + 1 (u)(an (u)t + bn (u)))
exp[-1 (u)(an (u)t + bn (u))]]]
= lim n exp[-1 (u)(an (u)t + bn (u))]
521 (u)(an (u)t + bn (u))
[1+--]
5 21 (u)(an (u)t + bn (u ))
= lim exp[-l(u)an (u)t]
exp[-1 (u)bn (u) + log ns 21 (u)bn (u)
[1 + +o(1)]][1 + o(1)] = exp[-t ],
t e (-<,<), u = 1,2,...,z,
which by Lemma 3.6 completes the proof.
Corollary 3.7. The reliability function of parallel system whit a mixed reserve of its components is given by
IRn (3)(t, u)
@ 1 - exp[ - exp[ -1 (ub(u) 11 + (l (u)bn (u) -1)],
bn (u )
t e (-œ,œ), u = 1,2,...,z. (14)
Proposition 3.8. If components of the homogeneous multi-state parallel system have improved component reliability functions i.e. its components failure rates l(u) is reduced by a factor p(u), p(u)e<0,1>, u = 1,2,...,z, and
= 5 2 n
an(u) =
1 t / x log n , 0
-, bn(u) =---, u = 1,2,...,z,
1(u)p(u) 1(u)p(u)
then
M(4)(t,u) = 1 -exp[-exp[-t]], t e (-<,<), u = 1,2,...,z,
is its limit reliability function.
Corollary 3.8. The reliability function of exponential parallel system whit improved reliability functions of its components is given by
IRn(4>(t, u) @ 1 - exp[- exp[-1(u)p(u)t + log n], (15)
t e (-<,<), u = 1,2,...,z.
4. Comparison of reliability improvement effects
The comparisons of the limit reliability functions of the systems with different kinds of reserve and such systems with improved components allow us to find the value of the components decreasing failure rate factor p(u), which warrants an equivalent effect of the system reliability improvement.
4.1 Series system
The comparisons of the system reliability improvement effects in the case of the reservation to the effects in the case its components reliability improvement may be obtained by solving with respect to the factor p (u) = p (t, u) the following equations
m (4) ((t -bn (u))/ (u))
= M (k) ((t - bn (u))/an (u)), u = 1,2,...,z, (16)
k = 1,2,3.
The factors p (u) = p (t, u) decreasing components failure rates of the homogeneous exponential multistate series system equivalent with the effects of hot, cold and mixed reserve of its components as a solution of the comparisons (16) are respectively given by
k = 1 p (u) = p (t, u) = 1 (u)t, u = 1,2,...,z,
k = 2 p (u) = p (t, u) =
k = 3 p (u) = p (t, u) =
1(u)t
u = 1,2,...,z,
2s, + s2
-1 (u)t, u = 1,2,...,z.
4.2. Parallel system
The comparisons of the system reliability improvement effects in the case of the reservation to the effects in the case its components reliability improvement may be obtained by solving with respect to the factor p (u) = p (t, u) the following equation
IÂ (4) ((t - bn (u))/ an (u))
= IÂ (k)((t-bn(u))/an(u)), u = 1,2,...,z, k = 1,2,3.
(17)
The factors p (u) = p (t, u) decreasing components failure rates of the homogeneous exponential multistate parallel system equivalent with the effects of hot, cold and mixed reserve of its components as a solution of the comparisons (17) are respectively given by
k = 1 p (u) = p (t, u) = 1 -
k = 2 p (u) = p (t, u) = 1 -u = 1,2,...,z,
log 2 1(u)t
, u = 1,2,...,z,
1 (u)bn (u) - log n 1(u)t
k =3
p (u) = p (t, u) =
1(u)bn(u) -1 1 (u )
1 (u)bn (u) -1 - log n
1(u)t
u = 1,2,...,z.
5. Conclusion
Proposed in the paper application of the limit multistate reliability functions for reliability of large systems evaluation and improvement simplifies calculations. The methods may be useful not only in the technical objects operation processes but also in their new processes designing, especially in their optimization. The case of series, parallel and "m out of n" (in part 2) systems composed of components having exponential reliability functions with double reserve of their components is considered only. It seems to be possible to extend the results to the systems having other much complicated reliability structures and components with different from the exponential reliability function. Further, it seems to be reasonable to elaborate a computer programs supporting calculations and accelerating decision making, addressed to reliability practitioners.
References
[1] Barlow, R. E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York.
[2] Chernoff, H. & Teicher, H. (1965). Limit distributions of the minimax of independent identically distributed random variables. Proc. Americ. Math. Soc. 116, 474-491.
[3] Kolowrocki, K. (2000). Asymptotic Approach to System Reliability Analysis. System Research Institute, Polish Academy of Sciences, Warsaw, (in Polish).
[4] Kolowrocki, K. (2001). On limit reliability functions of large multi-state systems with ageing components, Applied Mathematics and Computation, 121, 313-361.
[5] Kolowrocki, K. (2004). Reliability of Large Systems. Elsevier.
[6] Kolowrocki, K., Kwiatuszewska-Sarnecka, B. at al. (2003). Asymptotic Approach to Reliability Analysis and Optimisation of Complex Transportation Systems. Part 2, Project founded by Polish Committee for Scientific Research, Gdynia: Maritime University, (in Polish).
[7] Kwiatuszewska-Sarnecka, B. (2002). Analyse of Reserve Efficiency in Serie5 Sy5tem5. PhD thesis, Gdynia Maritime University, (in Poli5h).
[8] Kwiatuszewska-Sarnecka, B. (2006). Reliability improvement of large multi-state series-parallel systems. International Journal of Automation and Computing 2, 157-164.
[9] Kwiatuszewska-Sarnecka, B. On reliability improvement of large series systems with components in hot reserve. Applied Mathematic5 and Computation, (in review).
[10] Sarhan, A. (2000). Reliability equivalence of independent and non-identical components series systems. Reliability Engineering and Sy5tem Safety 67, 293-300.
[11] Xue, J. (1985). On multi-state system analysis, IEEE Tran5action5 on Reliability, 34, 329-337.
