ASYMPTOTIC
APPROACH TO RELIABILITY EVALUATION OF LARGE SYSTEMS IN VARIABLE OPERATION
CONDITIONS
Joanna Soszyńska
Maritime University,
e-mail: joannas@am.gdynia.pl
Keywords
reliability function, semi-markov
process, large multi-state system
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 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 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 ln- series 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 
as the process with
discrete states from the set
In practice a convenient assumption is that Z(t) is a semi-markov
process [1] with its conditional sojourn times 
at the operation state
when its next
operation state is 
In this case this
process may be described by:
- the vector of probabilities of the initial operation states 
- the matrix of the probabilities of its transitions between the states 
,
- the matrix of the conditional distribution functions 
of the sojourn times 
If the sojourn times , b, l 
have Weibull
distributions with parameters 
, i.e., if for 
= P(
< t) = 
then their mean values are determined
by
(1)
The unconditional distribution functions of the process 
sojourn times 
at the operation
states 
are given by
1 - exp[-
t]]
(2)
and, considering (1), their mean values are
E[
] =
, (3)
and variances are
D[] 
(4)
where, according to (2),
b = 1,2,...,v.
Limit values of the transient probabilities
at the operation states 
are given by
=
(5)
where 
are given by (3) and the
probabilities 
of the vector 
satisfy the system of
equations
3. Multi-state “m
out of 
”- 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},
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
One
of basic multi-state reliability structures with components degrading in time
are “m out of 
”- series systems.
To
define them, we additionally assume that Eij,
i = 1,2,...,kn, j =
1,2,...,li, kn, l1, l2,...,
Î N, are components of a system, Tij(u), i = 1,2,...,kn, j = 1,2,...,li,
kn, l1, l2,...,
Î N, are independent random variables representing the lifetimes of components Eij in the
state subset 
while they
were in the state z at the moment t = 0, eij(t) are components Eij states at the moment t, 
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, 
Definition
Rij(t
) = [Rij(t,0), Rij(t,1),..., Rij(t,z)],
where
Rij(t,u) = P(eij(t) ³ u | eij(0) = z) = P(Tij(u) > t)
for
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 
at the moment t, 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
R
(t) = [1, R(t,0), R(t,1),..., R(t,z)],
where
R(t,u) = P(s(t)
³ u | s(0) = z) = P(T(u)
> t)
for
, u = 0,1,...,z, is the probability that the system is in the reliability state
subset 
at the moment t, 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 
we have Rij(t,0) = 1 and R(t,0) = 1.
Definition ”- series if its lifetime T(u) in the state subset is given by
where
is mi-th maximal statistics in
the random variables set
,
Definition
”-series system is called regular if
l1 =
l2 = . . . = 
= ln
and m1 = m2 =...=
= m, ln ,
mÎ N,
m £ ln.
Definition ”-series system is called homogeneous if its component
lifetimes 
(u) have an
identical distribution function, i.e.
i.e. if
its components 
have the same reliability
function, i.e.
R(t,u)
= 
F(t,u), 
From the
above definitions it follows that the reliability function of the homogeneous
and regular “m out of ”- series system is given by [5]
(6)
where
, tÎ<0,¥), 
(7)
or by
(8)
where
, tÎ<0,¥), (9)
where 
is the number of “m out of ” series connected subsystems and
is the number of
components of the “m out of ” subsystems.
Under these definitions, if 
(t,u) = 1 for t £ 0, 
or 
= 1 for t £ 0, then
M(u) = 
u = 1,2,..., z, (10)
or
M(u) = 
u = 1,2,..., z,
(11)
is
the mean lifetime of the multi-state non-homogeneous regular “m out of ”- series system in the reliability state subset 
and the variance is given by
2
(12)
or by
2
(13)
The
mean lifetime 
of this system in the
particular states can be determined from the following relationships
(14)
Definition
r(t)
= P(s(t) < r | s(0) = z) = P(T(r) £ t),
that
the system is in the subset of states worse than the critical state r, r
Î{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 ”- series system.
Considering
Definition
6 and Definition 2, we have
r(t)
= 
(t,r), (15)
and
if t
is the moment when the system risk function exceeds a permitted level d, then
r
(16)
where
r
, if it exists, is the inverse function of the risk
function r(t).
4. Multi-state “m
out of ”- series system in
its operation process
We
assume that the changes of the process Z(t) states have an influence on the system
components reliability and the
system reliability structure as well. Thus, we denote the conditional
reliability function of the system component while the system is at
the operational state by
= [1, 
..., 
],
where for
and the conditional reliability function of the system while the system
is at the operational state by
= [1, 
,..., 
for 
where according to (7), we have
for
or by
= [1, 
,..., 
for
where according to (9), we have
for 
The
reliability function 
is the conditional probability that
the component lifetime 
in the reliability
state subset is not less than t, while the process Z(t)
is at the operation state 
. Similarly, the reliability function or
is the conditional probability
that the system lifetime 
in the reliability
state subset is not less than t, while the process Z(t)
is at the operation state 
In the case when the
system operation time 
is large enough, the
unconditional reliability function of the system
= [1, 
,..., 
],
where
for
or
= [1, 
,..., 
],
where
for
and 
is the unconditional
lifetime of the system in the reliability state subset is given by
(17)
or
(18)
for 
and the mean values and variances of the
system lifetimes in the reliability state subset are
for
(19)
where
Mb(u) = 
(20)
or
= 
(21)
and
2
(22)
or
2
(23)
for 
and 
are given by (4).
The
mean values of the system lifetimes in the particular reliability states 
by (14), are
, 
(24)
5. Large multi-state “m
out of ”- series system in
its operation process
Definition
where
is called a limit reliability
function of a multi-state homogeneous regular “m out of ”- series system in its operation process with reliability
function
= [1, 
,..., 
or
= [1, ,...,
where 
are given by (17) and
(18) if there exist normalising constants
such that for 
,
or
Hence, the following approximate
formulae are valid
(25)
or
(26)
Lemma 1
If
(i)
const, 
, 
, m= const, 
,
(ii) 
=
is
a non-degenerate reliability
function,
(iii) 
, tÎ(-¥,¥), is the reliability function
of a homogeneous regular
multi-state “m
out of 
”- series system, in variable
operation
conditions, where
t Î (-¥,¥),
where
, (27)
t Î (-¥,¥),
is its reliability function at the operational state ,
then
, t Î (-¥,¥),
is the multi-state limit reliability
function of that system if and only if
,
(28)
Proof. Since
, t Î (-¥,¥),
where
,
and 
defined by the
equation (27) is the reliability function of a
multi-state homogeneous regular “m
out of ”- series system at the
operational state , then according to
the Definition
7
, t Î (-¥,¥), (29)
where
=
(30)
and
=
is the multi-state limit reliability
function of that system if and only if
= for tÎ
, (31)
Above condition according to (30) and
Lemma 18.20 from [5] is holds if and only if
, (32)
which completes the proof. Œ
Proposition 1
If components of the multi-state
homogeneous, regular “m out of ”-series system at the operational state
(i)
have exponential reliability functions,
for 
for (33)
(ii)
const, , , 
= const, ,
(iii) 
= 
, 
= 
, (34)
then
, t Î (-¥,¥), (35)
where
=
, (36)
is the multi-state limit reliability
function of that system , i.e. for n
large enough we have
(37) for t Î (-¥,¥), 
Proof. Since
as 
for
then, according to (33) for n large enough, we obtain
for 
Hence, considering (28), it appears
that
for
which means that according to Lemma 1
the limit reliability function of that system is given by (35)-(36).
6. Conclusion
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 multi-state “m out of l”- series 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.
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