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7
TRELIABILITY ANALYSIS OF TWO-STATE CONSECUTIVE "M OUT OF L: F"-
SERIES SYSTEMS
S. GUZE
MARITIME UNIVERSITY, DEPARTMENT OF MARTHEMATICS, GDYNIA, POLAND
e-mail: sambor@am. gdynia.pl
ABSTRACT
A non-stationary approach to reliability analysis of two-state series and consecutive "m out of k: F" systems is presented. Further, the consecutive "m out of k: F"-series system is defined and the recurrent formulae for its reliability function evaluation are proposed.
1 INTRODUCTION
The basic analysis and diagnosis of systems reliability are often performed under the assumption that they are composed of two-state components. It allows us to consider two states of the system reliability. If the system works its reliability state is equal to 1 and if it is failed its reliability state is equal to 0. Reliability analysis of two-state consecutive "k out of n: F" systems can be done for stationary and non-stationary case. In the first case the system reliability is the independent of time probability that the system is in the reliability state 1. For this case the main results on the reliability evaluation and the algorithms for numerical approach to consecutive "k out of n: F" systems are given for instance in Antonopoulou & Papstavridis (1987), Barlow & Proschan (1975), Hwang (1982), Malinowski & Preuss (1995), Malinowski (2005). Transmitting stationary results to non-stationary time dependent case and the algorithms for numerical approach to evaluation of this reliability are presented in Guze (2007a, b). Other more complex two-state systems are discussed in Kolowrocki (2004). The paper is devoted to the combining the results on reliability of the two-state series and consecutive "m out of n: F" system into the formulae for the reliability function of the consecutive "m out of l: F"-series systems with dependent of time reliability functions of system components (Guze 2007a, b, c).
2 RELIABILITY OF A SERIES AND CONSECUTIVE "M OUT OF N: F" SYSTEMS
In the case of two-state reliability analysis of series systems and consecutive "m out of n: F" systems we assume that (Guze 2007b):
- n is the number of system components,
- Et, i = 1,2,...,n, are components of a system,
- Ti are independent random variables representing the lifetimes of a components Et, i = 1,2,...,n,
- Ri (t) = P(Ti > t), t e< 0, ro), is a reliability function of a component Ei, i = 1,2,...,n,
- Fi (t) = 1 - Ri (t) = P(Ti < t), t e< 0, ro), is the distribution function of a component Ei lifetime T, i = 1,2,...,n, also called an unreliability function of a component Et, i = 1,2,...,n.
In further analysis we will use one of the simplest system structure, namely a series system. Definition 1 A two-state system is called series if its lifetime T is given by
T = min{T }.
1<i<n
The scheme of a series system is given in Figure 1.
E1 E2
E
^n
Figure 1. The scheme of a series system
The above definition means that the series system is not failed if and only if all its components are not failed or equivalently the system is failed if at least one of its components is failed. It is easy to motivate that the series system reliability function is given by
R n (t)= nR (t), t g< 0, »>. (1)
i=1
Definition 2. A two-state series system is called homogeneous if its component lifetimes T have an identical distribution function
F(t) = P(Tt < t), t g< 0,ro), i = 1,2,...,n, i.e. if its components Et have the same reliability function
R(t) = 1 - F(t), t e< 0,ro). The above definition results in the following simplified formula
Rn (t) = [R(t)]n, t g< 0,»), (2)
for the reliability function of the homogeneous two-state series system.
Definition 3. A two-state system is called a two-state consecutive "m out of n: F" system if it is failed if and only if at least its m neighbouring components out of n its components arranged in a sequence of E}, E2, ..., En, are failed.
After assumption that:
- T is a random variable representing the lifetime of the consecutive "m out of n: F" system,
- CR(nm)(t) = P(T > t), t g< 0, ro), is the reliability function of a non-homogeneous consecutive "m out of n: F" system,
- CF(n m) (t) = 1 - CR(nm) (t) = P(T < t), t e< 0, ro), is the distribution function of a consecutive "m out of n: F" system lifetime T,
we can formulate the following auxiliary theorem (Guze 2007c).
Lemma 1. The reliability function of the two-state consecutive "m out of n: F" system is given by the following recurrent formula
CR(n m)(t) =
1 -n F (t)
for n < m,
for n = m,
i=1
(3)
Rn (t )CR(:-i (t) + ^ Rn_, (t )CR'n"j-i (t)
j=1
• n F (t)
i=n- j+1
for n > m,
for t e< 0, ro).
Definition 4. The consecutive "m out of n: F" system is called homogeneous if its components lifetimes Ti have an identical distribution function
F(t) = P(Tt < t), i = 1,2,... , n, t e<0,ro),
i.e. if its components Ei have the same reliability function R(t) = 1 - F(t), t e< 0, ro).
Lemma 1 simplified form for homogeneous systems takes the following form.
Lemma 2. The reliability of the homogeneous two-state consecutive "m out of n: F" system is given by the following recurrent formula
for n < m
CR(nm> (t) =
1 - [ F (t )]n
for n = m,
R(t )CR(:> (t)
+ R(t )£ Fj -1(t)
j=1
CRj(t) for n > m,
(4)
for t e< 0, ro).
3 RELIABILITY OF TWO-STATE CONSECUTIVE "M OUT OF L: F"-SERIES SYSTEM
To define a two-state consecutive "m out of l: F"-series systems, we assume that
1
Eij, i = 1,2,...,k, j = 1,2,...,//, are two-state components of the system having reliability functions
j) = P(Tj > t), t g< 0, rc), where
T/j, i = 1,2,...,k, j = 1,2,...,//,
are independent random variables representing the lifetimes of components Eij with distribution functions
Fij(t) = P(Ttj < t), t g< 0, rc).
Moreover, we assume that components Ei1, Ei 2,..., Ea , i=1, 2,... , k, create a consecutive 'W out of /i: F" subsystem Si, i=1, 2,. , k and that these subsystems create a series system.
Definition 5. A two-state system is called a consecutive "mi out of /■: F"-series system if it is failed if and only if at least one of its consecutive "mi out of /: F" subsystems Si, i = 1, 2,. , k, is failed.
According to the above definition and formula (4) the reliability function of the subsystem Si is given by
CRm )(t) =
1
for /, < m
i i'
1 -n Fj (t)
for / = m
i i -
j=1
rUi (t )crW- (t)+s Rlh - j (t )CR(mmh (t)
j=1
n F>(t)
for /• > m
i i
v=li - j+1
(5)
and its lifetime distribution function is given by CF(mi )(t) = 1 - CR(mi )(t), i = 1,2,... , k.
Hence and by (1), denoting by cr^'"2'"™'(t) = P(T>t), t e< 0,rc), the reliability function of the
k,hh.....k
consecutive "m out of /: F"-series system, we get the next result.
Lemma 3. The reliability function of the two-state consecutive "m/ out of F F"-series system is given by the following recurrent formula
m -1
CR-T^) = ncR^> (t)
i=1
n p -UF1 (t)]
j=1
n [ R^ (t )CR^(t)
i=1
for I, = m
2 2 '
for / < m4
i i -
m, -1
+ E Ri - j (t XRjt)
j=1
n Fv (t)]
V=1i - j+1
for I, > m
i i'
(6)
(7)
for t e< 0,
Motivation. Assuming in (1) that R, (t) = CR)(t), we get (6) and next considering (5), we get the formula (7).
Definition 6. The consecutive "m out of /: F"-series systems is called regular if
l1 = /2 = ... = /k = / and m1 = m2 = . . . = mk = m, where l , me m < l.
Definition 7. The consecutive "m; out of I;-: F"-series system is called homogeneous if its components lifetimes Ty have an identical distribution function
F(t) = P(Ttj < t), i = 1,2,... , k, j = 1,2,..., li, t e< 0, to), i.e. if its components Ej have the same reliability function R(t) = 1 - F(t), i = 1,2,. , k, j = 1,2,. ,/,, t e< 0,to).
Under Definition 6 and Definition 7 and formula (7), denoting by cR'Xt) = P(T > t), t e< 0, to), the
reliability function of a homogeneous and regular consecutive "m out of /: F"-series system, we get following result.
Lemma 4. The reliability function of the homogeneous and regular two-state consecutive "mt out of F"-series system is given by
1
1
for / < m,
[1 - Fl (t)]k for /=m,
k
(8)
m-1
+ R(t)SCR(,m)j-1(t)
j=1
Fj-1(t)]
for / > m,
t e< 0,rc). 4 CONCLUSIONS
The paper is devoted to a non-stationary approach to reliability analysis of two-state systems. Two recurrent formulae for two-state reliability functions, a general one for non-homogeneous and its simplified form for regular and homogeneous two-state consecutive "m out of /: F"-series system have been proposed.
The proposed methods and solutions may be applied to any two-state consecutive "m out of /: F"-series systems.
REFERENCES
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Guze S. - RELIABILITY ANALYSIS OF TWO-STATE CONSECUTIVE "M OUT OF L: F"-SERIES SYSTEMS r^&^I^f w(122
(Vol.2) 2009, March
