CUSTOMER-DRIVEN RELIABILITY MODELS
AND LOGICAL DIFFERENTIAL CALCULUS
FOR RELIABILITY ANALYSIS OF MULTI
STATE SYSTEM
Kapur K. C.*, Zaitseva E., Kovalik S., Matiasko
K.
*University of Washington, Industrial Eng., Box 352650, Seattle, WA 98195-2650, USA Department of Informatics, University of Zilina, Univerzitna 1, 01026, Zilina, Slovakia
Abstract: In this paper we investigated the relationship of two reliability measures of Multi-State
System (MSS): Dynamic Reliability Indices (DRI) and Customer-Driven Reliability Models. Combination two tools for reliability analysis of MSS allow improving measures of this system.
1. Introduction
Many well-known models for reliability are given in [1 – 5]. Discrete probability models are typically employed in reliability analysis. Multi-State System (MSS) is one of them model [1, 2]. In a MSS, both the system and its components may experience more than two states, for example, completely failed, partially working and perfect working. The description of real-world system by this model is better than the traditional reliability models, where the system and all of its components are assumed to have only two states of working efficiency (working perfectly or total failure).
Many methods for reliability analysis of MSS are considered in paper [1 – 5]. One group of these methods rests on MSS representation by structure function, which declares relation inter system components states and system reliability.
We propose the approach on the basis of structure function too. This approach evolves results that proposed in [6] and allows to evaluate of dynamic properties of the MSS reliability. Basic and theoretical concepts of this approach have been determined in [7]. Dynamic Reliability Indices (DRI) have been proposed in [7] as a measure of MSS reliability. These indices characterize the change of the MSS reliability that is caused by the change of a component state [7 – 8]. The calculation of DRI are realized by the Logical Differential Calculus of Multiple-Valued Logic (MVL).
In this paper we investigated the relationship of DRI and reliability measures of Customer-Driven Reliability Models that have been presented in [6] and have been named as Lower (Upper) Boundary Points. Combination two tools for reliability analysis of MSS allow improving measures of this system.
2. The direct partial logic derivative in reliability analysis of MSS
The structure function of MSS declares depending of the system reliability (system state) from its components state:(x1, …, xn) = (x): {0, …, m-1}n {0, …, m-1} (1)
where xi is the i-th system component; m is reliability states from the complete failure (it
is 0) to the perfect functioning (it is m-1); n is number of system component.
Every system component xi is characterized by probability of the performance rate form
zero to m-1:
1
,,
0
},
{Pr
,
x
s
s
m
p
si
i
i
i
i
. (2)There are assumptions for structure function (1) that are peculiar to reliability analysis [1, 2]:
the structure function is monotone (s)=s (s{0, …, m-1});
all components are s-independent and are relevant to the system.
The definition of the structure function (1) is well known as MVL function. It allows applying MVL tools for reliability analysis of the MSS [3, 4]. In particular, we use the Direct Partial Logic Derivatives (as the part of Logical Differential Calculus). These derivatives reflect the change in the value of the underlying function when the values of variables change
A direct partial logic derivative (jh)/xi(ab) of a MVL function (x) of n variables
with respect to variable xi reflects the fact of changing of function from j to h when the
(jk)/xi(ab)=
case
other
in the
,0
)
,
(
)
,
(
if
,1
a
j
b
k
m
ix
&
ix
where (ai, x) = (x1,…, xi-1, a, xi+1,…, xn); (bi, x) = (x1,…, xi-1, b, xi+1,…, xn); ai, bi {0,
…, m-1}.
For analysis of changes of several variables at the same time using a direct partial logic derivative with respect to variables vector is proposed in [9, 10]. In this paper, a direct partial logic derivative of a MVL function (x) of n variables with respect to variables vector x(p) = (x i1, xi2, …, xip) is defined as: (jh)/x(p)(a(p)b(p)) =
case
other
the
in
,0
)
,
(
)
,
(
if
,1
1 1a
j
b
b
k
a
m
p pi
i
i
i
x
x
&
(3)3. Lower (upper) boundary points and direct partial logic derivatives
In paper [6] for reliability analysis of MSS by the structure function the approach of Lower (Upper) Boundary Points has been proposed: xl is a Lower Boundary Points of system states level j iff (xl) j and xl > x implies
that (x) < j, j = 0, …, m-1;
xu is a Upper Boundary Points of system states level j iff (xu) j and xu < x implies
that (x) > j, j = 0, …, m-1
and x < x if xi xi for all i and xi < xi for at least one i; x > x if xi xi for all i and xi > xi
for at least one i; i = 1, …, n.
Conditions of definition of Lower Boundary Points by Direct Partial Logic Derivatives terminology (3):
xl {x (ql gl) /x(p)(a(p) b(p)) 0}, (4)
ql = j, ..., m-1; gl = 0, ..., j-1; a( p ) > b( p ) i.e. ai > bi and ai, bi {0, …, m-1};
Upper Boundary Points can be defined by Direct Partial Logic Derivatives with respect to vector x(p):
xu {x (qu gu)/x(p)(a(p) b(p)) 0}, (5)
qu = 0, ..., j; gu = j+1, ..., m-1; a( p )<b( p ) i.e. ai < bi and ai, bi{0, …, m-1}.
The calculation of Lower (Upper) Boundary Points by (4) and (5) is alternative algorithm concerning the well know algorithm that is considered in [6]. Note, the calculation of these points by algorithm from [6] needs to transform the initial structure function into binary function. But Lower (Upper) Boundary Points in (4) and (5) are determined on basis of initial structure function that is declared as in (1) and supplementary computation is not be required.
4. The dynamic reliability indices (DRI)
Some groups of DRI for the estimation of change influence one of system components on the change of system reliability are considered in paper [7]. One of these groups is Component Dynamic Reliability Indices (CDRI). CDRI are declared as a probability of the MSS failure or repair if state of the i-th system component changes in [7]. The conception of CDRI is defined for the MSS failure and repair that are caused by changes of states of some system components.
Definition 1 . CDRI are probabilities of MSS failure and repair that are caused by changes of states of some system components [7, 8]:
p j a i f f ρ pj j P 1 , 1 ) (x(p) , (6)
p j i r r pj P 1 0 , 0 ) (x(p) (7) where ρf is number of system states when the vector x(p) value change from a(p) to b(p)forces the system failure; 1 is number of system states “1”: (x)=1; pi,a is declared in (2);
r
ρ is number of system states when the vector x(p) value change from zero to b(p) forces
the system repair; 0 is number of zero system states ((x) = 0); pi , 0 is declared in (2).
The number ρf in (6) is calculated as numbers of nonzero Direct Partial Logic
Derivative (3):
f
because in Direct Partial Logic Derivative terminology the MSS failure is represented as the changing of the function value (x) from j into zero ((x): j 0) and as decrease of a system components availability vector x(p) from a(p) to b(p) according to equation (3):
(j0)/x(p)(a(p)b(p)),
where aij,bij {0,...,m1}, aij bij, j1,...,p.
The number ρ is related with the conception of MSS repair. The MSS repair can ber
considered in some versions that depend on refit of failed system components. It is attained by replacements or renewals of failed system components. In the first case the availability of p failed system components changes from zero into (m-1) and from zero into some level b {1, …, m-1} in the second case. We consider the first case in more detail. The MSS repair is declared in Direct Partial Logic Derivative terminology is defined as the structure function change from zero into h ((x): 0 h) and as p failed system components changes from zero into (m-1):
(0h)/x(p)(0(m-1)), (9) where h{1, …, m-1}; (0,...,0) p 0 and (( 1),..., ( 1)) p m m 1) (m . So, the number ρ is declared as a sum of nonzero elements of Direct Partial Logicr
Derivative (9):
1 1 ) ( m h h r r ρ
(0h)/x(p)(0(m-1)) 0; (10)5. Lower (upper) boundary points and CDRI
There numerically measures of MSS behavior form reliability point of view [1 – 5]. For the most part they relate to system failure or change of its reliability level, for example: the Reliability Function, the Mean Time to Failure (MTTF), the Mean Time Between Failures (MTBF) etc.
The Reliability Function R (j) is one of best known MSS reliability measures. It is probability that system reliability is greater than or equal to the level j:
R (j) = Pr {(x) j}, j = {0, m-1}. (11) The Customer-Driven Reliability Models for computation of Reliability Function (11) by Lower (Upper) Boundary Points has been proposed in [6]. But the probability of change
of MSS reliability level as a result of changes of some components states is important measure too. This probability is necessary above all for system failure and repair. This measure is CDRI (Figure 1).
R(2) R(1) R(0) S ys te m pe rf or m an ce i-th s ys te m co m po ne nt Pf(xi,xj) Pr(xi,xj) j-th s ys te m co m po ne nt Fig. 1. MSS Measures (m = 3)
Thus the Customer-Driven Reliability Models and CDRI are defined by Direct Partial Logic Derivative (3). Consider this relation in detail. Firstly, Lower (Upper) Boundary Points have new interpretation by Direct Partial Logic Derivative.
Definition 2 . Lower Boundary Points of system states level j xl are system states for which
a decrease at least component state causes the change of system state from j to j’ and j >j’. Definition 3 . Upper Boundary Points of system states level j xu are system states for which
a increase at least component state causes the improvement of system state from j to j’ and j < j’.
These definitions allow to use Lower (Upper) Boundary Points for calculation of traditional reliability measure by the Customer-Driven Reliability Models (it is Reliability Function R (j) (11)) and to consider new measure, that is declare relation of changes components states and system state. The algorithm of computation of Lower (Upper) Boundary Points in paper [6] needs to transform initial structure function of MSS (1) in binary function. The additional transformation isn’t necessary for calculation Lower (Upper) Boundary Points by (4) and (5).
Secondly, consider the definition of CDRI by Lower (Upper) Boundary Points. The conception of CDRI (definition 1) is used numbers ρf and ρ that are declaredr
interdependence of changes components states and system state. These numbers are computed by Direct Partial Logic Derivative with respect to variables vector (8) and (10). But this numbers can be declared by means of Lower (Upper) Boundary Points.
Definition 4. The number ρf of system states when the vector x(p) value change from a(p)
to b(p) forces the system failure is number of Lower Boundary Points x
l of system states
level j =1 for which xl > x if xli xi for all variables and xli > xi for at least p variables: f
ρ M{xl (xl)1 and (x)<1 for xl > x (xlixi, i=1,…,n
and xls>xs, s=1,…,p, p<n)} (12)
where M is cardinal number.
Definition 5. The number ρ is number of system states when the vector xr (p) value
change from zero to b(p) forces the system repair is number of Upper Boundary Points x u
of system states level j =0 for which xu < x if xui xi for all variables and xui < xi for at
least p variables:
r
ρ M{xl (xu)
0 and (x)>0 for xl < x (xui
xi, i=1,…,nand xus<xs, s=1,…,p, p<n)} (13)
where M is cardinal number.
Use conception of Lower (Upper) Boundary Points for declaration of CDRI allows to extend the definition of these indices and to consider for some system states level j, but not only for system failure or repair.
Definition 6. CDRI are probabilities of MSS reliability of level j change that are caused by changes of states of some system components:
s n s j i j u u j ρ ps s P( ) , , (14)
s n s j i j l l j ps s P( ) , (15)where ρ is Upper Boundary Points for level system reliability j; u j is number of system
states “j”: (x)=j; pi,j is declared in (2); ρ is Lower Boundary Points for level systeml
reliability j.
Therefore the interdependence of conception of the Customer-Driven Reliability Models and CDRI allows to do new interpretation of they and to extend they definitions. It is important for investigation of reliability of real-world systems in more detail.
6. Conclusion
The analysis of changes of some states of system components and the influence of these changes on the system reliability are considered. In this paper the correlation of two measures in reliability analysis is considered: Lower (Upper) Boundary Points [6] and CDRI [7, 8]. We declare a possibility of calculation these measures by Multiple-Valued Logic tools (in particular, Direct Partial Logic Derivatives) for the first time.
CDRI are probabilities of changes of the system states depending on modifications of components states and are defined by Direct Partial Logic Derivatives on the one hand. But these indices are declared by Lower (Upper) Boundary Points on the other hand. And Lower (Upper) Boundary Points are determined by Direct Partial Logic Derivatives in addition to the traditional definition form [6]. So, the Direct Partial Logic Derivatives approach becomes of connecting-link between measures of Lower (Upper) Boundary Points and CDRI (Figure 2). This condition permits to extend application of CDRI for real-world systems.
Acknowledgments
This work was partially supported by grant VEGA 1/3084/06 and Finland – Slovak MVTS of MS SR (Slovakia).
Direct Partial Logic Derivatives for calculation
(ql gl) /x(p)(a(p) b(p)) 0
(qu gu)/x(p)(a(p) b(p)) 0,
(10)/x(p)(a(p) b(p)) 0
(0h)/x(p)(0(m-1)) 0
Lower Boundary Points
xl
Upper Boundary Points
xu
CDRI of MSS failure
Pf(x(p))
CDRI of MSS repair
Pr(x(p))
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