=Paper=
{{Paper
|id=None
|storemode=property
|title=Direct Partial Logic Derivatives in Analysis of Boundary States of
Multi-State System
|pdfUrl=https://ceur-ws.org/Vol-1356/paper_64.pdf
|volume=Vol-1356
|dblpUrl=https://dblp.org/rec/conf/icteri/ZaitsevaLKK15
}}
==Direct Partial Logic Derivatives in Analysis of Boundary States of
Multi-State System==
https://ceur-ws.org/Vol-1356/paper_64.pdf
Direct Partial Logic Derivatives in Analysis of
Boundary States of Multi-State System
Elena Zaitseva1, Vitaly Levashenko1, Jozef Kostolny1, Miroslav Kvassay1
University of Zilina, Department of Infromatics, Univerzitna 8215/1,
1
010 26, Zilina, Slovakia
{elena.zaitseva, vitaly.levashenko, jozef.kostolny,
miroslav.kvassay}@fri.uniza.sk
Abstract. Multi-State System (MSS) is mathematical model that is used in
reliability engineering for the representation of initial investigated object
(system). In a MSS, both the system and its components may experience more
than two states (performance levels). One of possible description of MSS is a
structure function that is defined correlation between a system components
states and system performance level. The investigation of a structure function
allows obtaining different properties, measures and indices for MSS reliability.
For example, boundary system’s states, probabilities of a system performance
levels and other measures are calculated based a structure function. In this paper
mathematical approach of Direct Partial Logical Derivatives is proposed for
calculation of boundary states of MSS.
Keywords. Multi-State System, Multiple-Valued Logic, Direct Partial Logic
Derivatives, Boundary State
Key Terms. Reliability, Model, Approach, Methodology, ScientificField
1 Introduction
Reliability is a principal attribute for the operation of any modern technological
system. A principal issue in reliability analysis is the uncertainty in the failure
occurrences and consequences. With respect to the complexity of the system and the
modeling of their behavior, a distinctive feature of the system reliability analysis is a
comprehensive and integrated manner [1]. Focusing on safety, reliability engineering
methods aim at the quantification of the probability of failure of the system. In paper
[1] presents detail analysis of reliability engineering state and define principal
problems in this scientific discipline. According to [1] one direction of reliability
engineering is estimation of a complex system based on Multi-State System (MSS)
reliability analysis methods.
MSS is mathematical model used in reliability analysis for system with some
(more that two) levels of performance [1, 2]. This mathematical model has been
exploited in reliability engineering since 1975 [3-5]. Principal advantage of this
mathematical model is detail description of investigated object. MSS permits to
define and investigate several performance levels: from perfect function to fault. The
typical Binary-State System allows evaluating only two system states as functioning
and failure. Other states, for example, as partly functioning or functioning with
restrictions are not analyzed in case of Binary-State System use. But extra states and
performance levels in the mathematical model dramatically increase size of this
model and computational complexity of its analysis. Therefore the MSS has not been
used intensively in reliability analysis. There is other aspect to restrict the application
of MSS. It is absent of effective methods for MSS analysis.
According to analysis in paper [2] there are four groups of methods for MSS
analysis that are based on different mathematical approach: an extension of Boolean
models to the multi-valued case, the stochastic process as Markov process, the
universal generation function methods and the Monte-Carlo simulation techniques.
For example, Markov processes are used to analyze the system state transition process
[6]. The universal generation function application is useful in optimization problem
[7]. The Monte-Carlo simulation as a rule is used for reliability assessment of system
with large number of components [8]. The methods based on extension of Boolean
models to the multi-valued case were developed historically the first [9, 10].
According to these methods MSS is represented and defined by the structure function.
This function is defined the conformance of the system performance level and
components states. As a rule for the structure function definition and representation is
used Boolean functions [10, 11]. Only in separated publications the structure function
with some values has been considered [9, 12]. In papers [13, 14] the correlation of
Miltiple-Valued Logic (MVL) function and structure function was analyzed. The
interpretation of the structure function as the MVL function allowed using the
mathematical approach of MVL in the analysis of the MSS structure function. In
paper [14] the application of Logical Differential Calculus for MSS reliability
analysis has been proposed. The Logic Differential Calculus is used for analysis of
dynamic properties of MVL function and this approach can be applied for analysis of
dynamic behaviour of MSS that is determined by the structure function.
In the paper [14] the basic conception of application of Direct Partial Logic
Derivatives (it is part of Logical Differential Calculus) in MSS reliability analysis has
been considered. The proposed method permits to investigate the influence of one
system component state change to the system performance level. The new indices for
quantitative analysis of such influence have been defined. The application of these
derivatives for calculation of importance measures (Structural Importance, Birnbaum
Importance and Criticality Importance) has been investigated in paper [15, 16]. The
algorithm for calculation of Fussell-Vesely Importance based on Direct Partial Logic
Derivatives has been proposed in [17].
In this paper new application of Direct Partial Logic Derivatives for MSS
reliability analysis based on the structure function is considered. The new algorithm
for calculation of boundary states of MSS is proposed. This algorithm is based on
representation of MSS by the structure function (section 2). This section includes the
conception of boundary states of MSS for every performance level. The special
structures of MSS (parallel and series) and their structure function are considered in
the section 2 too. These types of system are typically employed in reliability analysis
[18]. The short description of conceptions of Direct Partial Logic Derivatives with
respect to one variable and with respect to variable vector are presented in the section
3. The calculation of MSS boundary states for every system performance level is
considered section 4.
2 MSS Structure Function
2.1 Structure Function of MSS
Consider the system of n components. Each component of this system has m
states: from the complete failure (it is 0) to the perfect functioning (it is m-1). The i-th
system component state is denoted as xi (i = 1, …, n). Suppose, that this system has m
performance level too: from the complete failure (it is 0) to the perfect functioning (it
is m-1). The dependence of the system performance level on components states is
defined by the structure function (x) identically:
(x1, x2,…, xn) = (x): {0, 1, …, m-1}n {0, 1, …, m-1}. (1)
The function (1) agrees with the definition of a MVL function [19]. Therefore the
mathematical approaches of MVL can be used in quantification analysis of MSS. But
the structure function (1) allows representing the very small class of real system for
which the number of system performance levels and number of every component
states are equal. As a rule, the real-world system has different numbers of states for
different components. And the number of performance levels can be different too.
Therefore the structure function of real-word system must be defined as:
(x): {0, 1, …, m1-1}…{0, 1, …, mn-1} {0, 1, …, M-1}, (2)
where mi is number of states for i-th system component (i = 1, …, n) and M is number
of a system performance levels.
The equation (2) can be interpreted as a MVL function. But some formal changes
allow transforming this structure function definition into an incompletely specified
MVL function. This transformation suppose the interpretation of the function (2) as
incompletely specified MVL function for maximal value of number mi and M:
mmax=MAX{m1, …, mn, M}. In this case, the structure function (2) is defined as:
`(x): {0, 1, …, mmax-1}n {0, 1, …, mmax -1}. (3)
The interpretation of the structure function (2) as an incompletely specified MVL
function (3) permits to use mathematical approaches of MVL without principal
restriction for analysis of properties of the structure function (2).
For example, consider the simple service system (Fig. 1) in a region from paper
[17]. This system consists of three components (n = 3) – service point 1 (x1), service
point 2 (x2) and infrastructure (x3). This system has four performance levels: 0 – non-
operational (no customer is satisfied), 1 – partially operational (some customers are
satisfied), 2 – partially non-operational (some customers are not satisfied), 3 – fully
Fig. 1. A simple service system.
Table 1. The structure function of the simple service system
Components states x3
x1 x2 0 1 2 3
0 0 0 0 0 0
0 1 0 1 1 2
1 0 0 1 1 2
1 1 0 2 3 3
Table 2. The structure function of the simple service system represented as an incompletely
specified MVL function
Components states x3
x1 x2 0 1 2 3
0 0 0 0 0 0
0 1 0 1 1 2
0 2
0 3
1 0 0 1 1 2
1 1 0 2 3 3
1 2
1 3
2 0
2 1
2 2
2 3
3 0
3 1
3 2
3 3
operational (all customers are satisfied). Next, we assume that the service points are
only functional (state 1) or dysfunctional (state 0). The infrastructure can be modelled
by 4 quality levels, i.e. from 0 (the quality of the infrastructure is poor) to 3 (the
quality is perfect). The structure function of this system according to (2) is defined in
Table 1 (m1 = m2 = 2, m3 = 4 and M = 4). The structure function of this system as an
incompletely specified MVL function is shown in Table 2 (mmax = 4).
2.2 Series and Parallel MSS
There are some typical structures in the reliability engineering: series, parallel, k-
out-of-n and bridge. Every system of these structures has single valued definition for
Binary-State System (Fig.2).
The series system: The k-out-of-n system:
n
(x) = i1 xi (x) = ( z k xi )
z
x1
x1 x2 … xn
x2 k/n
…
xn
The parallel system:
n
(x) = x
i 1 i
The bridge system:
(x) = (x1x2)(x4x5)(x1x3x5)(x2x3x4)
x1
x1 x2
x2
… x3
xn x4 x5
Fig. 2. Graphical and mathematical interpretation of typical structures of Binary-State System.
The extension of mathematical definition of the structure functions of these
system for the MSS has some variants. For example, in paper [24] the structure
functions of series, parallel and k-out-of-n MSS are defined based on the
interpretation of mathematical equations of these system in terms of MVL. Structure
functions of these MSS in MVL terms are declared by OR () and AND () MVL
functions: OR(a, b) = MAX(a, b) and AND(a, b) = MIN(a, b). According to [24]
structure functions parallel, series and k-out-of-n MSS are declared
- for parallel MSS as:
n
(4)
(x) = x = MAX(x1, x2, …, xn),
i 1 i
- for series MSS as:
n
(5)
(x) = i1 xi = MIN(x1, x2, …, xn),
- for k-out-of-n MSS as:
(x) = ( zk xi ) MAX(MIN1 ( xi , xi ,...,xi ),...,MIN w ( xi , xi ,...,xi )) ,
z 1 2 k 1 2 k
(6)
w n!/(k!(n k )!) .
In paper [27] other declaration of these MSS are presented in which a system
performance level depends on the number of functioning components. In Table 3
some structure functions of parallel MSS of two components (n = 2) with three states
(m1 = m2 =3) and three performance level (M = 3) are shown. All these structure
function in case of Binary-State System are parallel system. The series MSS can be
defined similarly.
Table 3. The structure functions of parallel MSS (n = 2, m1 = m2 = M = 3)
Components states Structure function of parallel MSS
x1 x2 1(x) 2(x) 3(x) 4(x)
0 0 0 0 0 0
0 1 1 1 1 1
0 2 2 1 1 1
1 0 1 1 1 1
1 1 1 2 1 1
1 2 2 2 1 2
2 0 2 1 1 1
2 1 2 2 1 2
2 2 2 2 2 2
Therefore the typical structures of MSS have no single valued mathematical
definition, because there are the set of structure functions of MSS that can be agreed
with one Binary-State System. The structure function allows defining MSS explicitly.
Therefore the structure function is preferable form of MSS mathematical
representation.
2.3 Boundary States of MSS
The conception of boundary states has been proposed for Binary-State System
firstly. The boundary state is defined as state for which the failure of one system
components or some components causes the fault of a system [20]. The boundary
state of MSS must be defined for every system performance level [2]. In papers [21,
22] the boundary states of MSS are interpreted as minimal cut/path sets. Authors of
[23] introduced conception of Lower (Upper) Boundary Points of MSS for system
performance level j (j =0, …, M-1). The boundary states for system performance level
j and component i (i = 1, …, n) (named as exact boundary states) has been proposed
and considered in papers [24, 25]. In paper [26] and [17] the correlations of these
boundary states with Lower (Upper) Boundary Points and minimal cut/path sets are
shown accordingly.
The exact boundary states have been considered in paper [25]. These states are
system states for which the change of the i-th component state from s to ~ s causes the
~ ~ ~ ~
system performance level change from j to j (s, s {0,…, mi -1}, s s and j, j
~
{0,…, M -1}, j j ). The exact boundary state is defined by the exact boundary
vector unambiguously. Illustrate the correlation of a system exact boundary state and
an exact boundary vector by the example for the service system in Fig.1.
Determine the exact boundary states of this service system for which the failure of
the first component causes the system failure as the change of the system performance
level from state “1” to “0”. According to Table 1, there are two situations that agree to
this condition. They are possible for the failure of the second component and the third
component state “1”or “2”. These exact boundary states can be presented as vector
states: x = (x1, x2, x3) = (1 0,0,1) and x = (x1, x2, x3) = (1 0,0,2). Note that the
boundary state x = (x1, x2, x3) = (1 0,0,3) does not satisfy the condition because the
system performance level changes from “1” to “2” depending on the failure of the
first component in this case.
One of possible mathematical approaches for the definition of the exact boundary
states in MVL is Logical Differential Calculus, in particular the Direct Partial Logical
Derivatives. Consider the application of this mathematical approach for analysis of
structure function of MSS.
3 Direct Partial Logical Derivatives
3.1 Direct Partial Logical Derivative with respect to one variable
The mathematical tool of Direct Partial Logic Derivatives has been proposed in
[25] for calculation of an exact boundary states of a MSS. The Direct Partial Logic
Derivative with respect to variable xi for the structure function (1) permits to analyze
~
the system performance level change from j to j when the i-th component state
changes from s to ~s :
1, if ( si , x ) j and (~ ~ (7)
~ ~ si , x ) j
( j j ) xi ( s s )
0, other ,
where (si , x) = (x1,…, xi-1, s, xi+1,…, xn); ( ~s , x) = (x1,…, xi-1, ~s , xi+1,…, xn); s,
~ ~ ~
s {0,…, mi -1}, s ~ s and j, j {0,…, M -1}, j j .
For example, investigate the influence of the first component failure to the fault of
the simple service system in Fig.1. The Direct Partial Logic Derivative
(10)/x1(10) allows to calculate the system state for which this failure causes
the system break down. The calculation of this derivative is shown in Fig.3 in form of
flow graph. The derivative (10)/x1(10) has two non-zero values that agrees
with state vectors: x = (x1, x2, x3) = (1 0,0,1) and x = (x1, x2, x3) = (1 0,0,2).
According to definition of the Direct Partial Logic Derivative (7) for these system
state the failure of the first system component causes the system failure too. Therefore
the service system fails after the failure of the first service point if the second service
point isn’t functioning and the functioning of the infrastructure conforms to state one
or two. The system states x = (x1, x2, x3) = (1 0,0,1) and x = (x1, x2, x3) =
(1 0,0,2) are exact boundary states for the first system component failure and the
system performance level 1.
x1 x2 x3 (x) (10)/x1(10)
0 0 0 0 0
0 0 1 0 1
0 0 2 0 1
0 0 3 0 0
0 1 0 0 0
0 1 1 1 0
0 1 2 1 0
0 1 3 2 0
1 0 0 0
1 0 1 1
1 0 2 1 x = (1 0,0,1)
1 0 3 2 x = (1 0,0,2)
1 1 0 0
1 1 1 2
1 1 2 3
1 1 3 3
Fig. 3. Calculation of the Direct Partial Logic Derivative (10)/x1(10).
The Direct Partial Logic Derivative (7) allows investigating boundary states of a
MSS for which component state xi change from s to ~ s causes the system
~
performance level change from j to j . Therefore, this derivative allows calculating
exact boundary states of the i-th system component for MSS performance level j that
agree to state vectors x = (x1, x2,…, xn). All possible changes of the i-th system
component and their influence to MSS performance level can be investigated based
on the Direct Partial Logic Derivative (7). But this derivative permits to investigate
the influence of one component only. The Direct Partial Logic Derivative with respect
of variable vector investigate the system state changes depending on changes of states
of some system components.
3.2 Direct Partial Logical Derivative with respect to variable vector
A Direct Partial Logic Derivatives of a structure function (x) of n variables with
respect to variables vector x(p) = (xi1, xi2, …, xip) reflects the fact of changing of
~
function from j to j when the value of every variable of vector x(p) is changing from
s to ~
s [15]:
1, if ( si ,...,si , x ) j and (~ ~ (8)
~ si ,...,~
si , x ) j
( j j ) xi ( s (p) ~
s (p)) 1 p 1 p
0, other .
In (8) a change of value of iq-th variable xi form si to ~si agrees with a change of
q q q
i -th MSS component state form s to ~s (q = 1, …, p and p < n). So, changes of
q iq iq
some components states correspond with change of a variables vector x(p) = (xi1, xi2,
…, xip). Every variable values of this vector changes form si to ~si . So, vector x(p) q q
can be interpreted as components states vector or components efficiencies vector.
For example, consider the simple service system (Fig.1) failure depending on fault
of the first service point and reduction of functioning of infrastructure from state 2 to
1. This system behavior can be presented by the Direct Partial Logic Derivative
(10)/x1(10) x3(21). The calculation of this derivative is shown in Fig.4 by
the flow graph.
The derivative (10)/x1(10) x3(21) has two values and one of them is
non-zero value that agrees with state vector: x = (x1, x2, x3) = (1 0, 0, 2 1). This
state vector define of the service system failure depending on the failure of the first
service point and deterioration of the infrastructure functioning from state 2 to state 1.
Therefore the system state x = (x1, x2, x3) = (1 0, 0, 2 1) can be interpreted as
exact boundary state for the first and the third system components of the system
performance level 1.
The Direct Partial Logic Derivative with respect to variable vector (8) allows
investigating boundary states of a MSS for which simultaneous changes of p
components states from si to ~si (q = 1, …, p and p < n) causes the system
q q
~
performance level change from j to j . Therefore, the Direct Partial Logic Derivative
with respect to variable vector allows calculating exact boundary states for MSS
performance level j of the i-th system component.
x1 x2 x3 (x)
0 0 0 0 (10)/x1(10)x3(21)
0 0 1 0 1
0 0 2 0
0 0 3 0
0 1 0 0
0 1 1 1 0
0 1 2 1
0 1 3 2
1 0 0 0
1 0 1 1
1 0 2 1 x = (10,0,21)
1 0 3 2 x = (10,1,21)
1 1 0 0
1 1 1 2
1 1 2 3
1 1 3 3
Fig. 4. Calculation of the Direct Partial Logic Derivative (10)/ x1(10) x3(21).
4 The Calculation and Estimation of Exact Boundary States of
MSS based on Direct Partial Logic Derivatives
The exact boundary state of MSS are defined based on the condition that fixed
system performance level change depending on the appointed change of one system
component state or specified changes of some components states. The Direct Partial
Logic Derivative with respect to one variable (7) and the Direct Partial Logic
Derivative with respect to variable vector (8) can be used to investigate change of the
~
system performance level from j to j that are caused by specified changes of one or
some system components states. These derivatives have non-zero values of the
structure function for system states that satisfy for specified condition: the system
~
performance level change from j to j depending on specified changes of one or some
system components states. Therefor the exact boundary states can be defined as
system states that conform to non-zero values of derivatives (7) and (8).
The exact boundary state for MSS performance level j depending on the i-th
j j
~
system component xi is indicated by vector state x = (x1,…, xi,…, xn) =
s ~s
~
(a1,…, si,…, an) for which (a1,…, si,…, an) = j and (a1,…, ~si ,…, an) = j . This
state is calculate as non-zero value of Direct Partial Logic Derivative (7). The exact
boundary state for MSS performance level j depending on p components xi1, xi2, …,
j ~j
xi p xi ...xi is indicated by vector state x = (x1,…, xi ,…, xi ,…, xn) =
s ~s s ~s
1 p 1 p
i1
i1 i p ip
(a1,…, si1,…, si p,…, an) for which (a1,…, si1,…, si p,…, an) = j and
~ ~ ~
(a ,, s ,, s ,, a ) j . This state is calculate as non-zero value of Direct Partial
1 i1 ip n
Logic Derivative (8).
Consider estimation of the system boundary states for coherent MSS. There are
next assumptions for structure function of coherent MSS [2]: (a) the structure
function is monotone and (s)=s (s{0, …, m-1}) and (b) all components are s-
independent and are relevant to the system.
Every system component is characterized by the probabilities of its state:
pi,s = Pr{xi = s}, s = 0, …, mi -1. (9)
.
The probability of every boundary state (a1, …, si, …, an) for MSS performance
level j depending on the i-th system component change from s to ~ s is calculated
based on the probabilities of components states:
j j
~
(10)
p( a ... s ... a ) xi p1,a pi 1,a pi ,s pi 1,a pn,a .
s~s
1 i n 1 i 1 i i 1 n
The probability of boundary state for MSS performance level j depending on the i-
th system component states changes is calculated as:
j j
~
j j
~
(11)
p xi p( a ... s ... a ) xi .
s~s s ,~s s~s
1 i n
Next measure is defined the probability of boundary state of MSS performance
level j depending of all component state change from s to ~
s :
j j n j j
~ ~
(12)
p x p xi .
s~s i1 s~s
The probability of MSS performance level change depending on the change of the
i-th system component from state s to ~
s is calculated according to:
j j
~
(13)
p xi p xi .
ss j , ~j s ~s
~
The probability of MSS performance level change depending on all change of the
i-th system component state is generalization of previous equation:
(14)
p xi p xi .
s ,~s s ~s
The similar measures to (10) - (14) can be defined for estimation of exact boundary
j ~j
state for MSS performance level j of p components xi1, xi2, …, xip xi ...xi .
s ~s s ~s 1 p
i1 i1 i p ip
Consider some examples for calculation of measures (10) - (14) for the simple
service system in Fig.1. The components states probabilities for this system are
defined in Table 4. Consider this system failure depending to the first components.
The Direct Partial Logic Derivative (10)/x1(10) represents this system
behavior (Fig.3). This derivative has two non-zero values that conform to two
boundary states x1 : x = (1,0,1) and x = (1,0,2).
1 0
10
Table 4. The components states probabilities of the simple service system
0 Components states
x1 x2 x3
0 1 0 1 0 1 2 3
pi,s 0.3 0.7 0.2 0.8 0.2 0.6 0.1 0.1
The probabilities of boundary states for the system failure depending the first
component break down x1 are calculate according to (10) and are:
1 0
10
(15)
p(101) x1 p1,1 p2, 0 p3,1 0.084 and p(102) x1 p1,1 p2, 0 p3, 2 0.014
10 10
10 10
The probability of boundary state for this system failure depending on the first
component is calculated based on (11) as:
(16)
p x1 p(101) x1 p(102) x1 0.098 .
10 10 10
10 10 10
Therefore according to (16) the service system failure depending on the breakdown
of the first service point is 0.098. By the similar way the probability of this system
failure depending on the breakdown of the second service point is calculated and this
probability is p x2 0.098 .
10
10
There are three boundary states for the system failure depending the fault of
infrastructure (the third component): p( 011) x3 0.144 , p(101) x3 0.084 and
10 10
10 10
p(111) x1 0.336 . Therefore the probability the service system fault caused by the
2 0
10
failure of the infrastructure is 0.564.
Other probabilities of this system failure or deterioration of the performance level
are calculated according to (10) – (14) similar.
5 Conclusion
The mathematical background for application of mathematical methods of MVL
for reliability analysis of MSS is considered in this paper. The correlation of the
structure function and MVL function are shown and proved by means of the
conception of incompletely specified MVL function. This background allows using
Direct Partial Logic Derivatives for analysis of MSS structure function.
Mathematical approach of Direct Partial Logic Derivatives in MVL is used for
investigation of dynamic properties of MVL function. The analysis of boundary
values of MVL function is possible based on these derivatives too. In this paper the
investigation of boundary values of the structure function of MSS and definition of
MSS exact boundary states based on these valued are considered. Conception of exact
boundary states is defined as boundary states for fixed MSS performance level change
depending on change of the appointed change components state. This conception is
extended for exact boundary states depending on changes of some components states.
New measures for estimation of MSS boundary states estimation are introduced
and considered in the paper. The analysis of MSS based on the exact boundary states
has not limits for the numbers of components (n) and states for every component (mi ),
and system performance levels (M) according to the theoretical background. But in
real-world applications these numbers have important influence to the structure
function dimension (number of structure function elements) that is calculated as:
Nstructure function dimension = m1 m1 … mn
As a rule the number of system performance levels (M) and number of component
states (mi ) are defined between two and seven. The increase of the structure function
dimension depending on the number of components (n) is illustrate in Fig.5.
According to the investigation in papers [15 – 17] the Direct Partial Logical
Derivatives is applicable for systems which have dimension less than ten millions
elements. So the proposed method can be used for the MSS at least ten components.
Fig. 5. Calculation of the Direct Partial Logic Derivative (10)/ x1(10) x3(21).
Acknowledgments. This work was partially supported by grant of Scientific Grant
Agency of the Ministry of Education of Slovak Republic (Vega 1/0498/14) and the
project "Innovation and internationalization of education - instruments for increasing
quality of the University of Žilina in the European educational space".
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