Reliability evaluation of multi-component cold-standby redundant systems
ABSTRACT A new methodology for the reliability evaluation of an l-dissimilar-unit non-repairable cold-standby redundant system is introduced in this paper. Each unit is composed of a number of independent components with generalized Erlang distributions of lifetimes, arranged in any general configuration. We also extend the proposed model to the general types of non-constant hazard functions. To evaluate the system reliability, we construct a directed stochastic network with exponentially distributed arc lengths, in which each path of this network corresponds with a particular minimal cut of the reliability graph of system. Then, we present an analytical method to solve the resulting system of differential equations and to obtain the reliability function of the standby system. The time complexity of the proposed algorithm is O(2n), which is much less than the standard state-space method with the complexity of O(3n2). Finally, we generalize the proposed methodology, in which the failure mechanisms of the components are different.
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ABSTRACT: The mission reliability of a multiple working mode system is subject to the changes of working modes adopted during phases of different missions, which can be improved through optimization of the working mode strategies under different system states. In this paper, a computing method is proposed for the mission reliability and the mission cost of a multiple working mode system within the planned time. Then, an optimization model for the mission reliability is put forward with the system working mode as its strategic variability, mission reliability as its optimization objective and mission cost for its constraint. Furthermore, the feasibility and practicability of the proposed model are testified through a case study.01/2011; - [Show abstract] [Hide abstract]
ABSTRACT: The aircraft environmental control system (ECS) is composed of several non-identical and non-dedicated subsystems working as warm–cold standby subsystems. Also, their state transition times are arbitrary distributed. This paper presents a flow-graph-based method to calculate time-to-failure data and failure probability of the ECS. The obtained data from the model may be used for maintenance optimization that employs the failure limit strategy for ECS. The model incorporates detectable failures such as hardware failures, critical human errors, common-cause failures, maintenance categories, and switch activation methods. A numerical example is also presented to demonstrate the application of the model.Reliability Engineering & System Safety. 01/2009; - SourceAvailable from: taylors.edu.my[Show abstract] [Hide abstract]
ABSTRACT: The aim of the present paper is to study and examine the various reliability characteristics of a global mobile satellite communication System (GMSCS) with the help of mathematical modelling. GMSCS basically comprises of four subsystems; Space segment, Land earth stations, Telecommunication terminals and Mobile earth stations. The system under consideration can have three different modes of working: normal, partial and complete failure. The system is characterized by determination of probabilities being in ‘up’ and ‘down’ states at any instant. Integro-differential equations are derived for these probabilities by identifying the system at suitable regeneration epochs. Based on the assumption that the failure rates of units are distributed exponentially while the repair rates are distributed arbitrarily, different reliability measures like operational availability, reliability, mean time to failure and cost effectiveness have been computed with the principle of Laplace transforms and supplementary variable technique. Considerable attention is devoted to illustrate the results with numerical examples to highlight the important features of reliability measures of the system.Journal of Engineering Science and Technology. 01/2011;
Page 1
Reliability evaluation of multi-component
cold-standby redundant systems
Amir Azaron*, Hideki Katagiri, Kosuke Kato,
Masatoshi Sakawa
Department of Artificial Complex Systems Engineering, Graduate School of Engineering,
Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima 739-8527, Japan
Abstract
A new methodology for the reliability evaluation of an l-dissimilar-unit non-repair-
able cold-standby redundant system is introduced in this paper. Each unit is composed
of a number of independent components with generalized Erlang distributions of life-
times, arranged in any general configuration. We also extend the proposed model to
the general types of non-constant hazard functions. To evaluate the system reliability,
we construct a directed stochastic network with exponentially distributed arc lengths,
in which each path of this network corresponds with a particular minimal cut of the reli-
ability graph of system. Then, we present an analytical method to solve the resulting sys-
tem of differential equations and to obtain the reliability function of the standby system.
The time complexity of the proposed algorithm is O(2n), which is much less than the stan-
dard state-space method with the complexity of O(3n2). Finally, we generalize the pro-
posed methodology, in which the failure mechanisms of the components are different.
? 2005 Elsevier Inc. All rights reserved.
Keywords: Reliability; Markov processes; Graph theory; Complexity
0096-3003/$ - see front matter ? 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2005.02.051
*Corresponding author.
E-mail address: azaron@msl.sys.hiroshima-u.ac.jp (A. Azaron).
Applied Mathematics and Computation 173 (2006) 137–149
www.elsevier.com/locate/amc
Page 2
1. Introduction
Many fielded systems use cold-standby redundancy as an effective system
design strategy. Cold-standby means that the redundant units cannot fail while
they are waiting. Space exploration and satellite systems achieve high reliability
by using cold-standby redundancy for non-repairable systems, see Sinaki [13].
Space inertial reference units are required to accurately monitor critical infor-
mation for extended mission times without opportunities for repair. Many
other systems use cold-standby redundancy as an effective strategy to achieve
high reliability including textile manufacturing systems, see Pandey et al.
[11], and carbon recovery systems used in fertilizer plants, see Kumar et al. [9].
Inthispaper,wepresentanewmethodologyforthereliabilityevaluationofan
l-dissimilar-unit multi-component non-repairable cold-standby redundant sys-
tem. Each unit is composed of a number of independent components, arranged
in any general configuration. The lifetimes of the components are assumed to
be independent random variables with generalized Erlang distributions. There-
fore, this methodology allows non-constant hazard functions.
To evaluate the system reliability, we construct a directed stochastic net-
work with exponentially distributed arc lengths, in which each path corre-
sponds with a particular minimal cut of the reliability graph of system.
Then, we prove that the system failure function is equal to the density function
of the shortest path, from the source node to the sink node, in the directed net-
work. Finally, we obtain a closed form for the system reliability.
Extensive research has been carried out on the reliability of redundant sys-
tems with similar/dissimilar units. Several methods and methodologies have
been discussed by Birolini [2] and Srinivasan and Subramanian [14]. Azaron
et al. [1] developed a new approach to evaluate the reliability function of a class
of dissimilar unit redundant systems with exponentially distributed lifetimes.
Multi-component systems have been analyzed by several authors including
Goel et al. [3] and Yamashiro [15]. Most of such studies deal with the analysis
Nomenclature
Ti
T
Cj
Xj
lifetime of the ith component of the system, i = 1,2,...,n
system lifetime
jth minimal cut of the reliability graph, j = 1,2,...,m
failure time of the jth minimal cut of the reliability graph,
j = 1,2,...,m
reliability function of the multi-component cold-standby system
distribution function of shortest path, from the source to the sink
node, in directed network
R(t)
F(t)
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A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
Page 3
of a single unit system. Gupta et al. [6] investigated a single server two-unit
multi-component cold-standby system under the assumption that the cold-
standby unit becomes operative instantaneously upon the failure of operative
unit. Gupta et al. [7] analyzed a two dissimilar unit multi-component cold-
standby system with correlated failures and repairs. There has been little
research toward the study of l-unit multi-component systems because of the
complexity in the equations and not getting the results in closed form. An l-unit
multi-component system is analyzed by Goel and Gupta [4] assuming the failed
unit is replaced by the leading standby unit with a constant replacement rate.
The major limitations in the reliability evaluation approaches for l-dissimi-
lar-unit cold-standby systems thus far are:
1. Most available algorithms assume that each unit is composed of a single
component, but they also cannot get the results in closed form, see Goel
and Gupta [5].
2. Available algorithms that do address dissimilar units multi-component cold-
standby systems assume that each unit is composed of a number of compo-
nents arranged in a series configuration. Although this is a start, there are
many more complicated system configurations that should be examined.
The problem lies in the difficulty of presenting more complicated structures.
3. Most of these studies have been carried out using the standard state-space
method. In this method, the reliability analysis becomes complex and time
consuming as the number of components increases, because of the large
number of states involved.
In this paper, we extend the work of Azaron et al. [1] to evaluate the reliabil-
ity function of an l-dissimilar-unit multi-component cold-standby redundant
system. Our methodology not only gets the reliability function in closed form
for the complex structures (l-dissimilar-unit multi-component cold-standby
systems with non-constant hazard functions), but also the size of the state-
space and the corresponding computational time, accordingly, are much less
than the standard state-space method. Finally, we generalize the proposed
methodology, in which the failure mechanisms of the components are different
or the distribution parameters of the components are considered as the combi-
natorial design variables.
2. Reliability evaluation of multi-component cold-standby systems
A very efficient method to compute the reliability of a system is to express it
as a reliability graph, see Shooman [12] for the details. A reliability graph con-
sists of a set of arcs. Each arc represents a component of the system, while the
nodes of the graph tie the arcs together and form the structure. Corresponding
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
139
Page 4
with the ith arc (component) of the reliability graph, i = 1,2,...,n, there is a
random variable Tias the lifetime of this component with generalized Erlang
distribution of order niand the infinitesimal generator matrix Gias
2
Gi¼
?ki1
0
ki1
0
???
???
???
???
???
00
?ki2
?
0
ki2
00
?
0
?
0
??
?kini
0
kini
0000
66666664
3
77777775
.
In this case, Tiwould be the time until absorption in the absorbing state. An
Erlang distribution of order ni is a generalized Erlang distribution with
ki1¼ ki2¼ ??? ¼ kini¼ ki. When ni= 1, the underlying distribution becomes
exponential with the parameter ki. Ti, i = 1,2,...,n, are independent random
variables, due to the fact that the components work independently.
By definition, a cut of the reliability graph is a set of components, which
interrupts all connections between input and output when removed from the
graph. A minimal cut is a cut with the minimum number of terms. Each system
failure can be represented by the removal of at least one minimal cut from the
graph.
As mentioned, we consider a cold-standby system, i.e., not all of its compo-
nents are set to function at time zero. Initially, only the components of the first
path of the reliability graph work. Upon failing one component of this path,
the system is switched to the next path and the connection between the input
and the output is established through this second path. This process continues
until no other connection between the input and the output of the reliability
graph exists. In that case, the system fails.
Lemma 1. For j = 1,2,...,m, the following relation holds:
X
Xj¼
i2Cj
Ti.
ð1Þ
Proof. Refer to Azaron et al. [1] for the details of proof.
h
To evaluate the reliability function, we construct a directed stochastic net-
work with exponentially distributed arc lengths. There are m paths in this net-
work, in which the jth path of this directed network corresponds with the jth
minimal cut of the reliability graph of the system, j = 1,2,...,m. Clearly, by
Lemma 1, the length of each path in this directed network is equal to the failure
time of the corresponding cut. For constructing this network, we use the idea
that if the lifetime of the ith component of the system is distributed according
to a generalized Erlang distribution of order niand the infinitesimal generator
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A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
Page 5
matrix Gi, it can be decomposed to niexponential serial arcs with parameters
ki1;ki2;...;kini. The following rule describes how to construct the proper direc-
ted network.
Rule 1. Arc i belongs to the jth path of the directed network, if and only if
i 2 Cj. If ni= 1, then the length of this arc is exponentially distributed with
parameter ki. Otherwise, if ni> 1, then this arc is replaced by niexponential
serial arcs with parameters ki1;ki2;...;kini.
Theorem 1. The system lifetime is given by
T ¼
min
j¼1;2;...;mfXjg.
ð2Þ
Proof. Upon the failure of the first minimal cut of the reliability graph of sys-
tem, all connections between the input and the output are interrupted, and con-
sequently the multi-component cold-standby system fails. Therefore, the
lifetime of the multi-component cold-standby system would be equal to the
failure time of the first minimal cut, which results in (2).
h
Corollary 1. The reliability function of the multi-component cold-standby system
is given by
RðtÞ ¼ 1 ? FðtÞ.
ð3Þ
Proof. Relation (3) follows from the definitions of R(t) and F(t).
h
3. Shortest path distribution in directed networks
Kulkarni?s method [8] is applied to obtain the distribution function of short-
est path, from the source to the sink node, in the directed network, and accord-
ingly the reliability function of the cold-standby system.
Let G = (V,A) be a directed network, in which V and A represent the sets of
nodes and arcs of the network, respectively. Let s and t represent the source
and the sink nodes of this network, respectively. The length of arc (u,v) 2 A
is indicated by T(u,v), which is an exponential random variable with parameter
k(u,v).
Definition 1. To describe the evolution of the stochastic process {X(t),t P 0},
for each X ? V, where s 2 X and t 2 X ¼ V ? X, we define the following
sets:
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
141
Page 6
1. X1? X, set of nodes not included in X with the property that each path
which connects any node of this set to the sink node t, contains at least
one member of X.
2. SðXÞ ¼ X [ X1.
Example 1. In the network depicted in Fig. 1, if we consider X = (1,2), then
X1¼ /, and S(X) = (1,2). However, if we consider X = (1,3,4), then the only
path that connects node ð2Þ 2 X to node (5) passes through node (4), which
belongs to X. Therefore, X1¼ ð2Þ, and S(X) = (1,2,3,4).
Definition 2
X ¼ fX ? V =s 2 X;t 2 X;X ¼ SðXÞg;
In Example 1, X* = {(1),(1,2),(1,3),(1,2,3),(1,2,4),(1,2,3,4),(1,2,3,4,5)}.
X?¼ X [ V .
ð4Þ
Definition 3. If X ? V such that s 2 X and t 2 X, then a cut is defined as:
CðX;XÞ ¼ fðu;vÞ 2 A=u 2 X;v 2 Xg.
There is a unique minimal cut contained in CðX;XÞ, denoted by C(X). If X 2 X,
then,
ð5Þ
CðX;XÞ ¼ CðXÞ.
It is shown that {X(t),t P 0} is a continuous-time Markov process with state
space X* and the infinitesimal generator matrix Q = [q(X,Y)](X,Y 2 X*), see
Kulkarni [8] for the details, where
P
?
ðu;vÞ2ðXÞ
0 otherwise.
qðX;YÞ ¼
ðu;vÞ2ðXÞ
kðu;vÞ
if Y ¼ SðX [ fvgÞ;
P
kðu;vÞ
if Y ¼ X;
8
>
>
>
>
>
>
>
>
:
<
ð6Þ
2
5 4
3
1
Fig. 1. Graph of Example 1.
142
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
Page 7
We assume that the states in X* are numbered 1,2,...,N = jX*j so that Q
matrix is upper triangular. State 1 is the initial state, and state N is the final
(absorbing) state. In Example 1, state 1 is (1), and state 7 is (1,2,3,4,5).
Let T represent the length of the shortest path in the directed network.
Clearly,
T ¼ minft > 0 : XðtÞ ¼ N=Xð0Þ ¼ 1g.
Therefore, the length of the shortest path in the directed network would be
equal to the time until {X(t),t P 0} gets absorbed in the final state N, starting
from state 1.
Chapman–Kolmogorov backward equations can be applied to compute
F(t) = P{T 6 t}. If we define
PiðtÞ ¼ PfXðtÞ ¼ N=Xð0Þ ¼ ig
then, F(t) = P1(t).
The system of linear differential equations is given by
ð7Þ
i ¼ 1;2;...;N;
ð8Þ
_PðtÞ ¼ Q ? PðtÞ;
Pð0Þ ¼ ½0;0;...;1?T;
where P(t) = [P1(t),P2(t),...,PN(t)]Trepresents the state vector of the system
and Q is the infinitesimal generator matrix.
Now, we present an efficient method to solve the system of differential equa-
tions with constant coefficients (9). Let M be the modal matrix of Q. That is, M
is the N · N matrix whose N columns are the eigenvectors of Q. Let
a1,a2,...,aNbe the eigenvalues of Q, which are the diagonal elements of Q
owning to its upper triangular nature. P(t) can be computed as follows:
ð9Þ
PðtÞ ¼ MeAtM?1Pð0Þ;
where eAtis the diagonal matrix as follows:
ea1t
0
ea2t
ð10Þ
eAt¼
?
?
?
?
0
0
?
??
0
?
?
eaNt
2
66664
3
77775
.
ð11Þ
As mentioned, F(t) = P1(t), and the reliability function of the multi-compo-
nent cold-standby system is computed from (3).
A system with multiple eigenvalues can be perturbed, by introducing a slight
change in some coefficients, to produce a system with distinct eigenvalues. In-
deed, the original system can be regarded as the limit of systems with distinct
eigenvalues, see Luenberger [10] for the details.
The proposed methodology is easily generalized, in which the distribution
parameters (kij,ni) are considered as the combinatorial design variables. In
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
143
Page 8
real-world problems, the designers sometimes use fundamentally different de-
signs or technologies with different distribution parameters, because the failure
mechanisms would be different. In this case, we first compute the reliability
function of the standby system, for all combinations of (kij,ni) for
j = 1,2,...,ni,
i = 1,2,...,n. Then,
i = 1,2,...,n, would be related to that combination, which results the maxi-
mum of the mean time to failure of the multi-component cold-standby redun-
dant system, considering the following equation:
Z1
It should also be noted that the infinitesimal generator matrix for each com-
bination of (kij,ni), for j = 1,2,...,ni, i = 1,2,...,n, would be different from the
other combinations, and this matter clearly increases the complexity of the
problem.
theoptimal
ðk?
ij;n?
iÞ,
j = 1,2,...,ni,
MTTF ¼
0
RðtÞdt.
ð12Þ
4. Numerical example
For controlling a spacecraft, there are 3 non-repairable dissimilar units in a
cold-standby redundancy scheme, depicted in Fig. 2.
At the beginning, the operating unit is unit 1, which is composed of a laptop
computer (component 1) and a power supply (component 2) arranged in a ser-
ies configuration. When this unit fails, the redundant unit 2, which is composed
of PC I (component 3), CD Drive I (component 4) and CD Drive II (compo-
nent 5), as the cold-standby redundant components, and also a monitor (com-
ponent 6), arranged in a general configuration, is put into operation. If unit 2
fails, then the redundant unit 3, which is composed of PC II (component 7) and
1 2
3
4
5
6
7
8
9
10
2
3
1
Fig. 2. Reliability graph of the multi-component cold-standby redundant system.
144
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
Page 9
Hard Drive I (component 8), Hard Drive II (component 9) and Hard Drive III
(component 10), as the cold standby redundant components, arranged in a gen-
eral configuration, goes into operation. The lifetimes of the components are
independent random variables with generalized Erlang distributions. There
are two combinations for the distribution parameters of this spacecraft control-
ler. Table 1 shows the parameters of the lifetime distributions, according to the
first combination. The time unit is in year. We are interested to find the reliabil-
ity function and the mean time to failure of this 3-dissimilar-unit multi-compo-
nent non-repairable cold-standbyredundant
combinations.
Fig. 3 shows the directed network, where each path is corresponding with a
particular minimal cut of the reliability graph. For example, the path 1–4–5–7
of this network is corresponding with the minimal cut (1,4,5,7) of the reliabil-
ity graph (2), which interrupts all connections between input and output of the
cold-standby system upon removing from the graph. Then, according to Rule
1, if ni> 1, we replace the particular arc by niexponential serial arcs with
parameters ki1;ki2;...;kini. The proper directed network of the first combina-
tion is shown in Fig. 4. The numbers above the arcs show the exponential
parameters of the corresponding arc lengths. For example, arc 1 in Fig. 3
system, considering both
Table 1
Parameters of the lifetime distributions (first combination)
ini
ðki1;ki2;...;kiniÞ
(2,4,3)
(1)
(2)
(5)
(3)
(1,6)
(3,2)
(4)
(5)
(6)
1
2
3
4
5
6
7
8
9
10
3
1
1
1
1
2
2
1
1
1
3
1 4 5 7
2 1
8 10
2 6 9
3
5
6
7 4
Fig. 3. Directed network.
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
145
Page 10
was replaced by 3 exponential serial arcs with parameters (2,4,3) in Fig. 4, be-
cause n1= 3, and (k11,k12,k13) = (2,4,3).
The stochastic process {X(t),t P 0} corresponding with the shortest path
analysis of the directed network, depicted in Fig. 4, has 14 states in the order
of
X* = {(1),(1,2),(1,2,3),(1,2,3,4),
(1,2,3,4,5,6,7),(1,2,3,4,5,6,7,8),
(1,2,3,4,5,6,7,8,10),(1,2,3,4,5,6,7,8,9,10),
Table 2 shows the corresponding infinitesimal generator matrix.
Then,weapplythe methoddescribed in Section 3,to solve the system of linear
differential equations_PðtÞ ¼ Q ? PðtÞ, P(0) = [0,0,...,1]T. Finally, the reliability
function and the mean time to failure of this multi-component cold-standby
redundant system, considering the first combination, are obtained as follows:
(1,2,3,4,5),(1,2,3,4,6),(1,2,3,4,5,6),
(1,2,3,4,5,6,7,9),(1,2,3,4,5,6,7,8,9),
(1,2,3,4,5,6,7,8,9,10,11)}.
RðtÞ ¼ 1 ? P1ðtÞ ¼ 106.66e?3t? 686.4e?4tþ 2911.7e?5tþ 707.15e?6t
? 4050te?6t? 4505e?7tþ 1419.96e?8tþ 673.92te?8tþ 46.37e?9t
þ 0.58e?11t? 0.02e?13t;
MTTF ffi 1.31.
Table 2
Matrix Q of the first combination
State12345678910 1112 1314
1
2
3
4
5
6
7
8
9
10
11
12
13
14
?320
4
1
1
4
0
0
0
5
0
0
0
1
0
0
0
0
0
1
5
0
0
0
2
5
8
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
3
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
5
3
0
0
0
0
0
0
0
0
0
2
2
6
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
?5
0
0
0
0
0
0
0
0
0
0
0
0
?4
0
0
0
0
0
0
0
0
0
0
0
?8
0
0
0
0
0
0
0
0
0
0
?6
0
0
0
0
0
0
0
0
0
?13
0
0
0
0
0
0
0
0
?11 11
?70
0
0
0
0
0
0
0
0
0
0
0
0
?8
0
0
0
0
0
?6
0
0
0
0
?7
0
0
0
?9
0
0
?8
0
2
2 4 3 5 3 3 2
2
3 4
1
1 6 4 6
5
8
1
5 9 7
6
10
11
Fig. 4. The proper directed network of the first combination, following Rule 1.
146
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
Page 11
Table 3 shows the parameters of the lifetime distributions, according to the
second combination.
The size of the state space, considering the second combination, is equal to
8. Finally, the reliability function and the mean time to failure of this multi-
component cold-standby redundant system, considering the second combina-
tion, are obtained as follows:
RðtÞ ¼ ?13e?3tþ 14e?2t? 2te?3t? 9te?2tþ 6.5t2e?2t? 0.17t3e?2tþ 0.42t4e?2t;
MTTF ffi 2.07.
Therefore, comparing the two mentioned combinations for the spacecraft
controller results that the best would be the second combination, which has
the maximum mean time to failure.
5. Conclusion
In this paper, we introduced a new methodology, by using continuous-time
Markov processes and shortest path technique, for the reliability evaluation of
an l-dissimilar-unit non-repairable cold-standby redundant system, where each
unit is composed of a number of independent components arranged in any gen-
eral configuration.The lifetime of each component was assumed to be a ran-
domvariablewith generalizedErlang
methodology allows non-constant hazard functions. Finally, we generalized
the proposed methodology, in which the failure mechanisms of the components
are different or the distribution parameters of the components are considered
as the combinatorial design variables.
Our approach is a powerful approach for the reliability evaluation of
these complex systems. Computing the reliability of these systems, if is not
distribution. Therefore, this
Table 3
Parameters of the lifetime distributions (second combination)
ini
ðki1;ki2;...;kiniÞ
(1)
(1,1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
1
2
3
4
5
6
7
8
9
10
1
2
1
1
1
1
1
1
1
1
A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149
147
Page 12
impossible, is at least so complicated for most real case problems, because
either the convolution integrals are intractable or the size of the state space
would be enormous. For example, this problem could be solved by using clever
complete enumeration of network states, see Shooman [12] for the details.
According to our methodology, for a complete constructed network with k
nodes and k(k?1) arcs representing the components of the system (the worst
case example), the size of the state space is 2k?2+ 1, refer to Kulkarni [8] for
details, but the size of the state space using clever complete enumeration of net-
work states would be equal to 3k(k?1), because each component can be in one of
these three states: work, fail and standby. Therefore, although the proposed
algorithm is still exponential, but its time complexity is decreased from
O(3n2), corresponding with the standard state space method, to O(2n). In the
numerical example, corresponding with the first combination, the state space
has only 14 states, but according to the state space method, mentioned above,
the size of the state space would be 310.
In the case of general distribution of lifetime, it could be possible to approx-
imate the lifetime distribution by an appropriate generalized Erlang distribu-
tion, by matching the first three moments, because the class of generalized
Erlang distributions is a special case of Coxian distributions, and each general
distribution can be easily approximated by a Coxian distribution. After
approximating the general lifetime distributions by the appropriate generalized
Erlang distributions, our methodology can be applied to obtain the reliability
function of the multi-component cold-standby system.
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