Applied Mathematics and Computation 173 (2006) 137–149 www.elsevier.com/locate/amc
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-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 stan2 dard state-space method with the complexity of O(3n ). Finally, we generalize the proposed methodology, in which the failure mechanisms of the components are different. 2005 Elsevier Inc. All rights reserved. Keywords: Reliability; Markov processes; Graph theory; Complexity
*
Corresponding author. E-mail address:
[email protected] (A. Azaron).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.051
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Nomenclature Ti T Cj Xj R(t) F(t)
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
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 information 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]. In this paper, we present a new methodology for the reliability evaluation of an l-dissimilar-unit multi-component non-repairable cold-standby redundant system. 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. Therefore, this methodology allows non-constant hazard functions. To evaluate the system reliability, we construct a directed stochastic network with exponentially distributed arc lengths, in which each path corresponds 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 network. Finally, we obtain a closed form for the system reliability. Extensive research has been carried out on the reliability of redundant systems 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
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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 coldstandby unit becomes operative instantaneously upon the failure of operative unit. Gupta et al. [7] analyzed a two dissimilar unit multi-component coldstandby 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-dissimilar-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 coldstandby systems assume that each unit is composed of a number of components 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 reliability 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 statespace 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 combinatorial 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 consists 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
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with the ith arc (component) of the reliability graph, i = 1, 2, . . . , n, there is a random variable Ti as the lifetime of this component with generalized Erlang distribution of order ni and the infinitesimal generator matrix Gi as 3 2 0 0 0 ki1 ki1 7 6 ki2 ki2 0 0 7 6 0 7 6 7 6 7. Gi ¼ 6 7 6 6 0 0 0 kini kini 7 5 4 0
0
0
0
0
In this case, Ti would be the time until absorption in the absorbing state. An Erlang distribution of order ni is a generalized Erlang distribution with ki1 ¼ ki2 ¼ ¼ kini ¼ 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 components 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 Xj ¼ T i.
ð1Þ
i2C j
Proof. Refer to Azaron et al. [1] for the details of proof.
h
To evaluate the reliability function, we construct a directed stochastic network with exponentially distributed arc lengths. There are m paths in this network, 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 ni and the infinitesimal generator
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matrix Gi, it can be decomposed to ni exponential serial arcs with parameters ki1 ; ki2 ; . . . ; kini . The following rule describes how to construct the proper directed 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 ni exponential serial arcs with parameters ki1 ; ki2 ; . . . ; kini . Theorem 1. The system lifetime is given by T ¼ min fX j g.
ð2Þ
j¼1;2;...;m
Proof. Upon the failure of the first minimal cut of the reliability graph of system, all connections between the input and the output are interrupted, and consequently 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 F ðtÞ. Proof. Relation (3) follows from the definitions of R(t) and F(t).
ð3Þ h
3. Shortest path distribution in directed networks KulkarniÕs method [8] is applied to obtain the distribution function of shortest path, from the source to the sink node, in the directed network, and accordingly 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:
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1. X 1 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. SðX Þ ¼ X [ X 1 . Example 1. In the network depicted in Fig. 1, if we consider X = (1, 2), then X 1 ¼ /, 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, X 1 ¼ ð2Þ, and S(X) = (1, 2, 3, 4). Definition 2 X ¼ X [ V .
X ¼ fX V =s 2 X ; t 2 X ; X ¼ SðX Þg;
ð4Þ
In Example 1, X* = {(1), (1, 2), (1, 3), (1, 2, 3), (1, 2, 4), (1, 2, 3, 4), (1, 2, 3, 4, 5)}. Definition 3. If X V such that s 2 X and t 2 X , then a cut is defined as: CðX ; X Þ ¼ fðu; vÞ 2 A=u 2 X ; v 2 X g.
ð5Þ
There is a unique minimal cut contained in CðX ; X Þ, denoted by C(X). If X 2 X, then, CðX ; X Þ ¼ CðX Þ. 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 8 P kðu;vÞ if Y ¼ SðX [ fvgÞ; > > > ðu;vÞ2ðX Þ > < P ð6Þ qðX ; Y Þ ¼ k if Y ¼ X ; > ðu;vÞ2ðX Þ ðu;vÞ > > > : 0 otherwise.
2 1
4 3
Fig. 1. Graph of Example 1.
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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 : X ðtÞ ¼ N =X ð0Þ ¼ 1g.
ð7Þ
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 P i ðtÞ ¼ P fX ðtÞ ¼ N =X ð0Þ ¼ ig
i ¼ 1; 2; . . . ; N ;
ð8Þ
then, F(t) = P1(t). The system of linear differential equations is given by P_ ðtÞ ¼ Q P ðtÞ; T
P ð0Þ ¼ ½0; 0; . . . ; 1 ;
ð9Þ
where P(t) = [P1(t), P2(t), . . . , PN(t)]T represents 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 equations 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, . . . , aN be the eigenvalues of Q, which are the diagonal elements of Q owning to its upper triangular nature. P(t) can be computed as follows: P ðtÞ ¼ MeAt M 1 P ð0Þ; where eAt is the diagonal matrix as follows: 2 a1 t 3 0 0 e 6 7 6 0 ea2 t 7 At 6 7. e ¼6 7 4 5 aN t 0 e
ð10Þ
ð11Þ
As mentioned, F(t) = P1(t), and the reliability function of the multi-component 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. Indeed, 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
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real-world problems, the designers sometimes use fundamentally different designs 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, the optimal ðkij ; ni Þ, j = 1, 2, . . . , ni, i = 1, 2, . . . , n, would be related to that combination, which results the maximum of the mean time to failure of the multi-component cold-standby redundant system, considering the following equation: Z 1 MTTF ¼ RðtÞ dt. ð12Þ 0
It should also be noted that the infinitesimal generator matrix for each combination 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.
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 series 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 (component 5), as the cold-standby redundant components, and also a monitor (component 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
1
4
2
2
3
6 5 8 7 3
9 10
Fig. 2. Reliability graph of the multi-component cold-standby redundant system.
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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 general 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 controller. 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 reliability function and the mean time to failure of this 3-dissimilar-unit multi-component non-repairable cold-standby redundant system, considering both 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 reliability 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 ni exponential serial arcs with parameters ki1 ; ki2 ; . . . ; kini . The proper directed network of the first combination 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
Table 1 Parameters of the lifetime distributions (first combination) i
ni
ðki1 ; ki2 ; . . . ; kini Þ
1 2 3 4 5 6 7 8 9 10
3 1 1 1 1 2 2 1 1 1
(2,4,3) (1) (2) (5) (3) (1,6) (3,2) (4) (5) (6)
3 1
1
2
4
3
5
7
4
7
8 2
6
Fig. 3. Directed network.
10 5
9
6
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1
4
2
3
3
4
5
5 1
3
7
6
2
9
4
11 6
8
6
1
3
5
10
Fig. 4. The proper directed network of the first combination, following Rule 1.
was replaced by 3 exponential serial arcs with parameters (2, 4, 3) in Fig. 4, because 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), (1, 2, 3, 4, 6), (1, 2, 3, 4, 5, 6), (1, 2, 3, 4, 5, 6, 7), (1, 2, 3, 4, 5, 6, 7, 8), (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, 10), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)}. Table 2 shows the corresponding infinitesimal generator matrix. Then, we apply the method described in Section 3, to solve the system of linear differential equations P_ ðtÞ ¼ Q P ðtÞ, 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: RðtÞ ¼ 1 P 1 ðtÞ ¼ 106.66e3t 686.4e4t þ 2911.7e5t þ 707.15e6t 4050te6t 4505e7t þ 1419.96e8t þ 673.92te8t þ 46.37e9t þ 0.58e11t 0.02e13t ; MTTF ffi 1.31. Table 2 Matrix Q of the first combination State
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 0 0 0 0 0 0 0 0 0 0 0 0 0
2 5 0 0 0 0 0 0 0 0 0 0 0 0
0 4 4 0 0 0 0 0 0 0 0 0 0 0
1 1 4 8 0 0 0 0 0 0 0 0 0 0
0 0 0 5 6 0 0 0 0 0 0 0 0 0
0 0 0 1 0 13 0 0 0 0 0 0 0 0
0 0 0 0 1 5 11 0 0 0 0 0 0 0
0 0 0 2 5 8 11 7 0 0 0 0 0 0
0 0 0 0 0 0 0 4 8 0 0 0 0 0
0 0 0 0 0 0 0 3 0 6 0 0 0 0
0 0 0 0 0 0 0 0 3 4 7 0 0 0
0 0 0 0 0 0 0 0 5 0 0 9 0 0
0 0 0 0 0 0 0 0 0 0 5 3 8 0
0 0 0 0 0 0 0 0 0 2 2 6 8 0
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Table 3 Parameters of the lifetime distributions (second combination) i
ni
ðki1 ; ki2 ; . . . ; kini Þ
1 2 3 4 5 6 7 8 9 10
1 2 1 1 1 1 1 1 1 1
(1) (1,1) (1) (1) (1) (1) (1) (1) (1) (1)
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 multicomponent cold-standby redundant system, considering the second combination, are obtained as follows: RðtÞ ¼ 13e3t þ 14e2t 2te3t 9te2t þ 6.5t2 e2t 0.17t3 e2t þ 0.42t4 e2t ; 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 general configuration.The lifetime of each component was assumed to be a random variable with generalized Erlang distribution. Therefore, this 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
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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(k1) arcs representing the components of the system (the worst case example), the size of the state space is 2k2 + 1, refer to Kulkarni [8] for details, but the size of the state space using clever complete enumeration of network states would be equal to 3k(k1), 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 2 O(3n ), 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 approximate the lifetime distribution by an appropriate generalized Erlang distribution, 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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