1. Introduction
The consecutive-k system was first studied around 1980, and it soon became a very popular subject [1-3].The reasons were many-folded, including: 1. This kind of system is simple and natural, which means that most people can understand it and many can do some analysis. However, it can be developed in many directions, and there is no lack of new topics [4-6]. 2. The consecutive system is simple enough. Thus, it has become a prototype for researchers to demonstrate their various ideas related to reliability theories or applications. For example, the interesting concept of component importance works best with the consecutive-k system [7-9]. 3. The system is supported by many applications. Over the course of nearly forty years, thousands of papers about this subject have been published [2,10-13]. For instance, in reaction to the problems of fixed point measurement of temperature and no measurement of humidity in the current cocoon drying process, a consecutive and dynamic measuring system of cocoon drying temperature and humidity has been developed [14]. The properties of consecutive energy-barrier systems are derived from the basic principles of deformation kinetics and from the rate theory [10]. As the importance and the popularity of the consecutive system keep increasing, its reliability is an important property that should be concerned.
There are some subclasses of the consecutive system. As a generalization of a k-out-of-n: F system, where “F” stands for “fail”, the consecutive k-out-of-n: F system has been studied by many researchers [7, 15-17]. A k-out-of-n: F system has a total of n components and fails if at least k components fail. This type of structure has been widely applied in voting systems, power systems, etc. Different from the k-out-of-n: F system, a consecutive k-out-of-n: F system fails if at least F consecutive components out of the total n components fail [18-19]. This type of structure has applications in telecommunication systems, oil pumping systems, etc. The failure of too many relay stations in the telecommunication system may make the signal transmission disconnected. Following the basic definitions of the two types of systems, some variants or generalizations of them have been studied [20-22]. Habib et al. [23] studied the reliability of a multi-state consecutive k-out-of-r-from-n: G system, where G stands for “good”. This type of system has n components having multiple states and works if there exist kj components out of any r consecutive components above state j for any j. Levitin [24] studied a system consisting of n components, which fails if the gap between any two groups of r consecutive components containing at least k failed components is less than m components. Peng et al. [25] studied the optimal component allocation in a linear multi-state consecutive connected system, of which the consecutive k-out-of-n: F system can be regarded as a special case.
None of them have considered the case where a system may fail if either any consecutive number of components fail or any total number of components fail. For example, the speech recognition system may be deemed as unsuccessful if too many consecutive words are recognized wrongly or the total number of wrongly recognized words is too many. In [26], a model is proposed to consider both failure modes. However, it shows only the iterative relationship, but not the detailed calculation process. In this paper, we consider a system with n linearly arranged components, which fails either when any k consecutive components fail or any v>k arbitrary components fail. The system is termed as a combined consecutive k and v-out-of-n system. The iterative approach proposed is somewhat different from the one in [26]. This paper focuses on the iterative relationship between the success probabilities, whereas [27] obtains the iterative relationship between the failure probabilities. Moreover, numerical examples are presented to show the detailed calculation processes.
The remaining of this paper is as follows. Section 2 elaborates on an iterative approach for evaluating the reliability of such a system. Section 3 provides a numerical example to illustrate the applications. Section 4 presents the conclusions.
Nomenclature
| n | Total numbers of components |
|---|---|
| p | Fault detection process |
| k | The number of the consecutive failed components that lead to the failure of the whole system |
| v | The number of the failed components in arbitrary position that lead to the failure of the whole system |
| $R(n,k,v,p)$ | The reliability of the whole system that have two types of failure modes |
| ${{R}_{C}}(n,k,p)$ | The reliability of the whole system that fails if k consecutive components fail |
| ${{R}_{A}}(n,v,p)$ | The reliability of the whole system that fails if v components in arbitrary positions fail |
| m | The index of the first component that does not fail |
| $Q(m)$ | The probability that the first working component is the No. m in the whole system |
| ${{N}_{C}}(j,k,n-m)$ | The number of ways to arrange j failed components within n-m linearly ordered components under the condition that no k or more failed components are consecutive |
| $j/k$ | The maximum integer no largerthan$j/k$ |
| $C(n,k,v)$ | The complexity of calculating the reliability of a combined consecutive k and v-out-of-n system |
2. The Model
In this study, we consider a combined consecutive k and v-out-of-n system with n linearly arranged homogeneous components, each with failure probability p. The reliability of the whole system is denoted as $R(n,k,v,p)$. The system fails if any $k<n$ consecutive components fail or any $v\text{ }(n>v>k)$ arbitrary components fail. The reliabilities for the two types of failure are indicated by ${{R}_{C}}(n,k,p)$ and ${{R}_{A}}(n,v,p)$ respectively. In order to get the form of $R(n,k,v,p)$, an iterative approach can be adopted. The system and the two types of failures are shown in Figure 1.
Figure 1
Figure 1.
The system and the failure types
Use m to denote the index of the first component that does not fail. It is easy to see that the system is possible to function only when m ≤ k. The probability associated with each m can be calculated as
This can be illustrated in Figure 2. For each realization of m, the conditional probability that the system functions can be denoted as $R(n-m,k,v-m+1,p)$. Furthermore, the system reliability can be obtained as
Figure 2
Figure 2.
Two failure types
In order to calculate $R(n-m,k,v-m+1,p)$, we need to distinguish a few cases based on the values of ($n-m$), k, and ($v-m+1$).In detail,if($n-m$) becomes smaller than both k and ($v-m+1$), this means the components left that may fail are less than the minimum number of the failed components that lead to the failure of the whole system. In this case, $R(n-m,k,v-m+1,p)$ equals 1.Otherwise, we should first determine whether the k is equal to($v-m+1$). If k is equal to ($v-m+1$), $R(n-m,k,v-m+1,p)$ becomes a k-out-of-(n-m) system where the system fails if any k arbitrary components fail. In the cases where the values of k and ($v-m+1$) are different and the ($n-m$)lies between k and ($v-m+1$), the system becomes either a consecutive [$k-(n-m)$] system or a v-out-of-(n-m) system. In either case, the form of $R(n-m,k,v-m+1,p)$ can be obtained. Finally, for the case where ($n-m$) is no smaller than both ($v-m+1$)and k, we still need to use the iterative approach.
The mathematical expressions for these cases are shown as follows:
$R(n-m,k,v-m+1,p)=1$,
if $n-m<k\text{ }\!\!\And\!\!\text{ }n-m<v-m+1$;
$R(n-m,k,v-m+1,p)={{R}_{C}}(n-m,k,p)$,
if $v-m+1>n-m\ge k$;
$R(n-m,k,v-m+1,p)={{R}_{A}}(n-m,v-m+1,p)$,
if $n-m\ge v-m+1\text{ }\!\!\And\!\!\text{ }k\ge v-m+1$;
$R(n-m,k,v-m+1,p)=\sum\nolimits_{1\le {m}'\le k}{Q({m}')R(n-m-{m}',k,v-m+1-{m}',p)}$,
if $n-m\ge k\text{ }\!\!\And\!\!\text{ }n-m\ge v-m+1\text{ }\!\!\And\!\!\text{ }k<v-m+1$.
For case (2), the system consisting of the last (n-m)components degenerates to a simple consecutive k-out-of-(n-m) system, the reliability of which can be calculated using the next formula:
Where ${{N}_{C}}(j,k,n-m)$ denotes the number of ways to arrange j failed components within (n-m) linearly ordered components under the condition that no k or more failed components are consecutive. ${{N}_{C}}(j,k,n-m)$is given as follows [12]:
Where $j/k$ denotes the maximum integer no largerthan $j/k$.
For case (3), the system consisting of the last n-m components degenerates to a simple (v-m+1)-out-of-(n-m) system, the reliability of which can be calculated using the next formula:
Where $C_{n-m}^{j}$ denotes the number of ways to arrange j failed components within (n-m) linearly ordered components in arbitrary position.
For case (4), the iteration process will be repeated again from the start. The whole process is shown in Figure 3.
Figure 3
Figure 3.
The flow chart for the calculation process of $R(n-m,k,v-m+1,p)$
Then, to determine the computational complexity of the system reliability, we can perform the complexity analysis for the reliability calculation. If the reliability of a system can be calculated directly without using Equation (2), we define the complexity in this case as 1 degree. The total complexity is obtained by the sum of the complexities of the systems that degenerate according to Equation (2).
We use $C(n,k,v)$ to represent the complexity of calculating the reliability of a combined consecutivek and v-out-of-n system. It can be calculated as follows:
3. Numerical Example
3.1. Example 1
In this section, a specific model configuration is shown for illustration. It is assumed that there are a total of two different kinds of debuggers. The simulation process is briefly described in the following flow chart:
Consider a system consisting of n=5 homogeneous components, each with failure probability p=0.1. The system fails if at least k=2 consecutive components fail or any three arbitrary components fail. The reliability of the system can be denotedas $R(5,2,3,0.1)$.
According to (2), we have
From(1), it can be calculated that Q(1)=0.9 and Q(2)=0.09. Thus, we have
Note that $R(4,2,3,0.1)$ belongs to case 4 discussed in section 2, and thus it should be further decomposed. Differently, $R(3,2,2,0.1)$ belongs to case(3) discussedin section 2, and it degenerates to ${{R}_{A}}(3,2,0.1)$. We first calculate ${{R}_{A}}(3,2,0.1)$. According to (5), we have
In addition, according to (2), we have
Furthermore, as Q(1)=0.9 and Q(2)=0.09, we have
It can be seen that $R(3,2,3,0.1)$ belongs to case(4) discussed in section 2. $R(2,2,2,0.1)$ belongs to case(3) discussed in section 2. It is easy to see that
Thus, we only need to calculate $R(3,2,3,0.1)$. According to (2), we have
Note that $R(2,2,3,0.1)$ belongs to case 2 discussed in section 2, and it degenerates to ${{R}_{C}}(2,2,0.1)$, which equals 0.99. On the other hand,$R(1,2,2,0.1)$ belongs to case 1 discussed in section 2, which equals 1. Thus, we have
Substituting (14) and (16) into (13), it can be obtained that
Substituting (12) and (13) into (6) gives
Thus, the system reliability is 0.96228.
The complexity of this example can be calculated as the following process:
3.2. Example 2
Consider a system consisting of n= 6 homogeneous components, each with failure probability p=0.1. The system fails if at least k= 3consecutive components fail or any four arbitrary components fail. The reliability of the system can be denotedas $R(6,3,4,0.2)$.
According to (2), we have
From (1), it can be calculated that Q(1)=0.8, Q(2)=0.16, and Q(3)=0.032. Thus, we have
Note that $R(5,3,4,0.2)$ belongs to case 4 discussed in section 2, and thus it should be further decomposed. Differently, $R(4,3,3,0.2)$ and $R(3,3,2,0.2)$ belong to case(3) discussedin section 2 and degenerate to ${{R}_{A}}(4,3,0.2)$ and ${{R}_{A}}(3,2,0.2)$ respectively. We first calculate $R(4,3,3,0.2)$ and $R(3,3,2,0.2)$. According to (3), we have
In addition, according to (2), we have
Similarly, as Q(1)=0.8, Q(2)=0.16, andQ(3)=0.032, we have
$R(3,3,3,0.2)$ and $R(2,3,2,0.2)$ belong to case(3) discussedin section 2, so they can degenerate to ${{R}_{A}}(3,3,0.2)$ and ${{R}_{A}}(2,2,0.2)$ respectively. We first calculate $R(3,3,3,0.2)$ and $R(2,3,2,0.2)$. According to (3), we have
For $R(4,3,4,0.2)$, it can continue to be decomposed according to Equation (2), as shown below:
$R(2,3,3,0.2)$ and $R(1,3,2,0.2)$ are directly equal to 1 according to the condition in case (1). $R(3,3,4,0.2)$ degenerates to ${{R}_{C}}(3,3,0.2)$ as shown in case (2), and it can be easily calculated as
Then, we can calculate the reliability of the whole system as the following process:
Thus, the reliability for this example is 0.966528.
The complexity of this example can be calculated as the following process:
4. Sensitivity Analysis
To investigate how the variation of the related parameters would influence the reliability of the system, the sensitivity analysis was conducted. We varied the values of k, v, and p to see how the reliability changes.
Table 1 shows the variations of the system reliability for different k. It is noted that the reliability of the system increases monotonically with k, which can be interpreted as that the system tends to be less prone to failure as the failure condition becomes rigorous. Especially when k = 1, the reliability is only about 0.59, which means the system is no longer a consecutive k-out-of-n system. This can largely influence the reliability.
Table 1. Variations of the reliability for different k when n=5, v=3, and p=0.1
| k | $R(5,k,3,0.1)$ |
|---|---|
| 1 | 0.59045 |
| 2 | 0.96228 |
| 3 | 0.99104 |
Table 2 presents the influence of the v on the system reliability. It can be seen that with an increase inv, the reliability of the whole system also increases. In this case, the number of failed components needed is increasing, which means the failure condition of the whole system also becomes rigorous.As a result, the reliability increases with an increase in v.
Table 2. Variations of the reliability for different v when n=5, k=2, and p=0.1
| v | $R(5,2,v,0.1)$ |
|---|---|
| 1 | 0.91845 |
| 2 | 0.96228 |
| 3 | 0.96309 |
Anopposite situation happens when considering the influence of failure probability for a single component on the system reliability. Table 3 shows the variations of the reliability for different p. The bigger p represents that the component is more likely to fail, which can increase the failure probability of the whole system. Thus, the reliability of the system is negatively correlated with the failure probability of its components.
Table 3. Variations of the reliability for different p when n=5, k=2, and v=3
| p | $R(5,2,3,p)$ |
|---|---|
| 0.1 | 0.96228 |
| 0.2 | 0.88704 |
| 0.5 | 0.495 |
| 0.8 | 0.103 |
In summary, the reliability of the system is positively associated with the minimum required quantities of the consecutive components and of the components in arbitrary position but negatively correlated with the failure probability of its components.
5. Conclusions
This paper considers a system that fails either due to the failure of too many components or too many consecutive components. An iterative approach has been proposed to calculate the system reliability. Different from previous research, which focused on the iterative relationship between the failure probabilities, this paper focuses on the iterative relationship between the success probabilities. Numerical examples are proposed for illustrative purposes. In the future, this work can be extended to the case where each component has more than two states.
Acknowledgements
This research was partially supported by the NSFC(No. 71671016, 71231001) and the Fundamental Research Fund of Central Universities (No. FRF-GF-17-B14).
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This paper proposes a model that generalizes the linear consecutive k-out-of- r-from- n: G system to multi-state case. In this model the system consists of n linearly ordered multi-state components. Both the system and its components can have different states: from complete failure up to perfect functioning. The system is in state j or above if and only if at least k j components out of r consecutive are in state j or above. An algorithm is provided for evaluating reliability of a special case of multi-state consecutive k-out-of- r-from- n: G system. The algorithm is based on the application of the total probability theorem and on the application of a special case taken from the [Jinsheng Huang, Ming J. Zuo, Member IEEE and Yanhong Wu, Generalized multi-state k-out-of- n: G system, IEEE Trans. Reliab. 49(1) (2000) 105 111.]. Also numerical results of the formerly published test examples and new examples are given.
Linear m-Gap-Consecutive k-out-of-r-from-n: F Systems
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Element Maintenance and Allocation for Linear Consecutively Connected Systems
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DOI:10.1080/0740817X.2011.649388
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This article considers optimal maintenance and allocation of elements in a Linear Multi-state Consecutively Connected System (LMCCS), which is important in signal transmission and other network systems. The system consists of N+1 linearly ordered positions (nodes) and fails if the first node (source) is not connected with the final node (sink). The reliability of an LMCCS has been studied in the past but has been restricted to the case when each system element has a constant reliability. In practice, system elements usually fail with increasing failure probability due to aging effects. Furthermore, in order to increase system availability, resources can be put into the maintenance of each element to increase the availability of the element. In this article, a framework is proposed to solve the cost optimal maintenance and allocation strategy of this type of system subject to an availability requirement. A universal generating function is used to estimate the availability of the system. A genetic algorithm is adopted for optimization. Illustrative examples are presented.
Reliability Evaluation of Combined k-out-of-n:F, Consecutive-k-out-of-n: F and Linear Connected-(r, s)-out-of--(m, n): F System Structures
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DOI:10.1109/24.855542
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). The algorithms are used for system reliability evaluation of furnace systems. The concept of the combined k-out-of-n:F and 1-dimensional and 2-dimensional consecutive-k-out-of-n:F systems can be extended to other variations of the consecutive-k-out-of-n:F systems, e.g., the consecutive-k-out-of-n:G system and 1-dimensional and 2-dimensional r-within-k-out-of-n:F systems. The concept of Markov chain imbeddable (MIS) systems is another excellent tool that can be used for analysis of such combined system structures
