Reliability modeling for consecutive k-out-of-n: F systems with local load-sharing components subject to dependent degradation and shock processes

Traditional k-out-of-n models assume that the components are independent, while recent research studies assume that the components are dependent caused by global load-sharing characteristic. In this paper, we investigate the consecutive k-out-of-n systems with dependent components by local load-sharing characteristic. The work load and shock load on failed components will be equally shared by adjacent components, so the components tend to fail consecutively. Consequently, the components degradation processes may be diverse, since their degradation rate (dependent on work load) and abrupt degradation (dependent on shock load) become unequal because of local load-sharing effect. Furthermore, the system failure will be path-dependent on the failure sequences of components, which results in that the same system states may have different system failure probabilities. This new dependence makes the system reliability model more complex. In this work, an analytical model that can be solved numerically is derived to compute the reliability with this complex dependence. The developed model is demonstrated by a cable-strut system in the suspension bridge. The results show that the reliability decreases significantly when the new dependence is considered.     

Reliability assessment models for dependent competing failure processes considering correlations between random shocks and degradations

This article develops reliability models for systems subject to two dependent competing failure processes, considering the correlation between additional damage size on degradation in soft failure process and stress magnitude of shock load in hard failure process, both of which are caused by the same kth random shock. The generalized correlative reliability assessment model based on copulas is proposed, which is then extended to three different shock patterns: (1) δ‐shock, (2) m‐shock, and (3) m‐run shocks. There are some statistical works to be introduced in reliability modeling, including data separation of total degradation amount, inferring the distribution of amount of aging continuous degradation at time t, and fitting copula to the specific correlation. The developed reliability models are demonstrated for an application example of a micro‐electro‐mechanical system.     

Load sharing in series configuration

In reliability engineering, load sharing is typically associated with a system in parallel configuration. Examples include bridge support structures, electric power supply systems, and multiprocessor computing systems. We consider a reliability maximization problem for a high‐voltage commutation device, wherein the total voltage across the device is shared by the components in series configuration. Here, the increase of the number of load‐sharing components increases component–level reliability (as the voltage load per component reduces) but may decrease system–level reliability (because of the increased number of components in series). We provide the solution for the 2 popular life‐load models: the proportional hazard and the accelerated failure time models with the underlying exponential and Weibull distributions for both a single and dual failure modes.     

Optimal design of multi-state weighted k-out-of-n systems based on component design

This paper presents a study on design optimization of multi-state weighted k-out-of-n systems. The studied system reliability model is more general than the traditional k-out-of-n system model. The system and its components are capable of assuming a whole range of performance levels, varying from perfect functioning to complete failure. A utility value corresponding to each state is used to indicate the corresponding performance level. A widely studied reliability optimization problem is the “component selection problem”, which involves selection of components with known reliability and cost characteristics. Less adequately addressed has been the problem of determining system cost and utility based on the relationships between component reliability, cost and utility. This paper addresses this topic. All the optimization problems dealt with in this paper can be categorized as either minimizing the expected total system cost subject to system reliability requirements, or maximizing system reliability subject to total system cost limitation. The resulting optimization problems are too complicated to be solved by traditional optimization approaches; therefore, genetic algorithm (GA) is used to solve them. Our results show that GA is a powerful tool for solving these kinds of problems.     

Efficient reliability analysis of dynamic k-out-of-n phase-AND mission systems

Motivated by real-world applications of satellites and wireless sensor networks, this paper models and evaluates a dynamic k-out-of-n phase-AND mission system (k/n-PAMS). The mission task conducted by a k/n-PAMS involves multiple consecutive phases; the mission is successful as long as the task is successful in any of the phases. Due to factors, such as scheduled maintenance, location changes in task execution during different phases, and resource sharing with other tasks, the total number of available components n for the considered mission task and the required number of working components k may change from phase to phase. In addition, due to varying load and working environments, component failure time distributions are also phase dependent. This paper proposes an analytical modeling approach based on multivalued decision diagrams (MDDs) for assessing reliability of the considered k/n-PAMS. The approach encompasses a new and fast MDD model generation algorithm that considers behaviors of all the mission phases simultaneously based on node labeling. As demonstrated through empirical studies on k/n-PAMSs with different sizes (different numbers of phases and different numbers of system components), the proposed algorithm is more efficient than the traditional phase-by-phase model generation method.     

Availability and reliability of k-out-of-(M+N):G warm standby systems

Redundancy or standby is a technique that has been widely applied to improving system reliability and availability in system design. In most cases, components in standby system are assumed to be statistically identical and independent. However, in many practical applications, not all components in standby can be treated as identical because they have different failure and repair rates. In this paper, one kind of such systems with two categories of components is studied, which is named k-out-of-(M+N):G warm standby system. In the system, one category of the components is of type 1 and the other type 2. There are M type 1 components and N type 2 components. Components of type 1 have a lower failure rate and are preferably repaired if there is one failed. There are r repair facilities available. By using Markov model, the system state transition process can be clearly illustrated, and furthermore, the solutions of system availability and reliability are obtained based on this. An example representing a power-generator and transmission system is given to illustrate the solutions of the system availability and reliability.     

Study of loading policies for unequal strength shared-load system

This paper presents three policies for load assignment among unequal strength units based on: (1) maximizing the mean time between failure (MTBF) of the system (Policy A); (2) requiring all active units to fail simultaneously (Policy B); and (3) simply taking the same load for all active units (Policy C). The traditional shared-load k-out-of-n: G system performs satisfactorily if at least k of the total n units are up, with load being shared among the surviving units. One of the advantages in such a system is that the active units play the role of both an actuator and a sensor. Thus we have enough time to make corrective replacement of failed units while the system is still kept on duty. We use examples to show that policies A and B have almost the same MTBF (the difference is within 1%), which is much more superior than Policy C. Two indices are used for the selection of valve k, with given number of total units. One is called the effectiveness factor, while the other is the tolerance factor. The former concerns with the effectiveness of load assignments among the surviving units. The latter deals with the time buffer when one more unit fails.     

Economic design of a circular consecutive--out-of-:F system with -step Markov dependence

A circular consecutive-k-out-of-n:F system consists of n components arranged along a circular path. The system fails if and only if at least k consecutive components in the system fail. The system reliability, the expected system life, and the expected number of failures are obtained under the assumption that the failure rate of a component depends on the number of consecutive failed components that follow it. A procedure to find the optimal k and a simulation procedure to search the near-optimal k are proposed with illustrative numerical examples. An expected cost per unit time is considered as the objective function to be minimized.     

Optimal design for resilient load‐sharing systems with nonidentical components

For system maintenance, strategic component restoration planning is an important conceptual framework for load‐sharing k‐out‐of‐n: G system. A cost‐effective treatment of failure events is imperative with the purpose of reinstating the system ability. This paper presents a new optimal design method for load‐sharing repairable k‐out‐of‐n: G system, in which a flowgraph is used in conjunction with multiresponse optimization. By introducing the concept of modular design, the system is partitioned into scalable and repairable maintenance modules. The determination of the optimal design depends on the type of system components, the module‐based system structure, and the repair rule setting. An extended flowgraph model, which links covariates into transition branches, is used for modeling the system failure evolution. With consideration of various system performance measurements as responses, multiresponse optimization with weighted principal component analysis is used to achieve an optimal design of maintenance modules as well as repair policy. The methodology presented in this paper provides an efficient way to design a system having nonidentical components and arbitrary repair time distributions with consideration of the variety of maintenance policies as well as the diversity of system operating conditions.     

Optimal allocation of elements in a linear multi-state sliding window system

This paper proposes a new model that generalizes the consecutive k-out-of-r-from-n:F system to multi-state case. In this model (named linear multi-state sliding window system) the system consists of n linearly ordered multi-state elements. Each element can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. The system fails if the sum of the performance rates of any r consecutive elements is lower than a demand W.An algorithm is suggested that finds the order of elements with different characteristics within linear multi-state sliding window system, which provides the greatest possible system reliability. The algorithm is based on using a universal generating function technique for system reliability evaluation. A genetic algorithm is used as the optimization tool. Illustrative examples are presented.     

On the Estimation of Safe Life When the Underlying Life Distribution Is Weibull

The problem of estimating Y ₍₁₎, the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered. A conditional confidence interval solution is proposed for the estimation of Y ₍₁₎. The results have direct application in reliability theory, where the time until the first failure in a group of k items in service provides a measure of assurance regarding the operation of the items.     

Bayesian reliability analysis of a k-out-of-m system and the estimation of sample size and censoring time

In the absence of lifetime data on a complete system, it is desirable to use operational experience on its components in the Bayesian analysis of the system. The present study deals with Bayesian reliability analysis of a k-out-of-m system using two types of censored failure information. In another development, Bayesian confidence intervals for unreliability have been used for estimating the sample size and the censoring time needed to get sufficient failure information.     

A new algorithm for generating minimal cut sets in k-out-of-n networks

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. A k-out-of-n network is a special network in that some nodes must receive at least k (>1) flows from all of their input edges (n). In real-life cases, many networks such as computer and telecommunications include k-out-of-n nodes for redundancy. The minimal-cut-node-set (MCN) is the major and fundamental tools for evaluating the k-out-of-n network reliability. In this study, a very simple algorithm based on some intuitive theorems that characterize the structure of the MCN is developed to solve the k-out-of-n network reliability. Compared to the existing algorithms, the proposed algorithm generates all k-out-of-n MCs without duplication based on fewer MCNs and fewer (k-out-of-n MC) candidates. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithms. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MCs are generated, verified, and implemented to evaluate the k-out-of-n network reliability using the proposed algorithm.     

预应力混凝土梁桥系统失效树分析

针对预应力混凝土梁桥,采用失效树的方法进行系统可靠性分析。运用全局临界强度分枝-约界准则识别结构系统主要失效模式,以JC法计算各主要失效模式的可靠指标,采用Ditlevsen上下界公式分析结构体系的失效概率。以某汉江公路大桥为例,分析了失效树图形,研究了主要失效模式的可靠指标、失效概率及桥梁结构体系可靠性。与实桥运营状态的对比分析表明,上述理论分析较好地预测了实际的结构行为,能用于桥梁结构安全性及可靠性分析评估。     

Minimal cut sets of s–t networks with k-out-of-n nodes

In this paper, we extend traditional directed s–t network by letting nodes have k-out-of-n property: To generate output flows, a node must receive at least k flows from its n input links, where k is an integer assigned for the node and its value can be any number from 1 to n. To evaluate the system reliability, minimal cut sets for the extended network are defined for nodes. Under this definition, an extended network and its sink node have the same minimal cut sets. A new algorithm is designed to generate minimal cut sets for all nodes, starting with the source node and ending with the sink node. With different initializations, the algorithm can be applied for extended s–t networks with or without node failures.     

A path-based algorithm for evaluating the k-out-of-n flow network reliability

Evaluating the network reliability is an important topic in the planning, designing and control of systems. The minimal path (MP, an edge set) set is one of the major, fundamental tools for evaluating network reliability. A k-out-of-n MP is a union of some MPs in a k-out-of-n flow network in which some nodes must receive flows from their k input distinctive edges (each input edge has one flow) to generate one flow, where k is an integer number between 2 and n. In this study, a very simple a-lgorithm based on some intuitive theorems that characterize the k-out-of-n MP structure and the relationship between k-out-of-n MPs and k-out-of-n MP candidates is developed to solve the k-out-of-n flow network reliability by finding the k-out-of-n MPs between two special nodes. The proposed algorithm is easier to understand and implement. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MPs are generated and verified in a k-out-of-n flow network using the proposed algorithm. The reliability of one example is then computing using the same example.     

Reliability evaluation of multi-state weighted -out-of- systems

The k-out-of-n systems have been extensively studied in recent years. A binary weighted k-out-of-n model has also been reported in the literature. In this paper, we first compare two approaches for reliability evaluation of binary weighted k-out-of-n systems. We then provide two models of multi-state weighted k-out-of-n system models. Recursive algorithms are presented for reliability evaluation of these new models. The universal generating function approach is also used for reliability evaluation of multi-state weighted k-out-of-n systems.     

A simple algorithm for evaluating the k-out-of-n network reliability

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The minimal cut set (MC, an edge set) is one of the major and fundamental tools for evaluating network reliability. A k-out-of-n MC is a special MC in a k-out-of-n network in which some nodes must receive at least k flows from their n input edges, where k is an integer number between 1 and n. In this study, an alternative method is given first to define a MC using a node set (called MCN) in k-out-of-n networks. A very simple algorithm based on some intuitive theorems that characterize the structure of the MCN and the relationship between MC and MCN is developed to solve the k-out-of-n network reliability by finding the k-out-of-n MCs between two special nodes. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithm. The correctness of the proposed algorithm will be analyzed and proven. Two examples are illustrated to show how all k-out-of-n MCs are generated and verified in a k-out-of-n network using the proposed algorithm. The reliability of one example is then computing using one example.     

Performability indicators for the traffic analysis of wide area networks

In this paper, a repairable circular consecutive-k-out-of-n:F system with one repairman is studied. It is assumed that the working time and the repair time of each component are both exponentially distributed and every component after repair is ‘as good as new’. Each component is classified as either a key component or an ordinary component. Key components have priority in repair when failed. By using the definition of generalized transition probability, the state transition probabilities of the system are derived. Important reliability indices are evaluated for an example.     

The k-out-of-n acyclic multistate-node networks reliability evaluation using the universal generating function method

In this article, a special node called the k⁻-out-of-n node, which cannot receive more than a certain amount of flows, is newly introduced. The acyclic multistate-node network (AMNN) that unsatisfied the flow conservation law is then extended to the k-out-of-n AMNN by including the k⁻-out-of-n and k⁺-out-of-n nodes. A very simple universal generating function method (UGFM) based on some intuitive properties that characterize the structure of the k-out-of-n AMNN is developed to solve the k-out-of-n AMNN reliability. The correctness of the proposed UGFM will be analyzed and proven. An example with three special cases illustrates how the k-out-of-n AMNN reliability is evaluated using the proposed UGFM. To show that the proposed UGFM can also solve the AMNN reliability, the first case of the example demonstrates that the proposed UGFM without needing to remove redundant terms and collecting like terms is more efficient and reasonable than the best-known UGFM.