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.     

Reliability of an Independent Component,s-Spare System with Exponential Life Times and General Repair Times

In this paper, the reliability of a system with N independent components supplied with a pool of s spares is investigated. The components have an exponential life time distribution and the failed components are repaired, one at a time, with a general repair time distribution. Numerical results are presented for the mean time to system failure (system is said to fail when a component fails with no spare available) and the reliability of the system in some special cases of interest.     

Reliability analysis of a repairable k-out-of-n system with some components being suspended when the system is down

A k-out-of-n system with independent exponential components is investigated. It is assumed that some working components are suspended as soon as the system is down, repair starts immediately when a component fails and repair times are independent and exponentially distributed. Formulas for various reliability indices of the system including mean time between failures, mean working time in a failure–repair cycle, and mean down time in a failure–repair cycle are derived.     

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.     

System-level load–strength interference based reliability modeling of k-out-of-n system

The issue of information loss in the process of system reliability modeling through conventional load–strength interference analysis is discussed first, and the reason why it is impossible to construct dependent system reliability model simply by means of component reliability index is demonstrated. Then, an approach to modeling the reliability of dependent system with common cause failure (CCF) is presented. The approach is based on system-level load–strength interference analysis and a concept of ‘conditional failure probability of component’ as well. With the opinion that load randomness is the direct cause of failure dependence, a discrete type system reliability model is developed via the conditional component failure probability concept. At last, the model's capabilities to estimate system reliability with CCF effect and to predict high multiplicity failure probability based on low multiplicity failure event data are proved.     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

On the inspection policy of a two-component parallel system with failure interaction

In this paper we study a two-component standby system which can successfully operate upon a demand if at least one component is not failed. We assume that failures can be detected only by periodic inspections. We consider that the failure of one component can modify the (conditional) failure probability of the component still alive with probability p and do not interact with probability 1−p. For that failure interaction scheme we obtain the system reliability function for the case of staggered inspections. We compare staggered and non-staggered inspections through numerical examples considering constant hazard rates.     

A model for Bayesian software reliability analysis

Two problems which are of great interest in relation to software reliability are the prediction of future times to failure and the calculation of the optimal release time. An important assumption in software reliability analysis is that the reliability grows whenever bugs are found and removed. In this paper we present a model for software reliability analysis using the Bayesian statistical approach in order to incorporate in the analysis prior assumptions such as the (decreasing) ordering in the assumed constant failure rates of prescribed intervals. We use as prior model the product of gamma functions for each pair of subsequent interval constant failure rates, considering as the location parameter of the first interval the failure rate of the following interval. In this way we include the failure rate ordering information. Using this approach sequentially, we predict the time to failure for the next failure using the previous information obtained. Using also the relevant predictive distributions obtained, we calculate the optimal release time for two different requirements of interest: (a) the probability of an in‐service failure in a prescribed time t; (b) the cost associated with a single or more failures in a prescribed time t. Finally a numerical example is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Performance analysis and optimization of a machine repair problem with warm spares and two heterogeneous repairmen

In this paper, we study a machine repair problem with warm spares and two heterogeneous repairmen from the view points of both queueing and reliability. In this system, the first repairman is always available for serving the failed units, while the second repairman leaves for a vacation of random length when the number of failed units is less than N. We obtain expressions for the steady-state probabilities of the system by solving the steady-state probability equations iteratively. Then, we obtain some performance measures for the system. We also obtain some performance measures of reliability for the system such as the steady-state availability, the steady-state failure frequency and the mean time between the system failures. Moreover, we present derivations of the mean time taken until the first failure of the system. A cost model is developed to determine the optimum value N while the system availability is maintained at a certain level. A sensitivity analysis is also performed.     

A multiple system governed by a quasi-birth-and-death process

The system we consider comprises n units, of which one has to operate for the system to work. The other units are in repair, in cold standby, or waiting for repair. Only the working unit can fail. The operational and repair times follow phase-type distributions. Upon failure, it is replaced by a standby unit and goes to the repair facility. There is only one repairman. When one unit operates the system is up and when all the units are in repair or waiting for repair, the system is down. This system is governed by a finite quasi-birth-and-death process. The stationary probability vector and useful performance measures in reliability, such as the availability and the rate of occurrence of failures are explicitly calculated. This model extends other previously considered in the literature. The case with an infinite number of units in cold standby is also studied. Computational implementation of the results is performed via a numerical example, and the different systems considered are compared from the reliability measures determined.     

Modelling and optimization of proof testing policies for safety instrumented systems

This paper introduces a new development for modelling the time-dependent probability of failure on demand of parallel architectures, and illustrates its application to multi-objective optimization of proof testing policies for safety instrumented systems. The model is based on the mean test cycle, which includes the different evaluation intervals that a module goes periodically through its time in service: test, repair and time between tests. The model is aimed at evaluating explicitly the effects of different test frequencies and strategies (i.e. simultaneous, sequential and staggered). It includes quantification of both detected and undetected failures, and puts special emphasis on the quantification of the contribution of the common cause failure to the system probability of failure on demand as an additional component. Subsequently, the paper presents the multi-objective optimization of proof testing policies with genetic algorithms, using this model for quantification of average probability of failure on demand as one of the objectives. The other two objectives are the system spurious trip rate and lifecycle cost. This permits balancing of the most important aspects of safety system implementation. The approach addresses the requirements of the standard IEC 61508. The overall methodology is illustrated through a practical application case of a protective system against high temperature and pressure of a chemical reactor.     

Quantification of sequential failure logic for fault tree analysis

Fault tree analysis (FTA) is generally accepted as an efficient method for analyzing system failures. It is well known that a fault tree (FT) is equivalent to a minimal cut set fault tree with all minimal cut-AND structures. The minimal cut-AND structure is an AND conjunction of an output and all inputs that compose a minimal cut set. For the structure, the failed state of the output becomes true when all failed states of inputs exist simultaneously. There are cases where the output of the minimal cut-AND structure depends not only on all failed states of inputs but also on the sequence of occurrences of those failures. This sequential failure logic (SFL) is equivalently expressed with Priority-AND gates in FTA, where inputs to the gates have constant failure and repair rates. A probabilistic model for analysis of SFL was proposed and equations with multiple integration for arbitrary number of inputs were derived from the model. However, it is usually difficult to solve the multiple integration when the number of inputs exceeds a certain range. This paper presents analytical solutions of the probability that the output is in a failed state at time t and the statistically expected number of failures of the output per unit time at time t for the special case where inputs are characterized by common failure and repair rates. In addition, the analysis of FT involving SFL is demonstrated by means of software Mathematica.     

Competing failure analysis considering cascading functional dependence and random failure propagation time

Functional dependence (FDEP) exists in many real‐world systems, where the failure of one component (trigger) causes other components (dependent components) within the same system to become isolated (inaccessible or unusable). The FDEP behavior complicates the system reliability analysis because it can cause competing failure effects in the time domain. Existing works have assumed noncascading FDEP, where each system component can be a trigger or a dependent component, but not both. However, in practical systems with hierarchical configurations, cascading FDEP takes place where a system component can play a dual role as both a trigger and a dependent component simultaneously. Such a component causes correlations among different FDEP groups, further complicating the system reliability analysis. Moreover, the existing works mostly assume that any failure propagation originating from a system component instantaneously takes effect, which is often not true in practical scenarios. In this work, we propose a new combinatorial method for the reliability analysis of competing systems subject to cascading FDEP and random failure propagation time. The method is hierarchical and flexible without limitations on the type of time‐to‐failure distributions for system components. A detailed case study is performed on a sensor system used in smart home applications to illustrate the proposed methodology.     

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.     

基于修理设备可更换和修理延迟策略的两不同型部件冷贮备可修系统

考虑具有修理设备可失效可更换和修理延迟策略且由两个不同型部件组成冷贮备可修系统,利用Markov更新过程理论和全概率分解技术,得到了系统的稳态可用度、首次故障前平均时间、稳态故障频度以及系统等待修理的概率,并且通过引入修理设备的“广义忙期”,获得了修理设备的稳态不可用度和稳态更换频度．最后以数值实例分析了修理延迟时间和修理设备的失效率对系统可靠性指标的影响．在工程应用中特别感兴趣的是稳态可靠性数量指标,希望本文所得结果对系统的优化设计和更换维修提供有用的信息．     

Modelling and Simulation of Scraper Reliability for Maintenance

A scraper conveyor is a kind of heavy machinery which can continuously transport goods and widely used in mines,ports and store enterprises.Since scraper failure rate directly affects production costs and production capacity,the evaluation and the prediction of scraper conveyor reliability are important for these enterprises.In this paper,the reliabilities of different parts are classified and discussed according to their structural characteristics and different failure factors.Based on the component’s time-to-failure density function,the reliability model of scraper chain is constructed to track the age distribution of part population and the reliability change of the scraper chain.Based on the stress-strength interference model,considering the decrease of strength due to fatigue failure,the dynamic reliability model of such component as gear,axis is developed to observe the change of the part reliability with the service time of scraper.Finally,system reliability model of the scraper is established for the maintenance to simulate and calculate the scraper reliability.     

Reliability modeling of multi‐state degraded repairable systems and its applications to automotive systems

In this paper, we presented a continuous‐time Markov process‐based model for evaluating time‐dependent reliability indices of multi‐state degraded systems, particularly for some automotive subsystems and components subject to minimal repairs and negative repair effects. The minimal repair policy, which restores the system back to an “as bad as old” functioning state just before failure, is widely used for automotive systems repair because of its low cost of maintenance. The current study distinguishes with others that the negative repair effects, such as unpredictable human error during repair work and negative effects caused by propagated failures, are considered in the model. The negative repair effects may transfer the system to a degraded operational state that is worse than before due to an imperfect repair. Additionally, a special condition that a system under repair may be directly transferred to a complete failure state is also considered. Using the continuous‐time Markov process approach, we obtained the general solutions to the time‐dependent probabilities of each system state. Moreover, we also provided the expressions for several reliability measures include availability, unavailability, reliability, mean life time, and mean time to first failure. An illustrative numerical example of reliability assessment of an electric car battery system is provided. Finally, we use the proposed multi‐state system model to model a vehicle sub‐frame fatigue degradation process. The proposed model can be applied for many practical systems, especially for the systems that are designed with finite service life.     

Time-redundant system reliability under randomly constrained time resources

The paper considers the time-redundant system where the system total task is a sequence of n phases and the total task must be executed during constrained time. For every phase, there is its own server, which executes the phase task during randomly distributed time. The server is not perfectly reliable and two types of failure (“open” and “closed”) are possible. Redundant servers may be used in any phase. The time–probability characteristics are introduced for any task, based on which the system reliability is treated as a probability that the system total task will be correctly completed during a corresponding time resource, which also may be randomly distributed. The adequate model is presented and a semi-Markov process is used as a mathematical technique. The closed-form solution was derived based on an acyclic Semi-Markov process. The numerical example of the elaborated approach is presented.     

Availability, reliability and downtime of systems with repairable components

Closed-form expressions are derived for the steady-state availability, mean rate of failure, mean duration of downtime and lower bound reliability of a general system with randomly and independently failing repairable components. Component failures are assumed to be homogeneous Poisson events in time and repair durations are assumed to be exponentially distributed. The results are expressed in terms of the mean rates of failure and mean durations of repair of the individual components. Closed-form expressions are also derived for the rates of change of the various probabilistic system performance measures with respect to the mean rate of failure and the mean duration of repair of each component. These expressions provide a convenient framework for identifying important components within the system and for decision-making aimed at upgrading the system availability or reliability, or reducing the mean duration of system downtime. Example applications to an electrical substation system demonstrate the use of the formulas developed in the paper.     

A hierarchical combinatorial reliability model for smart home systems

As an application of the Internet of Things, smart home systems have received significant attentions in recent years due to their precedent advantages, eg, in ensuring efficient electricity transmission and integration with renewable energy. This paper proposes a hierarchical and combinatorial methodology for modeling and evaluating reliability of a smart home system. Particularly, the proposed methodology encompasses a multi‐valued decision diagram‐based method for addressing phased‐mission, standby sparing, and functional dependence behaviors in the physical layer; and a combinatorial procedure based on the total probability theorem for addressing probabilistic competing failure behavior with random propagation time in the communication layer. The methods are applicable to arbitrary types of time‐to‐failure and time‐to‐propagation distributions for system components. A detailed case study of an example smart home system is performed to demonstrate applications of the proposed method and effects of different component parameters on the system reliability.