Search

In this paper, a general formula for computing the joint reliability importance of two components is obtained for a binary coherent system that consists of exchangeable dependent components. Using the new formula, the joint reliability importance can be easily calculated if the path sets of the system are known. As a special case, an expression for the joint reliability importance of two components is also obtained for a system consisting of independent and identical components. Illustrative numerical results are presented to compare the joint reliability importance of two components in the bridge system for the two cases when the components are exchangeable dependent and when the components are independent and identical.

Most practical systems consist of multiple types of components although the components perform the same task within the system. The analysis of such systems is more challenging than the systems with single type of components. In this paper, we present expressions for the survival function of the failure time, and mean time to failure of the system under the general case when the random failure times of components of the same type are exchangeable dependent and the random failure times of components of different type are dependent. As a case study, we apply our results to linear consecutive- $k$ -out-of- $n$ :F systems consisting of three different types of dependent components.

In this paper, we study a three-state ${\bf k}$ -out-of- $n$ system with $n$ independent components ( $ {\bf k} = ({k_{1}},{k_{2}}) $ ). Each component can be in a perfect functioning state (state “2”), partially working (state “1”), or failed (state “0”). We assume that, at time $ t = 0 $ , $ {n_{1}} $ components are in a partially working state while the rest $ {n_{2}} $ components are fully functioning ( $ n = {n_{1}} + {n_{2}} $ ). The system is considered to be at state “1” or above if at least $ {k_{1}} $ components are working (fully or partially). If at least $ {k_{1}} $ components are working and at least $ {k_{2}} $ components are in a perfect functioning state, we shall say that the system is at state “2”. In this paper, we develop formulae for the survival functions corresponding to the two different system's states described above. For illustration purposes, a numerical example which assumes that the degradation occurs according to a Markov process is presented.

This paper deals with two different shock models in a Markovian environment. We study a system from a reliability point of view under these two shock models. According to the first model, the system fails if the cumulative shock magnitude exceeds a critical level, while in the second model the failure occurs when the cumulative effect of the shocks in consecutive periods is above a critical level. The shock occurrences over discrete time periods are assumed to be Markovian. We obtain expressions for the failure time distributions of the system under the two model. Illustrative computational results are presented for the survival probabilities and mean time to failure values of the system.

This paper is concerned with the computation of the Barlow-Proschan importance measure for systems involving two common failure criteria, and consisting of statistically independent and identical components. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or the total number of failed or working components in the whole system. The results are applied to $(n,f,k):{\rm F}$ and $ :{\rm F}$ systems.

This paper is concerned with dynamic reliability modeling of three-state systems consisting of three-state $s$ -independent components. The components and the systems are assumed to be in three states: perfect functioning, partial performance, and complete failure. Survival functions of such systems are studied in different state subsets. It is shown that the survival function of a three-state system with a general structure can be represented as a mixture of the survival functions of the three-state $ {\bf k}$ -out-of- $n$ :G systems. The results are illustrated for the three-state consecutive- $ {\bf k}$ -out-of- $n$ :G systems whose components degrade according to a Markov process.

A generalized $k$ -out-of- $n$ system which is denoted by $((n_{1},\ldots,n_{N}),f,k)$ consists of $N$ modules ordered in a line or a circle, and the $i$ th module is composed of $n_{i}$ components in parallel $(n_{i}\geq 1,i=1,\ldots,N)$ . The system fails iff there exist at least $f$ failed components or at least $k$ consecutive failed modules. In this paper, we compute the signature of this system when $n_{1}=\ldots=n_{N}=n$ , and present illustrative examples to demonstrate its application. Simulation based computation of the signature is provided when the modules have different numbers of components.

Modeling statistical dependence between two systems or components is an important problem in reliability theory. Such a problem has been well studied for binary systems and components. In the present paper, we provide a way for modeling s-dependence between two multi-state components. Our method is based on the use of copulas which are very popular for modeling s-dependence. We obtain expressions for the joint state probabilities of the two components, and illustrate the results for the case when the degradation in both components follows a Markov process.

We study the joint reliability importance (JRI) of two components in Lin/m/Con/k/n:F systems. A Lin/m/Con/k/n:F system consists of n linearly ordered binary components, and the system fails iff there are at least m nonoverlapping runs of k consecutive failed components (n ≥ mk). In particular, we obtain expressions for the JRI in Lin/m/Con/k/n:F systems when the components are s-independent & identical, when the components are s-independent & nonidentical, and when the components are exchangeable & s-dependent. We present extensive numerical illustrations.

A k-out-of-n:G system consists of n components, and operates if at least k of its components operate. Its reliability properties have been widely studied in the literature from different perspectives. This paper is concerned with the reliability analysis of a k-out-of-n:G system equipped with a single warm standby unit. We obtain an explicit expression for the reliability function of the system for arbitrary lifetime distributions. Two different mean residual life functions are also studied for the system.

We propose a new model which generalizes the linear m-consecutive-k-out-of-n:F system to the case of the m-consecutive-k-out-of-n:F system with l-overlapping runs. The new system is called the m-consecutive-k, l-out-of-n:F system, and consists of linearly ordered components such that the system fails iff there are at least m l-overlapping runs of k consecutive failed components. The number of path sets including a specified number of working components is obtained via a combinatorial method, and this quantity is used to obtain the reliability and the signature of the system.

The consecutive k -out-of-n systems are important structures in reliability engineering due to their applications in various real life situations. In this paper, we study the residual lifetime of these systems under the condition that the total number of failed components at time t1 is r , and at time t2 (t2 >; t1) the system is still working. We obtain explicit expressions for the survival function of the corresponding residual life when the components are exchangeably dependent. We also obtain a signature-based mixture representation for a coherent system satisfying a certain property.

This paper is concerned with the number of components that are failed at the time of system failure. We study the corresponding quantity for a coherent structure via the system signature. Furthermore, we study the distribution of the number of failures after a specified time until the system failure. We illustrate the results for well-known general classes of coherent systems such as linear consecutive -within- -out-of- :F, and -consecutive- -out-of- :F.

A combined m-consecutive-k-out-of-n:F & consecutive k ;c ;-out-of-n:F systems consists of linearly ordered components, and fails iff there exist at least k ;c ; consecutive failed components, or at least m nonoverlapping runs of k consecutive failed components. This structure has applications for modeling systems such as infrared detecting and signal processing, and bank automatic payment systems. In this paper, we derive a combinatorial equation for the number of path sets of this structure including a specified number of working components. This number is used to derive a reliability function, and a signature based survival function formulae, for the system consisting of i.i.d. components. We also obtain a combinatorial equation for the reliability of a system with Markov dependent components.

Start-up demonstration testing is an effective method for illustrating the reliability of a unit before purchasing it. The test consists of starting-up the unit, and observing the outcomes, either success or failure. According to the total successes total failures (TSTF) test procedure, a unit under test is accepted when a specified number of successes is observed before a specified number of failures; otherwise, the unit is rejected. We study the TSTF procedure for dependent start-ups where the outcome of the present start-up depends on the total number of successful start-ups so far. The main characteristics of the TSTF test are obtained under this previous-sum dependent model, and numerical illustrations are presented.

Mean residual, and mean past lifetime functions are important reliability measures for evaluating systems' performance. These functions have been successfully used in binary state reliability analysis. This paper studies mean residual, and mean past lifetime concepts for multi-state systems. We evaluate mean residual, and mean past lifetime functions of one unit multi-state systems, and multi-state -out-of- :G systems under the assumption that the degradations in systems and components follow an acyclic Markov process which has a discrete state space.

The signature of a system is a useful tool in a variety of applications including the evaluation of the reliability characteristics of systems, and the comparison of the performance of competing systems. We study the evaluation and application of signatures of systems involving two common failure criteria which are common in real life applications. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or total number of failed or working components in the whole system. We provide a method for obtaining the signatures of such systems. Applications of the results are also presented.

We study the conditional lifetimes of consecutive k-out-of-n systems. We provide explicit representations for the conditional survival functions of these systems when 2k ? n. The mean residual life functions are obtained, and ordering results are provided. A signature-based evaluation of residual lifetimes is also presented, and some stochastic comparison results are obtained for particular consecutive systems.

Consecutive successes total failures (CSTF) is a well known start-up demonstration test procedure in which a unit under test is accepted when successive start-up attempts produce a specified number of consecutive successes before a specified number of failures; otherwise the unit is rejected, and in both cases testing (the experiment) is terminated. The CSTF procedure is studied here assuming that the probability of a successful start-up is a random variable. Under this assumption, the outcomes of the attempted start-ups are dependent random variables following an exchangeable model. The unconditional, and the conditional probability distributions of the waiting time (length of the test) are derived and studied in this situation. The first two moments of unconditional waiting time distribution are calculated. Numerical and graphical illustrations are provided, and comparisons are made with the corresponding results for the i.i.d. model.

A consecutive k-out-of-n:F system is a system of n components which fails iff k consecutive components fail. In this paper, we consider the lifetime of consecutive k-out-of-n:F systems and provide some stochastic ordering results. It is shown that the usual stochastic ordering is preserved under the formation of consecutive k-out-of-n:F systems of s-independent, nonidentical, and exchangeable components. A counter example is also given to show that the mean residual life ordering is not, in general, preserved by a consecutive k-out-of-n:F system