This article investigates the reliability and optimal replacement policy of a multistate system under the Markov renewal shock model, which has broad applications in engineering practice. The successive arrivals of shocks follow a continuous-time renewal process, and the system's state (damage) evolution is depicted as a homogeneous, irreducible Markov renewal process. In this context, the system incurs dual damages over time as follows: 1) one comes from the previous shocks' damage evolutions and 2) the other comes from the damage of successive arrivals of shocks. The reliability and optimal replacement time of the developed system are provided and the asymptotic evolution results for each shock, that is, the probabilities for each shock finally dissipating or causing the system's failure, are deduced. The optimal replacement policy for the developed shock model is discussed through minimizing the average cost rate function. Finally, numerical studies are given for the Markov renewal shock model, with interarrival times between adjacent shocks assumed to follow an exponential distribution.

Failures of safety-critical mission-based systems, such as aircraft and submarines, could result in significant losses and damage. To enhance the survivability of such systems, their missions may be aborted if the failure risk becomes too high. We investigate such mission abort policies under a completely observed two-stage degradation process that progresses stochastically from “normal” to “defective” to “failure.” Mission abort decisions are considered as a function of the duration of the defective stage. This mission abort problem is formulated as a discrete-time optimal stopping problem with the goal of minimizing the expected total cost of mission failure and system failure. In addition to deriving some structural properties, we also numerically evaluate several intuitive heuristic policies. Finally, a joint optimization problem is formulated to simultaneously identify the optimal mission abort policy and the optimal investment to delay system deterioration.

Compared to binary-state system models, multistate (MS) system models are more practical and useful when describing complex systems in applications, such as communication systems, oil pipeline systems, and grid or cloud computing infrastructures. This paper models a generalized sparsely connected MS consecutive-k-out-of-n: G system. The proposed model is regarded as a natural extension of the consecutive-k-out-of-n system. Moreover, the proposed system model could cover three special MS submodels, i.e., a decreasing MS: G system model, an increasing MS: G system model, and a nonmonotonic MS: G system model. Three corresponding real-world examples are given to illustrate the applications of the model. Meanwhile, by utilizing the well-known finite Markov chain imbedding method, the explicit system reliability and state distribution for the three kinds of submodels are presented, respectively. Finally, the developed indexes and analysis method are demonstrated by numerical examples.

In this paper, we consider four different kinds of multistate consecutive-k systems, namely a multistate linear m-consecutive-k-out-of-n: G system, a multistate linear consecutive-k-out-of-n: G system with sparse d, a multistate linear m-consecutive-k-out-of-n: G system with sparse d, and a multistate linear: G system. The reliability formulas for these systems are derived by using the finite Markov chain imbedding approach. Unlike some previous research, the vectors m, k, d, and f do not need to be monotonic in this paper. Some illustrative examples are also presented to show how the reliabilities of the four multistate consecutive-k systems are calculated.

Modern computing systems typically use a large number of independent, non-identical computing nodes to perform a set of coordinated computations in parallel. The computing system and its constituent computing nodes often exhibit more than two performance levels or states corresponding to different computing powers. This paper models and evaluates performability of largescale multi-state computing systems, which is the probability that a computing system performs at a particular performance level. The heterogeneity in the constituent components of different nodes (due to factors such as different model generations, model suppliers, and operating environments) makes performability analysis difficult and challenging. In this paper a specification method for system performance level (SPL) is first introduced. A multi-valued decision diagram (MDD) based approach is then proposed for performability analysis of multi-state computing systems consisting of nodes with different state occupation probabilities, which encompasses novel and efficient MDD model generation procedures. Example and benchmark studies are performed to show that the proposed approach can offer efficient performability analysis of large-scale computing systems.

In the field of reliability, there are many indexes, such as reliability, point availability, interval availability, and others, which have been widely used to describe the properties of repairable systems. However, some situations of interest cannot be covered by the existing indexes. In this paper, we introduce the definitions of multipoint covering availability and multi-interval covering availability, and formulas are derived in matrix forms for aggregated Markov repairable systems especially when the covering working intervals can overlap. The formulas of binary-point and binary-interval covering availabilities are given by an enumeration method. For multipoint and multi-interval problems, a recursive method is employed. Numerical examples are shown to illustrate the results derived in this paper and finally some conclusions and discussions are presented.

Observations on degradation performance are often used to analyze the underlying degradation process of highly reliable products. From the two-phase degradation path of the bearing performance observations, we observed that there exists an abrupt increase in degradation measurement at a change point. Then, the following degradation process started with the abrupt degradation measurement will degrade in a higher degradation rate. Here, a stochastic process-based degradation model is constructed to interpret the jump at the change point in the degradation process which is governed by the linear Wiener process. Meanwhile, the distribution of the first passage time over a prespecified threshold for the process is discussed. In addition, to get the estimates of the model parameter, the expectation-maximization algorithm is utilized since the change points are unobservable. Furthermore, to demonstrate the model's advantages over estimate, a comparison is made between the proposed and the existing known models from the literature. The results reveal that considering the jump in the degradation process can improve the accuracy of estimations in real applications.

In this paper, two degradation models are developed in terms of degradation signals of systems with some Wiener diffusion processes under pre-specified periodical calibrations. Some detailed results on two models are given by solving some recursive equations. The system reliability is defined as the probability that the degradation signals do not exceed a threshold value by time t. Two methods for getting the system reliability are presented: one is a stochastic process-based method in terms of recent results on the distribution of the first passage time, and the other is based on partial differential equations with absorbing boundary conditions. The moments of lifetime (first passage time) also are given based on both methods. Some extended questions and results are discussed for more general situations, and finally discussions and conclusions are presented.

Cascading failure models have been studied more recently as for electric power systems, epidemics, and interdependent networks. The current paper presents three cascading failure models in the point-view of system reliability based on the normalized CASCADE model. The corresponding system reliability indices are developed including system reliability, expectation lifetimes of systems, distributions of total number of failures, distributions of cascading failure propagation generations, and other related probabilities. Two numerical examples are discussed, and the conclusions are summarized.

We propose the consecutive-kr-out-of-nr:F linear zigzag structure and circular polygon structure, which extend the consecutive-k-out-of-n:F system model. By employing the finite Markov chain imbedding approach, we have proven the equations for calculating the reliability of consecutive-k-out-of-n:F system with different patterns given the state of the first and(or) the last component(s). Based on those reliability equations, we provide the rules for obtaining the reliability of the linear zigzag structure and circular polygon structure. Numerical examples are presented as an illustration of our new system model and demonstrate the use of the equations and rules proposed in this paper.

The performance of the system is often influenced by the performance of a particular component. Hence, it is important to identify the state changing of which component dominates the system performance changing in a maintenance process. Motivated by the Griffith importance measure (GIM) model, this paper proposed the generalized GIM, which extends the application of GIM to complex multi-state components, to evaluate the accurate contribution of the components in the changing of the system performance by considering the transition probabilities of the states of each component. Furthermore, the generalized GIM method is successfully applied in the continuous-state systems by extending the system structure function in GIM model. As a result, the expression of the performance of continuous system and the generalized GIM of the continuous-state components are proposed. A numerical example and an application to an oil transportation system are presented to illustrate how the proposed method works.

Here we present the structure functions for the linear and circular$m$-consecutive-$k$,$l$-out-of-$n$:F (G) systems, which consists of$n$linearly or circularly ordered components such that the system fails (works) iff there are at least$m$times$l$-overlapping runs of consecutive$k$failed (working) components. The duality of these F and G systems is obtained, and the proofs are given. The generalized reliability formulae for the systems are obtained by using the finite Markov chain imbedding approach for statistically independent and Markov-dependent cases. Some miscellaneous results on the reliability of these systems and numerical examples are provided to illustrate the results obtained in the paper.

A consecutive- k-out-of- n: F ( G ) system with sparse d consists of n components ordered in a line or a circle, while the system fails (works) iff, there exist at least k consecutive failed (working) components with sparse d for 0 ≤ d ≤ n - k. In this paper, a circular consecutive- k-out-of- n system with sparse d is considered. Some equations for system reliability and Birnbaum importance are derived by means of the finite Markov chain imbedding approach. Then the Birnbaum importance of components is compared in the situations where the system is under an IID model, and where one of the components is known to be failed, respectively. Finally, some numerical examples are followed to illustrate the results obtained in the paper.

Since Birnbaum importance was introduced in 1969, there have been more than twenty kinds of importance measures so far. Among the various measures, Birnbaum importance plays an extremely important role because many importance measures have been defined under its illumination and have relationships with it. A lot of work has been done for Birnbaum importance in consecutive- k systems since the systems were introduced. Because the problems in practice are increasingly complicated, in 2007, Zhao proposed consecutive- k systems with sparse d, which is an extension of the current consecutive- k systems. In this paper, we study Birnbaum importance for linear consecutive- k-out-of- n systems with sparse d. Some equations on Birnbaum importance are proposed. With these equations, the ranking of components in the system on the basis of Birnbaum importance is given; and then some patterns of ranking are presented. Finally, two numerical examples are given to illustrate the results obtained in this paper.

Importance measures are widely used to characterize the roles of components in systems. The system lifetime can be divided into different life stages. Traditionally, importance measures do not consider the possible effect of the expected number of component failures over a system's lifetime and over different life stages, which, however, has a great effect on the system performance changes, and should therefore be taken into consideration. This paper extends the integrated importance measure (IIM) from unit time to system lifetime, and to different life stages. Based on the renewal functions of components, this measure can evaluate the changes of the system performance due to component failures. This generalization of the IIM describes which component is the most important to improve the performance of the system during the system lifetime and at different life stages. An example of the application of an oil transportation system is presented to illustrate the use of the generalized IIM.

Availability is an important measure of performance for repairable systems because it can describe the probability of a repairable system being in working states. Single point and interval availability measurements (availabilities) have been widely used, but they cannot meet the requirements of all applications. In this paper, multi-point, multi-interval, and mixed multi-point-interval availabilities are introduced, which are extensions of the single point and interval availabilities. The multi-interval availability formula was given by Csenki in 2007 by using the induction method in which the multi-interval availability was called the joint interval reliability, but a new simple technique is provided to present the formula of multi-interval availability in this paper. We also consider the steady-state situations for all new availabilities. For Markov repairable systems, the calculation formulae are presented for the new availabilities, and some properties of new availabilities are discussed too. Finally, some special cases and numerical examples are given to illustrate the results obtained in the paper.

Abstract: In safety‐critical systems, components with high reliability are required. Once a component's reliability fails to meet the minimum safety requirements, either the component should be replaced or, transferred to another system which is allowed to operate on a lower reliability level. Safety standards are not only related to the cost of maintenance, but also to the system's (wear‐and‐tear) and repair degrees. In this paper, two different repair degrees are discussed, and the best safety virtual age criterion is studied in order to minimize financial cost. These results are supported by numerical examples.

A linear connected-$(r,s)$-out-of-$(m,n):F$system consists of$m\times n$components arranged in$m$rows by$n$columns, and it fails iff there exists a$r\times s$subsystem in which all components are failed. The linear connected-$(r,s)$-out-of-$(m,n):F$system can be used for modeling engineering systems such as temperature feeler systems, supervision systems, etc. In this paper, a general method is proposed based on the finite Markov chain imbedding approach to study the exact reliability of a linear connected-$(r,s)$-out-of-$(m,n):F$system. Then a new more efficient method, which reduces the size of the state space by combining some states into one state, is presented to reduce the computing time. Furthermore, three numerical examples are given. The first two numerical examples show that the proposed algorithm is efficient not only when the component states are i.i.d., but also when the component states are statistically independent and non-identically distributed. And the last numerical example shows that our method can be used to compute not only the reliability, but also the component importance.

A new system is defined based on a series Markov repairable system. In this new system, if a repair time of the system failure is too short (less than a given critical value) to cause the system to fail, then the repair interval may be omitted from the downtime record, i.e., the failure effect could be neglected. Otherwise, if a repair time is longer than the given critical value, then the system remains operating from the beginning of this repair until this repair time exceeding the critical value, i.e., the failure effect could be delayed. In Ion-Channel modeling, we call this situation the time interval omission problem. Incorporating this situation, the availability indices are presented as a measure of reliability. Some numerical examples are shown to illustrate the results obtained in the paper.

In 1986, Fu proposed the finite Markov chain imbedding approach. Since then, it has played an important role in the analyses of system reliability, such as on reliability computations, importance analyses of components, failure rate function computations, the birth of new reliability system models, and analyses of new reliability tests. We present a survey on the developments and applications of this approach in the reliability field.

As an excellent tool for reliability system research, the finite Markov chain imbedding (FMCI) approach can be used not only to evaluate the system reliability, but also to do other research in reliability systems such as generating functions, waiting time, and optimal arrangement problems. But the computation for system reliability based on the FMCI approach becomes difficult if the number of components in the system is comparatively large, because there would be more matrices that need to be multiplied. To overcome this drawback, the accelerated scan finite Markov chain imbedding (AS-FMCI) approaches with fixed step length, and alterable step, are introduced in this paper to decrease the complexity of computation for system reliability through decreasing the number of the matrixes that need to be multiplied. A numerical example is presented, and the results show that the new method can save computational time.

We introduce the system of consecutive failures with sparse d which is a natural extension of consecutive-k systems. Then a series of generalizations of consecutive-k systems are discussed, such as consecutive-k-out-n:F systems with sparse d, M consecutive-k-out-of-n:F systems with sparse d, and (n, f, k) :F systems with sparse d. We present the formulation for the system reliability of these generalized consecutive-k systems with various component settings in terms of the finite Markov chain imbedding idea, along with two numerical examples.

This paper introduces a new model for a single-unit Markov repairable system in which repair times that are sufficiently short (less than some critical value) do not result in system failure. We can say that such a repair interval is omitted from the downtime record. First we suppose that the critical repair time is a constant. The model is then generalized to allow the critical repair time to be a non-negative random variable. We calculate system availability for these new models as a measure of reliability. Some numerical examples are given to illustrate the results in the paper.

The effect of a repair of a complex system can usually be approximated by the following two types: minimal repair for which the system is restored to its functioning state with minimum effort, or perfect repair for which the system is replaced or repaired to a good-as-new state. When both types of repair are possible, an important problem is to determine the repair policy; that is, the type of repair which should be carried out after a failure. In this paper, an optimal allocation problem is studied for a monotonic failure rate repairable system under some resource constraints. In the first model, the numbers of minimal & perfect repairs are fixed, and the optimal repair policy maximizing the expected system lifetime is studied. In the second model, the total amount of repair resource is fixed, and the costs of each minimal & perfect repair are assumed to be known. The optimal allocation algorithm is derived in this case. Two numerical examples are shown to illustrate the procedures.