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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. [ABSTRACT FROM AUTHOR]

In maintenance field, there are many indexes such as instantaneous availability, steady state availability, interval availability and others, which have been widely used to describe the various properties and performance of repairable systems on maintenance. However, all the present availability indexes still cannot cover some situations which people are interested in. In the paper, two new interval availability indexes are introduced for Markov repairable systems, which is named as an availability with a given window length and containing a specified point or interval. The new interval availability is the probability that the system is working during a given window length and this window must contains a specified point or interval. Their calculation formulas are presented in matrix forms by using the Laplace transform. Some properties on the two indexes are discussed briefly, and numerical examples are shown to illustrate the features of the two new indexes, finally conclusions are given. These two new indices can become the conventional interval availability when the given window length is zero or equal to the specified interval length. The results in the paper may be used to measure the performance of repairable system more deeply and detail. [ABSTRACT FROM AUTHOR]

We propose the consecutive-$k_r $-out-of- $n_r $: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. [ABSTRACT FROM AUTHOR]

The environment in which a system operates can have a crucial impact on its performance; for example, a machine operating in mild or harsh environments or the flow of a river changing between seasons. In this article, we consider a dynamic reliability system operating under a cycle ofKregimes, which is modeled as a continuous-time Markov process withKdifferent transition rate matrices being used to describe the various regimes. Results for the availability of such a system and probability distributions of the first uptime are given. Three special cases, which occur due to situations where the durations of the regime are constant and where the number of up states in different regimes are identical or increasing, are considered in detail. Finally, some numerical examples are shown to validate the proposed approach. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

In this paper, Markov models of repairable systems with repair time omission are considered whose finite state space is grouped into two sets, the set of working states, W, and the set of failed states, F. If the system enters failed states from a working state at any instance, and sojourns at the failed states F less than a given nonnegative critical value τ, then the repair interval can be omitted from downtime records. Otherwise, If the system enters failed states from a working state at any instance, and sojourns at the failed states F more than the given nonnegative critical value τ, then the repair interval cannot be omitted from downtime records. In terms of the assumption, a new model is developed. The focus of attention is the new model's availability, interval reliability and interval unreliability. Several results are derived for these reliability indexes for the new model. Some special cases and numerical examples are given to illustrate the results obtained by using Maple software in the paper. [ABSTRACT FROM AUTHOR]

Abstract: A system consisting of supply and demand is considered in this paper. Both the supply patterns and demand patterns are random. Thus, both the supply and the demand are modeled by Markov processes. The performance measure considered here is the probability that the demand is met by the supply during given time interval . A close form expression for the performance measure is obtained by using aggregated stochastic process theory and Kronecker matrix operations. In the meanwhile, the performance measures in a general interval and multiple intervals have also been given. Finally, a numerical example is given to illustrate the results obtained in this paper. [Copyright &y& Elsevier]

This article analyzes a single-unit repairable system consisting of an operating subsystem and a maintenance subsystem that is solely used to repair the operating subsystem in the event of it breaking down. Formulae for reliability indexes such as availability and distributions concerned with visits to certain subsets of states are presented in terms of state aggregations in which two partitions are used. The situation with exponentially distributed operational times is discussed not only as a special case of the approach used in this article but also using a Markov model. A numerical example is given to illustrate the results obtained in this article. [ABSTRACT FROM PUBLISHER]

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-clwinnel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

A maintenance model is built under some assumptions which come from the real situation for storage products under periodical inspections. The similar model was discussed by Kijima14 when the periodical interval becomes zero, i.e., for the non-periodical inspection case. In Kijima's model, for the product's instantaneous availability, A(t), the probability that the storage products can work in time t, is one. However, it is not easy to get the instantaneous availability in our model, because of periodical inspections. After using the virtual age concept and several lemmas gotten, we, with closed-form for the storage products, present the instantaneous availability and the limiting average availability. Finally, an example is shown to illustrate the results. [ABSTRACT FROM AUTHOR]