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Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored $\delta$ -shock model, $\delta \ge 1$ , for which the system fails whenever no shock occurs within a $\delta$ -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob. 32 , 2000), we study the joint and marginal distributions of the system's lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system's lifetime follows a Markov geometric distribution of order $\delta$ (a geometric distribution of order $\delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system's lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored $\delta$ -shock model, for which the system fails when no shock occurs within a $\delta$ -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold $\gamma >0$. Similarly, for this mixed model, we study the joint and marginal distributions of the system's lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process. [ABSTRACT FROM AUTHOR]

δ$$ \delta $$‐shock model is one of the widely studied shock models in reliability theory and applied probability. In this model, the system fails due to the arrivals of two consecutive shocks which are too close to each other. That is, the system breaks down when the time between two successive shocks falls below a fixed threshold δ. In the literature, the δ‐shock model has been mostly studied by assuming that the time between shocks have continuous distribution. In the present paper, the discrete time version of the model is considered. In particular, a proper waiting time random variable is defined based on a sequence of two‐state Markov dependent binary trials and the problem of finding the distribution of the system's lifetime is linked with the distribution of the waiting time random variable, and we study the joint as well as the marginal distributions of the lifetime, the number of shocks and the number of failures associated with these binary trials. [ABSTRACT FROM AUTHOR]

In this article, a system that consists ofnindependent components each having two dependent subcomponents (Ai,Bi),i= 1, …,nis considered. The system is assumed to compose of components that have two correlated subcomponents (Ai,Bi), and functions iff both systems of subcomponentsA1,A2, …,AnandB1,B2, …,Bnwork under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented. [ABSTRACT FROM AUTHOR]

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T be a random variable that follows a geometric distribution of order k, and Y , Y ,... a sequence of independent and identically distributed discrete random variables which are independent of T . In the present article we develop some results on the distribution of the compound random variable $S_{k} =\sum_{t=1}^{T_{k}}Y_{t}$ . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-state \bf k-out-of-n system with n independent components ( \bf k = (k1,{k2}) ). 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 , n1 components are in a partially working state while the rest n2 components are fully functioning ( n = n1 + {n2} ). The system is considered to be at state “1” or above if at least k1 components are working (fully or partially). If at least k1 components are working and at least k2 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Let { Y i } i ≥ 1 be a sequence of { 0 , 1 } variables which forms a Markov chain with a given initial probability distribution and one-step transition probability matrix. Define N n to be the number of trials until the n th success (“1”) in { Y i } i ≥ 1 . In this paper, we study the distribution of the random variable T = ∑ i = 1 N n X i , where { X i } i ≥ 1 is a sequence of independent and identically distributed random variables having a common phase-type distribution. The distribution of T is obtained by means of phase-type distributions. [ABSTRACT FROM AUTHOR]

Multi-state systems have attracted great attention due to their wide applications in engineering. They have been effectively used in modeling various systems such as power supply systems and transportation systems. In this paper, phase type modeling is proposed for dynamic assessment of nonrepairable multi-state systems when the system degrades according to a Markov process. The utility of phase type modeling is demonstrated in the computation of mean lifetimes, mean residual lifetimes, and derivation of survival functions of series and parallel systems. A stochastic comparison result between two systems is also obtained using phase representations of survival functions. Extensive numerical results are presented to illustrate the applicability of the approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with dynamic reliability analysis of three-state k-out-of-n:G systems. It is assumed that the components and the systems can be in three states: perfect functioning, partial performance and complete failure. Using the concept of permanent, we study marginal and joint survival functions for the lifetime of two different three-state k-out-of-n:G systems that consist of independent and nonidentical components. Illustrative examples are also provided for the components which follow the Markov degradation process. [ABSTRACT FROM AUTHOR]

In this paper, we study a multi-state weighted k-out-of-n:G system model in a dynamic setup. In particular, we study the random time spent by the system with a minimum performance level of k. Our method is based on ordering the lifetimes of the system's components in different state subsets. Using this ordering along with the Monte-Carlo simulation algorithm, we obtain estimates of the mean and survival function of the time spent by the system in state k or above. We present illustrative computational results when the degradation in the components follows a Markov process. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we study a shock model in which the shocks occur according to a binomial process, i.e. the interarrival times between successive shocks follow a geometric distribution with mean . According to the model, the system fails when the time between two consecutive shocks is less than a prespecified level. This is the discrete time version of the so-called -shock model which has been previously studied for the continuous case. We obtain the probability mass function and probability generating function of the system’s lifetime. We also present an extension of the results to the case where the shock occurrences are dependent in a Markovian fashion. [Copyright &y& Elsevier]

A combined m-consecutive-k-out-of-n:F & consecutive-kc-out-of-n :F system consists of n linearly ordered components, and fails iff there exist at least kc consecutive failed components, or at least m non-overlapping 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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

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 k-out-of-n:G systems under the assumption that the degradations in systems and components follow an acyclic Markov process which has a discrete state space. [ABSTRACT FROM PUBLISHER]

A start-up demonstration test is a mechanism that can be used to demonstrate the reliability of an equipment to the customer. In this paper, we study CSTF (consecutive successes total failures) and TSTF (total successes total failures) procedures assuming that the individual start-ups follow a first order Markov dependence structure. Explicit (nonrecursive) expressions for the distributions of the test lengths are provided. Maximum likelihood and moments estimators of the expected test length are obtained and some numerical results are presented for an illustration. [ABSTRACT FROM AUTHOR]

Two new coherent system reliability models, which generalise k-out-of-n:F and consecutive k-out-of-n:G systems, are proposed and explicit formulae for the reliabilities of these systems are derived when the components are independent and identical and when they are Markov dependent. Our method of deriving reliability functions is based on the use of classical combinatorial arguments. These extensions consider an additional constraint on the number of working components between successive failures. More explicitly, in addition to the working conditions of k-out-of-n:F and consecutive k-out-of-n:G systems there must be at least d consecutive working components between any of two successive failures. This type of consideration might be useful in some situations including the analysis of constrained binary sequences arising in communications systems, and particular infrared detecting systems. [ABSTRACT FROM AUTHOR]

Reliability evaluation has a vital importance at all stages of processing and controlling modern engineering systems. In most of these modern systems the components have different contributions to the system, which are defined as weights. Among the other weighted systems, a linear consecutive-weighted-k-out-of-n:F system consists of ii components, where each component has its own weight and reliability. The system fails if and only if the total weight of the failed consecutive components is at least k. The aim of this paper is to study the reliability of linear consecutive-weighted-k-out-of-n:F system consisting of independent & nonidentical and nonhomogeneous Markov dependent components. Exact formulae are provided for computing the reliability for above mentioned cases. Approximation formulae for the reliability are also presented. [ABSTRACT FROM AUTHOR]

Abstract: We express the longest run statistic as the maximum of a sequence whose members are subject to a Markov chain condition. Via this setup some results for the exact expression and bounds for the distribution of the longest run in a sequence of Markov dependent trials are presented. Extension of the results to the second order Markov dependent trials are also provided. [Copyright &y& Elsevier]

The number of success runs for nonhomogeneous markov dependent trials are represented as the sum of Bernoulli trials and the expected value of runs are obtained by using this representation. The distribution and bounds for the distribution of the longest run are derived for markov dependent trials. [ABSTRACT FROM AUTHOR]