Search

For a system (or unit) whose lifetime is measured by the number cycles, according to the discrete time age replacement policy, it is replaced preventively after n cycles or correctively at failure, whichever occurs first. In this paper, discrete time age replacement policy is revisited when the lifetime of the system is modeled by a discrete phase-type distribution. In particular, the necessary conditions for the unique and finite replacement cycle which minimizes the expected cost per unit of time are obtained. The necessary conditions are mainly based on the behavior of the hazard rate. The results are illustrated for some special discrete phase-type lifetime distributions. Computational results are also presented for the optimal replacement cycle under specific real life setups.

Consider a k-out-of-n system which is subject to shocks that occur at random times. Each shock causes failure of random number of components, and hence the system’s lifetime corresponds to one of the arrival times of shocks. The reliability and mean time to failure of the system are studied when the times between shocks follow a phase type distribution. The optimal replacement time problem which is concerned with the minimization of the total long-run average cost per unit time is also defined and studied.

The wind power generated by a wind plant has a stochastic nature due to randomness in the wind speed. Although the empirical distribution of the wind power has been extensively studied by using data sets in different regions, several works focused on theoretical distribution of the wind power produced by wind turbines. In this paper, the theoretical distribution of the wind plant power is obtained. In the derivation of the distribution of the wind plant power, wind turbine reliability is taken into account. The wind plant power distribution can be effectively used if the wind speed probability distribution is known. Theoretical results are illustrated for Weibull and Birnbaum–Saunders wind speed distributions which have been found to be suitable for real data collected at two different locations.

A new mixed shock model is introduced and studied. According to the model, for two fixed critical values d 1 and d 2 such that d 1 d 2 , the system under concern fails upon the occurrence of k consecutive shocks of size at least d 1 or a single large shock of size at least d 2 . The new model combines run and extreme shock models. Reliability properties of the system are studied under two cases: when the interarrival time X i between the ( i − 1 ) th and i th shock, and the magnitude of the i th shock Y i are independent for all i , and when the interarrival time between the ( i − 1 ) th and i th shock, and the magnitude of the i th shock are dependent for all i .

In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i = 1 , 2 . Reliability and mean residual life functions of such components are studied when the times between shocks follow phase-type distribution.

This paper is concerned with a system consisting of multiple types of components and having ( k 1 , k 2 , … , k m ) -out-of- n structure. The ( k 1 , k 2 , … , k m ) -out-of- n system is a system consisting of n i components of type i , i = 1 , 2 , … , m , and functions if at least k 1 components of type 1, k 2 components of type 2, …, k m components of type m work, n = ∑ i = 1 n n i . The exact and approximate expressions are obtained for the survival function of the system under concern. The weighted- ( k 1 , k 2 , … , k m ) -out-of- n system is also defined and studied. This weighted model is applied to evaluate the wind power system that consists of two wind plants.

The number of failed components in a failed or operating system is a very useful quantity in terms of replacement and maintenance strategies. These quantities have been studied in several papers for a system consisting of identical components. In this paper, the number of failed components at the time when the system fails and the number of failed components when the system is working are considered for a well-known and widely applicable k-out-of-n structure. The system is assumed to have multiple types of components. That is, the system consists of components having nonidentical failure time distributions. Optimization problems are also formulated to find optimal values of the number of components of each type, and the optimal replacement time.

In most real life situations, the system’s components contribute differently in different performance levels. Such a situation can be modeled by systems with multi-state components having more than one working status, e.g. perfect functioning, and partial working. In this paper, a multi-state system that consists of two types of three-state components is defined and studied. An explicit formula for the probability that the performance of the system is at least a given level is obtained for the most general case when the components are statistically dependent. The model is applied to evaluate the wind power system that consists of two wind plants in different regions. An optimization problem is formulated to find the optimal number of wind turbines that must be installed in the wind plants by minimizing the total cost under specific power production.

A k -out-of- n system with weighted components is a system that consists of components contribute differently to the overall performance of the system, and functions if the total weight/contribution of all working components exceeds or equal to the level k . Such a system is useful and suitable for modeling capacity based systems such as power systems, transportation systems and manufacturing systems. This paper is concerned with the lost capacity by the weighted- k -out-of- n system at the time when the system fails. This random quantity is useful for making an optimal decision about the spare capacity that should be available to renew the system upon its failure. In particular, the distribution of this random quantity is derived and the theoretical results are illustrated for a power system consisting of a specified number of generating units.

This paper is concerned with properties of series–parallel systems when the component lifetimes have discrete failure time distribution. For a series–parallel system consisting of a specified number of subsystems, we particularly focus on the number of failed components in each subsystem at the time when the system fails. Each subsystem is assumed to have identical components while different subsystems have different types of components. Assuming all components within the system are independent, we obtain exact distributions of the number of failed components at the time when the system fails. For the special case when the components have phase-type failure time distributions, matrix-based expressions are derived for the quantities under concern. The results are used to obtain optimal configuration of the series–parallel system which is replaced at failure.

Shock models are of great interest in engineering reliability. Among the others, the δ-shock model has been widely studied in the literature. In this model, the system breaks down due to the arrivals of two successive shocks which are too close to each other. That is, the system fails when the time between two consecutive shocks falls below a fixed threshold δ. In the literature, the δ-shock model has been mostly studied by assuming that shocks arrive according to a renewal process so that the interarrival times between shocks are independent and identically distributed. In the current paper, we consider the case when the shock arrival process is described by a Polya process which has dependent interarrival times. In particular, we obtain survival function and mean lifetime of the system and study the optimal replacement policy for the δ-shock model based on Polya process.

The effect of adding cold standby redundancy to a system at system and component levels provides a useful information in reliability design. For a series (parallel) system adding cold standby redundancy at the component (system) level yields longer system lifetime. In this paper, the effect of adding cold standby redundancy to a general coherent structure at system and component levels is studied. In particular, signature-based expressions for the survival function of the system after standby redundancy at system and component levels are obtained. Thus for a given coherent structure with known signature, the survival functions and mean time to failure of new systems can be easily calculated and comparisons can be done in terms of stochastic ordering, and mean time to failure ordering. As a case study, circular consecutive-k-out-of-n:G system which can be used to analyze activities in a nuclear accelerator is considered.

In this paper we study a discrete time risk model based on exchangeable dependent claim occurrences. In particular we obtain expressions for the finite time non-ruin probability and the joint distribution of the time to ruin the surplus immediately before ruin and the deficit at ruin. An illustration of the results is given and some implications of the results are provided. Comparisons are made with the corresponding results for the classical compound binomial model of independent and identically distributed claim occurrences. (C) 2016 Elsevier E.V. All rights reserved.

Particular class of reliability shock models is studied.Phase-type modeling is proposed.Optimal replacement time and mean residual life are computed.Matrix-based efficient formulae are obtained. In this paper, matrix-based methods are presented to compute the optimal replacement time and mean residual lifetime of a system under particular class of reliability shock models. The times between successive shocks are assumed to have a common continuous phase-type distribution. The system's lifetime is represented as a compound random variable and some properties of phase-type distributions are utilized. Extreme shock model, run shock model, and generalized extreme shock model are shown to be the members of this class. Graphical illustrations and numerical examples are presented for the run shock model when the interarrival times between shocks follow Erlang distribution.

In this paper, we study the compound random variable S = ∑ t = 1 N Y t when there is a dependence between a random variable N and a sequence of random variables { Y t } t ≥ 1 . Such a compound random variable has been found to be useful in several fields including actuarial science, risk management, and reliability. In particular, we develop some results on distributional properties of the random variable S when N is a phase-type random variable that is defined on a sequence of binary trials and depends on { Y t } t ≥ 1 . We present illustrative examples and an application for the use of results in actuarial science.

This paper is concerned with reliability based long-term performance assessment of hybrid solar/wind power system. In particular, an analytical expression is obtained for the theoretical distribution of the power output of the hybrid system by taking into account the reliability values of renewable energy components. An expression for the expected energy not supplied (EENS) is also derived and used to compute the energy index of reliability (EIR) that is directly related to EENS. Because the derived expressions involve reliability values which are related to mechanical states of the renewable energy components, the results enable us to evaluate properly the performance of the hybrid system. A numerical example is included to illustrate the results.

For degraded multi-state systems, it has been assumed in the literature that, for any given system, the instantaneous degradation rates are fixed. This paper attempts to study a three-state degraded system that have random degradation rates among its states. In particular, a reliability model for such a three-state system is presented assuming that the degradation rates are random and statistically dependent. The dependence is modeled by copulas, and dynamic reliability analysis of the system is performed. Graphical illustrations are provided, and comparisons are made with the corresponding results for the classical fixed rates model.

Let {ξn,n≥1} be a sequence of independent trials with three possible outcomes 0, 1, 2 labeled as failure, success of type I and success of type II, respectively. Suppose that at each time a success of type I (type II) occurs in {ξn,n≥1} a random reward of type I (type II) is received. We obtain distributions of the number of trials until either the sum of consecutive rewards of type I is equal to or exceeds the level k1 or the sum of consecutive rewards of type II is equal to or exceeds the level k2 under two different schemes.

In this paper, a new class of lifetime distributions which is obtained by compounding arbitrary continuous lifetime distribution and discrete phase-type distribution is introduced. In particular, the class of exponential-phase type distributions is studied with some details.

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.

The number of components that are failed at the time of system failure is a useful quantity since it gives an idea of how many spares should be available to replace all failed components upon the system failure. In this paper, the number of failed components is considered at subsystem and system levels for the series–parallel system that consists of K subsystems. In particular, the joint behavior of the number of failed components in each subsystem is studied when each subsystem has identical components and different subsystems have different types of components. The results are then used to find the optimal number of components in each subsystem by minimizing an expected cost per unit of time upon the system failure.

This article presents a signature-based representation for the expected cost rate of age-based preventive maintenance policy for a binary coherent system consisting of independent exponential components, and then specializes the method to consecutive k-out-of-n system and its generalizations. According to the age-based preventive maintenance policy, the system is replaced at failure or before failure. For an arbitrary coherent system, the number of failed components at replacement time is a random variable. Thus, the expected cost per unit of time involves the mean number of failed components at replacement time. This mean is represented in terms of signature. Extensive numerical and graphical examples are presented for m-consecutive k-out-of-n:F and consecutive-k-within-m-out-of-n:F systems.

It is important to elicit information about the potential power output of a wind turbine and a wind farm consisting of specified number of wind turbines before installation of the turbines. Such information can be used to estimate the potential power output of the wind farm which will be built in a specific region. The output power of a wind turbine is affected by two factors: wind speed and turbine availability. As shown in the literature, the correlation between wind speed and wind turbine availability has an impact on the output of a wind farm. Thus, the probability distribution of the power produced by the farm depending on the wind speed distribution and turbine availability can be effectively used for planning and risk management. In this paper, the theoretical distribution of the wind farm power is derived by considering the dependence between turbine availability and the wind speed. The theoretical results are illustrated for real wind turbine reliability and wind speed data.

Integrating multiple wind farms into power systems may reduce the fluctuation in total power output of wind farms and hence it decreases the system risk resulting from the wind speed variability. In this paper, a wind power system consisting of two wind farms is modeled and analyzed considering the dependence between wind speeds at two sites. In particular, the system is modeled as a threshold system and reliability values of wind turbines are also taken into account in capacity based calculations. The results are illustrated for the available bivariate wind speed data in the literature.

According to the well-known age replacement policy, the system is replaced preventively at time t or correctively at system failure, whichever occurs first. For a coherent system consisting of components having common failure time distribution which has increasing failure rate, we present necessary conditions for the existence of the unique optimal value which minimizes the mean cost rate. The conditions are mainly based on the signature which only depends on the system’s structure. The results are illustrated for linear and circular consecutive type systems.

This paper is concerned with two optimization problems for a parallel system that consists of dependent components. First, the problem of finding the number of elements in the system that minimizes the mean cost rate of the system is considered. The second problem is concerned with the optimal replacement time of the system. Previous work assumes that the components are independent. We discuss the impact of dropping this assumption. In particular, we numerically examine how the dependence between the components affects the optimal number of units and replacement time for the system which minimize mean cost rates. We first consider the case when the components are exchangeable and dependent, i.e. the system consists of single type of dependent components. Subsequently, we consider a system that consists of multiple types of dependent components. Comparative numerical results are presented for particularly chosen dependence models.

The study of compound sums have always been very popular in the literature. Many models in insurance and engineering have been represented and solved by compound sums. In this paper, two different bivariate compound sums are proposed and studied. The phase-type distribution is applied to obtain the probability generating function of the bivariate sum.

In this paper, three different discrete time shock models are studied. In the first model, the failure occurs when the additively accumulated damage exceeds a certain level while in the second model the system fails upon the local damage caused by the consecutively occurring shocks. The third model is a mixed model and combines the first and second models. The survival functions of the systems under these models are obtained when the occurrences of the shocks are independent, and when they are Markov dependent over the periods.

A three-state k -out-of- n system with n independent components is considered, where the vector k of integers is determined by given fixed scalars k 1 and k 2 such that k 1 , k 2 ? n . The mean number of components of each type either in a perfect functioning state or in a partially working state at the time of the system failure and at a time while the system is working are studied. An optimization problem concerned with the most economical value of n is also formulated.

We consider two different sets of exchangeable samples which are assumed to be dependent. A single set of observations is obtained from these two dependent samples. The distribution of single order statistic, and the joint distribution of the minimum and an arbitrary order statistic are derived. The results are illustrated in the context of reliability problem.

A system is subject to random shocks over time. Let c 1 and c 2 be two critical levels such that c 1 c 2 . A shock with a magnitude between c 1 and c 2 has a partial damage on the system, and the system transits into a lower partially working state upon the occurrence of each shock in ( c 1 , c 2 ). A shock with a magnitude above c 2 has a catastrophic affect on the system and it causes a complete failure. Such a shock model creates a multi-state system having random number of states. The lifetime, the time spent by the system in a perfect functioning state, and the total time spent by the system in partially working states are defined and their survival functions are derived when the interarrival times between successive shocks follow phase-type distribution.

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.

This paper investigates the stress-strength reliability in the presence of fuzziness. The fuzzy membership function is defined as a function of the difference between stress and strength values, and the fuzzy reliability of single unit and multicomponent systems are calculated. The inclusion of fuzziness in the stress-strength interference enables the user to make more sensitive analysis. Illustrations are presented for various stress and strength distributions.

A binary weighted-k-out-of-n:G system is a system that consists of n binary components, and functions if and only if the total weight of working components is at least k. The performance of such a system is characterized by its total weight/capacity. Therefore, the evaluation of the capacity of the system is of special importance for understanding the behavior of the system over time. This paper is concerned with capacity loss and residual capacity in binary weighted-k-out-of-n:G systems. These measures are potentially useful for the purposes of preventive action. In particular, recursive and non-recursive equations are obtained for the mean capacity loss and mean residual capacity of the binary weighted-k -out-of-n:G system while it is working at a specific time. The mean residual capacity after the failure of the system is also studied.

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.

In this paper, we generalize geometric and binomial distributions of order k to q -geometric and q -binomial distributions of order k using Bernoulli trials with a geometrically varying success probability. In particular, we derive expressions for the probability mass functions of these distributions. For q = 1 , these distributions reduce to geometric and binomial distributions of order k which have been extensively studied in the literature.

In this paper, a multivariate copula based modeling methodology for dynamic reliability modeling of weighted- k -out-of- n systems is applied. The system under consideration is assumed to have n dependent components each having its own weight. It has a performance level of at least k when the total weight of operating components is k or above. Copula based expressions for the survival function and mean time to failure of such a system are obtained. Extensive numerical results are presented for Clayton and Gumbel type copulas. The behavior of survival function and mean time to failure are investigated with respect to the value of Kendall’s correlation coefficient.

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.

In this paper, we introduce and study geometric distribution of order k with a reward. In a sequence of binary trials, suppose that each time a success occurs a random reward is received. The distribution of the number of trials until the sum of consecutive rewards is equal to or exceeds the level k is called geometric distribution of order k with a reward. We obtain expressions for the probability mass function of this distribution.

A weighted- k -out-of- n :G system is a system that consists of n binary components, each with its own positive weight, and operates only when the total weight of working components is at least k . Such a structure is useful when the components have different contributions to the performance of the entire system. This paper is concerned with both marginal and joint Birnbaum, and Barlow–Proschan (BP) importances of the components in weighted- k -out-of- n :G systems. The method of universal generating function is used for computing marginal and joint Birnbaum importances. The method for computing BP-importance is based on a direct probabilistic approach. Extensive numerical calculations are presented. By the help of these calculations and illustrations, it is possible to observe how the marginal and joint importances change with respect to the weights of components.

The purpose of this paper is to show the usefulness of system signature for computing some important reliability indices of repairable systems. In particular, we obtain signature-based expressions for stationary availability, rate of occurrence of failure, and mean time to the first failure of repairable systems. Using these expressions we compute corresponding reliability indices of all systems with three and four components. Computational results are also presented for consecutive-k-within-m-out-of-n:F and m-consecutive-k-out-of-n:F systems. (C) 2013 Elsevier B.V. All rights reserved.

In this paper, we study parallel and consecutive-k-out-of-n:F systems consisting of components which are subject to random deterioration with time. The random deterioration in resistance of a component is defined through a stochastic process. We obtain lifetime distribution of a parallel system via classical probabilistic techniques. The lifetime distribution of a consecutive-k-out-of-n:F system is derived using the lifetime distribution of parallel systems and the concept of maximal signature. We also study the optimal replacement time for a parallel system. We present illustrative computational results using MATHCAD.

An extreme shock model when there is a change in the distribution of the magnitudes of shocks is defined and studied. Such a model is useful in practice since a sudden change in environmental conditions may cause a larger shock. In particular, the reliability and mean time to failure of the system is obtained by assuming that the times between arrivals of shocks follow phase-type distribution. The optimal replacement policy that is based on a control limit is also proposed. The results are illustrated when the number of shocks until the change point follows geometric distribution.

In this paper, we obtain an expression between the sums of the marginal distributions of the order statistics and the common marginal distribution of an exchangeable random sequence. We also derive an expression between the sums of the joint distribution of two order statistics and the two dimensional joint distribution of an exchangeable random sequence.

There are various systems consisting of components which may have different contribution to the performance of the system. Such systems can be modeled systems with weighted components. In this paper, we study the mean instantaneous performance of this type of systems after successive component failures. The mean instantaneous performance is a useful characteristic to take preventive action about the system. In particular, we obtain explicit expressions for the mean instantaneous performance of a system with weighted components that have arbitrarily distributed lifetimes. We illustrate the results when the lifetime distribution of components follow proportional hazard model. Some further results are also presented for the components having exponential lifetime distribution.

Consider a system consisting of n components each with its own positive integer-valued random weight (capacity). The system is assumed to have a performance level above c if there are at least k working components, and the total weight of all working components is above c. We study the reliability properties of such a system. A recursive formula is obtained for computing the system state probabilities. We present a Monte-Carlo simulation algorithm to observe the time spent by the system in state c or above. The algorithm is based on the use of ordered lifetimes of components. We illustrate the results with numerical computations.