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This paper deals with reliability assessment of the repairable two-unit cold standby system when the first, main unit has the better performance level than the second one. Therefore, after its repair, the main unit is always switched into operation. The new Laplace transform representation for the system’s lifetime is obtained for arbitrary operation and repair time distributions of the units. For some particular cases, the Laplace transform of the system is shown to be rational, which enables the use of the matrix-exponential distributions for obtaining relevant reliability indices. The discrete setup of the model is also considered through the corresponding matrix-geometric distributions, which are the discrete analogs of the matrix-exponential distributions.

This paper is concerned with the study of a discrete time repairable system consisting of one active and one standby component. The lifetime and repair time are assumed to have discrete phase-type distributions. The system’s lifetime is represented as a compound random variable. A matrix-based expression for the probability generating function of the system’s lifetime is obtained based on the phase characteristics of lifetime and repair time distributions. The probability generating function is then used to obtain the distribution of the system’s lifetime. Reliability and hazard rate functions are computed and evaluated for some particular choices of lifetime and repair time distributions. The limiting behavior of the hazard rates is also investigated.

Reliability assessment of system suffering from random shocks is attracting a great deal of attention in recent years. Excluding internal factors such as aging and wear-out, external shocks which lead to sudden changes in the system operation environment are also important causes of system failure. Therefore, efficiently modeling the reliability of such system is an important applied problem. A variety of shock models are developed to model the inter-arrival time between shocks and magnitude of shocks. In a cumulative shock model, the system fails when the cumulative magnitude of damage caused by shocks exceed a threshold. Nevertheless, in the existing literatures, only the magnitude is taken into consideration, while the source of shocks is usually neglected. Using the same distribution to model the magnitude of shocks from different sources is too critical in real practice. To this end, considering a system subject to random shocks from various sources with different probabilities, we develop a generalized cumulative shock model in this article. We use phase-type distribution to model the variables, which is highly versatile to be used for modeling quantitative features of random phenomenon. We will discuss the reliability characteristics of such system in some detail and give some clear expressions under the one-dimensional case. Numerical example for illustration is also provided along with a summary.

The reliability of a weighted-[Formula: see text]-out-of-[Formula: see text] system that consists of three-state components is studied. The system is assumed to comprise [Formula: see text] three-state components, namely, perfect functioning, partial working, and complete failure and functions if the total weight of all the working components is at least [Formula: see text]. Reliability expressions are presented when the times spent by components in perfect functioning and partial working states are dependent with a given joint distribution. Sufficient conditions are also provided to compare the expected total weights of two systems.

In this article, we develop a general statistical inference procedure for the probability of successful startup p in the case of startup demonstration tests when only the number of trials until termination of the experiment are observed. In particular, we define a class of startup demonstration tests and present expectation-maximization (EM) algorithm to get the maximum likelihood estimate of p for this class. Most of well-known startup testing procedures are involved in this class. Extension of the results to Markovian startups is also presented.

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 coherent system that consists of n independent components and equipped with r cold standby components is considered. A generalized mixture representation for the survival function of such a system is obtained, and it is used to examine reliability properties of the system. In particular, the effect of adding r standby components to a given set of original components is measured by computing mean time to failure of the system. The limiting behavior of the failure rate of the system is also examined using the mixture representation. The results are illustrated for a bridge system. A case study that is concerned with an oil pipeline system is also presented.

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.

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.

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.

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.

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.

Marginal and joint reliability importance measures have been found to be useful in optimal system design. Various importance measures have been defined and studied for a variety of system models. The results in the literature are mostly based on the assumption that the components within the system are independent or identical. The present paper is concerned with computation of marginal and joint reliability importance for a coherent system that consists of multiple types of dependent components. In particular, by utilizing the concept of survival signature, expressions for marginal and joint reliability importance measures are presented. We also introduce reliability importance for a system of which only the survival signature is known, which therefore can be regarded to be a black box system. (C) 2017 Elsevier Ltd. All rights reserved.

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 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.

A multistate k-out-of-n system model is an extension of binary k-out-of-n system model by allowing multiple performance levels for the system and its components. Various definitions of multistate k-out-of-n system model have been proposed in the literature. Previous studies on these systems mostly focus on reliability evaluation algorithms. The present paper investigates the lifetimes of multistate systems. In particular, the lifetimes of two different multistate k-out-of-n system models are represented in terms of order statistics, and bounds and approximations are presented using these representations. The results are illustrated for a multistate system whose components' degradation occurs according to a Markov process. Copyright © 2013 John Wiley & Sons, Ltd.

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.

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.

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 n 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.

Consecutive- k -out-of- n system and its generalisations have attracted substantial interest due to their applications to model telecommunications systems, oil pipeline systems, heating systems, etc. One of the most important generalisation of this system is an m -consecutive- k -out-of- n :F system. An m -consecutive- k -out-of- n :F system with overlapping runs is a system consisting of n linearly ordered components and fails if and only if there are at least m overlapping runs of k consecutive failures. For m = 1, the system is same with the usual consecutive- k -out-of- n :F system. In this paper, we study the reliability properties of such systems via system signature. In particular, we obtain a closed form expression for the signature of this system and use it to compute and evaluate several reliability characteristics of the system.