In many engineering systems, component failure arises from the interaction between time-varying resistance and random environmental stress. Classical formulations usually assume static resistance and independent components, assumptions often violated in practice. We develop a stochastic framework for parallel and consecutive 𝑘-out-of-𝑛:𝐹 systems in which component resistances follow linear degradation paths with random deterioration rates, while all components are exposed to a common random stress. Dependence among deterioration rates is modelled via a copula, yielding joint lifetime distributions for parallel systems; for consecutive 𝑘-out-of-𝑛:𝐹 structures, maximal signatures express system lifetimes in terms of parallel-system survival. Numerical illustrations with a survival Clayton copula and Weibull marginal deterioration (decreasing, constant, and increasing hazard) examine the impact of dependence strength and marginal failure behaviour on reliability and mean residual life. We also analyse an optimal replacement policy that minimises long-run average cost and compare maintenance decisions under independent and dependent deterioration. The results show that ignoring dependence systematically underestimates failure risk and cost, and that preventive replacement is economically justified only for components with increasing failure rates, whereas constant or decreasing hazard rates favour run-to-failure strategies.

In this paper, a polygonal k-out-of-n:F system reliability model is established, where subsystems may share components within the polygon. Components are exposed to a combination of extreme shocks and delta-shock. The reliability of each subsystem's components is modeled using a phase-type distribution and Markov chain approach, with component reliability being incorporated into subsystem reliability calculations. Disjoint cases are classified based on the number of failed components in shared regions to derive the shared component failure probability matrix and the initial probability distribution. Using the transfer matrix and transition rate matrix, the overall system reliability is determined through the Markov chain embedding approach. For this model, the total cost function of the system was determined by calculating the cost functions of different scenarios and different subsystems, and the optimal maintenance time for the system was obtained. The effectiveness of the method is verified by taking the unmanned aerial vehicle swarm working in a harsh environment as an example.

Preventive replacement is a standard strategy for redundant systems, yet most analytical cost models for k-out-of-n:F structures still assume independent components and continuous lifetimes. We develop a unified reliability-maintenance framework for heterogeneous systems whose component lifetimes may be dependent and continuous or discrete. Dependence is introduced via copulas, while discreteness and tied failures are handled through a signature-inspired representation built from ranking-pattern probabilities over weak orderings. This formulation yields general expressions for the system failure-time distribution and the long-run average cost rate under age-based replacement. For 2-out-of-3:F systems, the analysis leads to explicit cost-rate formulas in the continuous case, the discrete case with ties, and an exchangeable discrete setting that isolates the effect of dependence. Numerical experiments under Archimedean copulas show that dependence can substantially shift the optimal preventive age and alter the minimal cost rate, highlighting the bias induced when dependence or discreteness is ignored in maintenance decisions.

This paper addresses critical gaps in the reliability modeling of weighted k-out-of-n:G systems, which are widely used in safety-critical and fault-tolerant applications. Traditional models typically assume component homogeneity and independence, limiting their applicability to complex real-world systems. To overcome these limitations, we propose a generalized framework that (i) allows for an arbitrary number of distinct component types, (ii) models dependencies among components arising from shared resources, environmental conditions, or common-cause failures, and (iii) avoids computational inefficiencies by deriving a non-recursive reliability formula. Additionally, we introduce three practical optimization models to support system design decisions: cost minimization under reliability constraints, reliability maximization under budget constraints, and optimal system replacement timing. The proposed framework offers significant improvements in computational efficiency, modeling realism, and decision support for large-scale engineering systems. These contributions extend the applicability of weighted k-out-of-n systems to domains such as power grids, telecommunications, and industrial manufacturing where component heterogeneity and dependencies are prevalent.

This paper is about the reliability modeling of a linear consecutive k-out-of- n system that consists of two types of dependent components. The survival function and mean time to failure of such a system are expressed using copulas. Extensive numerical findings are provided for Clayton and Gumbel-type copulas. The survival and mean time to failure behaviors are explored in connection with the value of Kendall’s correlation coefficient.

This paper investigates a series-parallel system comprising N independent subsystems with interchangeable dependent components, a prevalent reliability structure in engineering and network design. The primary aim of this research is to derive the joint probability distribution of the number of failed components within these configurations, considering component dependence and varying distributions across subsystems. This approach reflects a more realistic scenario than previously explored in the literature. Initially, the analysis is conducted for systems with two subsystems and subsequently extended to encompass configurations with N subsystems. The study also evaluates key reliability metrics including the average number of failed components and the mean time to failure (MTTF) of the entire system, theoretically proving that the system’s MTTF increases with the number of components under certain sufficient conditions. In addition to probabilistic analysis, an optimization problem is addressed to determine the optimal allocation of components within each subsystem. The objective is to minimize the average cost associated with corrective maintenance, thereby enhancing the cost-effectiveness of system operation.

A system experiences random shocks over time, with two critical levels, d 1 and d 2 , where $d_{1} \lt d_{2}$ . k consecutive shocks with magnitudes between d 1 and d 2 partially damaging the system, causing it to transition to a lower, partially working state. Shocks with magnitudes above d 2 have a catastrophic effect, resulting in complete failure. This theoretical framework gives rise to a multi-state system characterized by an indeterminate quantity of states. When the time between successive shocks follows a phase-type distribution, a detailed analysis of the system’s dynamic reliability properties such as the lifetime of the system, the time it spends in perfect functioning, as well as the total time it spends in partially working states are discussed.

A generalized mixed shock model, which mixes two run shock models, is developed and analyzed. According to the model, the system subject to both internal degradation and external shocks fails upon the occurrence of k_1 consecutive shocks whose magnitude is between predefined critical values of d_1 and d_2 such that d_1

This paper is concerned with two optimization problems for a k-out-of-n system consisting of dependent components such as finding the number of elements in the system that minimize the system's mean cost rate and the system's optimal replacement time. In previous studies, either system consisting of independent components or parallel systems, a particular case of the present study, was examined. In particular, we numerically examine how the components' dependence affects the optimal number of units and replacement time for the system, minimizing mean cost rates. We consider when the components are exchangeable and dependent, that is, the system consists of dependent components. For three vastly used Clayton, Gumbel, and FGM copula functions, comparative numerical results are presented.

This paper is a short review of classical and recent results on Marshall–Olkin shock models and their applications in reliability analysis. The classical Marshall–Olkin shock model was introduced in Marshall and Olkin (J Am Stat Assoc 62:30–44, 1967). The model describes a joint distribution of lifetimes of two components of a system subjected to three types of shocks. The distribution has absolutely continuous and singular parts. The Marshall–Olkin copula also aroused the interest of researchers working on the theory of copulas as an example of a copula having absolutely continuous and singular parts. There are some recent papers considering general models and modifications constructed on the basic idea of Marshall and Olkin (1967). These works find wide applications in reliability analysis in the case of a general system having n ( $$n > 2$$ ) components and shocks coming from m ( $$m > 3$$ ) sources. Some applications can also be seen in the theory of credit risk, where instead of lifetimes of the components, one considers the times to the default of two counter-parties subject to three independent underlying economic or financial events. In this work, we analyze and describe the results dealing with the generalization and modification of the Marshall–Olkin model.

We consider an (n−k+1)-out-of-n concomitant system consisting of n components each having two subcomponents. This system functions if and only if at least (n−k+1) of the first subcomponents function, and the second subcomponents of working first components also function. The reliability of the proposed system is derived. The effect of dependent subcomponents on the system reliability relative to independent subcomponents is discussed. The system with two subcomponents is extended to the system with m subcomponents. Comparative numerical results and graphical representations are provided.

Combinatorial techniques have an important role to compute the joint reliability importance (JRI) of some coherent systems. We obtain combinatorial formula for calculation of the JRI of two components in a generalised version of consecutive type systems consisting of n linearly ordered components such that system fails if and only if (iff) there are at least m l-overlapping runs of k consecutive failed components (n >= m(k - l) + l, l < k). Overlapping runs mean having common elements which is denoted by l: We concentrate on both s-independent & identical components and exchangeable components. Explicit combinatorial formulae are provided for computing the JRI of the above mentioned cases. For both cases, we also compare the results with linear m-consecutive-k-out-of-n:F system (nonoverlapping case when l = 0). In addition, some numerical and illustrative examples are presented.

Coherent systems and Marshall-Olkin run shock models are combined. Coherent systems consisting of n components receive some kind of shocks from n+1 different sources similar to Marshall-Olkin type. More precisely, when the component j receives k consecutive fatal shocks from the source j or k consecutive fatal shocks from the source n+1, it fails, j = 1, …,n. When the interarrival time of shocks has phase-type distribution, reliability, mean time to failure (MTTF) and mean residual life (MRL) function of the coherent systems are studied. Numerical examples and graphical representations are provided.

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.

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.

In the classical Marshall-Olkin model, the system is subjected to two types of shocks coming at random times, and destroying components of the system. In statistics and reliability engineering literature, there are numerous papers dealing with various extensions of this model. However, none of these works takes into account the system structure, i.e., in existing shock models usually the system structure is not considered. In this work, we consider a new shock model involving the system structure. More precisely, we consider a coherent system which is subjected to Marshall-Olkin type shocks. We investigate the reliability, and mean time to failure (MTTF) of such systems subjected to shocks coming at random times. Numerical examples and graphs are provided, and an extension to a general model is discussed.

In this paper, the influence of a cold standby component on a coherent system is studied. A method for computing the system reliability of coherent systems with a cold standby component based on signature is presented. Numerical examples are presented. Reliability and mean time to failure of different systems are computed.

In classical Marshall–Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall–Olkin type shock model, we assume that if the magnitude of the shock exceeds some predefined threshold, then the component, which is subjected to this shock, is destroyed; otherwise it survives. More precisely, we assume that the shock time and the magnitude of the shock are dependent random variables with given bivariate distribution. This approach allows to meet requirements of many real life applications of shock models, where the magnitude of shocks is an important factor that should be taken into account. A new class of bivariate distributions, obtained in this work, involve the joint distributions of shock times and their magnitudes. Dependence properties of new bivariate distributions have been studied. For different examples of underlying bivariate distributions of lifetimes and shock magnitudes, the joint distributions of lifetimes of the components are investigated. The multivariate extension of the proposed model is also discussed.