System components usually attain marginal lifetimes with stochastic dependence in the context of load-sharing reliability structures. This study deals with the load-sharing parallel systems of two components. We prove that two marginal lifetimes are positively quadrant dependent when component lifetimes have continuous probability distributions, and such a stochastic dependence is upgraded to the total positive of order 2 in the setting of component lifetimes having an exponential distribution. In addition, we discuss how these findings shed light on related results for the load-sharing Ross model, the conditional residual lifetime, and the conditional inactivity time.

Abstract The stochastic ordering of order statistics plays a central role in many fields involving probability and statistics. Over the past decade, significant progress has been made in ordering the first and second order statistics from heterogeneous or statistically dependent observations. This paper further develops necessary and sufficient conditions for the third-largest and third-smallest order statistics from observations linked by an Archimedean copula and following, respectively, the proportional hazards, proportional reverse hazards, and scale models. The characterization results and sufficient conditions presented herein not only enrich the existing literature but also provide novel theoretical insights for applications in fields such as reliability systems, multiple outlier models, and the design of third-price auction mechanisms.

This paper studies the load-sharing parallel system with two components having statistically dependent lifetimes. We examine how the statistical dependence, the virtual age, the decelerating factor, the load allocation impact on system reliability through developing the usual stochastic order on system lifetime. Also, we present ordering results on system reliability to compare this system with other particular structures, including a single component, hot standby system and warm standby system. One of the new findings quantifies the role of the statistical dependence in terms of the usual stochastic order, and the others serve as interesting complements to the existing results in the recent literature on load-sharing reliability systems.

In the Markov model of degraded multi-state systems, instantaneous degradation rates are constant and their functioning state lifetimes are semi-independent. This note attempts to study systems of two independent three-state components with arbitrary degradation rates and any statistical dependence between functioning state lifetimes. In particular, we derive nonparametric reliability functions for functioning state lifetimes of both parallel and series systems of the two components. Moreover, we investigate how the statistical dependence impacts the system lifetime through conducting nonparametric stochastic comparison on system functioning state lifetimes. Some numerical examples are presented to illustrate the results as well.

Recently, the relevation transformation has received further attention from researchers, and some interesting results have been developed. It is well known that the active redundancy at component level results in a more reliable coherent system than that at system level. However, the lack of study of this problem with relevation redundancy prevents us from fully understanding such a generalization of the active redundancy. In this note we deal with relevation redundancy to coherent systems of homogeneous components. Typically, for a series system of independent components, we have proved that the lifetime of a system with relevation redundancy at component level is larger than that with relevation redundancy at system level in the sense of the usual stochastic order and the likelihood ratio order, respectively. For a coherent system with dependent components, we have developed a sufficient condition in terms of the domination function to the usual stochastic order between the system lifetime with redundancy at component level and that at system level.

In this note, we study a k-out-of-n system with component lifetimes linked by Archimedean (survival) copula and independent of the lifetime of a cold standby redundancy. We derive the survival function and the residual life function of the redundant system. Also, we examine how the redundancy and the system structure impact the system reliability through developing the usual stochastic order on system lifetime. In particular, for parallel systems we investigate the role played by the component lifetime, the generator of Archimedean copula, and the starting time of the redundancy as well. The present framework well relates to the real situation and thus the main results are instructive to reliability engineers.

Krakowski (Rev Fr Autom Inform Rech Opèr. 1973;7:107–120.) introduced the relevation transform for component and active redundancy with independent lifetimes, and except for Johnson and Kotz (Am J Math Manag Sci. 1981;1:155–165; Nav Res Logist. 1983;30:163–169.) most subsequent researches were conducted under this framework. However, it is not uncommon that a component and its active redundancy bear some common stresses due to the environment and thus they have dependent lifetimes. In this note, we equip the involved lifetimes with a survival copula and then clarify the potential difference between the new and classical versions through making stochastic comparison. Moreover, by ordering the lifetime of system with relevation redundancy we also study the way of allocating a relevation redundancy at component level to ultimately improve the system reliability. The present results on series and parallel systems serve as a generalization of the corresponding ones of Belzunce et al. (Appl Stoch Models Bus Ind. 2019;35:492–503.). Several numerical examples are presented to illustrate these findings as well.

For bivariate random lifetimes, we study the role of stochastic dependence through conducting stochastic comparison on the residual life, inactivity time and some aging behavior of one marginal conditioned by the survival or failure of the other. The main results serve as essential supplement to those related ones of Bassan and Spizzichino (1999) and Longobardi and Pellerey (2019). Applications in frailty, resilience and load-sharing models are presented as well.

In this article, we study a load-sharing redundancy system in the presence of statistical dependence between base and redundancy component lifetimes when they are under partial workload. We derive the system reliability function in terms of reliability functions and the copula function of the base and redundancy component lifetimes. Further, in the setting of some aging properties, the base component and absolute continuous copula, we develop the hazard rate order, the reversed hazard rate order, and the likelihood ratio order between the system lifetime and the base component lifetimes. Two typical scenarios with independence between the base and redundancy component lifetimes are discussed in particular. Several numerical examples are presented to illustrate the theoretical findings as well.

Distributed generators are being widely integrated into China's urban distribution network unevenly, which accounts for different load rates of distribution transformer cluster in spatial dimension and low net load simultaneity rate in temporal dimension. Therefore, it is necessary to carry out flexible dispatching for the distribution transformer cluster in active distribution network to effectively reduce the grid loss. Different from the existing research on dynamic reconfiguration of the network supplied by a single transformer or a single feeder, the grid loss of distribution transformer cluster is optimized in the paper. Firstly, a 0-1 integer optimization model is established. The model is characterized by calculating the total grid loss of distribution transformer cluster based on the connectivity of power supply path and considering the spatio-temporal constraints such as total change times of switches. Secondly, uncertainty of renewable energy output is considered with k-means clustering algorithm effectively. The reduced scenarios are used for flexible dispatching simulation. Then, the discrete monkey algorithm, characterized by the cross iteration of local optimization and global search, is improved for solving 0-1 integer programming problem. The improved method can deal with the temporal coupling constraints in the model. Finally, the effectiveness of the proposed method is identified by case study. Flexible dispatching of distribution transformer cluster can greatly reduce the grid loss of active distribution network, and the optimal dispatching scheme will change along with the change of distributed generator output.

As a relevant topic in reliability theory, the preservation of aging properties under the formation of various coherent structures contributes to improving system performance through better structure design and more effective system maintenance. The classical research in this line usually focuses upon coherent systems with independent component lifetimes. Recently, some authors discussed the preservation of IFR, NBU, and DMRL in the setting of dependent component lifetimes. This paper further investigates sufficient conditions for coherent systems with dependent component lifetimes to preserve aging properties including NBUC, NBU (2), DMRL, and their dual versions. Some examples are presented to illustrate coherent structures and typical copula functions fulfilling the present sufficient conditions as well.

This paper develops sufficient conditions for the preservation of WSAI, RWSAI and LWSAI under increasing transformations on the coordinates, respectively. Applications of the preservation in the asset allocation problem are presented as well.

In view of uncertainties caused by large-scale wind power integration, energy storage system (ESS) is being considered to stabilize the fluctuation of wind power. In this paper, the influence of ESS on power system operation with wind power is analyzed in detail, and an economic dispatch (ED) model with wind power and ESS is proposed based on scenario set. First, the initial scenario set of wind power output is generated by the Monte Carlo sampling. To overcome the shortcoming of heavy dependence on the initial clustering centers, which usually leads to unstable clustering results, the k-means clustering is improved by combining self-organizing feature map neural network and particle swarm optimization (PSO). Then, the initial scenario set is reduced based on this improved k-means clustering method. Finally, an ED model solved by PSO is used to minimize the comprehensive power generation cost based on the reduced scenario set. Taking IEEE-39 bus system as an example, the scenario-set-based ED model is implemented in this paper. The simulation results show that, when solving the ED problem with wind power and ESS, the proposed method considering scenario reduction makes not only the clustering index better, but also the results of ED more reasonable.

This paper deals with series and parallel systems of dependent components equipped with starters. We study the hazard rate order, the dispersive order and the usual stochastic order of system lifetimes in the context of component lifetimes having proportional hazard rates. The main results either generalize or extend corresponding conclusions of Joo and Mi (2010) and Da, Ding, and Li (2010).

Financial instruments traded in the market are usually subject to mutually dependent default risks, and a default does not always make a zero return for the concerned risky asset. This paper revisits the portfolio selection problem with assets exposed to dependent default risks. To better model the default mechanism, we generalize the threshold default model and the independence default model due to Cheung and Yang (2004) by introducing a smaller nonzero realizable return for a default risky asset. By utilizing stochastic arrangement increasing techniques, we develop sufficient conditions to enable actuaries to order the amount allocated to each asset in the two generalized models. Also, some examples of dependence structures fulfilling the sufficient conditions are presented as illustrations.

In this paper, we consider series systems and parallel systems with the dependence between the component lifetimes modelled by an Archimedean copulas. We obtain sufficient and necessary conditions of relative ageing orders between series (parallel) systems with different component numbers, which partially generalize some main results of Misra and Francis. When the component lifetimes follow the scale model, we also characterize the ordering properties between the series systems and (n-1)-out-of-n systems (parallel systems and 2-out-of-n systems) by mixture distribution.

This paper studies a Pareto-optimal reinsurance contract in the presence of negative statistical dependence between the insurance claim and the random recovery rate. In the context of symmetric information model and asymmetric information model, we investigate properties of the Pareto-optimal indemnity schedules. For risk neutral reinsurer with proportional cost and associated expense, we present possible forms of the Pareto-optimal indemnity schedule as well.

We develop sufficient conditions for the hazard rate order on minimums of sample with Archimedean survival copulas and having proportional hazard rates or scales, and the reversed hazard rate order on maximums of sample with Archimedean copulas and have proportional reversed hazard rates, respectively. Also, we present applications of the hazard rate order on sample minimums in engineering reliability and actuarial risk.

We review recent research results on stochastic arrangement increasing risks in financial and actuarial risk management, including allocation of deductibles and coverage limits concerned with multiple dependent risks in an insurance policy, the independence model and the threshold models for a portfolio of defaults risks with dependence, and the optimal capital allocation for a financial institute with multiple line of business.