Stochastic comparisons of two-unit repairable systems

Abstract

We consider two models of two-unit repairable systems: cold standby system and warm standby system. We suppose that the lifetimes and repair times of the units are all independent exponentially distributed random variables. Using stochastic orders we compare the lifetimes of systems under different assumptions on the parameters of exponential distributions. We also consider a cold standby system where the lifetimes and repair times of its units are not necessarily exponentially distributed.     

On optimal allocation of redundant components for series and parallel systems of two dependent components

In this paper we consider the problem of optimal allocation of a redundant component in the case of series and parallel systems of two components when all the components are dependent. Whereas this problem has been extensively treated for the case of independent components, the case of dependent components has not received too much attention. In this paper we show that for this problem the main tools are the joint stochastic orders introduced by Shanthikumar and Yao (1991).     

Conditional k-out-of-n systems with a cold standby component

ABSTRACT

In this article, we consider a k-out-of-n system with a cold standby component under the general condition that lth (0 < l ? n ? k + 1) component is working at time t. The survival function and mean residual life function of such system are derived. Some stochastic monotonic properties of the system lifetimes are presented as well. Numeric results are provided to illustrate the results. The main results obtained in this article complement and generalize related ones in Eryilmaz (2012 Eryilmaz, S. (2012). On the mean residual life of a k-out-of-n: G system with a single cold standby component. Eur. J. Oper. Res. 222:273–277.[Crossref], [Web of Science ®] , [Google Scholar]).     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

Allocation strategies of standby redundancies in series/parallel system

This article deals with the topic of optimal allocation of two standby redundancies in a two-component series/parallel system. There are two original components C₁ and C₂ which can be used to construct a series/parallel system, and two spares R₁ (same as C₁) and R₂ (different from both C₁ and C₂) at hand with them being standby redundancies so as to enhance the reliability level of the system. The goal for an engineer is to seek after the optimal allocation policy in this framework. It is shown that, for the series structure, the engineer should allocate R₂ to C₁ and R₁ to C₂ provided that C₁ (or R₁) performs either the best or worst among all the units; otherwise, the allocation policy should be reversed. For the parallel structure, the optimal allocation strategy is just opposed to that of series case. We also provide some numerical examples for illustrating the theoretical results.     

Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components

By adding a resilience parameter to the scale model, a general distribution family called resilience-scale model is introduced including exponential, Weibull, generalized exponential, exponentiated Weibull and exponentiated Lomax distributions as special cases. This paper carries out stochastic comparisons on parallel and series systems with heterogeneous resilience-scaled components. On the one hand, it is shown that more heterogeneity among the resilience-scaled components of a parallel [series] system with an Archimedean [survival] copula leads to better [worse] performance in the sense of the usual stochastic order. On the other hand, the [reversed hazard] hazard rate order is established for two series [parallel] systems consisting of independent heterogeneous resilience-scaled components. The skewness and dispersiveness are also investigated for the lifetimes of two parallel systems consisting of independent heterogeneous and homogeneous [multiple-outlier] resilience-scaled components. Numerical examples are provided to illustrate the effectiveness of our theoretical findings. These results not only generalize and extend some known ones in the literature, but also provide guidance for engineers to assemble systems with higher reliability in practical situations.     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

On stochastic behaviors of load-sharing parallel systems

Abstract

This paper mainly investigates a general load-sharing parallel system having two units. First, we construct some comparisons among a load standby system, a warm standby system, a hot standby system and a cold standby system. Moreover, some stochastic comparisons between the load-sharing parallel system and one of its two components are obtained in the sense of the usual stochastic order. Finally, the residual life of this system and its properties are examined.     

Copula-based reliability analysis for a parallel system with a cold standby

The traditional reliability models cannot well reflect the effect of performance dependence of subsystems on the reliability of system, and neglect the problems of initial reliability and standby redundancy. In this paper, the reliability of a parallel system with active multicomponents and a single cold-standby unit has been investigated. The simultaneously working components are dependent and the dependence is expressed by a copula function. Based on the theories of conditional probability, the explicit expressions for the reliability and the MTTF of the system, in terms of the copula function and marginal lifetime distributions, are obtained. Let the copula function be the FGM copula and the marginal lifetime distribution be exponential distribution, a system with two parallel dependent units and a single cold-standby unit is taken as an example. The effect of different degrees of dependence among components on system reliability is analyzed, and the system reliability can be expressed as the linear combination of exponential reliability functions with different failure rates. For investigating how the degree of dependence affects the mean lifetime, furthermore, the parallel system with a single cold standby, comprising different number of active components, is also presented. The effectiveness of the modeling method is verified, and the method presented provides a theoretical basis for reliability design of engineering systems and physics of failure.     

Comparisons between parallel systems with exponentiated generalized gamma components

Consider two parallel systems with their independent components’ lifetimes following heterogeneous exponentiated generalized gamma distributions, where the heterogeneity is in both shape and scale parameters. We then obtain the usual stochastic (reversed hazard rate) order between the lifetimes of two systems by using the weak submajorization order between the vectors of shape parameters and the p-larger (weak supermajorization) order between the vectors of scale parameters, under some restrictions on the involved parameters. Further, by reducing the heterogeneity of parameters in each system, the usual stochastic (reversed hazard rate) order mentioned above is strengthened to the hazard rate (likelihood ratio) order. Finally, two characterization results concerning the comparisons of two parallel systems, one with independent heterogeneous generalized exponential components and another with independent homogeneous generalized exponential components, are derived. These characterization results enable us to find some lower and upper bounds for the hazard rate and reversed hazard rate functions of a parallel system consisting of independent heterogeneous generalized exponential components. The results established here generalize some of the known results in the literature, concerning the comparisons of parallel systems under generalized exponential and exponentiated Weibull models.     

Reliability of weighted k-out-of-n: G systems consisting of two types of components and a cold standby component

In this article, the influence of a cold standby component to the reliability of weighted k-out-of-n: G systems consisting of two different types of components is studied. Weighted k-out-of-n: G systems are generalization of k-out-of-n systems that has attracted substantial interest in reliability theory because of their various applications in engineering. A method based on residual lifetimes of mixed components is presented for computing reliability of weighted k-out-of-n: G systems with two types of components and a cold standby component. Reliability and mean time to failure of different structured systems have been computed. Moreover, obtained results are used for defining optimal system configurations that can minimize the overall system costs.     

Some ordering properties of series and parallel systems with dependent component lifetimes

Abstract

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.     

Stochastic properties and parameter estimation for a general load-sharing system

We study here a general load-sharing parallel system in which the lifetimes of the components of the system are arbitrary continuous random variables. The system functions if at least one component in the system functions and the surviving unit shares the whole load. Some sufficient conditions are obtained for the usual stochastic order between two different load-sharing systems. We then consider the optimal allocation problem of one load standby in a series system with two independent components. Finally, the maximum likelihood estimation of the parameters for some specific systems is discussed.     

Signatures of series and parallel systems consisting of non disjoint modules

System signature is a useful tool to analyze coherent systems. In reliability theory, a large number of systems is actually the composition of disjoint subsystems (modules). However, in real life, there are situations in which the subsystems have common components. In this article, we consider the problem of obtaining signatures as well as minimal signatures of series and parallel systems that are composed of subsystems sharing a component. That is, these subsystems are no longer disjoint. Computational results are also presented to illustrate our findings.     

Stochastic comparisons of parallel and series systems of dependent components equipped with starting devices

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).     

Likelihood ratio order of parallel systems under multiple-outlier models

This paper studies the likelihood ratio ordering of parallel systems under multiple-outlier models. We introduce a partial order, the so-called θ-order, and show that the θ-order between the parameter vectors of the parallel systems implies the likelihood ratio order between the systems.     

On the effect of dependency in information properties of series and parallel systems

Information measures of reliability systems has been widely studied in the statistical and reliability literatures. These findings were obtained when lifetimes of components are independent and identically distributed. But, there is no context about the information properties of such systems when lifetimes of components are dependent. In this paper, we explore properties of the entropy and Kullback–Leibler discrimination information for series and parallel system’s lifetimes when lifetimes of components are dependent and identically distributed. Specifically, we give some results on the entropy of series systems when lifetimes of components are positive or negative quadrant dependence. Moreover, several results are obtained about the entropy ordering properties related to other well known stochastic orders. To illustrate the quality of the given results, some examples are also given.     

A lifetime distribution motivated by parallel and series structures

A new three-parameter distribution with decreasing, increasing, and bathtub-shaped hazard rates obtained by compounding geometric, power series, and exponential distributions is introduced. It includes some well-known distributions as particular cases. Various mathematical properties of the new distribution as well as details of the maximum likelihood estimation and a sensitivity analysis for its parameters are presented. Finally, two real data applications are presented.     

Predictive information and distance between past and future of a time series

A characterization for the nullity of the cosine angle between two subspaces of a Hilbert space is established. Given a time series x, we use this characterization in order to investigate the relationship between the notions of predictor space and distance between the information contained in the past and in the future of x. In particular, we prove that the predictor space of x coincides with the zero vector space {0} if and only if this distance achieves its maximum value.