On allocating redundancies to k‐out‐of‐ n reliability systems

For two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k ‐out‐of‐ n system with components’ lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic comparison of residual lifetime mixture models

In this paper, we consider stochastic comparisons of mixtures ($F^1$ and $G^1$) of residual life distributions ($F_{\theta }$ and $G_{\theta }$, $\theta >0$) arising out of different baseline distributions ($F$ and $G$) and different mixing distributions ($H_1$ and $H_2$). Under certain conditions on $F$, $G$, $H_1$ and $H_2$, we establish that the distributions $F^1$ and $G^1$ are stochastically ordered. Further, we make stochastic comparison of mixture distribution $F^1$ with the distribution of residual lifetime at random time ${\Theta }_1$.     

Optimal allocation of redundancies in series systems

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.     

Probabilistic mean value theorems for conditioned random variables with applications

In this paper, we propose a probabilistic analogue of the mean value theorem for conditional nonnegative random variables ordered in the hazard rate and reversed hazard rate order, upon conditioning on intervals of the form (t,∞) and [0,t]. This result is then specialized within the proportional hazards model and the proportional reversed hazards model with applications to series systems in reliability theory and to absorption random times of linear birth‐death processes. We also study the comparison of residual entropies and discuss some connections to Wasserstein and stop‐loss distances of random variables. A treatment for the additive hazard rate model is finally provided, with an application to life annuities.     

Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

In this paper, we obtain and discuss some general properties of hazard rate (HR) functions constructed via generalized mixtures of two members. These results are applied to determine the shape of generalized mixtures of an increasing hazard rate (IHR) model and an exponential model. In addition, we note that these kind of generalized mixtures can be used to construct bathtub‐shaped HR models. As examples, we study in detail two cases: when the IHR model chosen is a linear HR function and when the IHR model is the extended exponential‐geometric distribution. Finally, we apply the results and show the utility of generalized mixtures in determining the shape of the HR function of different systems, such as mixed systems or consecutive k‐out‐of‐n systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

Aging properties of the residual life length of k‐out‐of‐n systems with independent but non‐identical components

The k‐out‐of‐n structure is a very popular type of redundancy in fault‐tolerant systems, it has founded wide applications in industrial and military systems during the past several decades. This paper will investigate the residual life length of a k‐out‐of‐n system with independent (not necessarily identical) components, given that the (n?k)th failure has occurred at time t?0. Behaviour of PF₂, IFR, DRHR, DMRL, NBU(2) and NBUC classes of life distributions are derived in terms of the monotonicity of the residual life given the time of the (n?k)th failure. Copyright © 2004 John Wiley & Sons, Ltd.