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This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict aging property, some numerical results are presented as well. [ABSTRACT FROM AUTHOR]

Modeling cyber-attacks is a very attractive area of research because of its practical importance. However, most of the related research in the literature does not consider the simultaneous (or coordinated) attacks, which, in fact, is an important attack instrument in practice. This is mainly because of the complicated evolution of cyber-attacks over networks. In this paper, we propose a novel model, which can accommodate different types of simultaneous attacks with possible heterogeneous compromise probabilities. Our results show that simultaneous attacks have a significant effect on the reliability/dynamics of network systems. In particular, we present a sufficient condition for the epidemics dying out over the network, and upper bounds for the time to extinction. We also provide upper bounds for compromise probabilities of network systems when the evolution enters the quasi-equilibrium state. The effects of strength of simultaneous attacks and heterogeneity among successful attack probabilities on epidemic spreading are studied as well. The theoretical results are further validated by the simulation evidence. [ABSTRACT FROM AUTHOR]

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9]. [ABSTRACT FROM AUTHOR]

Let X1, ? , Xnbe independent random variables with Xihaving survival function F?i, i= 1, ? , n, and let Y1, ? ,Ynbe a random sample with common population survival distribution F, where c= ?i=1n?i/n. Let Xn:nand Yn:ndenote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:nis greater than Yn:nin terms of likelihood ratio order. It is also proved that the sample range Xn:n? X1:nis larger than Yn:n? Y1:naccording to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively. [ABSTRACT FROM AUTHOR]

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order. [ABSTRACT FROM AUTHOR]