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Dependence among cyber risks has been an essential and challenging component of risk management. The current study characterizes cyber dependence from both qualitative and quantitative perspectives based on L-hop propagation model. From the qualitative side, it is shown that cyber risks always possess positive association based on the proposed risk propagation model. From the quantitative side, an explicit formula for computing the fundamental dependence measure of covariance is provided for an arbitrary network. In particular, we study the impacts of factors—especially external and internal compromise probabilities, propagation depth, and network topologies—on dependence among cyber risks. We conclude by presenting some examples and applications.

This paper studies stochastic comparisons between a population and subpopulations in both multiplicative and additive frailty models. The comparisons between a population and its baseline in stocha...

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

This paper carries out stochastic comparisons of series and parallel systems with independent and heterogeneous components in the sense of the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. The main results extend and strengthen the corresponding ones by Misra and Misra [18] and by Ding, Zhang, and Zhao [8]. Meanwhile, the results on the hazard rate order of parallel systems and the reversed hazard order of series systems serve as nice supplements to Theorem 16.B.1 of Boland and Proschan [4] and Theorem 3.2 of Nanda and Shaked [20], respectively.

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.

As one of the criteria for comparing variabilities among distributions, the sample range has attracted considerable attention in past decades. In this paper, we establish stochastic comparison results of sample ranges arising from two sets of heterogeneous exponential samples. It is shown that the reversed hazard rate of the sample range is a Schur-convex function of the parameter vector while its distribution function is a Schur-concave function of the vector of logarithms of the coordinates of the parameter vector. Moreover, when samples follow the proportional hazard rates models, we prove that the distribution function of the sample range is Schur-concave in the parameter vector, thereby extending several results known in the literature including Kochar and Rojo (1996) [13], Kochar and Xu (2007) [14] and Zhao and Zhang (2012) [31].

In this paper, we study how to compute the signature of a k-out-of-n coherent system consisting of n modules. Formulas for computing the signature and the minimal signature of this kind of systems based on those of their modules are derived. Examples are presented to demonstrate the applications of our formulas. The main results obtained in this paper generalize some related ones in recent literature.

Stochastic comparisons of linear (circular) consecutive k-out-of-n:F systems with independent components are studied. A sufficient condition is given under which the lifetime of a circular consecutive k-out-of-n:F system with independent and nonidentically distributed components is stochastically decreasing in n for fixed k. Furthermore, the likelihood ratio orderings of the lifetimes of linear (circular) consecutive k-out-of-n:F systems with independent and identically distributed components are also established, and some counterexamples are given to show that these orderings are not true in general.

Let X 1,…, X n be independent exponential random variables with respective hazard rates λ1,…, λ n , and Y 1,…, Y n be i.i.d. exponential random variables with common hazard rate λ. It is proved that X 2: n , the second order statistic from X 1,…, X n , is larger than Y 2: n , the second order statistic from Y 1,…, Y n , with respect to the right spread order if and only if with and Λ i = Λ(1) − λ i , and X 2: n is smaller than Y 2: n with respect to the right spread order if and only if Further, the case with proportional decreasing hazard rate is also studied, and the results obtained here form nice extensions to some corresponding ones known in the literature.

This note builds a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent components having proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP2 dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.

This paper deals with the reversed hazard rate of general mixture models. Dependence and monotone properties of the reversed hazard rate are studied. Some lower bounds of the expected inactivity time of the overall population are presented with an application. Finally, preservation of DRHR under general mixture is discussed as well.

It is difficult to compute the signature of a coherent system with a large number of components. This paper derives two basic formulas for computing the signature of a system which can be decomposed into two subsystems (modules). As an immediate application, we obtain the formula for computing the signature of systemwise redundancy in terms of the signatures of the original system and the backup one. The formula for computing the signature of a componentwise redundancy system is also derived. Some examples are given to illustrate the power of the main results.