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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...

Computing the system signature is an attractive but challenging problem in system reliability. In this article, we propose a novel algorithm to compute the signature of a system with exchangeable c...

The computation of the signature of a complex system is often challenging, as it may involve a large number of components and a complex architecture. In this study, we propose a novel approach to computing the signature of a system consisting of subsystems with shared components. The proposed approach relies on a new concept called decomposed survival signatures. This new concept can be efficiently used to compute the signature of a modular system with an arbitrary organizing structure and to provide explicit formulas for computation. Applications in cyber systems, transportation systems, and branched oil pipeline systems are highlighted.

The online storage of sensitive data enjoys many benefits such as flexibility, cost-savings, scalability, and convenience but it also poses a big concern on the data confidentiality and availability. To ensure the confidentiality and availability of sensitive data over a network system, the data partition technique is often employed. We study the optimal data partition strategy over an arbitrary network under cyber threats. Both the outside attack and the risk propagation (i.e., inside attack) are considered for the data partition. The data breach probability and retrieve probability are discussed under both limited and unlimited risk propagation for various scenarios. It is discovered that the risk propagation can have much more impact on the optimal partition strategy than that of outside attacks, and the unlimited risk propagation leads to more severer cyber risk. The network topology significantly impacts the partition strategy which hints that the network topology should never be overlooked in practice. The corruption due to compromise can lead to different partition strategies. An optimal partition model is developed for determining the optimal strategy and the pareto non-dominated solutions are recommended for practical use.

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.

A well-known principle in engineering is that redundancy at the component level is generally more reliability effective than that at the system level. Here, the redundancy simply means that components are connected in parallel. Motivated by this principle, in this paper, a more general problem of assembling $K$ -out-of- $N$ systems optimally is studied. We consider two assembled systems: one is by assembling a $k$ -out-of- $n$ system and a $l$ -out-of- $m$ system at the system level, and the other is by assembling them at a component level. With some appropriate assumptions on $(n,k)$ and $(m,l)$ , some results on stochastic comparisons of the two assembly systems are derived. From this result, some useful principles for assembling $K$ -out-of- $N$ -systems optimally are proposed. Two numerical examples are given to illustrate our results as well.

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.

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.

Modeling and analyzing security of networked systems is an important problem in the emerging Science of Security and has been under active investigation. In this paper, we propose a new approach towards tackling the problem. Our approach is inspired by the {\em shock model} and {\em random environment} techniques in the Theory of Reliability, while accommodating security ingredients. To the best of our knowledge, our model is the first that can accommodate a certain degree of {\em adaptiveness of attacks}, which substantially weakens the often-made independence and exponential attack inter-arrival time assumptions. The approach leads to a stochastic process model with two security metrics, and we attain some analytic results in terms of the security metrics., Proceedings of the 2014 Symposium on the Science of Security (HotSoS'14)

Gertsbakh et al. (Journal of Applied Probability, 49, 416–429, 2012) proposed the concept of bivariate signature for a 3-state system. In this paper, we first give an equivalent definition for the bivariate signature in the sense of order statistics of component lifetimes and establish the formula for computing the bivariate signature of the dual of a 3-state system. A useful method for computing the bivariate signature based on the combinatorial meaning of the bivariate tail signature is presented. By this method, we derive formulas for computing the bivariate signatures of some systems consisting of independent modules. Some examples are also presented to illustrate our main results.

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.