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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.

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.