Fig. 10

Availability (\(\mathcal{A}\)) with sequential rejuvenations. Each sub-figure corresponds to an environment under a particular type of attack: parallel (∥) in sub-figure 10a; sequential (∴) in sub-figure 10b. In each sub-figure: each curve corresponds to a 〈n,f,k〉 system, where n is the total number of nodes, f is the threshold of tolerable intrusions, and k is the number of rejuvenating (offline) nodes at any given instant; the horizontal axis measures δ, the time offset between rejuvenations of different nodes; the vertical axis measures \(\mathcal{A}\), the expected proportion of time for which the number of intruded nodes is at most f; in the rightmost column of the auxiliary box in the upper right corner of each sub-figure, each value δ_max is the maximum value of parameter δ for which the \(\mathcal{A}\) of the respective 〈n,f,k〉 system is better (i.e., higher) than that of the reference system (curve e, ); the reference curve was obtained from the analytical expression (1−e^r)/r; all other curves (i.e., for systems with f>1) were obtained by joining pairs \((\langle\delta,\mathcal{A}^{\parallel} \rangle)\) or \((\langle\delta,\mathcal{A}^{\therefore} \rangle)\), with δ spaced in intervals of at most 0.01, and with \(\mathcal{A}^{\parallel}\) or \(\mathcal{A}^{\therefore}\), respectively, being an average over the result of 100 probabilistic simulations with a mission timeδ×10⁵