# Support Vector Machine-Assisted Importance Sampling for Optimal Reliability Design
## 2 Model Description  
In the light of the requirements of designers, ORD can be formulated either to maximize the system reliability under resource constraints or to minimize the resource under the minimum demand on system reliability. For brevity, we only take the former to illustrate. For the latter, the proposed algorithm is also applied. Put more clearly, the mathematical model of ORD is given by [26]:  
\[\begin{array}{rl}\max:&R_{s}(z)\\ \text{s.t. }&G_{i}(\mathbf{z})\leq \begin{matrix}\mathbf{z}^{L}\\ \mathbf{z}^{L}\end{matrix} (i=1,2,  
**Example 1**. : _Takethe Wi-Fi system shown in Figure 1 to illustrate. The whole area is covered by three signal networks, namely, Verizon, AT&T, and T-Mobile. Each carrier has four relay stations, and each relay station can send and receive signals in a specific block. Here, each block is covered by three consecutive staggered relay stations operated by these three carriers. The Wi-Fi system uses the strongest detection signals from different carriers. Unequivocally, the Wi-Fi signal loss in a particular area occurs if and only if the three consecutive staggered relay stations fail altogether. This Wi-Fi system can be abstracted into a Lin/Con/k/n:F system where \(k=3\) and \(n=12\). The Lin/Con/k/n:F system is a special two-terminal network that includes an ordered sequence of \(n\) components arranged in a line. The system fails if and only if at least \(k\) consecutive components fail. The Lin/Con/k/n:F system is important and has many applications, such as the pipeline system, streetlight system, and telecommunications system._  
To improve the reliability of this Wi-Fi system, we can increase the reliability of relay station or the number of relay stations. Without loss of generality, we consider a Lin/Con/\(k/n\):F system with redundant components, as shown in Figure 2, in which \(\mathbf{z}_{j}\) is the number of redundant components of the \(j\)th subsystem for \(j=1,2,\cdots,n\), and \(n\) is the number of subsystems. The system fails if the successive \(k\) subsystems fail. In this example, subsystem \(j\) contains an active-standby component and \(\mathbf{z}_{j}-1\) cold-standby redundant components. Suppose that the switch is required at all times, and there is a constant probability that the switching will be successful [8]. In addition, the following assumptions are made. (1) Each component/switch possesses only two states: normal and abnormal. (2) The performance of each component/switch is not affected by others. (3) There is no repair/maintenance during the whole service cycle. (4) The components or switches of a subsystem are of the same type. (5) There is imperfect switching to activate the cold-standby redundant components. (6) The time to failure of components is exponential. Figure 1: A Wi-Fi system. Figure 2: A Lin/Con/\(k/n\):F system. Then, following [27, 28], the reliability of subsystem \(j\) is:  
\[R_{j}(t)=r_{j}(t)+\sum_{s=1}^{z_{j}-1}\int_{0}^{t}\rho_{j}(u)r_{j}(t-u)f_{j}^{(s )}(u)du\qquad(j=1,2,\cdots,n) \tag{2}\]  
where \(r_{j}(t)\) is the component reliability at moment \(t\) for the \(j\)th subsystem, i.e., the probability that the lifetime of the component in the \(j\)th subsystem is larger than \(t\). \(\rho_{j}(t)\) is the reliability of the switching mechanism at moment \(t\). \(f_{j}^{(s)}(u)\) is the PDF for the \(s\)th failure in the \(j\)th subsystem, i.e., the probability that the \(s\)th failure in the \(j\)th subsystem arrives at moment \(u\). The first term of (2) indicates that the active-standby redundant component remains in a good state until moment \(t\): during this period, no cold-standby redundant components are put into operation. The summation term in (2) represents \(s\) cold-standby redundant components being sequentially activated through the switch. This implies that the initial active-standby redundant component and the first \(s-1\) cold-standby redundant components have failed before moment \(t\), and the \(s\)th cold-standby redundant component works until moment \(t\).