## 2 Background
### Reliability Block Diagram  
Consider a system S composed by a set of components, \(C=\{c_{i}|1\leq i\leq n\}\), where the state of the system S and its components could be either operational or failed. Let the discrete random variable \(x_{i}\) indicates the state of component \(i\), thus:  
\[x_{i}= \begin{cases}0&\quad\text{$if$ the component $i$ has failed}\\ 1&\quad\text{$if$ the component $i$ is operational}\end{cases} \]  
The vector \(\mathbf{x}=(x_{1},x_{2},...,x_{i},...,x_{n})\). Wherever need and the context is clear, \(\mathbf{x}\) may also be referred to as a set that represents the state of each component of the system, and it is named state vector. The system state may be represented by a discrete random variable \(\phi(\mathbf{x})=\phi(x_{1},x_{2},...,x_{i},...,x_{n})\), such that:  
\[\phi(\mathbf{x})= \begin{cases}0&\quad\text{$if$ the system has failed}\\ 1&\quad\text{$if$ the system is operational}\end{cases} \]  
\(\phi(\mathbf{x})\) is called the structure function of the system. For any component \(c_{i}\):  
\[\phi(\mathbf{x})=x_{i}\,\phi(1_{i},\mathbf{x})+(1-x_{i})\,\phi(0_{i},\mathbf{ x}), \tag{1}\]  
where \(\phi(1_{i},\mathbf{x})=\phi(x_{1},x_{2},...,1_{i},...,x_{n})\) and \(\phi(0_{i},\mathbf{x})=\phi(x_{1},x_{2},...,0_{i},...,x_{n})\). Equation (1) expresses the system structure function in terms of two conditions. The first term \((x_{i}\,\phi(1_{i},\mathbf{x}))\) represents a state where the component \(c_{i}\) is operational and the state of the other components are random variables \((\phi(x_{1},x_{2},...,1_{i},...,x_{n}))\). The second term \(((1-x_{i})\,\phi(0_{i},\mathbf{x}))\), on the other hand, states the condition where the component \(c_{i}\) has failed and the state of the other components are random variables \((\phi(x_{1},x_{2},...,0_{i},...,x_{n}))\). Equation (1) is known as factoring of the  
Figure 3: Power consumption model of the neSC assignment operator. structure function and very useful for studying complex system structures, since through its repeated application one can eventually reach a subsystem whose structure function is simple to deal with Equation (1). If one is interested in representing the system state at a specific time \(t\), the components' state variables should be interpreted as a random variables at time \(t\). Hence, \(\phi(\textbf{x}(t))\), where \(\textbf{x}(t)=(x_{1}(t),x_{2}(t),...,x_{i}(t),...,x_{n}(t))\). The most straightforward strategy for computing system reliability of system composed of independent components is through its respective structure function \(\phi(\textbf{x}(t))\). The system reliability is defined by \(R_{S}(t)==P\{\phi(\textbf{x}(t))=1\}\), where \(P\{c\}\) denotes the probability a given condition \(c\). Since \(\phi(\textbf{x}(t))\) is a Bernoulli random variable, then \(P\{\phi(\textbf{x}(t))=1\}=E[\phi(\textbf{x}(t))]\), where _E[X]_ is the expected value of the random variable \(X\), therefore, \(R_{S}(t)=E[\phi(\textbf{x}(t))]\). Reliability Block Diagram (RBD) is the most adopted notation to express and assess the reliability of systems [1,21,22]. RBDs enable us to represent and evaluate the reliability of a system by composing a set of blocks, where each block represents the reliability of an element of the system, e.g., the reliability of a router. As shown in Figure 4, the RBD has input and output points, which represent the beginning and the end of the system being evaluated, respectively. Blocks are arranged in series, parallel or combining them. These blocks have only two states (failure or working) and the failures are independent from each other. Each block has a reliability associated: block \(B_{i}\) has its reliability \(R_{i}(t)\) associated to \"working state\" at time \(t\). Similarly, its complement (\(I-R_{i}(t)\)) represents the \"failure state\" at time \(t\). If for a system correctly functioning, every of its components should properly operate, the respective system RBD should be represented as a series its system components.