## 2 Brief Description of the Lz-Transform Method and Markov Reward Approach
### Lz-Transform Method  
The Lz-transform method is implemented for the performance evaluation for an FMS aging system in the current article. [PERSON] [36] introduced the method where one can find detailed content and mathematical proofs. A brief introduction comes in the following paragraphs.  
A discrete-state continuous-time (DSCT) Markov process \(X(t)\in\{x_{1},x_{2},\ldots,x_{K}\}\), which has \(K\) possible states and the performance level associated with any state \(i\) is \(x_{i}\), \(i=1,\ 2,\ldots,K\). The Markov process is completely defined by the set of possible states \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{K}\}\), the transition intensities matrix depend on time \(\mathbf{A}=\begin{bmatrix}a_{ij}\end{bmatrix}\), \(i,j=1,\ 2,\ldots,K\) and by the initial states probability distribution given by \(\mathbf{p}_{0}=\{p_{10}=\Pr[X(0)=x_{1}],\ldots,p_{K0}=\Pr[X(0)=x_{K}]\}\). Therefore, according to [37], the Lz-transform of a DSCT Markov process \(X(t)\) is defined by the following expression:  
\[L_{z}[X(t)]=\sum_{i=1}^{K}p_{i}(t)z^{\mathbb{g}_{i}}, \tag{1}\]where \(p_{i}(t)\) is the probability that the process is in state \(i\) at time instant \(t\geq 0\) for a given initial states probability distribution \(\mathbf{p}_{0}\), \(\mathbf{g}_{i}\) is the performance level of state \(i\), and \(z\) is a complex variable.  
Without losing generality, we consider any element \(e\) in a multistate system (MSS) can have \(k_{e}\) different states corresponding to different performances, represented by the set \(\mathbf{g}_{e}=\left\{g_{e1},\ldots,g_{ek_{e}}\right\}\), where \(\mathbf{g}_{ei}\) is the performance rate of element \(e\) in the state \(i\), \(i\in\{1,\;2,\ldots,k_{e}\}\), and \(e\in\{1,\;2,\ldots,n\}\), where \(n\) is the number of elements in the MSS.  
The first step of Lz-transform is to build a Markov model of the stochastic process for each multistate element in the MSS. Based on this model, the state probabilities \(p_{ei}=\Pr\{G_{e}(t)=g_{ei}\}\), \(i\in\{1,\;2,\ldots,k_{e}\}\) for every MSS's element can be obtained as a solution of the corresponding system of differential equations under the given initial conditions. These probabilities define the output stochastic process \(G_{e}(t)\) for each element \(e\) in the MSS. Then, the individual Lz-transform for each element \(e\) can be found as  
\[L_{z}[G_{e}(t)]=\sum\
olimits_{i=1}^{k_{e}}p_{ei}(t)z^{g_{ei}},\;e=\{1,\;2, \ldots,n\}. \tag{2}\]  
Then the next step is based on the previously determined Lz-transform for each element \(e\) and system structure function \(f\), given by \(G(t)=f(G_{1}(t),G_{2}(t),\ldots,G_{n}(t))\), where the Lz-transform of the output stochastic process for the entire MSS should be defined. Using the Ushakov's operator \(\Omega_{f}\)[38] over all Lz-transforms of individual elements, one can obtain the resulting Lz-transform \(L_{z}[G(t)]\) associated with output performance stochastic process \(G(t)\) of the entire MSS:  
\[L_{z}[G(t)]=\Omega_{f}\{L_{z}[G_{1}(t)],L_{z}[G_{2}(t)],\ldots,L_{z}[G_{n}(t)]\}. \tag{3}\]  
The resulting Lz-transform is associated with output performance stochastic process for the entire MSS:  
\[L_{z}[G(t)]=\sum\limits_{k=1}^{K}p_{k}(t)z^{g_{k}}, \tag{4}\]  
and many performance measures of MSS can be easily derived from the resulting Lz-transform. For example, the mean instantaneous performance calculated in the following form:  
\[E(t)=\sum\
olimits_{g_{k}\geq 0}p_{k}(t)g_{k}. \tag{5}\]