## 4 Materials and Methods
### Reliability Assessment Model for Radial Type Distributed Client-Server Architecture CN
A fragment of the state graph is shown in Figure 3.  
For the stationary mode, a system of linear Equations (28)-(30) is obtained:  
\[-\left(n\mu^{srv}+\mu^{hub}+\sum_{i=1}^{N}m_{i}\mu_{i}^{cl}\right)P_{0,0,0,\ldots,0}+\lambda^{srv}P_{1,0,0,\ldots,0}+\\ +\lambda^{hub}P_{0,1,0,\ldots,0}+\lambda_{1}^{cl}P_{0,0,1,\ldots,0} +...+\lambda_{N}^{cl}P_{0,0,0,\ldots,1}=0\\ \cdots \tag{28}\]  
\[-\left[(n-\vec{j}^{srv})\mu^{srv}+\vec{j}^{srv}\lambda^{srv}+\left(1- \vec{j}^{hub}\right)\mu^{hub}+\right.\\ +\vec{j}^{hub}\lambda^{hub}+\sum_{i=1}^{N}\left((m_{i}-\vec{s}_{i} ^{cl})\mu_{i}^{cl}+\vec{s}_{i}^{cl}\lambda_{i}^{cl}\right)\right]\times\\ P_{\vec{j}^{srv},\vec{j}^{hub},\vec{s}_{1}^{cl},\ldots,\vec{s}_{N}^{cl }}+(n-\vec{j}^{srv}+1)\mu^{srv}P_{\vec{j}^{srv}-1,\vec{j}^{hub},\vec{s}_{1}^{ cl}\ldots\vec{s}_{N}^{cl}}\\ +\vec{j}^{hub}\mu^{hub}P_{\vec{j}^{srv},0,\vec{s}_{1}^{cl},\ldots, \vec{s}_{N}^{cl}}+\\ +(m_{1}-\vec{s}_{1}^{cl}+1)\mu_{1}^{cl}P_{\vec{j}^{srv},\vec{j}^{ hub},\vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}}+\\...+(m_{N}-\vec{s}_{N}^{cl}+1)\mu_{N}^{cl}P_{\vec{j}^{srv}, \vec{j}^{hub},\vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}}+\\ +(\vec{j}^{srv}+1)\lambda^{srv}P_{\vec{j}^{srv}+1,\vec{j}^{thv}, \vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}}+\\ +(1-\vec{j}^{hub})\lambda^{hub}P_{\vec{j}^{srv},\vec{j}^{thv}, \vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}}+\\ +(\vec{j}_{1}^{cl}+1)\lambda_{1}^{cl}P_{\vec{j}^{srv},\vec{j}^{ hub},\vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}}+\\...+(\vec{j}_{N}^{cl}+1)\lambda_{N}^{cl}P_{\vec{j}^{srv}, \vec{j}^{hub},\vec{s}_{1}^{cl}\ldots\vec{s}_{N}^{cl}+1}=0\cdots \tag{29}\]  
Figure 3: Fragment of the MSS state graph. \[-\left(n\lambda^{srv}+\lambda^{hub}+\sum_{i=1}^{N}m_{i}\lambda_{i}^{cl }\right)P_{n,1,m_{1},...,m_{N}}+\\ +\mu^{srv}P_{n-1,1,m_{1},...,m_{N}}+\mu^{hub}P_{n,0,m_{1},...,m_{N} }+\\ +\mu_{1}^{cl}P_{n,1,m_{1}-1,...,m_{N}}+...+\mu_{N}^{cl}P_{n,1,m_{1},...,m_ {N}-1}=0 \tag{30}\]  
The system of Equations (28)-(30) has a single solution subject to the normalization condition (31):  
\[\sum_{j^{srv}=0}^{n}\sum_{j^{hub}=0}^{1}\sum_{j_{1}^{cl}=0}^{m_{1}}...\sum_{j_{ N}^{cl}=0}^{m_{N}}P_{j^{srv},j^{hub},j_{1}^{cl},...,j_{N}^{cl}}=1 \tag{31}\]  
To solve the system of Equations (28)-(30) in general form, the following substitution (32) is made:  
\[P_{j^{srv},j^{hub},j_{1}^{cl},...,j_{N}^{cl}}=P_{j^{srv}}^{hub}\cdot P_{j^{hub} }^{i}\cdot\prod_{i=1}^{N}P_{j_{i}^{cl}}^{cl_{i}} \tag{32}\]  
The system of equations for the \(i\)-th type of CN elements is written, introducing the following notations:  
\[\text{When}\quad k\geq 1,\quad z_{k}=(n_{i}-k)\mu_{i}P_{k-1}^{i}+\lambda_{i}P_{ k+1}^{i}\]  
The equation in the general case (31) is given by the following:  
\[P_{j_{i}}^{i}=\frac{n_{i}!}{(n_{i}-j_{i})!}\rho_{i}^{j_{i}}P_{0}^{i} \tag{33}\]  
Backward substitution of (32) into (31) is performed, taking into account the binomial and polynomial coefficients. Additionally, the general designations of the characteristics of the CN elements are substituted. The solution of the system of Equation (28) is then obtained in general form (32):  
\[P_{j^{srv},j^{hub},j_{1}^{cl},...,j_{N}^{cl}} = \frac{\frac{n!}{(n-j^{srv})!}\rho_{srv}^{j^{srv}}\cdot P_{hub}^{ j^{hub}}\cdot\prod_{i=1}^{N}\frac{m_{i}!}{(m_{i}-j_{i}^{cl})!}\rho_{cl_{i}}^{j^{cl}}} {\sum_{j^{srv}=0}^{n}\sum_{j^{hub}=0}^{m_{1}}\sum_{j^{cl}_{1}=0}^{m_{1}}... \sum_{j^{cl}_{N}=0}^{m_{N}}\frac{n!}{(n-j^{srv})!}\rho_{srv}^{j^{rrv}}\cdot \rho_{hub}^{j^{hub}}\cdot\prod_{i=1}^{N}\frac{m_{i}!}{(m_{i}-j_{i}^{cl})!} \rho_{cl_{i}}^{j^{cl}}}\]  
The main indicator of aircraft operation reliability in steady-state mode is the availability factor \(K_{r}\). The availability factor is defined as the probability that the system will be in an operable state at any point in time, except for planned periods when the system is not intended to be used [69].