## 4 Materials and Methods
### Reliability Assessment Model for Radial Type Distributed Client-Server Architecture CN  
Analytical modeling is an effective tool for studying various characteristics of designed computing systems, enabling the identification of their most efficient structures. However, current analytical methods for assessing the reliability of DCSs, in relation to their configurations, remain underdeveloped [68]. CN with a radial architecture consists of an arbitrary number of client types, with each type containing an arbitrary number of clients, all interacting with a multiprocessor server. Let the network consist of \(N\) client types, each containing \(m_{i}\) clients \((i=1,...,N)\). Each client connects to the hub via communication channels, while the server is made up of \(n\) homogeneous processors. Radial computing theory posits that individual components of a computing system are prone to failure and will eventually require repairs. The failure rates of the various components in the system are clearly defined as follows:  
* \(\lambda_{\text{icl}}\)--failure rate of client nodes of type \(i\), \(i=1,\cdots,N\);
* \(\lambda_{\text{sv}}\)--the failure rate of the server processors;
* \(\lambda_{\text{hub}}\)--the failure rate of the concentrator. Recovery times for all elements exiting the operational state follow an exponential distribution with the following parameters:  
* \(\mu_{\text{icl}}\)-- recovery intensity of client nodes of the \(i\)-th type (\(i=1,N\));
* \(\mu_{\text{sv}}\)-- the intensity of server processor recovery;
* \(\mu_{\text{hub}}\)--the recovery intensity of the hub. When a new recovery request finds the service attendant free, it is accepted. Conversely, if the servicing device is busy, the incoming request must queue until it is serviced, following a random equal probability selection from the queue. The closed MSS can occupy a range of states depending on which components are operational or under repair. Key configurations include the following:  
* \(a_{0,\cdots,0}\): No servers, no hub, and no clients are available; all elements are down and undergoing restoration, halting computation. * \(a_{1,0,\cdots,0}\): Exactly one server processor is functional while the remaining \(n-1\) processors, the hub, and all clients are failed and being repaired; computation is suspended. * \(a_{0,1,0,\cdots,0}\): Only the central hub is up, with every server processor and client node failed and in repair; no processing occurs. * \(a_{0,0,1,0,\cdots,0}\) or \(a_{0,0,0,\cdots,1}\): A single client of type 1 (or type \(N\)) is operational, while its \(m_{i}-1\) peers, the hub, and all servers are down and repairing; computational tasks remain paused. * \(a_{j_{\text{sv}},0,j_{1,\text{cl}},\cdots,j_{N,\text{cl}}}\): A subset of \(j_{\text{sv}}\) server processors and \(j_{i,\text{cl}}\) clients of each type \(i\) are working, with the hub and the other components in failed-repair mode; processing is not active. * \(a_{j_{\text{sv}},1,j_{1,\text{cl}},\cdots,j_{N,\text{cl}}}\): The active group includes \(j_{\text{sv}}\) servers, the hub, and \(j_{i,\text{cl}}\) clients of each class, while the rest are under repair; computation proceeds. * \(a_{n,1,m_{1},\cdots,m_{N}}\): The entire network--\(n\) server processors, the hub, and all \(m_{i}\) clients of each type--is up and executing tasks. Figure 2 depicts a fragment of this state-transition graph.