## 4 Materials and Methods
### Performance Model for Heterogeneous Client-Server Networks  
Neuromorphic computing encompasses a wide range of information processing approaches and differs from traditional computing systems due to its neurobiological inspiration [58]. Presently, determining the appropriate type and amount of specialization is essential to ensure that a heterogeneous system remains flexible and programmable. For instance, the authors in study [59] established a framework for designing self-optimizing and self-programmable computing systems (SOSPCS), which enhance programmability and flexibility while exploiting the heterogeneity of computing resources. Empirical studies conducted by the authors in [60] reveal that new relationships can be predicted between elements related to network topology and the properties of the components involved. Predicting these connections will enable the determination of network link formation processes based on currently observed connections. As parallel systems gained popularity, many researchers faced the challenge of optimizing them and reducing computational costs [61]. Figure 1 illustrates the overall architecture of the heterogeneous client-server computing network under consideration. The computational problem is characterized by four key metrics [62]: \(NO_{\text{alg}}\), the average number of elementary machine operations performed per iteration of the primary solution algorithm; \(NO_{\text{ctrl}}\), the average operation count of the control subroutine per iteration; \(V_{\text{alg}}\), the mean volume of data (in bits) exchanged between client and server each iteration; and \(T_{\text{lim}}\), the maximum allowable end-to-end execution time for the problem. The operation of the CN can be expressed as a collection of states varying with time. If we describe this process within the framework of mass service theory [63], we can analogize the functioning of the CN to a closed mass service system (MSS) with waiting and random distribution of requests across each server processor. Assume that there are \(N\) types of clients, with an arbitrary number of each type \((m_{1},m_{2},\cdots,m_{N})\). Each client necessitates servicing at intervals determined randomly. Services are executed via \(n\) homogeneous processors. Client request arrivals for each user class \(i\) (\(i=1,\cdots,\overline{N}\)) follow a Dien-Wang distribution with rate parameter \(\lambda_{i}\). Service times for each request are exponentially distributed with rate \(\mu_{i}\)[64]. When one or more servers are idle, an arriving request is assigned uniformly at random to any idle server. If all servers are busy, incoming requests join a finite-capacity queue; once the queue reaches its limit, the system applies a random-replacement policy, selecting an existing queued request with equal probability for service admission or drop. MSS is described by states of the form  
\[a_{j_{1},\cdots,j_{N},k,l},\]  
Figure 1: Generalized structure of heterogeneous CN. where \(j_{i}\) is the number of requests from client type \(i\) (\(i=1,\cdots,N\)), \(k\) denotes the number of busy server processors, and \(l\) is the number of queued requests. The following special cases illustrate this notation:  
* \(a_{0,\cdots,0,0,0}\): no requests in the system, all \(n\) servers idle. * \(a_{1,0,\cdots,0,1,0}\): one request from type-1 clients is being served on one processor, with no queue. * \(a_{0,\cdots,0,1,0}\): one request from type-\(N\) clients is in service, with no queue. In the general state \(a_{j_{1},\cdots,j_{N},k,l}\), the total number of requests in the system is  
\[M=\sum_{i=1}^{N}j_{i},\]  
and the relationship between \(M\), \(k\), and \(l\) is given by  
\[k=\min(M,n),\quad l=M-k,\]  
where \(n\) is the total number of server processors. Figure 2 shows a fragment of the MSS state-transition graph. Figure 2: Fragment of the MS state-transition diagram.