## 4 Materials and Methods
### Performance Model for Heterogeneous Client-Server Networks
\[\tau_{i}=\tau_{trns_{i}}+\tau_{srv},\quad\tau_{trns_{i}}=Valg\cdot\
u_{i},\quad \tau_{srv}=NO_{ctrl}\cdot\omega_{srv},\]  
\[\tau_{i}=Valg\cdot\
u_{i}+NO_{ctrl}\cdot\omega_{srv}.\]  
Request arrival intensity is defined as the average number of requests arriving per unit of time:  
\[\lambda_{i}=\frac{I}{I\cdot T_{0i}-I\tau_{i}}=\frac{1}{T_{0i}-\tau_{i}},\]  
where \(I\) is the average number of iterations on one branch of the algorithm for solving the problem. The service intensity \(\mu_{i}\) is characterized by the inverse of the average request servicing time:  
\[\mu_{i}=\frac{1}{\tau_{i}}.\]  
The system of linear equations describing the functioning of the MSS, taking into account \(Q_{k,l}\)--the probability that the system being in the state \(a_{k,l}\), after servicing one of \(k+l\) requests, the total number of requests waiting for service in queues will not change [67]--has the following form:  
\[-\sum_{i=1}^{N}m_{i}\lambda_{i}P_{0,0,\cdots,0,0}+\mu_{1}P_{1,0, \cdots,1,0}+\mu_{2}P_{0,1,\cdots,1,0}+\\ \cdots+\mu_{N}P_{0,0,\cdots,1,0}=0 \tag{13}\]  
Figure 1:0  
where \(k+l-j_{1}j_{2}=0\) and \(j_{1}=0,\quad M-k,\quad l=0,\quad h=1\)  
The linear equation system (14) is complete and possesses a unique solution because in the equation system (15), \(\sum_{i=1}^{N}j_{i}=(k+l)\), which allows the indices \(k\) and \(l\) in the probability notation to be disregarded. For the equations in general form as established by the previous reference, we need to employ substitutions to express each equation as one for an MSS composed of multiple request sources (clients) of identical types and serving devices \(n\) (i.e., server processors). The resultant equations from this derived system can be sequentially solved. Consequently, utilizing induction method transformations, we derive general forms for subsequent solutions. Applying the normalization condition leads us to the following:  
Figure 1:1  
The resultant expression permits determining the probabilities \(P_{k,l}\) associated with the MSS states (18):  
Figure 1:2  
where  
\[h= \begin{cases}n&\text{if $m>n$,}\\ m&\text{if $m\leq n$.}\end{cases} \]  
The comprehensive expression for solving the system of equations represented in (14) while considering the constraints stipulated in conditions (16) and (18) is streamlined into (19)  
\[\tilde{P}_{ji}=\binom{k+l-1}{l}\frac{(n^{k})m_{i}!(m_{i}-j_{i})!}{n^{k+l}\rho _{i}^{j_{i}}}\tilde{P}_{0i} \tag{19}\]  
where  
\[\sum\tilde{P}_{ji}m_{i}j_{i}=0\
eq 1.\]  
By implementing inverse substitution of (19) within (14) with the number of paths \(\chi(k+l),j_{i}\) leading to state \(a_{j_{1},j_{2},\cdots,j_{N},k,l}\), which in the general case corresponds to the polynomial coefficient equal to the equation in [47], let us derive the general equation for the system (20):  
\[P_{k,\ell}=\sum_{j_{1}=0}^{k+\ell}\sum_{j_{2}=0}^{k+\ell-j_{1}}\cdots\sum_{j_ {N}-1=0}^{k+\ell-R}P_{j_{1},j_{2},\ldots,j_{N-1},(k+\ell-R)}^{k,\ell} \tag{20}\]  
In evaluating the performance of the client-server network, it is essential to consider the total number of requests in service and those queued, irrespective of client type. The stationary probability distribution \(P_{k,l}\) from (21) provides the probability of observing \(k\) busy servers and \(l\) queued requests. From this, the expected queue length  
\[L\ =\ \sum_{k=0}^{n}\sum_{l=0}^{L_{\text{max}}}l\,P_{k,l}\]  
as defined in (22), quantifies the average backlog. Moreover, client-type performance degradation is modeled by coefficients \(\theta_{i}\) in (23), which weight each type's sensitivity to waiting and service delays.