# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Accuracy Criteria for Solving the Problem  
The closeness of the mean values and the coefficients of variation for the two distributions does not mean that the output of the model, which is a queuing system, will yield results with acceptable accuracy [19]. Practical interest is associated with the quality of service indicators, standardized for multiservice networks [12]. Of which, it is appropriate to single out the average request delay time in the queuing system, \(S^{(1)}\), and the \(p\)-quantile of the same random variable, \(t_{p}\). The closeness of these values for different distributions becomes a criterion for the accuracy of solving the problem. To investigate the arising errors, it is appropriate to choose a model with one server of the \(G\)/\(M\)/\(I\) type in [PERSON]'s classification [20]. This model is convenient because it allows an analytical search of all the necessary characteristics of a random variable, such as the requests' delay time in the queuing system. The model assumes that the holding time of requests is a random variable distributed exponentially. The value of the intensity of servicing requests is equal to \(\mu\). The distribution law \(A(t)\) can be arbitrary. First, it is necessary to solve the equation given, for example, in [21]. Taking into account Expression (5), it can be represented in the following form:  
\[\sigma=\sum_{i=0}^{n}p_{i}e^{-i\tau\mu(1-\sigma)} \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline
**Value \(m\)** & **Value \(A^{(1)}\)** & **Value \(C_{A}\)** \\ \hline
11 & 2.36 & 0.70 \\ \hline
21 & 2.41 & 0.82 \\ \hline
31 & 2.45 & 1.00 \\ \hline
101 & 2.77 & 2.45 \\ \hline
501 & 4.56 & 7.31 \\ \hline \end{tabular}
\end{table} 
Table 3: Changes in the values of \(A^{(1)}\) and \(C_{A}\). The value of \(\sigma\) is the only root of Equation (10) within the range from zero to one. After finding the value of \(\sigma\), the values of \(S^{(1)}\) and \(t_{p}\) are calculated using the following formulas [21]:  
\[S^{(1)}=\frac{1}{\mu(1-\sigma)},\;t_{p}=-\frac{\ln(1-p)}{\mu(1-\sigma)} \tag{11}\]  
When using different histograms, different values of the parameter \(\sigma\) are obtained. This determines the occurring errors. Let us suppose that when analyzing two histograms, the values of \(\sigma_{1}\) and \(\sigma_{2}\) were obtained. Using Formula (11), it is easy to show that the relative errors in assessing the quantities \(S^{(1)}\) and \(t_{p}\) are equal. This makes it possible to denote the relative error with the letter \(\delta\) without using indices. Therefore, the following relationship holds:  
\[\delta=\frac{|\sigma_{1}-\sigma_{2}|}{(1-\sigma_{2})} \tag{12}\]  
The values of \(\sigma_{1}\) and \(\sigma_{2}\) depend on the load of the queuing system \(\rho\). The greater the load of the queuing system, the closer the values of the parameters \(\sigma_{1}\) and \(\sigma_{2}\) to one. This means, based on Expression (12), that the error \(\delta\) can noticeably increase at a large load unless the values of \(\sigma_{1}\) and \(\sigma_{2}\) become very close to each other. To analyze the operation of queuing systems under conditions of a dramatically increasing load, it is appropriate to investigate the dependence of \(\delta=f(\rho)\). For this purpose, the value of the service intensity, \(\mu\), should be changed within the limits that allow investigation of the dependence of \(\delta=f(\rho)\) in the high-load range of interest. When solving problems of designing networks and their individual components, the level of acceptable (permissible) load for which the investigation of the dependence of \(\delta=f(\rho)\) is rather irrelevant is selected. An important issue is the solution of the so-called inverse problem. For the _G/M/1_ model, the assessment of the values of \(S^{(1)}\) and \(t_{p}\) calculated using Formulas (11) is the simplest example of a direct problem. The inverse problem is to assess the value of \(\mu\) under the given limitations on the permissible levels of the values of \(S^{(1)}\) and \(t_{p}\).