# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Three Numerical Examples  
The first numerical example is related to the estimation of the error when discarding the last three readings. For this purpose, one should use formulas that allow the calculation of the values of errors obtained the investigated characteristics of a random variable. For the value \(A^{(1)}\), the relative error was about 26%, and for the \(C_{A}\) values it was about 25%. Typically, these error values are not considered acceptable. For this reason, the use of histograms of the form \(g(t)\) does not seem reasonable. The second numerical example is related to the choice of the \(m\) value. For this purpose, it is first required to solve two equations: \(A^{(1)}(a)=A^{(1)}(h)\) and \(C_{A}(a)=C_{A}(h)\), clause II of the previous section of the paper. This will allow the choice of the \(m\) value. For the histogram \(h(t)\), it is appropriate to choose the value of \(m=9\). Then, the values \(A^{(1)}\) and \(C_{A}\) for the histograms \(a(t)\) and \(h(t)\) will compose the following pairs: 3.08 and 3.08, 0.86 and 0.84. This means that the mean values coincide, and the relative error in calculating the coefficient of variation is about 2.3%. It is possible to choose another value of \(m\), for which the relative error in calculating the coefficient of variation will be zero. Then there will be differences in the mean values. If the error in estimating the coefficient of variation at the level of 2.3% seems to be acceptable, then the choice of the value of \(m=9\) should be considered justified. A more general methodological approach is based on the use of both \(m_{A}\) and \(m_{c}\) values, that is, the operation specified in clause III is not required. The \(m_{A}\) and \(m_{c}\) values allow the original histogram to be converted into two new functions. It should be noted that, in order to obtain guaranteed upper values of the studied characteristics, it is better not to neglect the operation provided for in clause III. In other words, it is advisable to use only one value of \(m\) for further analysis, which is the larger among the pair of \(m_{A}\) and \(m_{c}\). The third numerical example is related to the influence of the location of the last reading on the abscissa axis. The size of this reading is also significant. This is the last increment on the histogram \(h(t)\). Let us suppose that the last reading is shifted to the right along the abscissa axis at different distances. The last increment for the histogram \(h(t)\) is 0.04. Table 2 shows the dynamics of growth of characteristics \(A^{(1)}\) and \(C_{A}\) during the shift of the last reading of the measured random variable. The location of the last reading is indicated by the letter \"\(m\)\". For the second row in Table 2, the condition \(m=n\) is true. The data given in the table indicate that even small values of \(p_{m}\) that are located at a noticeable distance (along the abscissa axis) from the main part of the observed values will significantly affect the characteristics of the studied random variable. For this reason, ignoring the tail of the distribution function is fraught with dramatic distortions of the measurement results. The value of the last \(p_{n}\) increment in this example may seem significant. For this reason, it makes sense to repeat the calculations by artificially increasing the number of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Value \(m\)** & **Value \(A^{(1)}\)** & **Value \(C_{A}\)** \\ \hline
11 & 3.08 & 0.86 \\ \hline
21 & 3.48 & 1.19 \\ \hline
31 & 3.88 & 1.52 \\ \hline
101 & 6.68 & 2.90 \\ \hline
501 & 22.68 & 4.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dependence of the values of \(A^{(1)}\) and \(C_{A}\) on the value of \(m\). observations for the process under investigation to the \(n\tau\) point. If all the initial ordinate values (except for the last one) are increased ten times in the \(h(t)\) histogram, then the values of \(A^{(1)}\) and \(C_{A}\) will become different.