# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Handling of the Function \(a(t)\)  
The histogram \(a(t)\) built following the measurement results can be represented by its Laplace-Stieltjes transform \(\alpha(s)\):  
\[\alpha(s)=\sum_{i=0}^{n}p_{i}e^{-i\tau s}. \tag{5}\]  
In this formula, \(\tau\) is the period, with which the increments of the measured function equal to \(p_{i}\) are sampled. The value \(n\) defines a point on the abscissa axis for the last \(p_{i}\) value. The following condition always holds: \(n<\infty\). Let us note that most of the approximations of the function \(A(t)\) in the queuing theory are based on distributions defined in \([t_{0},\ \infty)\). Moreover, \(t_{0}=0\) for a significant part of the studied models. Typically, all such approximations lead to significant errors in the study of queuing systems. It is necessary to select the value of \(\tau\) before taking measurements. It is appropriate to define it as the greatest common divisor for the entire set of fixed values of time intervals between packet arrivals [9]. In this case, the CDF \(A(t)\) represents the law of packet arrival more accurately. The top of Figure 4 shows an example of a histogram \(a(t)\). Below are the transformed \(a_{U1}(t)\) and \(a_{U2}(t)\) histograms for which the period \(\tau\) is doubled and quadrupled, respectively. The subscript \"\(U\)\" (the first letter in the word \"upper\") indicates the fact that the ordinate values for the histograms \(a_{U1}(t)\) and \(a_{U2}(t)\) are calculated by summing them (within the time interval under consideration) to the nearest larger value of \(t/\tau\). This method of transforming the \(a(t)\) histogram, which is proposed for example in [10], is hereinafter referred to as option I. In the general case, the histograms describe asymmetric distributions of the form \(A(t)\). Figure 5 shows the histogram \(a(t)\) and two results of its transformation, as determined by option II. The subscript \"\(L\)\" (the first letter in the word \"lower\") indicates the fact that the ordinate values for the histograms \(a_{L1}(t)\) and \(a_{L2}(t)\) are calculated by summing them (within the time interval under consideration) to the nearest smaller value of \(t/\tau\). The two options for the transformation of the function \(a(t)\) make it possible to obtain the lower and upper limits for  
Based on the numerical values given in Table 1, the occurring errors are very significant. This allows two conclusions to be drawn on the reasonable choice of scales when processing measurement results: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multicolumn{5}{c}{**Types of the Studied Functions**} \\ \cline{2-6}  & \(\mathbf{a(t)}\) & \(\mathbf{a_{U1}(t)}\) & \(\mathbf{a_{U2}(t)}\) & \(\mathbf{a_{L1}(t)}\) & \(\mathbf{a_{L2}(t)}\) \\ \hline \(A^{(1)}\) & 3.444 & 4.000 & 2.889 & 4.889 & 2.000 \\ \hline \(C_{A}\) & 0.776 & 0.667 & 0.958 & 0.582 & 1.374 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the measured function. Figure 4: Function \(a(t)\) and its two transformations. Option I.  
Figure 5: Function \(a(t)\) and its two transformations. Option II. option I, in which \"enlarging\" the scale, provides an upper estimate for the mathematical expectation and a lower estimate for the coefficient of variation;
* option II, in which \"enlarging\" the scale, provides a lower estimate for the mathematical expectation and an upper estimate for the coefficient of variation. This means that, when processing the measurement results, it is not advisable to enlarge the scale along the abscissa, that is, it is recommended to use the minimum value of \(\tau\). In this case, the accuracy of the obtained results will be maximum (this aspect of processing the measurement results is described in more detail below).