# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Reducing the Number of Samples in the Histogram a(t)  
As a result of measurements of the request flow, a histogram with a very large number of readings can be generated; this number is determined by the value of \(n\). In this case, the clarity of the function \(A(t)\) is sometimes lost. The coefficient of variation, \(C_{A}\), will be used for a number of subsequent assessments. For Function (5), it is calculated as follows:  
\[C_{A}=\sqrt{\frac{\sum_{i=0}^{n}i^{2}p_{i}}{\left(\sum_{i=0}^{n}ip_{i}\right) ^{2}}-1} \tag{9}\]  
To construct the initial \(a(t)\) histogram, the number of values of the measured random variable, which is equal to \(n+1\), is used. An example of such a histogram is shown in the upper part of Figure 7. It is assumed that 25 values of the measured random variable, which are denoted, in the general case, as the set \(\{Z\}\), are obtained. In this case, the readings at the points \(7r\), \(9r\), and \(11r\) are considered rare phenomena that insignificantly affect the nature of the distribution \(A(t)\). In some cases, there is a desire to discard them, that is, to  
Figure 6: Probability of packet loss as a function of load.  
reduce the set \(\{Z\}\). Subsequently, a histogram \(g(t)\) is generated (option (b)). The third solution, as shown at the bottom of the same figure, is to replace the rare \(i\tau\) values with one increment observed \(x\) times and located at the \(m\tau\) point. In the example used, \(m=9\). Then, another histogram \(h(t)\) that is represented by option (c) is generated.  
First, let us consider an option of transforming the initial histogram, for which the results in the region \(\frac{t}{\tau}>5\) are neglected. In this case, to construct the histogram \(g(t)\) and the corresponding distribution function, the set \(\{Z\}\) will include 22 elements.  
The last reading for the histogram \(h(t)\) is denoted as \(x\). In the example under consideration, \(x=3\). It is necessary to choose a value m, for which the value of \(x\) is transferred to the \(m\tau\) point. The criteria for choosing the value of \(m\) are the proximity of the values of A\({}^{(1)}\) and \(C_{A}\) for both histograms. The value of \(m\) is estimated numerically using the following approach:  
I. Characteristics of a random variable \(A^{(1)}\) and \(C_{A}\) for the \(a(t)\), \(g(t)\), and \(h(t)\) histograms are denoted as \(A^{1}(a)\), \(C_{A}(a)\), \(A^{(1)}(g)\), \(C_{A}(g)\), \(A^{(1)}(h)\), and \(C_{A}(h)\), respectively.  
II. The values of \(m_{A}\) and \(m_{c}\) are found by numerical solution of the equations \(A^{(1)}(a)=A^{(1)}(h)\) and \(C_{A}(a)=C_{A}(h)\).  
III. One value of \(m\) that is equal to the maximum value of the pair of \(m_{A}\) and \(m_{c}\) is selected.  
In fact, you may choose a value for \(m\) that is not an integer. However, in the case of a small value of \(\tau\), such an approach will not lead to a noticeable increase in the accuracy of the main investigated characteristics of a random variable. It is possible to use the two  
Figure 7: Results of measurements and transformation of the obtained data.  
values of \(m\), which are denoted below as \(m_{1}\) and \(m_{2}\), respectively. The \(m_{1}\) value is obtained by discarding the significant digits after the period (comma) in the resulting \(m\) value. Then, \(m_{2}=m_{1}+1\).