# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Two Additional Tasks  
The results obtained make it possible to solve two additional tasks of great practical significance. The first task is related to the choice of the approximating function \(f(t)\), which is a composition of two or more known distribution laws of random variables. The meaning of the second task is to sample the histogram _a(t)_, which is a continuous function using Expression (5). Then the analysis of a number of models can be simplified.  
The measurement results show that, very often, the histograms \(a(t)\) have several extrema. For this reason, the measurement results are difficult to represent using one of the known distribution laws of random variables. It should be noted that such an approximation is useful for clarity but ineffective for the subsequent analysis of models due to the possibility of the occurrence of significant errors.  
Analysis of a large number of histograms showed that the composition of two distributions given at different intervals along the abscissa axis is used more often. Figure 8 shows the original histogram \(a(t)\) with two extrema, as well as two functions, \(\varphi_{1}(t)\) and \(\varphi_{2}(t)\). These functions approximate the histogram \(a(t)\). Figure 8 gives examples of the Erlang distribution of the order \(k\) and the Simpson distribution [22], respectively.  
The approximating \(f(t)\) function can be represented as follows [22]:  
\[f(t)=\left\{\begin{array}{c}\frac{\lambda^{k}}{(k-1)!}t^{k-1}e^{-\lambda t}, \ if\ 0\leq t<\infty\\ \frac{2}{t_{2}-t_{1}}\Big{[}1-\frac{[t_{1}+t_{2}-2t]}{t_{2}-t_{1}}\Big{]},\ if\ t_{1}\leq t\leq t_{2}\end{array}\right.. \tag{15}\]  
The illustration above and Relationships (15) allow a better understanding of the nature of the CDF \(A(t)\). In some cases, the physical nature of the process of request arrival in the queuing system becomes clear.  
For some types of the functions \(A(t)\), it is impossible to obtain relationships for analyzing the parameters of the request delay time. Then, the \(A(t)\) can be replaced by a stepwise function. The results of solving this task are given in [11]. They allow the sampling period, \(d\), to be determined, with which readings of the values of \(a(t)\) are required to be taken.  
Let us assume that the admissible relative error, \(\delta\), is given, which determines the accuracy of the estimation of the delay time parameters. If the coefficient of variation, \(C_{A}\) does not exceed one, then the numerical value of \(d\) is determined from the following inequality:  
\[d\leq\frac{2\delta}{\lambda} \tag{16}\]  
In some cases, the coefficient of variation, \(C_{A}\), significantly exceeds one. Therefore, it is required to apply another formula:  
\[d\leq\frac{2\delta}{\lambda C_{A}} \tag{17}\]  
Expression (17) allows more accurate values of the sampling period, \(d\), to be obtained for the functions \(A(t)\) with a very high coefficient of variation. These functions often reflect the actual processes of packet arrival to multiservice nodes. Moreover, high \(C_{A}\) values are typical for multiservice node congestion modes. Such phenomena are often observed in emergency situations that generate a dramatic increase in traffic [17].  
Figure 8: Example of approximation of the histogram \(a(t)\) by two functions.