# Study of the Process of Packet Arrival at a Multiservice Node
## 2 Mathematical Model of a Multiservice Node
### Choice of Function \(A(t)\) Based on a Reasonable Hypothesis  
Let us suppose that approximate assessments of \(A^{(1)}\) and \(C_{A}\) are known. However, there are no measurement results that allow the choice of the function \(A(t)\). It is appropriate to distinguish two classes of distributions \(A(t)\). The first one is set for a limited time interval, \([t_{0},\,t_{1}]\). In some cases, \(t_{0}=0\). Further, similar CDFs are denoted as \(A_{1}(t)\). The subscript \"\(l\)\" is the first letter in the word \"limited\". The subscript \"\(u\)\" (from the word \"unlimited\") is used to denote the \(A_{U}(t)\) as determined on the interval \([t_{0},\,\infty)\). Estimates of \(A^{(1)}\) and \(C_{A}\) allow the choice of two-parameter distributions: \(A_{1}(t)\) and \(A_{U}(t)\). A suitable example of a function from the \(A_{1}(t)\) family is the beta distribution [13; 14]. Of practical value is a case in which the coefficient of variation \(C_{A}\) exceeds one. For this condition, it is appropriate to choose the Weibull distribution from the functions of the \(A_{U}(t)\) family [14]. However, other types of the \(A_{U}(t)\) function lead to results close to those of the Weibull distribution. A particular interest in ensuring the normalized quality of service indicators for multiservice traffic is associated with the operation of queuing systems under conditions of increased load, \(\rho\). Taking into account the accepted designations, the value of \(\rho\) is determined by the ratio of \(\lambda\) to the service intensity \(\mu\):  
\[\rho=\frac{\lambda}{\mu} \tag{6}\]  
Approximate formulas for assessing the average request delay time \(S^{(1)}\) in a queuing system at a high load are given for example in [15; 16] for a model with arbitrary CDFs \(A(t)\) and \(B(t)\). For \(A_{U}(t)\) and \(A_{l}(t)\), the results, which are given in [15; 16], can be represented in the following form [17], under the condition of a constant request holding time:  
\[ \begin{split} S^{(1)}&\approx B\bigg{[}1+\frac{\rho ^{2}C_{A}^{2}}{2(1-\rho)}\bigg{]},\,\,\,\text{for}\,\,A_{U}(t),\,\,if\,\,1> \rho\geq 0.7;\\ S^{(1)}&\approx B\bigg{[}1+\frac{\rho C_{A}^{2}}{2(1 -\rho)}\bigg{]},\,\,\,\text{for}\,\,A_{l}(t),\,\,if\,\,1>\rho\geq 0.7.\end{split}  \tag{7}\]  
The choice of a model with constant holding time is justified by the fact that all packets in the switching nodes of the multiservice network are processed in the same way. This allows the introduction of a reasonable assumption about the constancy of the request processing time [18]. Clearly, the average value of the packet delay time for distributions of the form \(A_{l}(t)\) will be greater in comparison with that of the model, which is characterized by the function \(A_{U}(t)\). A significant difference between the \(A_{l}(t)\) and \(A_{U}(t)\) is also observed for the probability of packet loss, \(\pi\). Figure 6 shows the dependences of the probability of packet loss on the load for two distributions of the form \(A_{l}(t)\) and \(A_{U}(t)\). The coefficient of variation, \(C_{A}\), for both types of distribution is chosen to be equal to ten. More significant packet losses for the functions \(A_{l}(t)\) are associated with the fact that the duration of the idle period in the corresponding queuing model is limited to \(t_{1}\). There is no such limitation for the functions \(A_{U}(t)\). To assess the upper limit \(\pi\), which corresponds to the model with the function \(A_{l}(t)\), an approximate formula was proposed in [17] if the packet processing time is constant and the capacity of the queue is limited to \(r\):  
\[\pi\approx(1-\rho)\rho^{\frac{2r}{C_{A}}}. \tag{8}\]  
Formulas (7) and (8) make it possible to assess the characteristics of the quality of service with an error of no more than 20%. This statement applies to a wide class of distributions included in the \(A_{U}(t)\) family. It is also true for the beta distribution in the \(A_{l}(t)\) family.