## 5 Model of Traffic That Overflows from Primary Resources
### Methods for Determining the Parameters of Overflow Traffic
In the traffic change scheme under consideration, the group with capacity \(V=\Delta v_{c,s}^{\rm X}\) is offered equivalent traffic of the Erlang type with intensity \(A=A_{c,ss}^{\rm X}\). From this group, traffic overflows with the average value \(R=A_{c,s}^{\rm X}\) and the variance \(\left[\sigma_{c,\Delta s}^{2}\right]^{\rm X}\). Therefore, in the notation adopted for this article, [PERSON]'s formulas can be written in the following way:  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]3  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]4  
Formulas (19) and (20) allow us to determine the parameter pair (\(A_{c,ss}^{\rm X}\), \(\Delta v_{c,s}^{\rm X}\)) on the basis of known values of the parameter pair (\(A_{c,ss}^{\rm X}\), \(\left[\sigma_{c,\Delta s}^{2}\right]^{\rm X}\)), which depending on the type of traffic under consideration, can be described by Formulas (16)-(18). Following the determination of the value of the parameters (\(A_{c,ss}^{\rm X}\), \(\Delta v_{c,s}^{\rm X}\)), it is possible to estimate the capacity of the equivalent fictitious primary resource \(s_{c}^{\rm X}\), that is, the parameter \(v_{c,ss}^{\rm X}\)[3]:  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]5  
Note that the parameter pair (\(A_{c,ss}^{\rm X}\), \(v_{c,ss}^{\rm X}\)) for each of the equivalent fictitious primary resources \(s_{c}^{\rm X}\) is determined by the Erlang model for a full-availability group [41]. Overflow traffic of class \(c\) that overflows from the primary resource \(s\) is equivalent to traffic that overflows from the equivalent fictitious primary resource \(s_{c}\) and will be characterized by two parameters: the average value of traffic intensity \(R_{c,s}^{\rm X}\) and the variance \([\sigma_{c,s}^{2}]^{\rm X}\). Since the equivalent fictitious primary resource \(s_{c}^{\rm X}\) is determined by a single-service Erlang model, the parameters \(R_{c,s}^{\rm X}\) and \([\sigma_{c,s}^{2}]^{\rm X}\) can be determined on the basis of [PERSON]'s formulas and, in the adopted notation, will take on the following form:  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]6  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]7  
The peakedness coefficients (factors) \(Z_{c,s}^{\rm X}\) of traffic of class \(c\) that overflows from the equivalent fictitious primary resource \(s_{c}^{\rm X}\) are defined as the ratio between the variance and the average value:  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]8  
Accuracy Analysis of Glabowski-Kmiecik-Stasiak 2018 Method for Determining the Parameters of Overflow Traffic  
The method for determining the parameters of overflow traffic in [35], based on the concept in [54], was subjected to in-house validation experiments. As a result of our simulation studies, we established that the main element of the method in [35] that influenced its inaccuracy was the procedure by which Pascal traffic classes were changed into equivalent classes of the Erlang type [62]. The problem is that both the average value and the variance of traffic offered to the primary resources, described by Formula (18), take into consideration the following dependency:  
\[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]9  
We note that in the above dependency, the value of traffic offered by one free source \(\alpha=1\) returns a value that is not allowed. An increase in \(\alpha\), within the range \(\langle 0,1\rangle\), leads to an exponential increase in the above dependency, followed by negative values that tend toward zero, so that this range will be exceeded. Values exceeding the range in question, that is, for \(\alpha>1\), in successive stages of changing the fictitious resources into equivalent fictitious resources (Section 5.1), will cause the evaluation of the average value of overflow traffic to be near 0.