## 5 Model of Traffic That Overflows from Primary Resources
### New Method for Determining the Parameters of Multi-Service Overflow Systems  
The analysis of the factors that decrease the accuracy in modeling systems with multi-service overflow traffic prompted us to develop a new method for the decomposition of primary resources that would help determine the first two traffic moments of traffic that overflows to the secondary resources. In order to present the most important results of this method, we note that for each class \(c\) that is serviced in the primary resource \(s\), the difference between the average intensity of traffic \(A^{\rm X}_{c,s}\) and the value of carried traffic \(Y^{\rm X}_{c,s}\) is the average value of the intensity of traffic \(R^{\rm X}_{c,s}\) that overflows from this resource:  
\[R^{\rm X}_{c,s}=A^{\rm X}_{c,s}-Y^{\rm X}_{c,s}. \tag{26}\]  
At the same time, the value of \(R^{\rm X}_{c,s}\) can be estimated as the part of offered traffic \(A^{\rm X}_{c,s}\) that is moved to the resource \(s\) when calls of class \(c\) are blocked:  
\[R^{\rm X}_{c,s}=A^{\rm X}_{c,s}E^{\rm X}_{c,s}(A^{\rm X}_{c,s}), \tag{27}\]  
where \(E^{\rm X}_{c,s}\) is the blocking probability for calls of class \(c\) of type X in the primary resource \(s\) (Formula (4)). The value of this probability can be determined on the basis of the method in [60]. At the same time, this method allows the average value \(Y^{\rm X}_{c,s}\) of carried traffic of class \(c\) of type X in the resource \(s\) to be precisely determined.  
Having accurate values of \(E^{\rm X}_{c,s}\) and \(Y^{\rm X}_{c,s}\) and using Formulas (26) and (27), we can determine, for any traffic type X, the value of the equivalent Erlang traffic \(A^{\rm X}_{c,s}\) that leads to the same value of carried traffic of type X:  
\[A^{\rm X}_{c,s}=A^{\rm X}_{c,s}=\frac{Y^{\rm X}_{c,s}}{1-E^{\rm X}_{c,s}(A^{ \rm X}_{c,s})}. \tag{28}\]  
The average value of offered traffic \(A^{\rm X}_{c,s}\) determined by Formula (28), defines the first parameter of the parameter pair (\(A^{\rm X}_{c,s}\), \(V^{\rm X}_{c,s}\)) for the equivalent fictitious primary resource \(s^{\rm X}_{c}\). The other parameter from the pair of parameters is determined by a juxtaposition of the blocking value \(E^{\rm X}_{c,s}\), determined on the basis of Formula (4), and the values that result from the Erlang B formula (Formula (13)). After taking into consideration the description of the parameter pair of the equivalent fictitious primary resource, the dependency that results from Formula (13) can be written in the following form:  
Figure 6: Relative error of the average value of overflow traffic, System 2, Resource 2 (\(\alpha\) values specified directly in the figure).  
\[E_{c,s}^{X}=\frac{\frac{(A_{c,s}^{X})_{c,s}^{V_{cs}^{X}}}{V_{cs,s}^{X}}}{\sum_{n=0 }^{V_{cs}^{X}}\frac{(A_{cs}^{X})^{n}}{n!}}. \tag{29}\]  
Thus, the values of the average value parameters \(A_{c,s}^{X}\) of offered traffic and the capacity \(V_{c,s}^{X}\) allow us to determine, using [PERSON]'s Formulas (22) and (23), the average value of traffic that overflows to the secondary resources \(R_{cs}^{X}\) as well as its variance \((\sigma^{2})_{c_{S}}^{X}\).  
The discussed model was incorporated into a method based on the matching criterion for the blocking probability, and its accuracy was validated by simulations for many configurations of overflow systems.