## 5 Model of Traffic That Overflows from Primary Resources
### Methods for Determining the Parameters of Overflow Traffic
\[\forall_{c\in\text{M}_{s}^{\text{X}}}\left[E_{c}^{\text{X}}\right]_{p_{c,s}^{ \text{X}}}=\left[E_{c}^{\text{X}}\right]_{V_{s}}=\left[P_{p_{c,s}^{\text{X}}} \right]_{p_{c,s}^{\text{X}}}. \tag{12}\]  
In this way, the capacities of the fictitious resources to which single-service Erlang, Engset, and Pascal traffic is offered can be determined on the basis of appropriate formulas that determine the blocking probability in the Erlang, Engset and Pascal model:  
\[\left[E_{i}^{\text{Er}}\right]_{v_{i,s}^{\text{Er}}}=\left[P_{p_{i,s}^{\text{Er }}}\right]_{v_{i,s}^{\text{Er}}}=E_{p_{i,s}^{\text{Er}}}\Big{(}A_{i,s}^{\text{ Er}}\Big{)}=\frac{\frac{\left(A_{i,s}^{\text{Er}}\right)^{v_{i,s}^{\text{Er}}}}{ \left(v_{i,s}^{\text{Er}}\right)!}}{\sum\limits_{n=0}^{v_{i,s}^{\text{Er}}} \frac{\left(A_{i,s}^{\text{Er}}\right)^{n}}{n!}}, \tag{13}\]  
\[\left[E_{j}^{\text{En}}\right]_{v_{j,s}^{\text{En}}}=\left[P_{p_{j,s}^{\text{ En}}}\right]_{v_{j,s}^{\text{En}}}=\frac{\frac{\left(S_{j,s}^{\text{En}} \right)\left(\text{a}_{j,s}^{\text{En}}\right)^{v_{j,s}^{\text{En}}}}{\sum \limits_{n=0}^{v_{j,s}^{\text{En}}}\left(S_{j,s}^{\text{En}}\right)\left( \text{a}_{j,s}^{\text{En}}\right)^{n}}}{\sum\limits_{n=0}^{v_{j,s}^{\text{En}} }\left(S_{j,s}^{\text{En}}\right)\left(\text{a}_{j,s}^{\text{En}}\right)^{n}}, \tag{14}\]  
\[\left[E_{l}^{\text{Pa}}\right]_{v_{l,s}^{\text{Pa}}}=\left[P_{p_{l,s}^{\text{ Pa}}}\right]_{v_{l,s}^{\text{Pa}}}=\frac{\left(S_{l,s}^{\text{Pa}}+v_{l,s}^{\text{ Pa}}-1\right)\left(v_{l,s}^{\text{Pa}}\right)^{v_{l,s}^{\text{Pa}}}}{\sum \limits_{n=0}^{v_{j,s}^{\text{Pa}}}\left(S_{l,s}^{\text{Pa}}+n-1\right)\left( \text{a}_{l,s}^{\text{Pa}}\right)^{n}}. \tag{15}\]  
Formulas (13)-(15) deal with single-service systems, in which admission of a new call denotes the occupancy of a single AU. In keeping with [35], the next step in the decomposition of the primary resource \(s\) was to change the fictitious primary resource \(s_{c}^{\text{X}}\), servicing Engset or Pascal traffic, into an equivalent fictitious primary resource that services equivalent Erlang traffic. Equivalent Erlang traffic \(A_{j,s}^{\text{En}}\) and \(A_{l,s}^{\text{Pa}}\) is such traffic offered to certain fictitious additional resources with capacities \(\Delta v_{j,s}^{\text{En}}\), \(\Delta v_{l,s}^{\text{Pa}}\) that overflows from these resources is equal, with regard to the average value and variance, to Engset traffic \(\left(A_{j,s}^{\text{En}},\,\left[\sigma_{j,\Delta s}^{2}\right]^{\text{En}}\right)\) and Pascal traffic \(\left(A_{l,s}^{\text{Pa}},\,\left[\sigma_{l,\Delta s}^{2}\right]^{\text{Pa}}\right)\). Note that in the notation of the equivalent traffic, for example, in \(A_{j,s}^{\text{En}}\), the symbol \(\text{En}\), which indicates the primary nature of this traffic, has been preserved in the superscript. The asterisk inserted into the subscript indicates that it is already equivalent Erlang traffic. For primary Erlang traffic, we then have \(A_{l,s}^{\text{Er}}=A_{l,s}^{\text{Er}}\), \(\left[\sigma_{l,\Delta s}^{2}\right]^{\text{Er}}=A_{l,s}^{\text{Er}}\) and \(\Delta v_{l,s}^{\text{Er}}=0\). The parameters of the equivalent fictitious primary resource \(s_{c}^{\text{X}}\) can be determined on the basis of the equivalent random technique (ERT) method [35]. Average values and variances of Erlang, Engset and Pascal traffic can be determined on the basis of the following dependencies:  
* Erlang traffic: \[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]0 (16)
* Engset traffic: \[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]1 (17)* Pascal traffic: \[Y_{c,s}=A_{c,s}(1-[E_{c}]_{V_{s}}) \tag{9}\]2 (18)  
In keeping with the ERT method, the average value \(R\) and the variance \(\sigma^{2}\) of traffic that overflows from the primary resource with capacity \(V\) to which traffic \(A\) of the Erlang type is offered can be determined on the basis of [PERSON]'s Formulas (5) and (6).