## 4 Optimal Number of ABSs and Default Parameters and Assumptions  
### Optimal Number of ABSs Estimation  
The UE traffic activity can be modeled as an exponentially distributed continuous time Poisson process since sessions or call arrivals can be modeled as a Poisson process [24; 25; 26]. Let \(\lambda_{1}\) and \(\lambda_{2}\) denote respectively the average rate of arrivals of user group 1 and user group 2 to a system and are served during \(T_{\text{ABS}}\) and \(T_{\text{non-ABS}}\) of an APP \(T_{\text{APP}}\), respectively. Considering a fair allocation of time resources to each group of users, the allocation of \(T_{\text{ABS}}\) and \(T_{\text{non-ABS}}\) is defined in proportionate with \(\lambda_{1}\) and \(\lambda_{2}\) respectively. In such cases, an optimal value of \(T_{\text{non-ABS}}\) can be found by solving the following optimization problem:  
\[\begin{array}{ll}\min&T_{\text{non-ABS}}\\ \text{subject to:}&\text{(a)}\ \lambda_{2}/\lambda_{1}=T_{\text{non-ABS}}/T_{ \text{ABS}}\\ &\text{(b)}\ T_{\text{APP}}=T_{\text{ABS}}+T_{\text{non-ABS}}\end{array} \tag{55}\]  
The optimal solution of (55) in favor of \(\lambda_{2}\) is given by:  
\[T_{\text{non-ABS}}^{*}=[\lambda_{2}/(\lambda_{1}+\lambda_{2})]\times T_{\text {APP}} \tag{56}\]  
**Proof 1.** Using constraints (55) (a) and (b):  
\[\lambda_{1}/\lambda_{2}=T_{\text{ABS}}/T_{\text{non-ABS}}\]  
Solving the above equation, we get the following:  
\[T_{\text{ABS}}=[1/(\lambda_{2}/\lambda_{1}+1)]\times T_{\text{APP}}\]Since \(T_{\text{ABS}}\) and \(T_{\text{non-ABS}}\) are strictly integers, by allowing a favor to \(\lambda_{2}\), the optimal value of \(T_{\text{non-ABS}}\) is given by:  
\[T_{\text{ABS}}{}^{*}=\lfloor\lambda_{1}/(\lambda_{1}+\lambda_{2})\rfloor\times T _{\text{APP}}\]  
\[T_{\text{non-ABS}}=T_{\text{APP}}-\lfloor\lambda_{1}/(\lambda_{1}+\lambda_{2}) \rfloor\times T_{\text{APP}}\]  
\[T_{\text{non-ABS}}{}^{*}=\lfloor\lambda_{2}/(\lambda_{1}+\lambda_{2})\rfloor \times T_{\text{APP}}\]  
\(\Box\)  
Let \(\lambda_{\text{MNO},1,\text{SU}}\), \(\lambda_{\text{MNO},2}\), \(\lambda_{\text{SPS}}\), and \(\lambda_{\text{MNO},1,\text{ IMU}}\) denote respectively the average rate of arrival of small cell UEs of MNO 1, UEs of MNO 2, satellite UEs of SPS, and indoor macro UEs of MNO 1 into a building. Now, applying the above analogy for the solution \(T_{\text{non-ABS}}{}^{*}\), an optimal value of \(T_{\text{non-ABS}}\) for dedicated, DySP, and LSA techniques can be found respectively as follows.  
For the dedicated access technique:  
\[T_{\text{non-ABS},\text{DedA}}{}^{*}=\lfloor\lambda_{\text{MNO},1,\text{SU}}/ (\lambda_{\text{MNO},1,\text{IMU}}+\lambda_{\text{MNO},1,\text{SU}})\rfloor \times T_{\text{APP},\text{DedA}} \tag{57}\]  
For the DySP technique:  
\[T_{\text{non-ABS},\text{DySP}}{}^{*}=\lfloor\lambda_{\text{MNO},1,\text{SU}}/ (\lambda_{\text{MNO},2}+\lambda_{\text{MNO},1,\text{SU}})\rfloor\times T_{ \text{APP},\text{DySP}} \tag{58}\]  
For the LSA technique:  
\[T_{\text{non-ABS},\text{LSA}}{}^{*}=\lfloor\lambda_{\text{MNO},1,\text{SU}}/ (\lambda_{\text{SPS}}+\lambda_{\text{MNO},1,\text{SU}})\rfloor\times T_{ \text{APP},\text{LSA}} \tag{59}\]  
where \(T_{\text{APP},\text{DedA}}\), \(T_{\text{APP},\text{DySP}}\), and \(T_{\text{APP},\text{LSA}}\) denote the value of \(T_{\text{APP}}\) for the dedicated, DySP and LSA techniques, respectively.