## 3 Problem Formulation
### Dedicated Access  
For a single building, i.e., \(L=1\), the aggregate capacity of all macro UEs of MNO 1 can be expressed as:  
\[\sigma_{\text{MNO,1}}^{\text{MC,WIM}}=\sum\
olimits_{t=1}^{Q}\sum\
olimits_{ i=1}^{M_{\text{MNO,1}}}\sigma_{t,i}(\rho_{t,i}) \tag{4}\]  
where \(\sigma\) and \(\rho\) are the responses over \(M_{\text{MNO,1}}\) RBs in \(t\in\mathbf{T}\) such that indoor macro UEs are scheduled only during \(t_{\text{ABS}}\in\mathbf{T_{\text{ABS}}^{\text{MNO,1}}}\) and all other macro UEs are scheduled during \(t_{\text{non-ABS}}\in\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\).  
Since in dedicated access, SBSs per building operate at the same spectrum of MNO,1 in non-ABSs \(t_{\text{non-ABS}}\in\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\), the capacity served by an SBS is then given by:  
\[\sigma_{s,\text{DedA}}=\sum\
olimits_{t=t_{\text{non-ABS}}\in\mathbf{T}}\mathbf{T_{ \text{ABS}}^{\text{MNO,1}}}\sum\
olimits_{i=1}^{M_{\text{MNO,1}}}\sigma_{t, i}(\rho_{t,i}) \tag{5}\]  
Then for all SBSs per building, the aggregate capacity can be expressed as:  
\[\sigma_{\text{MNO,1}}^{\text{SC,DedA}}=\sum\
olimits_{s=1}^{S_{\text{F}}} \sigma_{s,\text{DedA}} \tag{6}\]  
Hence, using Equation (4), the system level capacity of MNO 1 for the dedicated access technique is given by:  
\[\sigma_{\text{MNO,1}}^{\text{SYS,DedA}}=\sigma_{\text{MNO,1}}^{\text{MC,WIM}}+ \sigma_{\text{MNO,1}}^{\text{SC,DedA}} \tag{7}\]  
Using (7), the average spectral efficiency of MNO 1 for \(L=1\) is given by:  
\[\sigma_{\text{MNO,1}}^{\text{SE,DedA}}=\sigma_{\text{MNO,1}}^{\text{SYS,DedA} }/(M_{\text{MNO,1}}\times Q) \tag{8}\]  
Likewise, the average energy efficiency of MNO 1 in Joules/bit (J/b) for \(L=1\) is given by:  
\[\sigma_{\text{MNO,1}}^{\text{EE,DedA}}=\left(\begin{array}{ll}\left(S_{\text {F}}\times\left(\left|\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\right|/\left| \mathbf{T}\right|\right)\times P_{\text{SC,1}}\right)+\\ +\left(S_{\text{P}}\times P_{\text{PC}}\right)+\left(S_{\text{M}}\times P_{ \text{MC}}\right)\end{array}\right)/\left(\sigma_{\text{MNO,1}}^{\text{SYS,DedA }}/Q\right) \tag{9}\]  
where \(\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\) represents a set of non-ABSs when SBSs are scheduled to serve small cell UEs.  
For \(L>1\), we assume that the indoor propagation characteristics and the distance of UEs from their respective SBSs in each of the \(L\) buildings do not deviate significantly from one another [26] such that by linear approximation the average aggregate capacity for the dedicated access technique is roughly given by for \(L>1\):  
\[\sigma_{\text{MNO,1}L}^{\text{SYS,DedA}}=\sigma_{\text{MNO,1}}^{\text{MC,WIM}} +\left(L\times\sigma_{\text{MNO,1}}^{\text{SC,DedA}}\right) \tag{10}\]Using (10), for \(L\) buildings, the spectral efficiency is given by:  
\[\sigma_{\text{MNO,1},L}^{\text{SE,DedA}}=\sigma_{\text{MNO,1},L}^{\text{SYS,DedA}} \left/\left(M_{\text{MNO,1}}\times Q\right)\right. \tag{11}\]  
Similarly, the energy efficiency is given by:  
\[\sigma_{\text{MNO,1},L}^{\text{EE,DedA}}=\left(\begin{array}{c}\left(L\times S _{\text{F}}\times\left(\left|\mathbf{T_{non-ABS}^{MNO,1}}\right|/\left|\mathbf{ T}\right|\right)\times P_{\text{SC,1}}\right)+\\ +\left(S_{\text{P}}\times P_{\text{PC}}\right)+\left(S_{\text{M}}\times P_{ \text{MC}}\right)\end{array}\right)/\left(\sigma_{\text{MNO,1},L}^{\text{SYS,DedA}} \left/Q\right) \tag{12}\]