## 3 Problem Formulation  
### Preliminaries  
Denote \(N\) as the total number of macro UEs of which a certain percentage of macro UEs denoted as \(\mu_{\text{MI}}\) is considered inside a number of buildings over the coverage of a macrocell of MNO 1. The maximum number of buildings and the number of small cells, i.e., femtocells, per building are denoted respectively by \(L\) and \(S_{\text{F}}\). We assume that \(S_{\text{F}}\) is the same for all buildings and each small cell serves exactly one UE in any TTI. Though in general, the number of small cell UEs in one building is independent of the other, for simplicity, we assume that in each of the \(L\) buildings, the same number of small cells is deployed.  
Denote \(\mathbf{T}\) as the simulation run time such that \(\mathbf{T}=\{1,2,3,\ldots,Q\}\) where \(Q\) represents the maximum time. Let \(\mathbf{T_{\text{ABS}}}\) denote the number of ABSs in every APP where an APP consists of eight subframes such that \(\mathbf{T_{\text{ABS}}}=\{i\colon\,t=8v+z;\,v=0,\,1,\,2,\,\ldots,\,Q/8;\,z=1,\, \ldots,\,T_{\text{ABS}}\}\). Note that \(T_{\text{ABS}}=1,\,2,\,\ldots,\,8\) that corresponds to ABS patterns \(\varphi=1/8,\,2/8,\,\ldots,\,8/8\), respectively. Let \(t_{\text{ABS}}\) and \(t_{\text{non-ABS}}\) denote respectively an ABS and a non-ABS such that \(t_{\text{non-ABS}}\in\mathbf{T_{\text{ABS}}}\) and \(t_{\text{non-ABS}}\in\mathbf{T}\backslash\mathbf{T_{\text{ABS}}^{\text{SPS}}}\).  
Let \(M_{\text{MNO},1}\) denote the number of RBs in the spectrum bandwidth of MNO 1 where an RB is equal to 180 kHz. Let \(S_{\text{P}}\) and \(S_{\text{M}}\) denote respectively the number of picocell BSs (PBSs) per MBS and the number of MBSs in the system. Recall that there are \(S_{\text{F}}\) SBSs per 3D building such that \(s\in\{1,2,\,\ldots,S_{\text{F}}\}\). Let \(P_{\text{SC},1}\) and \(P_{\text{SC},1}\) denote respectively the transmitting power of transceiver 1 and transceiver 2 of an SBS.  
The downlink received signal-to-interference-plus-noise ratio for a UE at resource block (RB) = \(i\) in TTI = \(t\) can be expressed as:  
\[\rho_{t,i}=\left(P_{t,i}/\left(N_{t,i}^{s}+I_{t,i}\right)\right)\times H_{t,i} \tag{1}\]  
where \(P_{t,i}\) is the transmit power, \(N_{t,i}^{s}\) is the noise power, \(I_{t,i}\) is the total interference signal power, and \(H_{t,i}\) is the link loss for a link between a UE and a BS at RB = \(i\) in TTI = \(t\).  
\(H_{t,i}\) can be expressed in dB as:  
\[H_{t,i}(\text{dB})=(G_{\text{t}}+G_{\text{r}})-(L_{\text{F}}+PL_{t,i})+(LS_{t,i}+ SS_{t,i}) \tag{2}\]  
where (\(G_{\text{t}}+G_{\text{r}}\)) and \(L_{\text{F}}\) are respectively the total antenna gain and connector loss. \(LS_{t,i}\), \(SS_{t,i}\), and \(PL_{t,i}\), respectively, denote the large scale shadowing effect, small scale Rayleigh or Rician fading, and distance-dependent path loss between a BS and a UE at RB = \(i\) in TTI = \(t\)[23].  
Let \(\beta\) denote the implementation loss factor. Using [PERSON]'s capacity formula, a link throughput at RB = \(i\) in TTI = \(t\) in bps per Hz is given by [24; 25]:  
\[\sigma_{t,i}(\rho_{t,i})=\left\{\begin{array}{ll}0,&\rho_{t,i}<-10\text{ dB}\\ \beta\log_{2}\!\left(1+10^{(\rho_{t,i}(\text{dB})/10)}\right),&-10\text{ dB}\leq\rho_{t,i}\leq 22\text{ dB}\\ 4.4,&\rho_{t,i}>22\text{ dB}\end{array}\right\} \tag{3}\]