## 2 Analysis
### Analytical Model
#### 2.2.1 Network Spectral Efficiency  
The proposed analytical model considers the spectral efficiency \(SE_{k}\) for each tier (\(k=1\dots\ N_{T}\)), as well as \(SE_{TOT}\) for the whole HetNet. The technique for selecting serving or candidate-serving cells in the LTE-A standard is, among other factors, based on the cell range extension (CRE), which has been proposed for pico-cell BSs to enable traffic load balancing and to prevent inter-cell RF interference in the areas of overlapping coverage. In this regard, the important parameter is the range expansion bias (REB), taking the values of \(\tau\) within the range \(\theta\). It is applied to the mean received pilot powers at the UE, coming from the candidate-serving macro and pico BSs to make the optimal selection of the small-cell tier to serve the UE according to two scenarios described in the following:  
In the first case, macro tier \(i\) is the serving tier, while pico tier \(j\) is the candidate-serving tier, whereas in the second case, the UE is served by pico tier \(j\), while having macro tier \(i\) as the candidate-serving one [8]. The distance from the UE to the candidate-serving (i.e., the nearest) macro BS is \(\mathcal{R}_{i}\), whereas the distance between the UE and the candidate-serving (the nearest) pico BS is \(\mathcal{R}_{j}\). In order to derive a simplified model of HetNet SE, we consider the instantaneous transmit power of any macro BS to be a random variable equal to either zero during the ABS state, or to \(P_{1}^{\text{tx}}\), otherwise, while the instantaneous transmit power of the serving BS is denoted as \(P_{2}^{\text{tx}}\). In this way, the selection bias for tier \(k\) is:  
\[\tau=\sqrt[4]{0.063\cdot\theta\cdot\frac{P_{2}^{\text{tx}}}{P_{1}^{\text{tx}}}} \tag{5}\]In the first case, we consider an arbitrarily located UE served by micro tier \(i\), and the probability of its SINR, herewith defined as \(\Gamma_{i}\), to be greater than the threshold--the SINR expected value \(\gamma\). According to (1) and (2), the spectral efficiencies \(SE_{i}\) and \(SE_{j}\) of these two scenarios, respectively, can be expressed in the following form [8]:  
\[ \begin{array}{c}\mathcal{P}_{i}=\mathcal{P}\left\{\Gamma_{i}>\gamma\parallel \mathcal{R}_{i}=r_{i},\mathcal{R}_{j}=r_{j}\right\}=\\ =(1-n_{ABS})\cdot\mathcal{P}\left\{\Gamma_{i}>\gamma\parallel\mathcal{R}_{i}= r_{i},\mathcal{R}_{j}=r_{j}\right\}=\\ =0.75\cdot e^{-\pi\cdot\lambda_{i}r_{1}^{2}\cdot\sqrt{\gamma}[\cot^{-1}(\frac{1 }{\sqrt{\gamma}})+\beta\cot^{-1}(\frac{\pi\cdot\lambda_{j}r_{1}^{2}}{\beta\cdot \pi\cdot\lambda_{i}r_{1}^{2}\cdot\sqrt{\gamma}})]}\end{array}  \tag{6}\]  
where \(r_{i}\) and \(r_{j}\) are instantaneous distances from UE to the serving macro BS and serving pico BS, respectively, whereas \(\beta\) is:  
\[\beta=\frac{\lambda_{2}}{\lambda_{1}}\sqrt[4]{0.063\cdot\theta\cdot\frac{P_{2 }^{\mathrm{tx}}}{P_{1}^{\mathrm{tx}}}} \tag{7}\]  
For the second case, the probability that UE is served by the pico tier, having the satisfactory level of SINR, denoted as \(\Gamma_{j}\), is as follows:  
\[ \begin{array}{c}\mathcal{P}_{j}=\mathcal{P}\left\{\Gamma_{j}>\gamma\parallel \mathcal{R}_{i}=r_{i},\mathcal{R}_{j}=r_{j}\right\}=\\ =(1-n_{ABS})\cdot\mathcal{P}\left\{\Gamma_{j}>\gamma\parallel\mathcal{R}_{i}= r_{i},\mathcal{R}_{j}=r_{j}\right\}+\\ +n_{ABS}\cdot\mathcal{P}\left\{\Gamma_{j}^{ABS}>\gamma\parallel\mathcal{R}_{i }=r_{i},\mathcal{R}_{j}=r_{j}\right\}\\ =0.75\cdot e^{-\pi\cdot\lambda_{i}r_{i}^{2}\cdot\sqrt{\gamma}[\cot^{-1}(\frac {1}{\sqrt{\gamma}})+\beta\cot^{-1}(\frac{\pi\cdot\lambda_{j}r_{i}^{2}}{\beta \cdot\pi\cdot\lambda_{i}r_{1}^{2}\cdot\sqrt{\gamma}})]}+0.25\cdot e^{-\pi\cdot \lambda_{j}r_{i}^{2}\cdot\sqrt{\gamma}\cot^{-1}(\frac{1}{\sqrt{\gamma}})}\end{array}