## 2 Analysis
### Analytical Model
#### 2.2.2 Network Deployment Efficiency Analysis  
In order to estimate the 2-tier HetNet DE according to the described scenarios, we derive a simple cost-oriented power model using the data sourced from incumbent network operators in Bosnia and Herzegovina, which we broke down into capital expenditure (CAPEX), operating expenses (OPEX), and indirect costs [29], as presented in Table 1.  
The linear regression model (of the form \(y=ax^{b}+c\)) for the sum \(C_{\text{tot}}\) of all costs [29] for a single BS with transmission power \(P_{BS}\), in \(k\)-th tier, and within a limited time period:  
\[ \begin{array}{l}C_{\text{tot}}=\sum\limits_{k=1}^{N_{\text{T}}}\sum\limits_{m= 1}^{N_{k}}C_{\text{BS}_{k,m}}=\sum\limits_{k=1}^{N_{\text{T}}}\sum\limits_{m= 1}^{N_{k}}\left(a\cdot P_{\text{BS}_{k,m}}^{b}+c\right)\\ N_{\text{T}}=2,\ N_{k}\epsilon\{N_{1},N_{2}\}\\ N_{1}=\lambda_{i}A,\ N_{2}=\lambda_{j}A\\ \{a,b,c\}=\{8.807\cdot 10^{4},0.305,-4.53\cdot 10^{4}\}\end{array}  \tag{14}\]  
was determined by means of the curve fitting feature of the MATLAB tool, providing the values of the coefficients \(a\), \(b\), and \(c\) in (14). Here \(C_{\text{tot}}\) includes purchasing \(k=1\), \(2\),..., \(N_{\text{T}}\) macro-tier BSs and \(m=1\), \(2\),..., \(N_{k}\) small-tier BSs calculated from density of serving and candidate-serving tier BSs, electrical power costs, rental costs, and operation and maintenance costs. Most often, a ten-year period is adopted [30], whereas for any other, the coefficients \(a\), \(b\), and \(c\) would be different. Note that indirect costs such as, e.g., the license, are not taken into account, as these are country- and time-dependent, but fixed within the observation period, which makes them not so relevant for \(DE\). Furthermore, network \(DE\) is defined as the ratio between the maximal data rate, i.e., network \(SE\) in bandwidth \(B\), and the total cost of the network:  
\[DE=\frac{B\cdot SE_{\text{TOT}}}{C_{\text{tot}}} \tag{15}\]  
where (15) can be developed by taking into account (11), (12), (13), and (14) as:  
\[DE=B\cdot\frac{\frac{\lambda_{j}}{\ln 2}\int_{0}^{\infty}\int_{0}^{\infty} \int_{\tau_{i}}^{\infty}\frac{f_{\text{R}_{i}}(r_{i})\cdot f_{\text{R}_{j}}( r_{j})\cdot P_{j}}{1+\gamma}d\gamma dr_{i}dr_{j}+\frac{\lambda_{j}}{\ln 2}\int_{0}^{\infty}\int_{0}^{ \infty}\int_{\gamma}^{\infty}\frac{f_{\text{R}_{i}}(r_{i})\cdot f_{\text{R}_ {j}}(r_{i})\cdot P_{j}}{1+\gamma}d\gamma dr_{j}dr_{i}}{\sum_{k=1}^{N_{\text{T} }}\sum_{m=1}^{N_{k}}\left(a\cdot P_{\text{BS}_{k,m}}^{b}+c\right)} \tag{16}\]  
The above expression is valid for the scenario of network deployment which envisages two tiers (\(N_{\text{T}}=2\); pico and macro), each consisting of \(N_{k}\) BSs. The relationship between the total deployment cost \(C_{\text{tot}}\) and the ratio \(P_{2}^{\text{tx}}/P_{1}^{\text{tx}}\) of the relative transmit powers is visualized in Figure 2(a), confirming that the benefits of spatial reuse of resources in a dense small-cell tier are significant. Comparing the scenarios of standalone pico and macro tiers emphasizes the strong potential of pico-tier deployments, especially for energy-efficient HetNets. Moreover, according to (15), the trade-off between the total \(DE\) and \(SE\) is presented in Figure 2(b), exhibiting three different cases, broken down by their \(\theta\) values. The graph shows \(SE\)-\(DE\) improvement, while preserving the mean transmitter power within the range of a single BS power. In the region of lower \(SE\), the expansion range bias has some impact on \(SE\)-\(DE\), whereas in the region of high spectral density, it has low or no influence at all.