## 2 Analysis
### Analytical Model  
The PPP models for representing BS deployment are widely accepted for various HetNet scenarios. While hexagonal models do not well represent real networks, the PPP ones are much more appropriate for this purpose [26]. Actually, among the PPP variants, such as the Matern Hard-Core Process (MHCP) [27], the one with non-uniformly distributed BSs is most often used. It is important to note that the PPP distribution of locations determines the SINR values for each user, i.e., that SINR depends on the UE-BS distance. This is sufficient for the analysis, as the other PPP variants consider the minimization of locations' distances, which matters in academic terms, but for real networks, increases the analysis complexity to the extent of becoming a problem in other scenarios that are themselves complex, especially in 5G networks. Thus, the overall HetNet efficiency has been analyzed through joint analysis of SE and DE by assessing SINR at an arbitrary UE, and considering network topology as a stochastic geometry form. Hence, we observe the signal at the receiver of an arbitrary end-user UE located in a _k_-tier heterogenous cellular network consisting of \(N_{T}\) tiers overall. Moreover, it is quite justifiable to assume that the signal-to-noise ratio (_SNR_) is high, and that the (RF) interference from other radio signal sources is a dominant impairment. Each tier is considered to be a homogeneous PPP \(\Phi_{k}\), characterized by triplet (\(P_{k}\), \(\lambda_{k}\), \(\tau_{k}\)) which denotes:  
* total transmit power in each tier,
* density of BSs and
* SINR threshold \(\tau\) (below defined as bias) at UE to restore data, respectively. The tiers are ranked in ascending order by the density of access points, i.e., \(\lambda_{1}\leq\lambda_{2}...\lambda_{k-1}\leq\lambda_{k}\). Given the density \(\lambda_{k}\), the number of access points belonging to tier \(k_{i}\) (\(i\) = 1,2,... \(N_{T}\)) in the network area \(\mathcal{A}\) [m\({}^{2}\)] represents a random Poisson variable, with the mean density \(\mathcal{A}\cdot\lambda_{k}\), which is independent of other tiers. Furthermore, all access points in the _k_-tier have the same transmission power \(P_{k}\). Each downlink is modeled by the Rayleigh fading channel, where \(P_{i}^{tx}\) and \(P_{i}^{rx}\) are the transmitted (from BS) and the received power (at an arbitrary UE\({}_{i}\) at \(R_{i}\) distance from BS), respectively. The model considers the path-loss exponent to be equal to 4 [2]. There are \(n_{ABS}\) Almost Blank Subframes (ABS) in which macro BSs do not transmit any power [3]. For each tier, the factor of frequency reuse equals 1, and the RF band is allocated in a way that one real channel is skipped between two tiers of the same wireless technology standard. Thus, for a typical UE connected to tier \(k\), the set of interfering BSs include all BSs in tier \(k\) except the serving one. We assumed the scenario where the user is only allowed to access the BSs in tiers 1,2,...,\(K_{open}\), which belong to macro/pico cells with Open Access (OA), whereas the Closed Subscriber Group (CSG) femtocells are not allowed to serve the users under consideration, which is the most common case [8]. For example, HetNet would be represented by the count of tiers: \(N_{T}=3\) and the count of tiers with OA: \(N_{open}=2\), with tier 1 representing the macro cells, tier 2 the OA pico cells, and tier 3 the CSG femtocells. Furthermore, we assume that each BS in the observed tier transmits with maximal power allowed for BSs in that tier (We do not analyze here the scenario where the user moves quickly through service zones of two and more BSs, so initiating handoffs [28]). To demonstrate the relationship between \(DE\) and \(SE\), it is necessary to firstly analyze the relationship between the network \(SE\) and the total power in such a way that the base stations are distributed within the tiers in the form of PPP. The BS \(b_{k}\) of any serving tier \(k_{i}\) transmits only to a specific subset of the users \(U_{b}\) served by \(b_{k}\in\Phi_{k}\).