Decoupled Downlink-Uplink Coverage Analysis with Interference Management for Enriched Heterogeneous Cellular Networks

Heterogeneous cellular networks (HetCNets) offer a promising solution to cope with the current cellular coverage crunch. Due to the large transmit power disparity, while following maximum power received (MPR) association scheme, a larger number of users are associated with macro-cell BS (MBS) than small-cell BSs (SBSs). Therefore, an imbalance load arrangement takes place across the HetCNets. Hence, using cell range expansion-based cell association, we can balance the load across the congested MBS. However, using MPR association scheme, users' offloading leads to two challenges: 1) macro-cell interference, in which the MBS interferes with the offloaded users, and 2) coupled downlink-uplink cell association, in which a random user associates with a single tier's base station (BS) both in uplink (UL) and downlink (DL) directions. This paper aims to address these problems while considering a two-tier scenario consisting of small-cell and macro-cell tiers. For the MBS interference mitigation, we employ a reverse frequency allocation (RFA) scheme. Besides coupled DL-UL association (Co-DUA), this paper also highlights the notion of decoupled DL-UL association (De-DUA). In De-DUA, a random user associates with two different tiers' BSs, i.e., with one tier's BS in the DL direction and with the other tier's BS in the UL direction. Our results illustrate that, in comparison with the Co-DUA, De-DUA with RFA employment achieves a better coverage performance.

SECTION I. Introduction

To meet the current wireless data deluge, cellular networks enter into a new paradigm where, the low power small-cells, i.e., pico-cells and femto-cells, deployed within the existing infrastructure of high power macro-cells result in heterogeneous cellular networks (HetCNets) [1]. Correspondingly, different types of BSs differ from each other in terms of their transmit powers, BS densities, user densities associated to each tier’s BS, and cell selection offsets. Due to the higher transmit power, coverage regions of macro-cells are larger as compared with the low power small-cells. Consequently, fewer number of users associate with small-cells. Cell range expansion (CRE) [2], [3] is one of the critical load balancing methods, wherein a cell selection offset (also regarded as the association bias) is added to small-cells’ reference signals to increase their coverage range in order to push a fraction of users from macro-cells towards small-cells. Load balancing brings dual benefits to the HetCNets, i.e., it reduces macro-cells congestion and offloads more users to small-cells to utilize their resources efficiently. However, besides these benefits, two critical challenges are faced by HetCNets, i.e., the macro-cells interference towards the offloaded users and downlink cell association.

Furthermore, using a larger association bias causes high macro-cell interference (MCI) in downlink (DL) association. This is one of the key issues that limits the performance gain of HetCNets. Several approaches have been proposed in the literature to mitigate the MCI. In [4], the transmission of macro-cell BS (MBS) is muted in some of the available time slots, also termed as “almost blank sub frames (ABSs)”. ABSs are completely assigned to the offloaded users to diminish the MCI. However, this scheme results in decrease of macro-cells resources, since these ABSs are used by small cells. Frequency-based resource partitioning is presented in [5], wherein the MBS is muted on a fraction of available sub-bands, which are allocated exclusively to offloaded users. In [6], the authors investigate signal to interference plus noise ratio (SINR) based analysis for resource partitioning. However, one of the key parameters, i.e., offloading, is not considered in their work. Leveraging stochastic geometry framework, the authors in [7] extend the work presented in [6], and observe the affect of joint load balancing and time-based resource partitioning on the performance metrics. More importantly, however, the previous studies [4]–​[7] consider DL user association, in which a random user is associated both in UL and DL directions with a single tier BS [8], [9]. Furthermore, [10]–​[12] specify that DL user association is not necessarily optimal for the UL association. In these models, the users associate with one tier BS in DL direction while connect to another tier BS in UL direction. This is referred to as a decoupled access. In [13], the authors consider DL and UL user association distinctly. The authors in [14] analyze coverage and rate performances while considering different UL and DL association strategies. The studies in [15] and [16] validate their theoretical results by simulations showing that decoupled access accomplishes better performance gains in term of capacity. However, our proposed model aims to study the impact of decoupled downlink-uplink association (De-DUA) with load balancing, while considering reverse frequency allocation (RFA)¹ [17] to mitigate MCI and accomplish better network performance gain.

In this paper, we consider a two-tier HetCNets model, where both tiers’ BSs are distributed via two independent homogeneous Poisson point processes (PPPs) [18], [19]. For the analysis of our proposed model, we use tractable stochastic geometry framework [20]–​[23], which is one of the best analytical tools used thus far in the state-of-the-art. It is interesting to observe that the PPP and hexagonal models provide results with the same accuracy [24], and the stochastic geometry [25] tool captures the randomness of the SBSs and users. Hence, for our proposed network analysis, we use stochastic geometry.

The main contributions of this paper can be summarized as follows:

1. We consider a two-tier HetCNets model with resource partitioning, i.e., RFA, with two access modes: coupled downlink-uplink association (Co-DUA) and De-DUA. We probabilistically characterize the coverage performances of the proposed access modes and observe the effect of user and SBSs densities, SINR threshold and different values of association biases on the coverage probability.

2. Our results show that De-DUA with the RFA scheme surpasses coverage performance of other methods, such as Co-DUA with and without RFA scheme, and the conventional load balancing.

The rest of the paper is organized as follows. The system model is presented in Section II. Section III presents the SINR distribution and user-BS association rules. User association probabilities are described in Section IV. Section V presents the coverage performance. Numerical results are presented in Section VI. Finally, the paper is concluded in Section VII.

SECTION II. System Model

A. Base Station and User Distribution

Consider a two-tier HetCNets model, where BSs of each tier differ in terms of densities and transmit power. BSs of both tiers are deployed through independent PPPs, $\Theta _{\nu }$ , with densities $\lambda _{\nu }$ , where $\nu \in \{M,S\}$ . $M$ and $S$ imply MBSs and SBSs, respectively. Users across the HetCNet are also distributed via another homogeneous PPP, $\Theta _{\text {U}}$ , with density $\lambda _{U}$ . According to Slvinyak’s Theorem [20], the statistics of a PPP remain unchanged if the analysis is carried out for a typical user equipment (TUE) situated at the origin.

B. Channel Model and User Association

DL received power by TUE from $\nu$ BS, and UL received power by $\nu$ BS from TUE are ${P}_{rx,\text {DL}}^{\nu }= \text {T}_{\nu } {P}_{tx}^{\nu }H_{\nu } \lVert y_{\nu }\rVert ^{-\alpha _{\nu }}$ and ${P}_{rx,\text {UL}}^{\nu }= {P}_{tx}^{U} H_{\nu }\lVert y_{\nu }\rVert ^{-\alpha _{\nu }}$ , respectively. Here, ${P}_{tx}^{\nu }$ and ${P}_{tx}^{U}$ , express the transmit power of $\nu$ BSs and the associated users, respectively, and $H_{\nu }$ is the channel gain of $\nu$ BS. $\text {T}_{\nu }$ and $\lVert y_{\nu }\rVert$ denote the biasing factor of the $\nu$ BS and the minimum distance between the $\nu$ BS and TUE, respectively. The path loss exponent is represented by $\alpha$ . Rayleigh fading with unitary mean is assumed for all channels, e.g., the desired and interfering channels $H_{x}=H_{y}\sim \text {exp(1)}$ , where $x$ and $y$ represent the arbitrary desired and interferer channels, respectively. $H_{x}=H_{y}=H\sim \text {exp(1)}$ , since the fading is independent of the BSs’ site. For the ease of access, a few of the notations used in this paper are summarized in Table 1.

TABLE 1 Notation Summary

C. Reverse Frequency Allocation Scheme Employment

In the proposed co-channel deployed multi-tier setup, transmissions in DL and UL direction of a tier co-occur with the transmissions of the other tier, which causes sever MCI. Hence, the mitigation of such type of interference is one of the challenging issues in HetCNets. Several approaches are proposed in the state-of-the-art (e.g., see [4] and references therein) to address this issue. However, in our proposed work, we use the RFA scheme to abate MCI. Besides the RFA scheme, we also consider De-DUA to achieve maximum performance gain. To employ RFA, we split the available region $\mathcal {R}$ into two disjoint subregions, that is, an inner subregion, $\mathcal {R}^{(in)}$ , and an outer subregion, $\mathcal {R}^{(out)}$ , s.t. $\bigcap _{i=in,out}^{}\mathcal {R}^{(i)}=\emptyset$ , where $\mathcal {R}^{(in)}$ is defined as the subregion in which the MBS provides reasonable coverage to its associated users. However, $\mathcal {R}^{(out)}$ is regarded as cell edge region where MBS provides poor coverage to its associated users. Likewise, the total spectrum, BW, is divided into two sub-spectra, i.e., $\rm {BW}_{1}$ and $\rm {BW}_{2}$ , s.t. BW = $\bigcup _{j=1,2}^{}\text {BW}_{j}$ , where $\rm {BW}_{1}$ and $\rm {BW}_{2}$ express sub-spectra assigned to the MBS in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ , respectively. $\rm {BW}_{1}$ and $\rm {BW}_{2}$ , are further split into UL and DL sub-carriers, where $\rm {BW}_{1} =\text {BW}_{1,\text {UL}}+\text {BW}_{1,\text {DL}}$ and $\rm {BW}_{2} =\text {BW}_{2,\text {UL}}+\text {BW}_{2,\text {DL}}$ are the MBS sub-carriers in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ , respectively. The MBS sub-spectra in $\rm {BW}_{1}$ and $\rm {BW}_{2}$ are employed as the SBS sub-spectra as $\rm {BW}'_{1}$ and $\rm {BW}'_{2}$ in $\mathcal {R}^{(out)}$ and $\mathcal {R}^{(in)}$ , respectively, s.t. $\rm {BW}^{'}$ = $\bigcup _{j=1,2}^{}\text {BW}^{'}_{j}$ . Whereas, the UL and DL sub-carriers of the SBS in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ are denoted as $\rm {BW}^{'}_{2} =\text {BW}'_{2,\text {UL}}+\text {BW}'_{2,\text {DL}}$ and $\rm {BW}'_{1} =\text {BW}'_{1,\text {UL}}+\text {BW}'_{1,\text {DL}}$ , respectively. The SBS uses the MBS UL and DL sub-carriers in $\mathcal {R}^{(in)}$ in a reverse direction than in $\mathcal {R}^{(out)}$ . The spectrum partitioning of the proposed two-tier setup in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ is shown in Fig. 1.

FIGURE 1.

RFA scheme’s spectrum partitioning in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ with DL and UL sub-carriers.

The layout of the spectrum partitioning of the proposed two-tier network setup along with Co-DUA and De-DUA is shown in Fig. 2. The ranges of $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ are $0 \leq D\leq d_{1}$ and $d_{1} \leq D\leq d_{2}$ , respectively. Various DL and UL desired (solid-lines) and interfering channels (dashed-lines) between $\nu$ UEs and $\nu$ BSs $\forall \nu =\{M,S\}$ are shown for Co-DUA and De-DUA in Fig. 2. For example, U5 in $\mathcal {R}^{(in)}$ is associated with SBS-2 both in UL and DL directions based on the MPR strategy. For DL transmission, it receives DL interference from SBS-1 as $\text {I}_{\text {DL}\_{}2}$ in the same region, and UL interference from U1 located in $\mathcal {R}^{(out)}$ as $\text {I}_{\text {UL}\_{}1}$ .

FIGURE 2.

Network layout with Co-DUA and De-DUA.

Similarly, U7 in $\mathcal {R}^{(out)}$ follows De-DUA, where it is associated with SBS-5 in UL direction in the same region based on path loss (PL) based association scheme, and is associated to MBS-1 in DL direction in $\mathcal {R}^{(in)}$ under the DL association strategy². For UL transmission, it receives UL interference from U4 (and all other users associated with SBS in UL direction in $\mathcal {R}^{(out)}$ ) as $\text {I}_{\text {UL}\_{}4}$ in the same region and DL interference from MBS-1 located in $\mathcal {R}^{(in)}$ as $\text {I}_{\text {DL}\_{}3}$ . Besides MCI mitigation, the RFA scheme also enhances the spectral efficiency since no additional spectrum is required for the SBSs.

SECTION III. Sinr Statistics With User-BS Coupled/Decoupled Downlink and Uplink Association

In this section, we analyze the SINR distribution and also define downlink and uplink user-BS association rules. Based on this, coverage performance of the proposed model is analyzed.

A. Interferences Distribution With Coupled/Decoupled Downlink and Uplink Association Given That Tue$\in \mathcal {R}^{(j)},~~\forall j\in \{in,out\}$

In this subsection, the DL and UL interference experienced by a TUE in $\mathcal {R}^{(j)},~~\forall j\in \{in, out\}$ is presented.

1) Interference Distribution With Coupled Downlink-Uplink Association $\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}$ (Which is Shown as (1))

When a $\text {TUE}\in \mathcal {R}^{(j)}$ is associated with $\nu$ BS in both DL and UL direction, it experiences interference from the $\nu$ BSs located in $\mathcal {R}^{(j)}$ in DL direction (except from the serving BS) and from the users associated with $\tau$ BS, i.e., $\tau$ UEs, located in $\mathcal {R}^{(i)}$ in UL direction, $\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}$ ³ s.t., $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ . Moreover, $j\in \{in, out\}$ , $i\in \{in, out\}$ and $j\neq i$ .

2) Interference Distribution With Decoupled Downlink-Uplink Association $\text {I}_{\nu ,\tau ,decoup}^{\mathcal {R}^{(j)}}$ (Which is Shown as (2))

If a TUE in $\mathcal {R}^{(j)}$ is associated with $\nu$ BS in DL and with $\tau$ BS in UL direction, in DL transmission it receives interference from all $\nu$ BSs (except from the serving $\nu$ BS) in $\mathcal {R}^{(j)}$ , and UL interference from $\tau$ UEs in $\mathcal {R}^{(i)}$ . However, in UL transmission, TUE receives interference from all $\tau$ UEs in $\mathcal {R}^{(j)}$ as well as interference from $\nu$ BSs in $\mathcal {R}^{(i)}$ where $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ . Moreover, $j\in \{in, out\}$ , $i\in \{in, out\}$ and $j\neq i$ .

B. Signal-to-Interference-Plus-Noise Ratio Analysis With Coupled/Decoupled Downlink-Uplink Association Given That Tue$\in \mathcal {R}^{(j)}, ~~\forall j\in \{in,out\}$

Based on the interference distributions (1) and (2), as shown at the top of the next page,

for the proposed model, now we analyze the SINR distributions for Co-DUA and De-DUA, given that TUE is situated in $\mathcal {R}^{(j)}$ .

1) SINR Analysis With Coupled Downlink-Uplink Association, $\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}$ (Which is Shown as (3))

When $\text {TUE}\in \mathcal {R}^{(j)}$ is associated with $\nu$ BS both in UL and DL, SINR is modeled as (3), as shown at the top of the next page,

where $\sigma ^{2}$ represents the noise power received at TUE. In DL transmission, interference received from the $\nu$ BSs in $\mathcal {R}^{(j)}$ in DL direction is expressed as ${\sum _{x_{l}\in \Theta _{\nu }^{\mathcal {R}^{(j)}}\backslash y_{\nu }}\text {T}_{\nu } {P}_{tx}^{M}H_{x_{l}}^{\mathcal {R}^{(j)}}\lVert x_{l}\rVert ^{-\alpha }}$ and interference from $\tau$ UEs in UL direction in $\mathcal {R}^{(i)}$ , is denoted as ${\sum _{y_{n}\in \Theta _{\tau }^{\mathcal {R}^{(i)}}} {P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(i)}}\lVert y_{n}\rVert ^{-\alpha }}$ . In UL transmission, interference received from the $\tau$ UEs in $\mathcal {R}^{(j)}$ in UL direction, and from $\nu$ BSs in $\mathcal {R}^{(i)}$ in DL direction can be expressed as ${\sum _{y_{n}\in \Theta _{\nu }^{\mathcal {R}^{(j)}}\backslash y_{\nu }} {P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(j)}}\lVert y_{n}\rVert ^{-\alpha }}$ and ${\sum _{x_{l}\in \Theta _{\tau }^{\mathcal {R}^{(i)}}}\text {T}_{\tau } {P}_{tx}^{\tau }H_{x_{l}}^{\mathcal {R}^{(i)}}\lVert x_{l}\rVert ^{-\alpha }}$ , respectively. $H_{x_{l}}^{\mathcal {R}^{(j)}}$ and $H_{y_{n}}^{\mathcal {R}^{(j)}}$ are the channel gains of $\nu$ BS and $\tau$ BS in $\mathcal {R}^{(j)}$ , respectively, where $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ . Moreover, $j\in \{in, out\}$ , $i\in \{in, out\}$ and $j\neq i$ .

2) SINR Analysis With Decoupled Downlink-Uplink Association $\Xi _{\nu ,\tau ,decoup}^{\mathcal {R}^{(j)}}$ (Which is Shown as (4))

When $\text {TUE}\in \mathcal {R}^{(j)}$ is associated with $\nu$ BS in DL and with $\tau$ BS in UL. Using (2), SINR distribution with decoupled access is modeled as (4), as shown at the top of the next page.

For DL transmission, interference received at TUE from the $\nu$ BSs in $\mathcal {R}^{(j)}$ in DL direction is written as ${\sum _{x_{l}\in \Theta _{\nu }^{\mathcal {R}^{(j)}}\backslash y_{\nu }}\text {T}_{\nu } {P}_{tx}^{M}H_{x_{l}}^{\mathcal {R}^{(j)}}\lVert x_{l}\rVert ^{-\alpha }}$ , and interference received from $\tau$ UEs in $\mathcal {R}^{(i)}$ in UL direction can be written as ${\sum _{y_{n}\in \Theta _{\tau }^{\mathcal {R}^{(i)}}} {P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(i)}}\lVert y_{n}\rVert ^{-\alpha }}$ . For UL transmission, interference received from the $\tau$ UEs in $\mathcal {R}^{(j)}$ in UL direction is modeled as ${\sum _{y_{n}\in \Theta _{\tau }^{\mathcal {R}^{(j)}}\backslash y_{\tau }} {P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(j)}}\lVert y_{n}\rVert ^{-\alpha }}$ and from $\nu$ BSs in $\mathcal {R}^{(i)}$ in DL direction is ${\sum _{x_{l}\in \Theta _{\nu }^{\mathcal {R}^{(i)}}} \text {T}_{\nu } {P}_{tx}^{\nu }H_{x_{l}}^{\mathcal {R}^{(i)}}\lVert x_{l}\rVert ^{-\alpha }}$ . $H_{x_{l}}^{\mathcal {R}^{(j)}}$ and $H_{y_{n}}^{\mathcal {R}^{(j)}}$ are the channel gains of $\nu$ BS and $\tau$ BS in $\mathcal {R}^{(j)}$ , respectively, where $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ . Moreover, $j\in \{in, out\}$ , $i\in \{in, out\}$ and $j\neq i$ .

C. User-BS Association Scheme With Coupled/Decoupled Downlink-Uplink Association

User-BS association in Co-DUA, wherein the TUE is associated with a particular tier BS in both DL and UL directions, is based on the biased MPR association scheme [5], [7], [18]. The location of such type of a BS (to which the TUE associates) of tier $\omega$ is expressed as

$$\begin{equation} \omega =\arg ~ \mathop {max}_{\nu \in {\{M,S\}}}~ \text {T}_{\nu }{P}_{tx}^{\nu }\|y_{\nu }\|^{-\alpha _{\nu }}, \end{equation}$$ where $\|y_{\nu }\|$ represents distance of the TUE from the nearest BS for $\nu$ th-tier.

However, in the De-DUA, TUE associates to one particular tier BS in DL direction based on MPR association scheme, and with other tier BS in UL direction based on the PL association scheme. The location of the serving BS associated in DL direction based on MPR association scheme in De-DUA is the same as given in (5). However, the location of the serving BS associated in UL direction with the TUE of tier $\omega$ in De-DUA is expressed as

$$\begin{equation} \omega =\arg ~ \mathop {max}_{\nu \in {\{M,S\}}}~ \|y_{\nu }\|^{-\alpha _{\nu }}. \end{equation}$$

In the proposed two tier HetCNet model, the TUE associates with a particular tier BS under the following DL/UL association rules.

Downlink Association Rule: A TUE is associated with $\nu$ BS in DL direction if the power received by the TUE from the $\nu$ BS is the maximum, i.e., ${P}_{tx}^{\nu } \lVert y_{\nu }\rVert ^{-\alpha _{\nu }} > {P}_{tx}^{\tau } \lVert y_{\tau }\rVert ^{-\alpha _{\tau }}$ .

Uplink Association Rule: A TUE is associated with $\nu$ BS in UL direction when the $\nu$ BS receives maximum power from the TUE compared with all other BSs, i.e., ${P}_{tx}^{U} \lVert y_{\nu }\rVert ^{-\alpha _{\nu }} > {P}_{tx}^{U} \lVert y_{\tau }\rVert ^{-\alpha _{\tau }}$ . Otherwise, the TUE associates with a $\tau$ BS following the DL association rule, where $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ , and $\tau \neq \nu$ .

Here, {${P}_{tx}^{\nu }$ , ${P}_{tx}^{\tau }$ , ${P}_{tx}^{U}$ } are the $\nu$ BS, $\tau$ BS and user transmit power and {$\alpha _{\nu },\alpha _{\tau }$ } and {$\lVert y_{\nu }\rVert , \lVert y_{\tau }\rVert$ } are the sets of path loss exponents, and distances between the serving $\nu$ BS or $\tau$ BS and TUE, respectively. {$\text {T}_{\nu } , \text {T}_{\tau }$ } represents association bias of $\nu$ BS and $\tau$ BS, respectively.

SECTION IV. User Association Probability

In this section, we describe association probability which is defined as the probability that a TUE is served by a particular tier. To derive the association probability, we first measure the distribution of distance $y_{\nu }$ between the TUE and the serving BS, given that the TUE associates to a ${\nu }$ BS. Consider $\text{Y}_{\nu }$ as the random variable which denotes the statistical distance from the TUE to the closest associated BS of $\Theta _{\nu }$ . As the BSs deployment is through PPP, using null probability property of PPP [20], the probability distribution function (PDF) of $\text{Y}_{\nu }$ in $\mathbb {R}^{2}$ is given as

$$\begin{equation} f_{\text {Y}_{\nu }}(y_{\nu })=2\pi \lambda _{\nu }y_{\nu } exp(-\pi \lambda _{\nu }y_{\nu }^{2}), \quad \forall \nu \in \{M,S\}. \end{equation}$$ We employ RFA scheme in a multi-region environment to reduce MBS interference. Therefore, we split the available regions into two subregions, i.e., $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ . Hence, the TUE associates to one of these regions. The probabilities that TUE$\in \mathcal {R}^{(out)}$ and TUE$\in \mathcal {R}^{(in)}$ are $P[\text {TUE} \in \mathcal {R}^{(out)}]=e^{-\lambda _{M}\pi d_{1}^{2}}$ and $P[\text {TUE} \in \mathcal {R}^{(in)}]=1- e^{-\lambda _{M}\pi d_{1}^{2}}$ , respectively. Conditional distribution of the distances between serving $\nu$ BS and TUE in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ are described, respectively, as follows.

• Distance distribution between serving $\nu$ BS $Y_{\nu }$ , located at $y_{\nu }$ , and $\text {TUE}\in \mathcal {R}^{(in)}$ is given as

  $$\begin{align} f_{Y_{\nu }\lvert {\text {TUE}\in \mathcal {R}^{(in)}}}(y_{\nu })\equiv&\dfrac {f_{Y_{\nu }}(y_{\nu })}{P[\text {TUE} \in \mathcal {R}^{(in)}]}\notag \\=&\dfrac {2\pi \lambda _{\nu }y_{\nu }e^{-\pi \lambda _{\nu } y_{\nu }^{2}}}{1-e^{-\lambda _{M}\pi d_{1}^{2}}}, ~~\forall \nu \in \{M, S\}.\notag \\ {}\end{align}$$

• Distance distribution between serving $\nu$ BS $Y_{\nu }$ , located at $y_{\nu }$ , and $\text {TUE}\in \mathcal {R}^{(out)}$ is given as

  $$\begin{equation} f_{Y_{\nu }\lvert {\text {TUE}\in \mathcal {R}^{(out)}}}(y_{\nu })=\dfrac {2\pi \lambda _{\nu }y_{\nu }e^{-\pi \lambda _{\nu } y_{\nu }^{2}}}{e^{-\lambda _{M}\pi d_{1}^{2}}}. \end{equation}$$

The above conditional distributions are useful in characterizing association probabilities of a randomly located user in $\mathcal {R}^{(in)}$ and $\mathcal {R}^{(out)}$ of our proposed model.

A. Association Probability of a Tue With Coupled/Decoupled Downlink-Uplink ACCESS

1) Association Probabilities Of A Tue With Co-DUA

The association probability via which a TUE is associated with $\nu$ BS, i.e., $U_{\nu ,\nu }^{coupled}$ , both in UL and DL direction⁴ can be calculated as

$$\begin{align} U_{\nu ,\nu }^{coupled}=&P( {P}_{tx}^{\nu } \lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }} >{P}_{tx}^{\tau } \lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }},\notag \\&\quad ~\lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }} > \lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }}), \end{align}$$ where ${P}_{tx}^{\nu } \lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }} >{P}_{tx}^{\tau } \lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }}$ and $\lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }} > \lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }}~\forall \nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ are the DL and UL association events, respectively.

There are two cases to simplify (15).

• CASE-I:

  If we consider that a TUE associates with MBS, i.e., $\nu \in \{M\}$ , then $\tau \in \{S\}$ holds $\forall {P}_{tx}^{M}>{P}_{tx}^{S}$ . The intersection of the events given in (10) is $\lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} > \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}$ , because $\dfrac {P_{tx}^{S}}{P_{tx}^{M}} <1$ . Therefore, the association probability, $U_{M,M}^{coupled}$ , is written as

  $$\begin{equation} U_{M,M}^{coupled}=\dfrac {\lambda _{M}}{\sum _{\Omega =M,S}{\lambda _{\Omega }}}. \end{equation}$$

  Proof:

  $$\begin{align*} U_{M,M}^{coupled}=&P \left ({\lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} > \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}}\right ) \\=&\text {E}_{\text {Y}_{M}}\left [{P\left({\text {Y}_{S} > \text {Y}_{M}^{\left({\frac {\alpha _{M}}{\alpha _{S}}}\right)}}\right)}\right ] \\=&\displaystyle \int _{0}^{\infty }\left [{P\left({\text {Y}_{S} > \text {Y}_{M}^{\left({\frac {\alpha _{M}}{\alpha _{\S }}}\right)}}\right)}\right ]f_{\text {Y}_{M}}(y_{M})dy_{M}. \tag{12}\end{align*}$$ Assuming $\alpha _{\nu } =\alpha _{\tau }=\alpha$ and substituting (7) into (10), along with mathematical manipulations complete the proof of (11).

• CASE-II:

  TUE associations with SBS are of two types, i.e., biased association and unbiased association.

  1. Unbiased SBS’ association: If a TUE associates with unbiased-SBS, i.e., $\nu \in \{S\}$ , then $\tau \in \{M\} ~~$ holds $\forall ~ {P}_{tx}^{M}>{P}_{tx}^{S}$ . As a consequence, the intersection of events given in (10) is ${P}_{tx}^{S} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}} > {P}_{tx}^{M} \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}}$ , because $\dfrac {P_{tx}^{M}}{P_{tx}^{S}} >1$ . Therefore, the association probability, $U_{uB-S,S}^{coupled}$ , is written as

     $$\begin{equation*} U_{uB-S,S}^{coupled}=\dfrac {\lambda _{S}}{\sum _{\Omega =S,M}{\lambda _{\Omega }(\hat {P}_{tx}^{\Omega })^{\frac {2}{\alpha }}}},\quad ~~\forall ~~ \dfrac {P_{tx}^{M}}{P_{tx}^{S}} >>1. \tag{13}\end{equation*}$$ Here $\hat {P}_{tx}^{\Omega }$ is the ratio of the interfering MBS to the serving SBS transmit power.

  2. Biased SBS association: If a TUE associates with biased-SBS, i.e., $\nu \in \{S\}$ and $\tau \in \{M\} ~~$ hold $\forall ~ {P}_{tx}^{M}>{P}_{tx}^{S}$ , then the intersection of events given in (10) is $\text {T}_{S} {P}_{tx}^{S} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}> {P}_{tx}^{M} \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} >{P}_{tx}^{S} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}$ . Assuming identical path loss exponents i.e., $\alpha _{M}=\alpha _{S}=\alpha$ , the association probability, $U_{B-S,S}^{coupled}$ , is given as

     $$\begin{align*} U_{B-S,S}^{coupled}=\dfrac {\lambda _{S}}{\sum _{\Omega =M,S}{\lambda _{\Omega }(\hat {\text {T}}_{\Omega }\hat {P}_{tx}^{\Omega })^{\frac {2}{\alpha }}}}-\dfrac {\lambda _{S}}{\sum _{\Omega =M,S}{\lambda _{\Omega }(\hat {P}_{tx}^{\Omega })^{\frac {2}{\alpha }}}}. \\ \tag{14}\end{align*}$$ Eq.(14) leads to an interesting intuition that by increasing small-cell biasing factor, i.e., $\text {T}_{S}$ , the average number of offloaded users increases as $U_{B-S,S}^{coupled}$ increases.

2) Association Probabilities Of De-DUA

In De-DUA a TUE can associate with two different tiers BSs in UL and DL direction. Let us assume that the TUE associates with $\nu$ BS in DL direction and $\tau$ BS in UL direction. Then the association probability can be written as

$$\begin{align*} U_{\nu ,\tau }^{decoup}=&P(\text {T}_{\nu } {P}_{tx}^{\nu } \lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }} > \text {T}_{\tau } {P}_{tx}^{\tau } \lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }}, \\&~~\lVert \text {Y}_{\tau }\rVert ^{-\alpha _{\tau }} > \lVert \text {Y}_{\nu }\rVert ^{-\alpha _{\nu }}). \tag{15}\end{align*}$$ Here, $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ and $\nu \neq \tau$ . There are two ways to simplify (15).

1. If the user associates with MBS in DL direction, i.e., $\nu \in \{M\}$ and SBS in UL direction, i.e., $\tau \in \{S\}$ , then

   $$\begin{align*} U_{M,S}^{decoup}=&P \Big ( {P}_{tx}^{M} \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} > \text {T}_{S} {P}_{tx}^{S} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}, \\&\quad \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}} > \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} \Big ) \\=&P \Big ( \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} >\frac {\text {T}_{S}}{\hat {P}_{tx}^{M}} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}, \\&\quad \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}} > \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} \Big ). \tag{16}\end{align*}$$ Hence the intersection of both the above events can also be written as $\lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}>\lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} >\frac {\text {T}_{S}}{\hat {P}_{tx}^{M}} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}$ . Using the above event the association probability $U_{M,S}^{uncoupled}$ is given as $$\begin{equation*} U_{M,S}^{decoup}=\dfrac {\lambda _{S}}{\sum _{\Omega =M,S}{\lambda _{\Omega }}}-\dfrac {\lambda _{S}}{\sum _{\Omega =M,S}{\lambda _{\Omega }(\hat {\text {T}}_{\Omega }\hat {P}_{tx}^{\Omega })^{\frac {2}{\alpha }}}}. \tag{17}\end{equation*}$$

2. If the user associates with SBS in DL direction, i.e., $\nu \in \{S\}$ and MBS in UL direction, i.e., $\tau \in \{M\}$ , then

   $$\begin{align*} U_{S,M}^{decoup}=&P(\text {T}_{S} {P}_{tx}^{S} \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}} > {P}_{tx}^{M} \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}}, \\&\quad \lVert \text {Y}_{M}\rVert ^{-\alpha _{M}} > \lVert \text {Y}_{S}\rVert ^{-\alpha _{S}}). \tag{18}\end{align*}$$ Since the above events do not intersect, therefore, the association probability of the above case is 0, i.e., $U_{S,M}^{decoup}=0$ .

SECTION V. Coverage Probability Analysis

SINR coverage or coverage probability (equivalently CCDF of SINR $\Xi$ ) is used to measure the successful connectivity of a randomly chosen user with a serving BS. SINR coverage is defined as the probability that a randomly selected user experiences SINR greater than a target SINR $\beta$ . The interference distribution significantly affects the SINR coverage statistics. Based on the interference distribution for Co-DUA and De-DUA derived in Section III-A, a tractable analysis of the coverage performance of our proposed scheme, i.e., CRE-based cell association with RFA employment while considering both Co-DUA and De-DUA assumptions, is presented in this section.

Theorem 1 (SINR coverage):

The SINR coverage of a TUE associated with a $\nu$ BS located at $y_{\nu }$ can be written as

$$\begin{equation*} {P}_{\text {cov}}{(\beta _{\nu })}\buildrel \triangle \over = P(\Xi (y_{\nu })>\beta _{\nu }) =\text {E}[P(\Xi (y_{\nu })>\beta _{\nu })]. \tag{19}\end{equation*}$$

Using the interference distribution (as discussed in Section III-A) along with Theorem 1, the coverage probabilities for Co-DUA and De-DUA, given that TUE is situated in $\mathcal {R}^{(j)}$ , are described in Lemmas 1 and 2 as follows.

A. Coverage Probability of Tue With Co-DUA

Lemma 1:

Base on the Co-DUA, coverage probability of a the TUE associated with $\nu$ BS in $\mathcal {R}^{(j)}$ , i.e., ${P}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}{(\beta _{\nu })}$ , is given by (20), as shown at the top of this page.

Here the integral limits, i.e., [$D_{L}$ , $D_{U}$ ], depend on the region in which TUE is located⁵ , $\nu \in \{M,S\}$ , $j \in \{in,out\}$ , $i \in \{in,out\}$ , $j \neq i$ , $\text {G}_{coup}^{\text {DL}}=\biggl [\lambda _{\nu }^{\mathcal {R}^{(j)}} \text {V}_{\nu }^{\mathcal {R}^{(j)}} +\lambda _{\tau }^{\mathcal {R}^{(i)}}\text {V}_{\tau }^{\mathcal {R}^{(i)}}\biggr ]$ , and $\text {G}_{coup}^{\text {UL}}=\biggl [\lambda _{\nu }^{\mathcal {R}^{(j)}} \text {V}_{\nu }^{\mathcal {R}^{(j)}} +\lambda _{\tau }^{\mathcal {R}^{(i)}}\text {V}_{\tau }^{\mathcal {R}^{(i)}}\biggr ]$ .

Proof:

See Appendix for proof of (20).

The following corollary provides the coverage probability for a randomly selected user in the proposed model, while considering Co-DUA.

Corollary 1:

The CCDF of the SINR achieved in Co-DUA at the TUE can be expressed as

$$\begin{align*} {P}_{\nu ,\nu }^{coup}{(\beta _{\nu })}=\sum _{\nu \in \{M,S\}}\sum _{j\in \{in,out\}}{P}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}{(\beta _{\nu })}P[\text {TUE} \in \mathcal {R}^{(j)}], \\ \tag{21}\end{align*}$$ where $\nu \in \{M,S\}$ .

B. Coverage Probability of a Tue With De-DUA.

Lemma 2:

The coverage probability of a TUE associated with $\nu$ BS in DL direction and $\tau$ BS in UL direction in $\mathcal {R}^{(j)}$ , i.e., ${P}_{\nu ,\tau ,decoup}^{\mathcal {R}^{(j)}}{(\beta _{\nu }, \beta _{\tau })}$ , is given as (22), as shown at the top of this page.

In (22), $\nu \in \{M,S\}$ , $\tau \in \{M,S\}$ , $\nu \neq \tau$ , $j \in \{in,out\}$ , $i \in \{in,out\}$ , $j \neq i$ , $\text {G}_{decoup}^{\text {DL}}=\bigl [\lambda _{\nu }^{\mathcal {R}^{(j)}} \text {V}_{\nu }^{\mathcal {R}^{(j)}} +\lambda _{\tau }^{\mathcal {R}^{(i)}}\text {V}_{\tau }^{\mathcal {R}^{(i)}}\bigr ]$ and $\text {G}_{decoup}^{\text {UL}}=\bigl [\lambda _{\tau }^{\mathcal {R}^{(j)}} \text {V}_{\tau }^{\mathcal {R}^{(j)}} +\lambda _{\nu }^{\mathcal {R}^{(i)}}\text {V}_{\nu }^{\mathcal {R}^{(i)}}\bigr ]$ , $\text {V}_{\tau }^{\mathcal {R}^{(j)}}=\text {V}_{\nu }^{\mathcal {R}^{(j)}}$ if and only if $y_{\nu }^{''}=y_{\tau }^{''}$ . Similarly, $\text {V}_{\nu }^{\mathcal {R}^{(i)}}=\text {V}_{\nu }^{\mathcal {R}^{(i)}}$ if and only if $s\text {T}_{\tau }{P}_{tx}^{\tau }=s\text {T}_{\nu }{P}_{tx}^{\nu }$ .

The following corollary gives the coverage performance for a TUE with De-DUA.

Corollary 2:

The CCDF of the target SINR $\beta$ achieved at the TUE with De-DUA in $\mathcal {R}^{(j)}$ , i.e., ${P}_{\nu ,\tau }^{decoup}{(\beta _{\nu },\beta _{\tau })}$ , is given as

$$\begin{align*} &\hspace {-2pc} {P}_{\nu ,\tau }^{decoup}{(\beta _{\nu },\beta _{\tau })} \\=&\sum _{\nu \in \{M,S\}}\sum _{\tau \in \{M,S\}}\sum _{j\in \{in,out\}}\left [{ {P}_{\nu ,\tau ,decoup}^{\mathcal {R}^{(j)}}{(\beta _{\nu },\beta _{\tau })}}\right ] \\&\times P[\text {TUE} \in \mathcal {R}^{(j)}], \tag{23}\end{align*}$$ where $\nu \in \{M,S\},\tau \in \{M,S\},\nu \neq \tau$ . The total SINR coverage of a randomly chosen user is given as $$\begin{align*} {P}_{\text {cov}}^{\text {total}} {(\beta _{\nu }, \beta _{\tau })}\!\!\buildrel \triangle \over =\!\!\! \sum _{\nu \in \{M,S\}}\sum _{\tau \in \{M,S\}}\text {P}_{\nu ,\nu }^{coup}{(\beta _{\nu })}\!+\! \text {P}_{\nu ,\tau }^{decoup}{(\beta _{\nu }, \beta _{\tau })}. \\ \tag{24}\end{align*}$$

SECTION VI. Numerical Results and Discussion

In this section, we present numerical results of our proposed model and demonstrate the impact of RFA with De-DUA on the coverage performance, and compare it with Co-DUA, with and without RFA scheme. We validate our proposed model and evaluate analytical results through simulations using two tier setup as discussed in Section II, with the parameters listed in Table 2.

TABLE 2 Simulation Parameters

The variation of coverage probability is plotted for different values of coverage/SINR threshold $\beta$ in Fig. 3, while considering both Co-DUA and De-DUA. The coverage performance decreases with the increase in $\beta$ . As shown in the figure, at lower values of $\beta$ , the coverage performance is better. This is due to the fact that in CRE-based cell association in HetCNets, users prefer to associate with BSs of lower SINR. It can also be observed that for the proposed HetCNet, De-DUA along with RFA outperforms all the other methods. The above figure is a clear evident that simulations results for the proposed two tier setup nearly match the evaluated analytical results.

FIGURE 3.

Effect of Co-DUA and De-DUA with RFA scheme.

In Fig. 4, we study the effect of different values of association bias $\text {T}_{S}$ on the coverage probability, while considering both Co-DUA and De-DUA with and without RFA scheme. The lower set of plots reveals that the coverage probability monotonically decreases with the increase in $\text {T}_{S}$ . This means that under biased MPR strategy, if no proper interference management scheme is employed, users prefer to associate with BSs offering lower SINR. Hence user offloading via biased MPR strategy is suboptimal with larger values of $\text {T}_{S}$ since all the users associated to MBS are offloaded to SBSs. The upper set of plots describe that optimal coverage probability is achieved with RFA scheme since offloaded users are not affected by MCI. It can also be seen that the optimal SINR coverage range decreases with the De-DUA as compared to the Co-DUA, since users prefer to associate with MBS in DL and SBS in UL directions.

FIGURE 4.

Effect of Co-DUA and De-DUA with RFA scheme.

Fig. 5 depicts the coverage performance with the increasing range of $\mathcal {R}^{(in)}$ . The plots show how RFA affects the coverage of CRE-based cell association scheme while considering both Co-DUA and De-DUA. Without the RFA scheme, the coverage probability of HetCNets with Co-DUA and De-DUA increases very slowly as $\mathcal {R}^{(in)}$ coverage range increases. This improvement is 0.2% and 0.4%, respectively, till the optimal coverage extension which is 80% here for the proposed two tier setup. However, the coverage improvement of the proposed De-DUA with the RFA scheme is 5.9% for the optimal $\mathcal {R}^{(in)}$ coverage extension, which is 70%. This distinctly reveals that HetCNet with the RFA scheme, while considering De-DUA, outperforms all other methods in terms of coverage performance.

FIGURE 5.

Coverage probability vs inner subregion coverage range.

Fig. 6 shows the relation between the user association and different values of SBS density, while considering Co-DUA and De-DUA. It is observed that the average fraction of users associated with MBS in DL direction and SBS in UL direction grows with the increase in SBS density. The maximum user association is achieved at $\lambda _{S}/\lambda _{M}=8$ . It can also be observed that user association with SBS in DL direction and MBS in UL direction is 0.

FIGURE 6.

User association vs density of SBSs.

In Fig. 7, the coverage of MBS user is compared with the SBS user for different values of $\lambda _{U}$ and association bias $\text {T}_{S}$ . It is observed that the coverage of unbiased SBS users increases as the association bias increases. This is because of the fact that the number of offloaded users increases and load of the MBS decreases. As a consequence, the effect of MCI on the unbiased SBS users decreases. It can also be observed that the coverage of MBS users increases as the association bias increases. This is due to the fact that more users are offloaded to SBSs, thus, fewer number of MBS users share the available resources. It can also be seen that the coverage performance of each user type increases as $\lambda _{U}$ decreases, since user density directly affects the UL interference. In addition, the RFA scheme further improves the coverage performance.

FIGURE 7.

Impact of $\text {T}_{S}$ , $\lambda _{U}$ on coverage performance of different user types.

SECTION VII. Conclusion

In this paper we consider a two-tier HetCNets model, and probabilistically measure the coverage performance with the joint effect of RFA employment and De-DUA. We observe the effect of association bias and RFA employment on the coverage, while considering De-DUA. Numerical results clearly indicate that the coverage performance improves with the employment of RFA since it reduces the MCI on offloaded users. It is also observed that, as SBS density increases, larger number of users are associated with MBS and SBS in DL and UL directions, respectively. By selecting appropriate values of $\mathcal {R}^{(j)}$ coverage range and association bias, the coverage performance can be significantly improved.

Appendix

Proof:

$$\begin{align*}\text {P}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}{(\beta _{\nu })}=&\text {E}\left [{ P[\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}(y_{\nu })>\beta _{\nu }\lvert \text {TUE}\in \mathcal {R}^{(j)}]}\right ] \\[-0.5pt]=&\int _{0}^{d_{1}}P[\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}(y_{\nu })>\beta _{\nu }\lvert \text {TUE}\in \mathcal {R}^{(j)}] \\&\times f_{Y_{\nu }\lvert {\text {TUE}\in \mathcal {R}^{(j)}}}(y_{\nu })dy_{\nu }, \tag{25}\end{align*}$$ where $\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}$ and $\beta _{\nu }$ are the SINR and SINR threshold of the $\nu$ BS, respectively. $P[\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}(y_{\nu })>\beta _{\nu }\lvert \text {TUE}\in \mathcal {R}^{(j)}]$ is the probability of successful communication for a $\nu$ BS associated $\text {TUE}\in \mathcal {R}^{(j)}$ , and can be calculated as $$\begin{align*} P[\Xi _{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}(y_{\nu })>&\beta _{\nu }\lvert \text {TUE}\in \mathcal {R}^{(j)}] \\[-0.5pt]\stackrel {a}{=}&exp{\left ({ \dfrac {-\beta _{\nu }}{\text {SNR}}}\right )} \text {E}_{\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}}\left [{ e^{\left ({ -s{\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}}}\right )}}\right ] \\[-0.5pt]\stackrel {b}{=}&exp{\left ({ \dfrac {-\beta _{\nu }}{\text {SNR}}}\right )}\text {M}_{\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}}(s), \tag{26}\end{align*}$$ where SNR is the signal to noise ratio and $s =\dfrac {\beta _{\nu }\lVert y_{\nu }\rVert ^{-\alpha }}{P_{tx}^{\nu }}$ . Step $a$ in (26) arrives due to the Rayleigh assumption for all the associated channels (both desired and interferer) with mean value of 1, i.e., $H_{y}^{\mathcal {R}^{(j)}}(.) \sim exp(1)$ and interference independence. In Step $b$ , $\text {M}_{\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}}(s)$ is the moment generating function (MGF) [20] for both DL and UL transmission of the aggregate interference $\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}$ (1), which can be further written as (27), as shown at the top of this page,

where ${P}_{rx,\text {DL}}^{\nu ,\mathcal {R}^{(j)}}=\text {T}_{\nu } {P}_{tx}^{\nu }H_{x_{l}}^{\mathcal {R}^{(j)}}\lVert x_{l}\rVert ^{-\alpha }~ x_{l}$ and ${P}_{rx,\text {DL}}^{\tau ,\mathcal {R}^{(i)}}=\text {T}_{\tau } {P}_{tx}^{\tau }H_{x_{l}}^{\mathcal {R}^{(i)}}\lVert x_{l}\rVert ^{-\alpha }$ are the DL received power at the TUE from $\nu$ BS in $\mathcal {R}^{(j)}$ and from $\tau$ BS in $\mathcal {R}^{(i)}$ , respectively. ${P}_{rx,\text {UL}}^{\tau ,\mathcal {R}^{(j)}}={P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(j)}}\lVert y_{n}\rVert ^{-\alpha }$ and ${P}_{rx,\text {UL}}^{\tau ,\mathcal {R}^{(j)}}={P}_{tx}^{U}H_{y_{n}}^{\mathcal {R}^{(j)}}\lVert y_{n}\rVert ^{-\alpha }$ are the power received at $\nu$ BS from TUE in $\mathcal {R}^{(i)}$ , and at $\tau$ BS from TUE in $\mathcal {R}^{(j)}$ , respectively; and $x_{l}$ and $y_{n}$ are the locations of the BSs (except the serving BS) associated with users in DL and UL transmissions, respectively.

Assuming independence of the PPPs in DL and UL transmissions, i.e., $\{\Theta _{\nu }^{\mathcal {R}^{(i)}} , \Theta _{U}^{\mathcal {R}^{(j)}}\}$ , $\{\Theta _{U}^{\mathcal {R}^{(j)}} , \Theta _{\tau }^{\mathcal {R}^{(i)}}\}$ , and the fading channels, i.e., $H_{x_{l}}^{\mathcal {R}^{(j)}}$ and $H_{y_{n}}^{\mathcal {R}^{(i)}}$ , (27) is further simplified in Step $c$ as (28), as shown at the top of this page,

where $\text {M}_{H_{y_{n}}^{\mathcal {R}^{(j)}}}(.)$ and $\text {M}_{H_{y_{n}}^{\mathcal {R}^{(i)}}}(.)$ are the MGFs of the interfering channels in $\mathcal {R}^{(j)}$ and random users in $\mathcal {R}^{(i)}$ , respectively.

Assuming Rayleigh distribution for fading channels, (28) is further depicted as (29), as shown at the top of this page.

Here, $y_{\nu }^{'}$ and $y_{\nu }^{''}$ represent the nearest BS interferer distance and are obtained as follows: ${\underline {For ~y_{\nu }^{'}}}$ : $$\begin{align*} & {P}_{rx,\text {DL}}^{\nu ,\mathcal {R}^{(i)}}>{P}_{rx,\text {DL}}^{\tau ,\mathcal {R}^{(i)}} ~~ \\ \implies & \text {T}_{\nu }{P}_{tx}^{\nu }y_{\nu }^{-\alpha }>\text {T}_{\tau }{P}_{tx}^{\tau }y_{\nu }^{'-\alpha } \\ \implies & y_{\nu }^{'}>\left ({ \dfrac {{\text {T}}_{\tau }{P}_{tx}^{\tau }} {{{\text {T}}_{\nu }{P}_{tx}^{\nu }}}}\right ) ^{\frac {1}{\alpha }}{y_{\nu }}, \tag{30}\end{align*}$$ ${\underline {For y_{\nu }^{''}}}$ : $$\begin{align*} & {P}_{rx,\text {DL}}^{U,\mathcal {R}^{(i)}}>{P}_{rx,\text {DL}}^{U,\mathcal {R}^{(i)}} \\ \implies & y_{\nu }^{-\alpha }>y_{\nu }^{''-\alpha } \\ \implies & y_{\nu }^{''}>{y_{\nu }}, \tag{31}\end{align*}$$ $\text {where }~~ \{\text {T}_{\nu },~\text {T}_{\tau }\geq 1\}, \nu \in \{M,S\},~ \tau \in \{M,S\}$ , $~\text {and } \nu \neq \tau$ . Substituting $\{ y_{\nu }^{'}, y_{\nu }^{''}\}$ , $a=(s\text {T}_{\nu }{P}_{tx}^{\nu })^{\frac {2}{\alpha }}x_{l}^{2}$ , $b=(s{P}_{tx}^{U})^{\frac {2}{\alpha }}y_{n}^{2}$ , $c=(s{P}_{tx}^{U})^{\frac {2}{\alpha }}y_{n}^{2}$ and $d=(s\text {T}_{\tau }{P}_{tx}^{\tau })^{\frac {2}{\alpha }}x_{l}^{2}$ into (29), it is further simplified as: $$\begin{equation*} \text {M}_{\text {I}_{\nu ,\nu ,coup}^{\mathcal {R}^{(j)}}}(s) = \begin{cases} exp\biggl (-2\pi \biggl [\lambda _{\nu }^{\mathcal {R}^{(j)}} \text {V}_{\nu }^{\mathcal {R}^{(j)}} +\lambda _{\tau }^{\mathcal {R}^{(i)}}\text {V}_{\tau }^{\mathcal {R}^{(i)}}\biggr ]\biggr )\\ exp\biggl (-2\pi \biggl [\lambda _{\nu }^{\mathcal {R}^{(j)}} \text {V}_{\nu }^{\mathcal {R}^{(j)}} +\lambda _{\tau }^{\mathcal {R}^{(i)}}\text {V}_{\tau }^{\mathcal {R}^{(i)}}\biggr ]\biggr ), \end{cases} \tag{32}\end{equation*}$$ where $$\begin{equation*} \text {V}_{\nu }^{\mathcal {R}^{(j)}}=\int _{y_{\nu }^{'}}^{d_{1}} \dfrac {x_{l} dx_{l}}{1+(s\text {T}_{\nu }{P}_{tx}^{\nu })^{-1}\lVert x_{l}\rVert ^{\alpha }} = y_{\nu }^{\frac {2}{\alpha }}\int _{y_{\nu }^{\frac {-2}{\alpha }}}^{d_{1}}\dfrac {a}{1+a^{\frac {\alpha }{2}}}da, \\ \text {V}_{\tau }^{\mathcal {R}^{(i)}} =\int _{d_{1}}^{d_{2}} \dfrac {y_{n} dy_{n}}{1+(s{P}_{tx}^{U})^{-1}\lVert y_{n}\rVert ^{\alpha }} \quad \,\, = y_{\nu }^{\frac {2}{\alpha }}\int _{d_{1}}^{d_{2}}\dfrac {b}{1+b^{\frac {\alpha }{2}}}db, \\ \text {V}_{\nu }^{\mathcal {R}^{(j)}}=\int _{y_{\nu }^{''}}^{d_{1}} \dfrac {y_{n}}{1+(s{P}_{tx}^{U})^{-1}y_{n}^{\alpha }} dy_{n}\quad = y_{\nu }^{\frac {2}{\alpha }}\int _{x_{l}^{\frac {-2}{\alpha }}}^{d_{2}}\dfrac {c}{1+c^{\frac {\alpha }{2}}}dc, \end{equation*}$$ and $$\begin{equation*} \text {V}_{\tau }^{\mathcal {R}^{(i)}}=\int _{d_{1}}^{d_{2}} \dfrac {x_{l}}{1+(s\text {T}_{\tau }{P}_{tx}^{\tau })^{-1}x_{l}^{\alpha }} dx_{l}= y_{\nu }^{\frac {2}{\alpha }}\int _{d_{1}}^{d_{2}}\dfrac {f}{1+f^{\frac {\alpha }{2}}}df. \end{equation*}$$ Substituting (32) into (26), coverage probability is obtained as (33), as shown at the top of this page.

By substituting (33) into (25) completes the proof of (22).