#### 2.1. Two-Tier Heterogeneous Network Model

We consider a downlink two-tier HetNet based on the orthogonal frequency-division multiple-access (OFDMA) technique; that is, the intra-cell interference is not considered. In this paper, the HetNets consist of open access MBSs and FBSs, also termed "tier 1" and "tier 2", respectively. The locations of the BSs in the ith tier are modeled as a two-dimensional homogeneous PPP (HPPP), ${\mathrm{\Phi}}_{i}$, with density ${\lambda}_{i}$. Each BS in the ith tier transmits with the same power ${P}_{i}$. Furthermore, the users are located according to another HPPP, ${\mathrm{\Phi}}_{u}$, with density ${\lambda}_{u}$, which is independent of ${\mathrm{\Phi}}_{i}$.

Without any loss of generality, we consider a typical user located at the origin of the coordinate system. For tractability, the standard power loss propagation model is applied in the ith tier with a path-loss exponent of ${\alpha}_{i}>2$. As far as random channel fluctuations, Rayleigh fading with a unit mean (denoted as ${h}_{x}\sim \mathrm{exp}\left(1\right)$) is applied at each channel. The noise is assumed to be additive with power ${\sigma}^{2}$.

#### 2.2. Location-Aware Cross-Tier Cooperation Scheme

For a typical user located at the origin, we let ${R}_{i}$ denote the distance of the typical user from their nearest BS in the ith tier. ${MBS}_{0}$ and ${FBS}_{0}$ are denoted as the nearest BSs in each tier, which can provide the maximum long-term received power from each tier.

We consider a user association scheme on the basis of each user’s location information. The macro tier has an inner region ${\mathcal{A}}_{in}$, which is defined as the union of locations for which the distance to the nearest MBS is no larger than a prescribed value D, whereas the outer region is given by ${\mathcal{A}}_{out}={\mathbb{R}}^{2}\backslash {\mathcal{A}}_{in}$. The FBSs within ${\mathcal{A}}_{in}$ are deactivated because of their poor offloading effect from the MBSs. The users in ${\mathcal{A}}_{in}$, that is, macro inner users (MIUs), consequently are only served by an MBS in this region.

We denote

$\mathcal{B}$ as the serving BS set of the typical user, which can be expressed as follows:

where

D is the radius of inner region

${\mathcal{A}}_{in}$, and

$\beta \ge 0$ dB is the cooperation threshold for tier 2. Without any loss of generality, we let

${\beta}_{i}$ denote the cooperation threshold for the

ith tier. For simplicity, it is assumed that the cooperation threshold for tier 1 is the unity (

${\beta}_{1}=0\phantom{\rule{4.pt}{0ex}}$ dB) in this paper, while

${\beta}_{2}=\beta \ge 0$ dB for notational brevity. Hence, the typical user

$u\in U$ can lie in one of the following four disjoint sets:

which satisfy

${U}_{M,in}\cup {U}_{M,out}\cup {U}_{F}\cup {U}_{C}=U$. Clearly, the set

${U}_{M,in}$ is the set of MIUs and the set

${U}_{M,out}$ is the set of macro users within the outer region. The set

${U}_{F}$ is the set of femto non-CoMP users, which is independent of the cooperation threshold. The set

${U}_{C}$ is the set of femto CoMP users who are liable to experiencing excessive cross-tier interference. For brevity, we define a mapping

$\mathcal{K}:\left\{M,in;\phantom{\rule{4.pt}{0ex}}M,out;\phantom{\rule{4.pt}{0ex}}F;\phantom{\rule{4.pt}{0ex}}C\right\}\to \left\{1,\phantom{\rule{4.pt}{0ex}}2\right\}$ from the user-set index to the serving tier index, that is,

$\mathcal{K}\left(M,in\right)=\mathcal{K}\left(M,out\right)=1$,

$\mathcal{K}\left(F\right)=2$ and

$\mathcal{K}\left(C\right)=\left\{1,\phantom{\rule{4.pt}{0ex}}2\right\}$.

To elaborate, the proposed LA-CTC scheme is shown in

Figure 1. User 1 located in the inner region

${\mathcal{A}}_{in}$ of

${MBS}_{0}$, and is only associated with

${MBS}_{0}$.

${MBS}_{0}$ is close enough to guarantee the user 1’s quality of service (QoS). On the other hand, a user located in the outer region

${\mathcal{A}}_{out}$ of

${MBS}_{0}$ will be associated with

${MBS}_{0}$,

${FBS}_{0}$ or two tiers of BSs. User 2’s received power from

${MBS}_{0}$ is stronger than that from

${FBS}_{0}$ plus the cooperation threshold, and user 3’s received power from

${FBS}_{0}$ is stronger than that from

${MBS}_{0}$; thus user 2 and user 3 are only associated with

${MBS}_{0}$ and

${FBS}_{0}$, respectively. Although user 4’s received power from

${MBS}_{0}$ is larger than that from

${FBS}_{0}$, the received power from

${FBS}_{0}$ plus the cooperation threshold is larger. User 4 will be cooperatively served by

${MBS}_{0}$ and

${FBS}_{0}$ by means of jointly transmitting the user’s data; hence the user can be referred to as the CoMP user. Note that the prescribed value

D is decided by the MBSs. In the current operation of cellular networks, a user will periodically feed back the measurement reports, including the user’s location information, to its serving BS for assisting the serving BS selection procedure [

35]. Thus the MBSs classify the users as MIUs and MEUs through this procedure. Meanwhile, the FBSs use the pilot signals combined with a positive cooperation threshold to convince the vulnerable users to connect to both tiers of the BSs. The BSs use backhaul links (e.g., digital subscriber line (DSL)) [

3,

36] to exchange/share the users’ data and the control information, namely, the prescribed value

D and the cooperation threshold. It is assumed that the users are stationary in this paper, that is, handoffs (HOs) do not occur for the BSs. This assumption is widely adopted in the extensive literature. When taking the mobility of users into account, the HO procedure will occur between different BSs such that different user association schemes are adopted. However, the high mobility users, such as users driving cars on the highways or taking high-speed trains, will have higher HO interruptions, which will degrade the network performance. To mitigate the HO effect, a location-aware HO skipping scheme for the macro tier can be adopted, which is similar to the scheme proposed in [

37]. Additionally, our proposed LA-CTC scheme should be revised. In detail, it is assumed that the trajectory within the target MBS footprint is a priori via some trajectory estimation techniques [

38,

39]. Thus the HO skips associating to the target MBS if the user trajectory passes through the outer region of its target MBS; that is, when the minimum distance between the user trajectory and its target MBS exceeds the same predefined threshold value

D, the HO occurs. However, the high-mobility users will skip HOs to the entire femto tier to avoid the excessive HOs because the FBS coverage area is too small [

40]. The high-mobility users will be served directly by their un-skipped MBSs which are their trajectories pass through the inner region of the target MBSs.

We let ${\mathcal{Q}}_{M,in}$, ${\mathcal{Q}}_{M,out}$, ${\mathcal{Q}}_{F}$ and ${\mathcal{Q}}_{C}$ denote the probabilities that a typical user belongs to each of the above four disjoint user sets, respectively. Mathematically, this is ${\mathcal{Q}}_{l}=\mathbb{P}\left(u\in \phantom{\rule{4.pt}{0ex}}{U}_{l}\right)$, where $l\in \left\{M,in;\phantom{\rule{4.pt}{0ex}}M,out;\phantom{\rule{4.pt}{0ex}}F;\phantom{\rule{4.pt}{0ex}}C\right\}$. On the basis of the ergodicity of the PPP, these probabilities are derived in the following lemma.

**Lemma** **1.** ${\mathcal{Q}}_{M,in}$, ${\mathcal{Q}}_{M,out}$, ${\mathcal{Q}}_{F}$ and ${\mathcal{Q}}_{C}$ are given as follows: If the path loss exponents are equal, that is, ${\alpha}_{i}\equiv \alpha $, ${\mathcal{Q}}_{M,out}$, then ${\mathcal{Q}}_{F}$ can be reduced to a closed-form expression as:where ${\lambda}_{1}^{\prime}={\lambda}_{1}+{\lambda}_{2}{\left(\frac{\beta {P}_{2}}{{P}_{1}}\right)}^{2/\phantom{2\alpha}\phantom{\rule{0.0pt}{0ex}}\alpha}$ and ${\lambda}_{1}^{\u2033}={\lambda}_{1}+{\lambda}_{2}{\left(\frac{{P}_{2}}{{P}_{1}}\right)}^{2/\phantom{2\alpha}\phantom{\rule{0.0pt}{0ex}}\alpha}$.