Investigation on the Critical Densification Levels for Coupled and Decoupled User Association in Ultra-dense Networks

Abstract

Network densification and heterogeneity has attracted attention as an enabling technology for Fifth Generation (5G) communications due to the potential to enhance capacity using aggressive spatial spectrum reuse and flexibility for deployment. In the framework of Heterogeneous Networks (HetNets), densification is heavy on the pico- or femto-tiers. Therefore, the relative intensity of nodes at each tier impacts the network performance added to the different transmit powers. It could be asked for which densification levels and relative intensity of nodes can we use aggressive offloading with the established interference coordination techniques or decoupled association? In this paper, the concept of Poisson random networks were used to analytically obtain the relative densification levels corresponding to fair load distributions across tiers and intensity levels for which we need the coupled or decoupled User Association UA. The association window, where users choose to use decoupled association in terms of the relative intensity, transmit powers at each tiers and the path loss exponent of the propagation environment, is derived. Further, the ergodic rate expressions in order to study throughput performances in different densification regions, which can be computed numerically, are formulated. To validate the theoretical analysis, numerical, system level simulation and realistic network analysis were used. The analytical, simulation, and realistic test case results provide insights for the operators about the densification ranges, where to use coupled or decoupled association.

Appendix

1.1 Proof of Lemma 3—UL Ergodic Rates

When a typical UE is associated to the MC in the UL, the ergodic rate is given by:

$$\begin{aligned} \begin{aligned} R_{UL}^m =&\frac{1}{\ln (2)}\textbf{E}_{r,\psi }[\ln (1 + \psi _{UL}^m)]\\ =&\frac{1}{\ln (2)}\int _{0}^{\infty }\textbf{E}_\psi [\ln (1 + \frac{P_uh_mr^{-\gamma _m}}{I})]\cdot f(r,1)dr, \end{aligned} \end{aligned}$$

(21)

where $I = \sum _{k \in \Phi _u\setminus u} P_ug_kx^{-\gamma _k}$ is the interference from users except the typical user at the origin and f(r, 1)dr is the distance distribution of the serving node given in (1). The expectation of the spectral efficiency term in right-hand side (RHS) of (21) can be obtained as in [23].

$$\begin{aligned} \begin{aligned} R_{UL}^{m*}=&\textbf{E}_\psi [\ln (1 + \frac{P_uh_mr^{-\gamma _m}}{I})] \\ =&\int _{0}^{\infty }\textbf{P}\{\ln (1 + \frac{P_uh_mr^{-\gamma _m}}{I})> y\}dy\\ =&\int _{0}^{\infty }\textbf{P}\{h_m > IP_u^{-1}r^{\gamma _m}(e^y - 1)\}dy \\&{\mathop {=}\limits ^{a}} \int _{0}^{\infty }\exp \{- \mu IP_u^{-1}r^{\gamma _m}(e^y - 1)\}dy\\ =&\int _{0}^{\infty }\mathcal {L}_I\{\mu P_u^{-1}r^{\gamma _m}(e^y - 1)\} dy \end{aligned} \end{aligned}$$

(22)

Where (a) follows from the exponentially distributed $h_m$ with mean $1/\mu$ and the Laplace Transform (LT) of the interference can be expressed as:

$$\begin{aligned} \begin{aligned}&\mathcal {L}_I(s) = \textbf{E}_{\Phi _u,g_k}[e^{-sI}] \\&\quad = \textbf{E}_{\Phi _u,g_k}[\exp \{-s\sum _{k \in \Phi _u\setminus u} P_ug_kx^{-\gamma _k}\}]\\&\quad = \textbf{E}_{\Phi _u,g_k}[\prod _{k \in \Phi _u\setminus u}\exp \{-s P_ug_kx^{-\gamma _k}\}]\\&\quad {\mathop {=}\limits ^{a}} \textbf{E}_{\Phi _u}[\prod _{k \in \Phi _u\setminus u} \textbf{E}_{g_k}[\exp \{-s P_uh_kx^{-\gamma _k}\}]] \end{aligned} \end{aligned}$$

(23)

(a) follows from the independence between $\Phi _u$ and $g_k$. With help of Probability Generating Functional (PGFL) [24] and [25] of the PPP, which states for some function f(x) that $\textbf{E}[\prod _{x\in \Phi }f(x)] = \exp \{-\lambda \int _{R^2}(1-f(x))dx\}$, the equation in (23) becomes:

$$\begin{aligned} \begin{aligned}&\mathcal {L}_I(s) = \textbf{E}_{\Phi _u,g_k}[e^{-sI}] \\&\quad = \exp \{-2\pi \lambda _u \int _{r}^{\infty }(1-\textbf{E}_{g_k} \left[\exp \{-s P_ug_kx^{-\gamma _k}\}\right])xdx\}\\&\quad {\mathop {=}\limits ^{a}} \exp \{-2\pi \lambda _u^* \int _{r}^{\infty } \left(1-\frac{\mu }{sP_ux^{-\gamma _k} + \mu }\right)xdx\} \end{aligned} \end{aligned}$$

(24)

Where (a) follows from exponential distribution of $g_k$. Substituting $s = \mu P_u^{-1}r^{\gamma _m}(e^y - 1)$ and putting (24) in (22) and (21) with simplification gives the result.

The same procedure can be followed to obtain the ergodic user rate when a typical UE is associated to the LPN in the UL, the ergodic rate is given by (19).

1.2 Proof of Lemma 4—DL Ergodic Rates

When a typical UE is associated to the MC in the DL, the ergodic rate is given by:

$$\begin{aligned} \begin{aligned} R_{DL}^m =&\frac{1}{\ln (2)}\textbf{E}_{r,\psi }[\ln (1 + \psi _{DL}^m)]\\ =&\frac{1}{\ln (2)}\int _{0}^{\infty }\textbf{E}_\psi [\ln (1 + \frac{P_mh_mr^{-\gamma _m}}{I})]\cdot f(r,1)dr, \end{aligned} \end{aligned}$$

(25)

where $I = \sum _{k \in \Phi _m\setminus m} P_mg_kr^{-\gamma _k} + \sum _{k \in \Phi _l} P_lg_kr^{-\gamma _k}$ is the interference from MCs and LPNs to a typical user at the origin which being served by MC m and f(r, 1)dr is the distance distribution of the serving node. The expectation of the spectral efficiency term in RHS of (25) can be obtained as follows.

$$\begin{aligned} \begin{aligned} R_{DL}^{m*}=&\mathbf {E}_\psi \left[\ln \left(1 + \frac{P_mh_mr^{-\gamma _m}}{I}\right)\right]\\ =&\int _{0}^{\infty }\mathbf {P} \{\ln \left(1 + \frac{P_mh_mr^{-\gamma _m}}{I}\right)> y\}dy\\ =&\int _{0}^{\infty }\mathbf {P}\{h_m > IP_m^{-1}r^{\gamma _m}(e^y - 1)\}dy \\&{\mathop {=}\limits ^{a}} \int _{0}^{\infty }\exp \{- \mu IP_m^{-1}r^{\gamma _m}(e^y - 1)\}dy\\ =&\int _{0}^{\infty }\mathcal {L}_I\{\mu P_m^{-1}r^{\gamma _m}\left(e^y - 1\right)\} dy, \end{aligned} \end{aligned}$$

(26)

where (a) follows from the exponentially distributed $h_m$ with mean $1/\mu$. The LT of the interference can be expressed as:

$$\begin{aligned} \begin{aligned} \mathcal {L}_I(s) =&\mathbf {E}_{\Phi _m,\Phi _l,g_k}\left[e^{-sI}\right]\\ =&\mathbf {E}_{\Phi _m,\Phi _l,g_k}[\exp \{-s(\sum _{k \in \Phi _m\setminus m} P_mg_kr^{-\gamma _k}\\ +&\sum _{k \in \Phi _l} P_lg_kr^{-\gamma _k})\}]\\ =&\mathbf {E}_{\Phi _m,\Phi _l,g_k}[\prod _{k \in \Phi _m\setminus m}\exp \{-s P_mg_kr^{-\gamma _k}\}\\ \times&\prod _{k \in \Phi _l}\exp \{-s P_lg_kr^{-\gamma _k}\}]\\&{\mathop {=}\limits ^{a}} \mathbf {E}_{\Phi _m}[\prod _{k \in \Phi _m\setminus m} \mathbf {E}_{g_k}[\exp \{-s P_mg_kr^{-\gamma _k}\}]] \\&\times \mathbf {E}_{\Phi _l}[\prod _{k \in \Phi _l} \mathbf {E}_{g_k}[\exp \{-s P_lg_kr^{-\gamma _k}\}]] \end{aligned} \end{aligned}$$

(27)

(a) follows from the independence between $\Phi _u, \Phi _l$ and $h_k$. With help of PGFL [24] and [25] of the PPP, which states for some function f(x) that $\textbf{E}[\prod _{x\in \Phi }f(x)] = \exp \{-\lambda \int _{R^2}(1-f(x))dx\}$, and considering exponential distribution of $g_k$ equation in (27) becomes:

$$\begin{aligned} {\mathcal{L}_I}(s) = & {{\mathbf{E}}_{{\Phi _m},{\Phi _l},{h_k}}}[{e^{ - sI}}] \\ & = \exp \left\{ { - 2\pi {\lambda _m}\int_r^\infty {\left( {1 - \frac{\mu }{{s{P_m}{x^{ - {\gamma _k}}} + \mu }}} \right)xdx} } \right\} \\ & \times \exp \left\{ { - 2\pi {\lambda _l}\int_r^\infty {\left( {1 - \frac{\mu }{{s{P_l}{z^{ - {\gamma _k}}} + \mu }}} \right)zdz} } \right\} \\ \end{aligned}$$

(28)

Substituting $s = \mu P_m^{-1}r^{\gamma _m}(e^y - 1)$ and putting (28) in (26) and (25) with simplification gives the result.

Similarly, the same procedure can be followed to obtain the ergodic user rate when a typical UE is associated to the LPN in the DL, the ergodic rate is given by (20).