Nearest Distance Based Downlink Rate Modeling in Hyper-dense Heterogeneous Wireless Cell Networks

Abstract

Different kinds of low transmit power, flexible cell extension and hyper dense base stations (BSs) overlaying on the well established macro cells, forming the hyper-dense multilayer heterogeneous wireless cell networks (HetNets), where the BSs in each tier are characterized by the differences in BSs transmit power, path loss exponent, tier-biasing factor and BSs spatial density, is the promising technology for 5th generation mobile networks and to support hot spots and cell edge coverage. The HetNets downlink rate modeling is one of the key and challenge problems for the random deployment characteristics of BSs in lower layer. In this paper, a three tier HetNets, which includes macro tier, Pico tier and Femto tier, is first modeled with the spatial point process in stochastic geometry theory; then, considering the BSs low transmit power, flexible cell extension and hyper BSs spatial density, a tractable downlink rate model based on the nearest distance and bias accepted factor is proposed, which adopt stochastic geometry theory and is easy to compute; last, the proposed downlink rate model is validated by numerical examples to confirm that the flexible cell extension BSs deployment in complex high traffic area with different path loss exponent is an effective scheme to improve quality of service.

Appendix

1.1 Proof of Theorem

In this appendix, we will provide the proof of theorem in part three and establish the average communication rate model of $j_{ 0} {\text{th}}$ tier in HetNets based on the nearest distance, according to the Shannon formula in information theory. Denoting the communication rate in $j_{ 0} {\text{th}}$ tier as $\tau_{j}$, which is the mean over distance $r$ and distributed channel power.

$$\begin{aligned} \tau_{{j_{0} }} & \triangleq {\mathbb{E}}\left[ {\ln \left( {1 + SINR} \right)} \right] \\ \text{ } & = \int_{r \succ 0} {{\mathbb{E}}\left[ {\ln \left( {1 + \frac{{P_{{j_{0} }} g_{{j_{0} }} r^{{ - \alpha_{{j_{0} }} }} }}{{\sum\limits_{\kappa = 1,}^{K} {\sum\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }}^{{}} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } + W/L_{0} } }}} \right)} \right]f_{r} \left( r \right)} dr \\ \end{aligned}$$

(7)

where $SINR$ is the ratio of signal to interference noise, which could be obtained from (4) in Lemma 1, $f_{r} \left( r \right)$ is the distribution function of the nearest between user and BS, which comes from (5) in Lemma 2, $B_{{j_{0} }}$ denotes the base station which the user associates with in $j_{ 0} {\text{th}}$ tier. According to ${\mathbb{E}}\left( X \right) = \int_{0}^{\infty } {{\mathbb{P}}\left[ {X > x} \right]dx}$ for $X > 0$, we could obtain,

$$\begin{aligned} \text{ }\tau_{{j_{0} }} & = \int_{r \succ 0} {2\lambda_{{j_{0} }} \pi re^{{ - \lambda_{{j_{0} }} \pi r^{2} }} {\mathbb{E}}\left[ {\ln \left( {1 + \frac{{P_{{j_{0} }} g_{{j_{0} }} r^{{ - \alpha_{{j_{0} }} }} }}{{\sum\limits_{\kappa = 1,}^{K} {\sum\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }}^{{}} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } + W/L_{0} } }}} \right)} \right]} dr \\ & \text{ } = \int_{r \succ 0} {2\lambda_{{j_{0} }} \pi re^{{ - \lambda_{{j_{0} }} \pi r^{2} }} \int_{t \succ 0} {{\mathbb{P}}\left( {\ln \left( {1 + \frac{{P_{{j_{0} }} g_{{j_{0} }} r^{{ - \alpha_{{j_{0} }} }} }}{{\sum\limits_{\kappa = 1,}^{K} {\sum\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }}^{{}} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } + W/L_{0} } }}} \right) \succ t} \right)} dt} dr \\ & = \int_{r \succ 0} {2\lambda_{{j_{0} }} \pi re^{{ - \lambda_{{j_{0} }} \pi r^{2} }} \int_{t \succ 0} {{\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right)}dt} dr \\ \end{aligned}$$

(8)

In (8), $Q = \sum\limits_{\kappa = 1}^{K} {\sum\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } + W/L_{0} }$ is the sum of interference and noise. For ${\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right)$, we could deduced it as follows,

$$\begin{aligned} {\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right) & = \int_{0}^{\infty } {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)q} \right)f_{Q} \left( q \right)dq} \\ & \text{ } = E_{Q} \left\{ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)q} \right)} \right\} \\ & = E_{Q} \left\{ {\exp \left[ { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\left( {\sum\limits_{\kappa = 1}^{K} {I_{\kappa } } + W/L_{0} } \right)} \right]} \right\} \\ \end{aligned}$$

(9)

$$\begin{aligned} & \text{ = }E_{Q} \left\{ {\exp \left[ { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{\kappa = 1}^{K} {I_{\kappa } } - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)W/L_{0} } \right]} \right\} \\ & \text{ = }\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)W/L_{0} } \right)E_{Q} \left\{ {\exp \left[ { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{\kappa = 1}^{K} {I_{\kappa } } } \right]} \right\} \\ \text{ } & \text{ = }\exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{{{\raise0.7ex\hbox{${P_{{j_{0} }} L_{0} r^{{\alpha_{{j_{0} }} }} }$} \!\mathord{\left/ {\vphantom {{P_{{j_{0} }} L_{0} r^{{\alpha_{{j_{0} }} }} } W}}\right.\kern-0pt} \!\lower0.7ex\hbox{$W$}}}}} \right)E_{Q} \left\{ {\exp \left[ { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{\kappa = 1}^{K} {I_{\kappa } } } \right]} \right\} \\ & \text{ = }\exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{SNR}} \right)E_{Q} \left\{ {\exp \left[ { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{\kappa = 1}^{K} {I_{\kappa } } } \right]} \right\} \\ \end{aligned}$$

(10)

$$\begin{aligned} & = \exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{SNR}} \right)E_{Q} \left\{ {\prod\limits_{\kappa = 1}^{K} {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)I_{\kappa } } \right)} } \right\} \\ & = \exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{SNR}} \right)\prod\limits_{\kappa = 1}^{K} {{\mathbb{E}}_{{I_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)I_{\kappa } } \right)} \right]} \\ \end{aligned}$$

(11)

In (9), $I_{\kappa } = \sum\limits_{{i \in \phi_{\kappa } /B_{j0} }} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} }$ denotes the interference from BSs in kth tier; $SNR = \frac{{P_{{j_{0} }} L_{0} r^{{\alpha_{{j_{0} }} }} }}{W}$ in (10) is the signal to noise ration of user in jth tier. ${\mathbb{E}}_{{I_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)I_{\kappa } } \right)} \right]$ in (11) could be deduced as follows, $\begin{aligned} & {\mathbb{E}}_{{I_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)I_{\kappa } } \right)} \right] \\ \text{ } & \quad \text{ } = {\mathbb{E}}_{{\phi_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }}^{{}} {P_{\kappa } h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } } \right)} \right] \\ \text{ } & \quad \text{ } = {\mathbb{E}}_{{\phi_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} P_{\kappa } r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)\sum\limits_{{i \in \phi_{\kappa } /B_{j0} }}^{{}} {h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } } \right)} \right] \\ \text{ } & \quad \text{ } = {\mathbb{E}}_{{\phi_{\kappa } }} \left[ {\prod\limits_{{i \in \phi_{\kappa } /B_{{j_{0} }} }} {\exp \left( { - P_{{j_{0} }}^{ - 1} P_{\kappa } r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)h_{\kappa i} \left| {Y_{\kappa i} } \right|^{{ - \alpha_{\kappa } }} } \right)} } \right] \\ \end{aligned}$

$$= \exp \left\{ {1 - 2\pi \lambda_{\kappa } \int_{0}^{\infty } {\left[ {1 - {\mathbb{E}}_{{\phi_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} P_{\kappa } r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)h_{\kappa i} \left| y \right|^{{ - \alpha_{\kappa } }} } \right)} \right]} \right]ydy} } \right\}$$

(12)

$$\begin{aligned} & = \exp \left\{ {1 - 2\pi \lambda_{\kappa } \int_{{z_{k} }}^{\infty } {\left[ {1 - \int_{0}^{\infty } {\exp \left( { - P_{{j_{0} }}^{ - 1} P_{\kappa } r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)h_{\kappa i} \left| y \right|^{{ - \alpha_{\kappa } }} } \right)} f\left( {h_{\kappa i} } \right)dh_{\kappa i} } \right]ydy} } \right\} \\ \text{ } & \text{ = }\exp \left\{ {1 - 2\pi \lambda_{\kappa } \int_{{z_{k} }}^{\infty } {\left[ {1 - \frac{ 1}{{r^{{\alpha_{{j_{0} }} }} \hat{P}_{\kappa } \left( {e^{t} - 1} \right)y^{{ - \alpha_{\kappa } }} { + 1}}}} \right]ydy} } \right\} \\ \end{aligned}$$

(13)

$$\begin{aligned} & \text{ } = \exp \left\{ {1 - 2\pi \lambda_{\kappa } \int_{{z_{\kappa } }}^{\infty } {\frac{y}{{ 1 { + }\left( {r^{{\alpha_{{j_{0} }} }} \hat{P}_{\kappa } \left( {e^{t} - 1} \right)} \right)^{ - 1} y^{{\alpha_{\kappa } }} }}dy} } \right\} \\ & = \exp \left\{ {1 - \pi \lambda_{\kappa } r^{{\frac{2}{{\hat{\alpha }_{\kappa } }}}} \left( {\hat{P}_{\kappa } \left( {e^{t} - 1} \right)} \right)^{{\frac{2}{{\alpha_{\kappa } }}}} \int_{{\left( {{\raise0.7ex\hbox{${\hat{\delta }_{\kappa } }$} \!\mathord{\left/ {\vphantom {{\hat{\delta }_{\kappa } } \tau }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\tau $}}} \right)^{{\frac{2}{{\alpha_{\kappa } }}}} }}^{\infty } {\frac{1}{{ 1 { + }\mu^{{{\raise0.7ex\hbox{${\alpha_{\kappa } }$} \!\mathord{\left/ {\vphantom {{\alpha_{\kappa } } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}dy} } \right\} \\ \text{ } & \text{ = }\exp \left\{ {1 - \pi \lambda_{\kappa } r^{{\frac{2}{{\hat{\alpha }_{\kappa } }}}} \hat{P}_{\kappa }^{{\frac{2}{{\alpha_{\kappa } }}}} \mathcal{Z}\left( {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right)} \right\} \\ \end{aligned}$$

(14)

In (12) ${\mathbb{E}}\left\{ {\prod\limits_{x \in \phi } {f\left( x \right)} } \right\} = \exp \left\{ {1 - \lambda \int_{{R^{2} }} {\left( {1 - f\left( x \right)} \right)dx} } \right\}$ follows from the probability generating functional of the PPP, which states for some function $f\left( x \right)$ that ${\mathbb{E}}\left\{ {\prod\limits_{x \in \phi } {f\left( x \right)} } \right\} = \exp \left\{ {1 - \lambda \int_{{R^{2} }} {\left( {1 - f\left( x \right)} \right)dx} } \right\}$ [5]; $Z_{\kappa } = \left( {\hat{P}_{\kappa } \hat{\delta }_{\kappa } } \right)^{{1/\alpha_{\kappa } }} r^{{\hat{\alpha }_{\kappa } }}$ in (13) denotes the at least distance of the closet interference between tagged user and inference BSs in $\kappa$ tier [8], and in (14), $\mathcal{Z}\left[ {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right] = \left( {e^{t} - 1} \right)^{{1/\alpha_{\kappa } }} \int_{{\left[ {\hat{\delta }_{\kappa } /\left( {e^{t} - 1} \right)} \right]^{{2/\alpha_{\kappa } }} }}^{\infty } {\frac{1}{{1 + \mu^{{\alpha_{\kappa } /2}} }}} d\mu$. Plugging (14) in (11), then ${\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right)$ is,

$$\begin{aligned} {\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right) & = \int_{0}^{\infty } {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)q} \right)f_{Q} \left( q \right)dq} \\ & \text{ } = \exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{SNR}} \right)\prod\limits_{\kappa = 1}^{K} {{\mathbb{E}}_{{I_{\kappa } }} \left[ {\exp \left( { - P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} \left( {e^{t} - 1} \right)I_{\kappa } } \right)} \right]} \\ & \text{ } = \exp \left( { - \frac{{\left( {e^{t} - 1} \right)}}{SNR}} \right)\prod\limits_{\kappa = 1}^{K} {\exp \left\{ {1 - \pi \lambda_{\kappa } r^{{\frac{2}{{\hat{\alpha }_{\kappa } }}}} \hat{P}_{\kappa }^{{\frac{2}{{\alpha_{\kappa } }}}} \mathcal{Z}\left( {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right)} \right\}} \\ & \text{ } = \exp \left\{ {1 - \frac{{\left( {e^{t} - 1} \right)}}{SNR} - \sum\limits_{\kappa = 1}^{K} {\pi \lambda_{\kappa } r^{{\frac{2}{{\hat{\alpha }_{\kappa } }}}} \hat{P}_{\kappa }^{{\frac{2}{{\alpha_{\kappa } }}}} \mathcal{Z}\left( {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right)} } \right\} \\ \end{aligned}$$

(15)

substituting the (15) formula into the (9) form, then the communication rate $\tau_{{j_{0} }}$ in jth tier is,

$$\begin{aligned} \tau_{j} & \triangleq {\mathbb{E}}\left[ {\ln \left( {1 + SINR} \right)} \right] \\ & \text{ } = \int_{r \succ 0} {2\lambda_{{j_{0} }} \pi re^{{ - \lambda_{{j_{0} }} \pi r^{2} }} \int_{t \succ 0} {{\mathbb{P}}\left( {g_{{j_{0} }} \succ P_{{j_{0} }}^{ - 1} r^{{\alpha_{{j_{0} }} }} Q\left( {e^{t} - 1} \right)} \right)} {\ominus }dt} dr \\ & \text{ } = \int_{r \succ 0} {2\lambda_{{j_{0} }} \pi re^{{ - \lambda_{{j_{0} }} \pi r^{2} }} \int_{0}^{\infty } {\exp \left\{ {1 - \frac{{\left( {e^{t} - 1} \right)}}{SNR} - \sum\limits_{\kappa = 1}^{K} {\pi \lambda_{\kappa } r^{{\frac{2}{{\hat{\alpha }_{\kappa } }}}} \hat{P}_{\kappa }^{{\frac{2}{{\alpha_{\kappa } }}}} \mathcal{Z}\left( {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right)} } \right\}} dt} dr \\ \end{aligned}$$

and,

$$\begin{aligned} \mathcal{Z}\left( {\left( {e^{t} - 1} \right),\alpha_{\kappa } ,\hat{\delta }_{\kappa } } \right) & & = \left( {e^{t} - 1} \right)^{{2/\alpha_{\kappa } }} \int_{{\left[ {{\raise0.7ex\hbox{${\hat{\delta }_{\kappa } }$} \!\mathord{\left/ {\vphantom {{\hat{\delta }_{\kappa } } {\left( {e^{t} - 1} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {e^{t} - 1} \right)}$}}} \right]^{{2/\alpha_{\kappa } }} }}^{\infty } {\frac{1}{{1 + \mu^{{\alpha_{\kappa } /2}} }}d\mu } \\ \text{ } & \text{ } = \frac{{2\left( {e^{t} - 1} \right)\hat{\delta }_{\kappa }^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {\alpha_{\kappa } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\alpha_{\kappa } }$}} - 1}} }}{{\alpha_{\kappa } - 2}}{}_{2}F_{1} \left[ {1,\text{ }1 - \frac{2}{{\alpha_{\kappa } }};\text{ }1 - \frac{2}{{\alpha_{\kappa } }};\text{ }\frac{{1 - e^{t} }}{{\hat{\delta }_{\kappa } }}} \right] \\ \end{aligned}$$

where ${}_{2}F_{1} \left[ {1,\text{ }1 - \frac{2}{{\alpha_{\kappa } }};\text{ }1 - \frac{2}{{\alpha_{\kappa } }};\text{ }\frac{{1 - e^{t} }}{{\hat{\delta }_{\kappa } }}} \right]$ is a hyper geometry function.

And the proof is completed.