Stochastic geometry modeling and energy efficiency analysis of millimeter wave cellular networks

Abstract

Current wireless networks face unprecedented challenges because of the exponentially increasing demand for mobile data and the rapid growth in infrastructure and power consumption. This study investigates the optimal energy efficiency of millimeter wave (mmWave) cellular networks, given that these networks are some of the most promising 5G-enabling technologies. Based on the stochastic geometry, a mathematical framework of coverage probability is proposed and the optimal energy efficiency is obtained with coverage performance constraints. Numerical results show that increasing the base station density damages coverage performance exceeding the threshold. This work demonstrates an essential understanding of the deployment and dynamic control of energy-efficient mmWave cellular networks.

Appendix

1.1 Derivation of laplace transform of interference

The derivation process is shown in as follows:

$$\begin{aligned} {\mathcal {L}}_{I_L}(s) \mathop {=}\limits ^{\triangle }&{\mathbb {E}}\left[ {\mathrm{e}}^{-s(I_L+I_N) }\right] \\ \mathop {=}\limits ^{(a)}&{\mathbb {E}}\left[ {\mathrm{e}}^{-sI_L}\right] {\mathbb {E}}\left[ {\mathrm{e}}^{-sI_N}\right] \\ \mathop {=}\limits ^{(b)}&{\mathbb {E}}\left[ exp\left( -s\sum _{i\in \phi _{b_L} \backslash b_u} P_t g_i D_i \ell _L(r_i) \right) \right] \\&{\mathbb {E}}\left[ exp\left( -s\sum _{i\in \phi _{b_N} \backslash b_u} P_t g_i D_i \ell _N(r_i) \right) \right] \\ \mathop {=}\limits ^{(c)}&{\mathbb {E}}\left[ \prod _{i\in \phi _{b_L} \backslash b_u}{\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i D_i \ell _L(r_i)} \right] \right] \\&{\mathbb {E}}\left[ \prod _{i\in \phi _{b_N} \backslash b_u}{\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i D_i \ell _N(r_i)} \right] \right] \\ \mathop {=}\limits ^{(d)}&exp\left\{ \left( - \sum _{k=1}^{4}b_k \int _{r}^{\infty } \left( 1 - {\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i a_k \ell _L(t)} \right] \right) \right. \right. \\&tdt -\sum _{k=1}^{4}b_k \int _{\psi _L(r)}^{\infty } \left( 1 - {\mathbb {E}}_g\left[ {\mathrm{e}}^{-s P_t g_i a_k \ell _N(t)} \right] \right) \\&\left. \left. tdt \right) 2\pi \lambda _b \right\} \\ \mathop {=}\limits ^{(e)}&exp\left\{ -2\pi \lambda _b \sum _{k=1}^{4}b_k \left( \int _{r}^{\infty } \frac{t}{1+\frac{\tau }{sP_t \ell _L(r)a_k}}dt \right. \right. \\&\left. \left. + \int _{\psi _L(r)}^{\infty } \frac{t}{1+\frac{\tau }{sP_t \ell _N(t)a_k}}dt \right) \right\} \\ =&exp\left\{ -\pi \lambda _b \sum _{k=1}^{4}b_k \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \end{aligned}$$

where (a) is due to the independence of $\varPhi _{b_L}$ and $\varPhi _{b_N}$, (b) (c) follows the independence within the PPP of $\varPhi _{b_L}$ or $\varPhi _{b_N}$. (d) follows the Laplace transform of PPP [32], and (e) follows the integral transform.

1.2 Derivation of conditional coverage probability

For a LOS link with the radius of r, the conditional coverage probability $P_{c,L}(\lambda _b | r)$ can be derived from Eq. (10) by adding the condition of radius of r, it is given as:

$$\begin{aligned} P_{c,L}(\lambda _b | r) =&{\mathrm{e}}^{-sN_u}{\mathbb {E}}\left[ {\mathrm{e}}^{-sI}\right] \\ \mathop {=}\limits ^{(a)}&exp\left\{ -\frac{\tau \rho N_u}{P_t Q_r Q_t}(\mu _L r)^{\alpha _L} -\pi \lambda _b \sum _{k=1}^{4}b_k \right. \\&\left. \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \\ =&exp\left\{ -\frac{\tau \rho N}{P_t Q_r Q_t} \frac{\lambda _b}{\lambda _u}(\mu _L r)^{\alpha _L} -\pi \lambda _b \sum _{k=1}^{4}b_k \right. \\&\left. \left( r^2 I(\rho ,\alpha _L) + \psi _L^2(r)I(\rho ,\alpha _N) \right) \right\} \end{aligned}$$

where (a) follows the Laplace transform of PPP [32].

1.3 Derivation of the SINR coverage probability

When considering blockage effects, the SINR coverage probability should be the weighted summation of LOS and NLOS circumstances, which is given as:

$$\begin{aligned} P_c(\lambda _b) \mathop {=}\limits ^{(a)}&A_L P_{c,L}(\lambda _b) + A_N P_{c, N}(\lambda _b) \nonumber \\ \mathop {=}\limits ^{(b)}&A_L \int _0^\infty \hat{f_L} P_{c, L}(\lambda _b|r) dr + A_N \int _0^\infty \hat{f_N}P_{c, N}(\lambda _b|r)dr \end{aligned}$$

According to the Ref. [6], these equations can be introduced as follows:

$$\begin{aligned} A_L= & {} \int _{0}^{\infty }{\mathrm{e}}^{-2\pi \lambda \int _{0}^{\psi _{L}(r)}t(1-p(t))dt }f_L(r)dr\\ {\hat{f}}_L(r)= & {} \frac{B_L f_L(r)}{A_L} {\mathrm{e}}^{-2\pi \lambda _b \int _{0}^{\psi _L(r)} (1-p(t))tdt}\\ {\hat{f}}_N(r)= & {} \frac{B_N f_L(r)}{A_N} {\mathrm{e}}^{-2\pi \lambda _b \int _{0}^{\psi _N(r)} p(t)tdt} \end{aligned}$$

where $A_L$ is the probability that a user is associated with an LOS BS. The probability density functions of the distance to a serving BS are denoted as $\hat{f_L}$ and $\hat{f_N}$ at the conditions that the typical user is associated with a LOS BS and a NOS BS respectively. (a) follows the weighted summation of LOS and NLOS circumstances, $P_{c,L}(\lambda _b)$ and $P_{c, N}(\lambda _b)$ are the SINR probabilities under LOS and NLOS circumstances, respectively. The probability that a user is associated with an NLOS BS is denoted as $A_N$. Then $A_N=1-A_L$. (b) follows the expansion with conditional probability density function.