Performance Analysis of Energy Harvesting in Dense Urban Millimeter-Wave Cellular Network Using Stochastic Geometry

Abstract

Energy harvesting (EH) in millimeter-wave (mm-wave) cellular networks has recently gained considerable interest due to the extensive use of massive antenna arrays and the dense deployment of base stations (BSs). Solid objects can easily block mm-wave signals, leading to high path losses, which result in non-line-of-sight (NLOS) conditions and signal outages. This paper presents an analytical framework to evaluate the energy coverage probability (ECP) performance of typical user equipment (TUE) in mm-wave cellular networks employing stochastic geometry. We utilize a line-of-sight (LOS) ball model to incorporate the blockage effects and derive a closed-form expression for the ECP. We compare the ECP derived from the LOS ball model with that from random shape blockage models, which are adapted to urban building data for Austin and Los Angeles. We also investigate the impact of varying the LOS ball radius on ECP performance. The derived ECP expression proves analytically tractable, enabling further analysis. The results show that the LOS ball model effectively characterizes the effect of blockages, similar to the random shape model. Furthermore, the findings demonstrate that increasing the density of BSs leads to an increasing trend in the ECP, making the influence of NLOS links negligible compared to that of LOS links. The reason is that dense BS deployment enhances the likelihood of having an LOS link between the BS and the UE. This study provides valuable insights for developing efficient wireless EH systems in mm-wave networks by leveraging higher BS density and enhancing LOS conditions.

Appendices

Appendix A: Proof of Theorem 1

The ECP of the TUE is given by $p(\tau )=\mathbb {P}({\mathcal {E}_h}>\tau )$, where $\tau$ represents a predefined threshold and ${\mathcal {E}_h}$ is defined in (9). Based on (5) and (7), and according to the approximation in (4), which is given in [33], the inequality in $\mathbb {P}({\mathcal {E}_h}>\tau )$ can be approximated as

$$\begin{aligned} p(\tau ) = \sum _{k=0}^{V} (-1)^k \left( {\begin{array}{c}V\\ k\end{array}}\right) \mathbb {E} \left[ \exp \left( -\hat{\mathcal {Y}} \left( P^L + I^L + I^N \right) \right) \right] , \nonumber \\ \end{aligned}$$

(A1)

where $\hat{\mathcal {Y}}=\frac{\mathcal {Y}k}{\tau }$, $\mathcal {Y}={V\left( V!\right) }^{-\frac{1}{V}}$, and to evaluate the expectation in (A1) such that

$$\begin{aligned}&\mathbb {E} \left[ \exp \left( -\hat{\mathcal {Y}} \left( P^L + I^L + I^N \right) \right) \right] \nonumber \\&\, {=} \mathbb {E} \left[ \exp \left( {-}\hat{\mathcal {Y}} P^L \right) \cdot \mathbb {E} \left[ \exp \left( {-}\hat{\mathcal {Y}} I^L \right) \right] \cdot \mathbb {E} \left[ \exp \left( {-}\hat{\mathcal {Y}} I^N \right) \right] \right] . \end{aligned}$$

(A2)

The inner expectation $\mathbb {E}[\exp (-\hat{\mathcal {Y}} I^L)] \cdot \mathbb {E}[\exp (-\hat{\mathcal {Y}} I^N)]$ in (A2) represents the product of the expectations of the LOS and NLOS interference signals. Each of these expectations can be computed separately. The expectation $\mathbb {E}[\exp (-\hat{\mathcal {Y}} I^L)]$ is derived as follows

$$\begin{aligned} f_{I}^L&= \mathbb {E} \left[ \exp \left( -\hat{\mathcal {Y}} I^L \right) \right] \nonumber \\ &{\mathop {=}\limits ^{(a)}} \mathbb {E}_{\phi } \left[ \prod _{i \in \Phi \setminus \{0\}} \mathbb {E}_{H_i^L} \left[ \exp \left( -\hat{\mathcal {Y}} p_t H_i^L D_i C r_i^{-\alpha L} \right) \right] \right] \nonumber \\ &{\mathop {=}\limits ^{(b)}} \exp \left( -\lambda _I \int _{\mathbb {R}^2} \left( 1 - \mathbb {E}_{H_i^L}\right. \right. \nonumber \\ &\quad \left. \left. \left[ \exp \left( -\hat{\mathcal {Y}} p_t H_i^L D_i C r_i^{-\alpha L} \right) \right] \right) r_i \, dr_i \right) \nonumber \\ &{\mathop {=}\limits ^{(c)}} \exp \left( {-}2 \pi \lambda _I \int _{r_0}^{R_B} \left( 1 {-} \mathbb {E}_{H_i^L}\right. \right. \nonumber \\ &\quad \left. \left. \left[ \exp \left( {-}\hat{\mathcal {Y}} p_t H_i^L D_i C r_i^{{-}\alpha L} \right) \right] \right) r_i \, dr_i \right) \nonumber \\ &{\mathop {=}\limits ^{(d)}} \exp \left( -2 \pi \lambda _I \int _{r_0}^{R_B} \left( 1 - \mathbb {E}_{D_i}\right. \right. \nonumber \\ &\quad \left. \left. \left[ \left( \frac{1}{1 + \hat{\mathcal {Y}} p_t D_i C r_i^{-\alpha L} g_L^{-1}} \right) ^{g_L} \right] \right) r_i \, dr_i \right) , \nonumber \\ \end{aligned}$$

(A3)

where (a) is obtained by substituting (7) and simplifying it further. (b) is derived by conditioning on $r_i$ and employing the probability-generating functional (PGFL) of the PPP $\Phi ^L$, where $\lambda _I = \frac{\lambda _{BS}}{\sqrt{n_t}}$ [32]. In (c), we switch to polar coordinates, and (d) is derived using the fact that $H_i^L$ follows a Nakagami distribution. We use the moment-generating function (MGF) of $H_i^L$ with parameter $g_L$, where the MGF of $H_i^L$ is $M_{H_i^L}(t) = \left( 1 - \frac{t}{g_L}\right) ^{-g_L}$, where $t=\hat{\mathcal {Y}} p_t m C r_i^{-\alpha L}$.

In (A3), $\mathbb {E}_{D_i}[\cdot ]$ denotes the expectation of the array gain of the i-th interfering link, which depends on the random variable $D_i$. It is given by

$$\begin{aligned}&\mathbb {E}_{D_i} \left[ \left( \frac{1}{1 {+} \hat{\mathcal {Y}} p_t D_i C r_i^{-\alpha L} g_L^{-1}}\right) ^{g_L}\right] \nonumber \\ &\quad {=} p_1 \left( \frac{1}{1 + \hat{\mathcal {Y}} p_t M C r_i^{-\alpha L} g_L^{-1}}\right) ^{g_L} \nonumber \\&\qquad {+} p_2 \left( \frac{1}{1 + \hat{\mathcal {Y}} p_t m C r_i^{-\alpha L} g_L^{-1}}\right) ^{g_L}, \end{aligned}$$

(A4)

where $p_1$ and $p_2$ are the probabilities associated with the different possible values of $D_i$, which are M and m, respectively. By substituting (A4) into (A3), the final expression for the expectation of the LOS interference signal is given by (11).

For the expectation $\mathbb {E}[\exp (-\hat{\mathcal {Y}} I^N)]$, it can be obtained as follows

$$\begin{aligned} f_{I}^N&= \mathbb {E} \left[ \exp \left( -\hat{\mathcal {Y}} I^N \right) \right] \nonumber \\ &{=} \mathbb {E}_{\phi } \left[ \prod _{i \in \Phi ^N} \mathbb {E}_{H_i^N} \left[ \exp \left( -\hat{\mathcal {Y}} p_t H_i^N D_i C r_i^{-\alpha N} \right) \right] \right] \nonumber \\ &{=} \exp \left( -\lambda _I \int _{\mathbb {R}^2} \left( 1 - \mathbb {E}_{H_i^N} \left[ \exp \left( -\hat{\mathcal {Y}} p_t H_i^N D_i C r_i^{-\alpha N} \right) \right] \right) r_i \, dr_i \right) \nonumber \\ &{=} \exp \left( {-}2 \pi \lambda _I \int _{R_B}^{\infty } \left( 1 {-} \mathbb {E}_{H_i^N} \left[ \exp \left( {-}\hat{\mathcal {Y}} p_t H_i^N D_i C r_i^{{-}\alpha N} \right) \right] \right) r_i \, dr_i \right) \nonumber \\ &{\mathop {=}\limits ^{(a)}} \exp \left( -2 \pi \lambda _I \int _{R_B}^{\infty } \left( 1 - \mathbb {E}_{D_i} \left[ \frac{1}{1 + \hat{\mathcal {Y}} p_t D_i C r_i^{-\alpha N}} \right] \right) r_i \, dr_i \right) , \end{aligned}$$

(A5)

where (a) is derived using the fact that $H_i^N$ follows an exponential distribution with mean 1. The Laplace transform of $H_i^N$ is given by

$$\begin{aligned} \mathbb {E}_{H_i^N} \left[ \exp (-s H_i^N) \right] =&\int _{0}^{\infty } \exp (-s h) \exp (-h) dh \, \\ =&\int _{0}^{\infty } \exp \left( -(s+1)h\right) \, dh = \frac{1}{s+1}, \end{aligned}$$

where $s$ is the Laplace variable. In this context, $s = \hat{\mathcal {Y}} p_t D_i C r_i^{-\alpha N}$.

In (A5), $\mathbb {E}_{D_i}[\cdot ]$ can be obtained as follows

$$\begin{aligned} \mathbb {E}_{D_i} \left[ \frac{1}{1 + \hat{\mathcal {Y}} p_t D_i C r_i^{-\alpha N}}\right]&= \left( \frac{p_1}{1 + \hat{\mathcal {Y}} p_t M C r_i^{-\alpha N}}\right) \nonumber \\&\quad + \left( \frac{p_2}{1 + \hat{\mathcal {Y}} p_t m C r_i^{-\alpha N} }\right) . \end{aligned}$$

(A6)

By substituting (A6) into (A5), the final expression for the expectation of the NLOS interference signal is given by (12).

Similarly, we can derive the expectation of the signal received from the serving LOS BS by the TUE as follows

$$\begin{aligned} f_{BS}^L&= \mathbb {E}\left[ \exp \left( -\hat{\mathcal {Y}} P^L\right) \right] \nonumber \\&{\mathop {=}\limits ^{(a)}} \mathbb {E}_{r_0, H_0^L} \left[ \exp \left( -\hat{\mathcal {Y}} p_t M H_0^L C r_0^{-\alpha L}\right) \right] \nonumber \\&{\mathop {=}\limits ^{(b)}} \mathbb {E}_{r_0} \left[ \left( \frac{1}{1 + \hat{\mathcal {Y}} p_t M C r_0^{-\alpha L} g_L^{-1}}\right) ^{g_L}\right] \nonumber \\&{\mathop {=}\limits ^{(c)}} 2 \pi \lambda _{BS} \int _{r_0=0}^{R_B} \left( \frac{1}{1 + \hat{\mathcal {Y}} p_t M C r_0^{-\alpha L} g_L^{-1}}\right) ^{g_L} \nonumber \\&\quad \times \frac{r_0 \exp \left( -\lambda _{BS} \pi r_0^2\right) }{1 - \exp \left( -\lambda _{BS} \pi R_B^2\right) } \, dr_0, \end{aligned}$$

(A7)

where (a) is obtained by substituting (5), (b) is derived using the fact that $H_0^L$ follows a Nakagami distribution and the moment-generating function (MGF) of $H_0^L$ with parameter $g_L$, and (c) is obtained by determining the expectation $\mathbb {E}_{r_0}\left[ \cdot \right]$ and substituting the pdf of $r_0$ defined in (4). By substituting (11), (12), and (A7) into (A2), and subsequently into (A1), we obtain the final expression for the ECP of the TUE, as given in (10).

Appendix B: Proof of the closed-form expression of ECP

Assuming $\alpha L = 2$ and $g_L = 1$, and that the TUE can receive interference signals with the main beam (resulting in the power gain $D_i = M$), while excluding the NLOS-interfering signals $f_{I}^N$ from (10), we can rewrite (10) as

$$\begin{aligned} \hat{p}(\tau ) = \sum _{k=0}^{V} (-1)^k \left( {\begin{array}{c}V\\ k\end{array}}\right) \mathcal {Q}, \end{aligned}$$

(B8)

where

$$\begin{aligned} \mathcal {Q} = \int _{0}^{R_B} f_{BS}^L(r_0) f_{I}^L(r_0) dr_0, \end{aligned}$$

(B9)

We begin by simplifying $f_{I}^L(r_0)$ as follows

$$\begin{aligned} \mathcal {F}_{I}^L&= f_{I}^L(r_0) \nonumber \\ &{\mathop {=}\limits ^{(a)}}\exp \left( -\frac{AB}{\sqrt{n_t}} \int _{r_0^2 + A}^{R_B^2 + A} \frac{1}{v} \, dv \right) \nonumber \\ &{\mathop {=}\limits ^{(b)}} \left( \frac{R_B^2 + A}{r_0^2 + A} \right) ^{-\frac{AB}{\sqrt{n_t}}}, \end{aligned}$$

(B10)

where (a) is obtained by substituting $A = \hat{\mathcal {Y}} p_t M C$ and $B = \pi \lambda _{BS}$, and (b) is obtained by using the substitution $v = r_i^2 + A$ and $dv = 2 r_i \, dr_i$.

Now, let simplify $f_{BS}^L(r_0)$ given in (A7) as follows

$$\begin{aligned} \mathcal {F}_{BS}^L&=f_{BS}^L(r_0)\nonumber \\&{\mathop {=}\limits ^{(a)}}\frac{2B r_0^3}{(1 - e^{-B R_B^2})(r_0^2 + A)} e^{-B r_0^2}. \end{aligned}$$

(B11)

where (a) is obtained by substituting $A$, $B$ and simplifying further. By substituting (B10) and (B11) into (B9) and simplifying it further by performing a change of variables $t=B\left( r_0^{\ 2}+A\right)$, $dt=2Br_0dr_0$, so we can rewrite (B9) as

$$\begin{aligned} \mathcal {Q} = \frac{(R_B^2 + A)^{-\frac{AB}{\sqrt{n_t}}} e^{AB}}{(1 - e^{-B R_B^2})} \left( B^{-\frac{AB}{\sqrt{n_t}}} I_1 - AB^{\frac{\sqrt{n_t} - AB}{\sqrt{n_t}}} I_2\right) , \nonumber \\ \end{aligned}$$

(B12)

where expression of $I_1$ and $I_2$ in (B12) can be derived, respectively, as

$$\begin{aligned} I_1= & \int _{t=AB}^{B(R_B^2+A)} t^{\frac{AB}{\sqrt{n_t}}} e^{-t} \, dt, \end{aligned}$$

(B13)

$$\begin{aligned} I_2= & \int _{t=AB}^{B\left( {R_B}^2+A\right) } t^{\left( \frac{AB}{\sqrt{n_t}}-1\right) } e^{-t} dt. \end{aligned}$$

(B14)

Recognizing that the integrals in (B13) and (B14) align with the definition of the incomplete gamma function, $\int _{\mathcal {U}}^{\mathcal {V}}t^{s-1}e^{-t}dt=\left| \Gamma (s\,\mathcal {V})-\Gamma (s\,\mathcal {U})\right|$. The integrals in (B13) and (B14) can be, respectively, expressed as in (14) and (15) using the definition of the incomplete gamma function. By substituting (14) and (15) into (B12), and then into (B9), we can express the closed-form of ECP in (10) as in (13).