Performance analysis of UAV multiple antenna-assisted small cell network with clustered users

Abstract

The emerging unmanned aerial vehicles (UAVs) are playing an important role to assist cellular networks and provide ubiquitous coverage for cellular networks. UAVs can eventually increase capacity and facilitate line-of-sight connections. In this paper, we develop an analytical framework to analyze coverage probability for UAVs-assisted small base stations (SBSs) with clustered users, taking into consideration mm-Wave and directional beamforming. Moreover, the contributions mainly include the consideration of multiple-input-multiple-output (MIMO) transmission. Additionally, the clustered user’s locations are modeled as Matern cluster process (MCP). Using tools from stochastic geometry, we derive a general expression for coverage probability according to the signal-to-interference-plus-noise ratio, by considering interference between UAVs and SBSs. The analytical results are validated using Monte-Carlo simulations in which the transmit power of SBS and UAV are considered to be 30 and 24 dBm respectively along with a carrier frequency of 28 GHz. The association probability is calculated as a function of cluster size and UAV density with different values of UAV height. Similarly, the coverage probability is computed as a function of SINR. It is observed from the results that the association probability depends on the altitude of the deployed UAVs and the typical height of a UAV should be between 40–120 m in order to achieve maximum association with a user. Further, it is observed that an upsurge in the density of UAVs and cluster size also affects the association criterion.

Appendices

Appendix A

$$\begin{aligned} F_{R_U}&= P(r>\chi ) \\&= E_D[(P(r>\chi )\mid k,D) P(k\mid D)] \\&= W_U^{k^{-1}}\\&\quad \times E_D \left[ P\left( \sqrt{D^2 + H^2}> \chi \mid k,D\right) P_U^k\left( \sqrt{D^2 + H^2}\right) \right] \\&= W_U^{k^{-1}} \times E_D \left[ P\left( D > \sqrt{\chi ^2 - H^2}\right) P_U^k\left( \sqrt{D^2 + H^2}\right) \right] \\&= W_U^{k^{-1}} \int _l^\infty P_U^k(\sqrt{D^2 + H^2}) f_D(d) {\mathbf {d}}d \end{aligned}$$

where $W_U^k=1-e^{2 \pi \lambda \int _{H_z}^\infty t P_U^k(t) dt}$ is the probability that the typical UE has at least one LOS/NLOS UAV [28]

Appendix B

Association probability for UAV $(U\ne U')$.

$$\begin{aligned} {\mathbb {A}}_U^k&= P(P_UG_UB_UPL_U^{k^{-1}} \ge P_{U'}G_{U'}B_{U'}PL_{U'}^{-1}) \\&= P(P L_U^{k'^{-1}} \ge P L_{U}^{K^{-1}}) \times P(P_UG_UB_UPL_U^{k^{-1}} \\&\ge P_{U'}G_{U'}B_{U'}PL_{U'}^{b^{-1}}) \\&= P(P L_U^{k'^{-1}} \ge P L_{U}^{K^{-1}}) \\&\quad \times P\left( P L_{U'}^{b}\ge \frac{P_{U'}G_{U'}B_{U'}}{P_UG_UB_U} P L_{U}^{K}\right) \\&= E_{R^k_U}\Bigg [P(P L_U^{k'^{-1}} \ge P L_{U}^{K^{-1}})P(\Phi _{U'}^b(\tau _{U'}^2 + H_{z,U^2})^{\frac{\alpha ^b_{U'}}{2}} \\&> \frac{P_{U'}G_{U'}B_{U'}}{P_UG_UB_U} \Phi ^k_U(\tau _{U}^2 + H_{z,U^2})^{\frac{\alpha ^b_{U}}{2}}\Bigg ] \\&= E_{R^k_U}\\&\quad \times \left[ P\left( \tau _U>\sqrt{\left( \frac{\Phi _U^k}{\Phi _{U'}^{k'}}\left( \tau ^2_U + H_{z,U^2}\right) ^{\frac{\alpha ^b_{U}}{2}}\right) ^{\frac{2}{\alpha ^k_{U}}} - H_{z,U}^2}\right) \right. \\&\left. \quad \times P\left( \tau _{U'} > \sqrt{\left( \frac{P_{U'}G_{U'}B_{U'}\Phi _U^k}{P_UG_UB_U\Phi _{U'}^b}\left( \tau ^2_U + H_{z,U^2}\right) ^{\frac{\alpha ^k_{U}}{2}}\right) ^{\frac{2}{\alpha ^b_{U'}}} - H_{z,U}^2 }\right) \right] \\&= E_{R^k_U} \left[ \prod F_U^{k'} (C_{UU}^{kk'}(\tau _U)) F_{R_U^b}(C_{U'U}^{kb}(\tau _U)) \right] \end{aligned}$$

where $C_{UU}^{kk'}(\tau _U) = \sqrt{\left( \frac{\Phi _U^k}{\Phi _{U'}^{k'}} \left( \tau ^2_U + H_{z,U^2}\right) ^{\frac{2}{\alpha _U^k}} - H_{z,U^2} \right) }$ and $C_{U'U}^{kb}(\tau _U) = \sqrt{\left( \frac{P_{U'}G_{U'}B_{U'}\Phi _U^k}{P_{U}G_{U}B_{U}\Phi _{U'}^k} \left( \tau ^2_U + H_{z,U^2}\right) ^{\frac{2}{\alpha _{U'}^k}} - H_{z,U^2} \right) }$

Association Probability for SBS $(S\ne S')$

$$\begin{aligned} {\mathbb {A}}_S^k&= P(P_SG_SB_SPL_S^{k^{-1}} \ge P_{S'}G_{S'}B_{S'}PL_{S'}^{-1}) \\&= P(P L_S^{k'^{-1}} \ge P L_{S}^{K^{-1}}) \times P(P_SG_SB_SPL_S^{k^{-1}} \\&\ge P_{S'}G_{S'}B_{S'}PL_{S'}^{b^{-1}}) \\&= P(A_S^{k'} \ge A_{S}^{K} \tau ^{\alpha ^k_S}) \times P\left( P_SG_SB_SA_{S'}^b\tau _{S'}^{\alpha ^b_{S'}}\right. \\&\left. \ge P_{S'}G_{S'}B_{S'}A_{S}^k\tau _{S}^{\alpha ^k_{S}} \right) \\&= E_{R^k_S}\left[ P\left( A_{S}^{K'} \tau _{S}^{\alpha ^{k'}_{S}}\right) \right. \\&\quad \left. \times P\left( \tau _{S'}^{\alpha ^{b}_{S'}}> \frac{P_{U'}G_{U'}B_{U'}\Phi ^k_U }{P_UG_UB_U \Phi ^k_{U'}} (\tau _{S}^{\alpha ^k_{S}} \right) ^{\frac{1}{\alpha ^b_{S'}}} \right] \\&= E_{R^k_S}\left[ P\left( \left( \frac{A_{S}^{K}}{A_{S}^{K'}}\right) ^{\frac{1}{\alpha ^k_S}}\right) \right. \\&\quad \left. \times P\left( \tau _{S'}^{\alpha ^{b}_{S'}} > \frac{P_{U'}G_{U'}B_{U'}\Phi ^k_U }{P_UG_UB_U \Phi ^k_{U'}} (\tau _{S}^{\alpha ^k_{S}} \right) ^{\frac{1}{\alpha ^b_{S'}}} \right] \\&= E_{R^k_S} \left[ \prod F_S^{k'} (Q_{SS}^{kk'}(\tau _S)) F_{R_S^b}(Q_{S'S}^{kb}(\tau _S))\right] \end{aligned}$$

where $Q_{SS}^{kk'}(\tau _S) = \left( \frac{A_S^k}{A_{S}^{k'}} \tau ^{\alpha _S^k}_S\right) ^{\frac{1}{\alpha _S^k}}$ and $Q_{S'S}^{kb}(\tau _S) = \left( \frac{P_{U'}G_{U'}B_{U'}\Phi _U^k}{P_{U}G_{U}B_{U}\Phi _{U'}^b} \tau ^{\alpha _S^k}_S\right) ^{\frac{1}{\alpha _{S'}^b}}$

Appendix C

$$\begin{aligned}&P_{C,U}^k = P(\gamma _U> \tau ) = P\left( \frac{N_tP_{U}G_{U}h_{U}PL_U^k(r)^{-1}}{I_{U'}+I_S+\sigma ^2}> \tau \right) \\&\quad = P\left( h_U > \frac{\tau PL_U^k(r)}{N_tP_{U}M_rM_t} (I_{U'}+I_S+\sigma ^2)\right) \\&\quad = E_{M_rM_t} \left[ 1- E_{I_{U'},I_S} \left[ \left( 1- e^{-\frac{\tau PL_U^k(r)}{N_tP_{U}M_rM_t} (I_{U'}+I_S+\sigma ^2)}\right) ^{N_k}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) E_{I_{U'},I_S} \left[ e^{-\mu (I_{U'}+I_S+\sigma ^2)}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) E_{I_{U'}} \left[ e^{-\mu \sigma ^2I_{U'})}\right] E_{I_{S}} \left[ e^{-\mu \sigma ^2I_{S})}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) e^{-\mu \sigma ^2)} {{\textbf {L}}}_{I_{U'}}(\mu ) {{\textbf {L}}}_{I_{S}}(\mu )\right] \end{aligned}$$

(C.1) is from [32], (C.2) follows the Binomial theorem and the assumption that $N_k$ is an integer. (C.3) follows from the independence between the interference and (C.4) is from the definition of the Laplace transform [33].

$$\begin{aligned}&P_{C,S}^k = P(\gamma _S> \tau ) = P\left( \frac{P_{S}G_{S}h_{S}PL_S^k(r)^{-1}}{I_{S'}+I_U+\sigma ^2}> \tau \right) \\&\quad = P\left( h_S > \frac{\tau PL_S^k(r)}{M_rM_tP_S} (I_{S'}+I_U+\sigma ^2)\right) \\&\quad \approx E_{M_rM_t} \left[ 1- E_{I_{S'},I_U} \left[ \left( 1- e^{-\frac{\tau PL_S^k(r)}{M_rM_tP_S} (I_{S'}+I_U+\sigma ^2)}\right) ^{N_k}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) E_{I_{S'},I_U} \left[ e^{-q(I_{S'}+I_U+\sigma ^2)}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) E_{I_{S'}} \left[ e^{-q \sigma ^2I_{S'})}\right] E_{I_{U}} \left[ e^{-\mu \sigma ^2I_{U})}\right] \right] \\&\quad = E_{M_rM_t} \left[ \sum _{n=1}^{N_k} (-1)^{n+1} \left( \begin{array}{c} N_{k}\\ n \end{array}\right) e^{-q \sigma ^2)} {{\textbf {L}}}_{I_{S'}}(\mu ) {{\textbf {L}}}_{I_{U}}(\mu )\right] \end{aligned}$$

Appendix D

$$\begin{aligned} {{\textbf {L}}}_{I_{U'}}(\mu )&= E[e^{-\mu I_{U'}}] \\&= E\left[ \prod _{U'\in \frac{\Phi _{UAV}}{b(0,X)}} exp\left( -\mu N_t P_{U'} G_{U'} h_{U'} PL_{U'}^b(r)\right) \right] \\&= \textit{exp}\left( -2\pi \lambda _U E_{M'_rM'_t} \left[ \left( \int _x^\infty 1 - E_{h_U} \bigg [exp\bigg (\right. \right. \right. \\&\quad -\mu N_tP_{U'}G_{U'}h_{U'}PL_{U'}^b(r)\bigg )\bigg ]\bigg ) r P_{U'}^b(r) dr\bigg ]\bigg ) \\&= \textit{exp}\left( -2\pi \lambda _U E_{M'_r M'_t} \left[ \left( \int _x^\infty 1 \right. \right. \right. \\&\left. \left. \left. \quad - \frac{1}{1 + \frac{\mu N_t P_{U'}G_{U'}h_{U'}PL_{U'}^b(r)}{N_k}}\right) r P_{U'}^b(r) dr\right] \right) \end{aligned}$$

$$\begin{aligned} {{\textbf {L}}}_{I_{S'}}(\mu )&= E[e^{-qI_{S'}}] \\&= E\left[ exp\left( -q \sum _{S' \ne S} P_{S'}G_{S'}h_{S'}PL_{S'}^b(r)\right) \right] \\&= E\left[ \prod _{S'\in \frac{\Phi _{UAV}}{b(0,X)}} exp\left( -q P_{S'}G_{S'}h_{S'}PL_{S'}^b(r)\right) \right] \\&= {\textit{exp}}\left( \left[ \left( \int _x^\infty 1 - E_{h_S} \left[ {\textit{exp}}\left( -q P_{S'}G_{S'}h_{S'}PL_{S'}^b(r)\right) \right] \right) \right. \right. \\&\quad \left. \left. r P_{S'}^b(r) dr\right] \right) \\&= {\textit{exp}}\left( \left[ \left( \int _x^\infty 1 - \frac{1}{1 + \frac{q P_{S'}G_{S'}h_{S'}PL_{S'}^b(r)}{N_k}}\right) r P_{S'}^b(r) dr\right] \right) \end{aligned}$$

Appendix E

$$\begin{aligned} \gamma _U&= \frac{N_tq P_{U}G_{U}h_{U}PL_{U}^{-1}(r)}{I_{U'}+I_S+\sigma ^2} \\&= \frac{\frac{N_t P_{U}G_{U}h_{U}PL_{U}^{-1}(r)}{\sigma ^2}}{\frac{I}{\sigma ^2}+1} \\&= \frac{\bar{\gamma }_U h}{\bar{\gamma }_I h + 1} \\&= \frac{V}{Y} \end{aligned}$$

where $h = h_U = h_S$, $\bar{\gamma }_U = \frac{N_t P_{U}G_{U}h_{U}PL_{U}^{-1}(r)}{\sigma ^2}$ and $\bar{\gamma }_I = \frac{(N_t P_{U'}G_{U'}h_{U}PL_{U'}^{b}(r))+(P_SG_SPL_S^k(r))}{\sigma ^2}$ Now, $f_V(v) = \frac{f_h(\frac{v}{\bar{\gamma }_U})}{\bar{\gamma }_U}$ and $f_Y(y) = \frac{f_h(\frac{y}{\bar{\gamma }_I} - \frac{1}{\bar{\gamma }_I})}{\bar{\gamma }_I}$ By applying the confluent hypergeometric function [34], we obtain:

$$\begin{aligned} f_{\gamma U}(\gamma _U; \bar{\gamma }_U, \bar{\gamma }_I, m)&= \int _1^\infty y f_Y(y) f_V(y \gamma _U) dy \\&= \frac{1}{2 \pi \gamma _U \Gamma (m)} m^{2m} \left( \frac{1}{\bar{\gamma }_I}\right) ^n \left( \frac{\gamma _U}{\bar{\gamma }_U}\right) ^m \\&\quad \times {\textit{exp}}\left( \frac{m(\bar{\gamma }_U - \bar{\gamma }_I \gamma _U)}{2 \bar{\gamma }_U \bar{\gamma }_I } \right) \\&\quad \times m \left( \frac{\gamma _U}{\bar{\gamma }_U} + \frac{1}{\bar{\gamma }_I} \right) ^{\frac{1}{2}-m}\\&\quad \times \left( K_{\frac{1}{2}-m}\left( \frac{m(\bar{\gamma }_U - \bar{\gamma }_I \gamma _U)}{2 \bar{\gamma }_U \bar{\gamma }_I }\right) \right. \\&\left. \quad + K_{-\frac{1}{2}-m}\left( \frac{m(\bar{\gamma }_U - \bar{\gamma }_I \gamma _U)}{2 \bar{\gamma }_U \bar{\gamma }_I }\right) \right) \end{aligned}$$