Effects of 3D Position Fluctuations on Air-to-Ground mmWave UAV Communications

This subsection models the position and position fluctuations of UAVs. Since hovering UAVs will fluctuate randomly, the real-time position of each UAV fluctuates randomly around its central position [5, 25]. We denote $(\mu_{x_{i}},\mu_{y_{i}},\mu_{h_{i}})$ and $({x_{i}},{y_{i}},{h_{i}})$ as the central position and the real-time position for the $i$-th UAV, respectively. The central positions on the two-dimensional (2D) plane is modeled as a homogeneous Poisson point process (PPP) with density $\lambda_{T}$ [11]. We assume that the maximum service radius of the UAV is $R$. Therefore, the number of all potential serving UAVs $M$ for the typical user is Poisson distributed i.e., $\mathbb{P}_{M}(m)=\frac{{[\lambda_{T}\pi{R}^{2}]}^{m}}{m!}\exp\left({\!-\!\lambda_{T}\pi{R}^{2}}\right)$. Then, due to the real-time position of the hovering UAV fluctuates randomly, and numerous literature has modeled UAV position fluctuations as Gaussian distributions [27, 7, 5, 28], the exact position of the $i$-th UAV can be modeled as $P_{i}\sim N(\mu_{P_{i}},\sigma^{2}),P\in\{x,y,h\}$, where $\sigma^{2}$ is the variance of the UAV position fluctuations. It is clear that the greater the $\sigma$ is, the greater the fluctuation is. Therefore, we regard $\sigma$ as a measure of UAV position fluctuation strength.

Remark 1: From Theorem 1, (38) and (39), we can observe that the mean $\mu_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ is independent of the variance $\sigma^{2}$ of the UAV position fluctuations. However, the variance $\sigma^{2}_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ is an increasing function of $\sigma^{2}$. Therefore, the higher the fluctuation strength of the UAV, the larger the deviation of $\phi_{i}(h_{i},r_{i})$ from its mean $\mu_{\phi_{i}}$.

Then, the relationship between $\mu_{\phi_{i}}$, $\sigma_{\phi_{i}}$ and average 2D distance $\mu_{r_{i}}$ is given as follows:

Remark 2: According to Corollary 1, we can conclude that if the UAV is deployed closer to the user, i.e., $\mu_{r_{i}}$ is smaller, the mean $\mu_{\phi_{i}}$ of $\phi_{i}(h_{i},r_{i})$ will be smaller. However, in this case, the variance $\sigma_{\phi_{i}}^{2}$ of $\phi_{i}(h_{i},r_{i})$ will become larger. Therefore, there may be an appropriate $\mu_{r_{i}}$ to obtain a trade-off between $\mu_{r_{i}}$ and $\sigma_{\phi_{i}}^{2}$ so that the system can achieve a better QoS.

Using Corollary 2, we can obtain the mean and variance of $\tilde{\psi_{i}}(h_{i},r_{i})$ separately as

Remark 3: From (13) and (14), we can infer that the mean and variance of $\psi_{i}(h_{i},r_{i})$ may also be affected by the position fluctuation of UAV. However, in our system assumption, we can obtain: $\mu_{\tilde{\psi_{i}}}\!\approx\!\exp{\left(-(\epsilon\mu_{r_{i}}\!+\!\epsilon_{0}\right))}$ and $\sigma_{\tilde{\psi_{i}}}^{2}\approx 0$, where the detailed proof can be seen in Appendix C. The results indicate that the $\psi_{i}(h_{i},r_{i})$ hardly changes with the position fluctuation of the UAV, which matches the intuition that the dimension of the position fluctuation of the UAV is very small compared to the size of buildings. Therefore, the impact of UAV position fluctuations on the static blockage probability of the link is negligible, and only when the UAV is located at the edge of the building will position fluctuations of the UAV affect the link’ static blockage state.

We use an indicative random variable $C_{i}$ to indicate that the $i$-th UAV is available, $\mathbb{P}(C_{i})$ denotes the probability of $C_{i}$, which is also the probability that the $i$-th link is not blocked by static and self-blockage. Then, the distribution of the number of available UAVs is obtained as:

Then, we derive the closed-form expression of reliable service probability for single and multiple UAV cases according to Definition 1. It indicates that the greater the fluctuation strength $\sigma$ of the UAV, the lower the $\mathbb{P}_{\mathrm{rel}}$, and the worse the QoS. The specific details are given as follows.

In this case, we assume that the typical user can only link to one UAV, and denote the UAV as the $i$-th UAV. Let $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ represent the reliable service probability for the single UAV case, it indicates the probability that the $i$-th UAV is available and the dynamic blockage probability of the link is lower than the threshold $p_{\mathrm{th}}$. Then, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ can be obtained as shown in Theorem 2.

According to Theorem 2, we can obtain the effect of $\sigma$ on the QoS. In (17), $\mathbb{P}(C_{i})$ and $\mu_{\phi_{i}}$ are not affected by $\sigma$ according to (47) and (38), respectively. So only $\sigma_{\phi_{i}}$ is related to $\sigma$, and $\sigma_{\phi_{i}}$ is an increasing function of $\sigma$, which can be observed from (39), we first analyze the effect of $\sigma_{\phi_{i}}$ on $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. Since error function erf$(\cdot)$ is a monotonically function and erf$(0)\!=\!0$, $\sigma_{\phi_{i}}$ will show two different effects on $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$. For $p_{\mathrm{th}}>\mu_{\phi_{i}}$, erf$(\cdot)>0$ and $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ decreases as $\sigma_{\phi_{i}}$ increases. For $p_{\mathrm{th}}<\mu_{\phi_{i}}$, erf$(\cdot)\!<\!0$ and $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ increases as $\sigma_{\phi_{i}}$ increases. However, in the latter situation, the value of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is lower, even less than 0.5, which indicates that the QoS is evil, further analyze the effect of $\sigma$ on the QoS is almost meaningless. Hence, throughout this paper, we pay more attention to the former case ($p_{\mathrm{th}}>\mu_{\phi_{i}}$). We can conclude that the greater the $\sigma$ is, the larger the $\sigma_{\phi_{i}}$ is, the lower the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is, and the worse the QoS is. It is worth noting that from the derivation of (IV-B), it can be found that $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is similar to the CDF of $\phi_{i}(h_{i},r_{i})$, which is normally distributed according to Theorem 1. Therefore, when $\sigma_{\phi_{i}}$ changes, the variation of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ is similar to that of the CDF of $\phi_{i}(h_{i},r_{i})$. The above analysis is also consistent with this phenomenon. Moreover, for an open area scenario (park/stadium/square)[11], we can rewrite $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$ as follows.

In this case, we assume that the typical user has more than one UAVs that can be connected, and it will immediately switch to any other available UAV when the currently serving link is blocked. We define $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as the reliable service probability for this case, it represents the probability that at least one UAV is available with a dynamic blockage probability lower than the predefined threshold $p_{\mathrm{th}}$. Then, we obtain the approximate $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as shown in Theorem 3.

Although Theorem 3 gives an approximate expression of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, the expression is not tractable due to $n\in[0,\infty]$. Fortunately, since it is unrealistic to have an infinite number of UAVs in practice, we can assume the maximum number of UAVs is $K$. Then, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ can be rewritten as:

which is solvable. Furthermore, we can get the lower bound of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ as shown in Corollary 4.

According to Theorem 3 and Corollary 4, we can conclude that the larger the $\sigma$ is, the smaller the $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ is, and the lower the QoS is, which is consistent with the single UAV case. Similarly, when $\mu_{\phi}=\min(\mu_{\phi_{i}})$ and $\sigma_{\phi}=\min(\sigma_{\phi_{i}})$, $i=1,\cdots,n$, we can obtain an upper bound of $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, which is similar to (IV-C) and is omitted for brevity. For an open area scenario, $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$ can be obtained by setting $\mathbb{P}(C_{i})\!=\!1\!-\!\frac{\theta}{2\pi}$, the reason is consistent with the proof of Corollary 3.

Finally, a graphical example of the impact of $\sigma$ on the blockage and the QoS is shown in Fig. 2. This example depicts two snapshots of the links’ blockage status. As we can see, the blockage status and reliable service in the two snapshots is different even though the blockers are the same, which is due to the position fluctuations of the UAV. Moreover, as discussed in Theorem 2 and Corollary 4, the function of reliable service probability has almost similar variation characteristics to the CDF of normal distribution. Therefore, when $p_{\mathrm{th}}>\mu_{\phi_{i}}$, the higher the UAV’s fluctuation, the smaller the probability of reliable service. In conclusion, the larger the $\sigma$ is, the lower the reliable service probability is, and the worse the QoS is.

Since UAV-user link is only dynamically blocked when the UAV is available, the LoS probability of the $i$-th link can be expressed as follows:

and we assume that the channel between UAVs and the target user is based on the dominant LoS. Therefore, the channel gain $g_{i}$ of the $i$-th UAV-user link is given by [30]:

where $\beta_{0}$ is the path loss (PL) at unit distance, $\alpha$ is parameter of the PL, $d_{i}\!=\!\sqrt{r_{i}^{2}\!+\!(h_{i}\!-\!h_{R})^{2}}$ is the 3D distance from the $i$-th UAV to the typical user. Then, the SNR at the user is given by

where $P_{t}$ is the transmit power of the UAV and $N_{0}$ is the noise power. It is clear that $\gamma_{i}$ is a function of random variables $r_{i}$ and $h_{i}$, which means that the position fluctuations of UAVs will affect the $\gamma_{i}$. Therefore, the coverage probability given in Definition 2 can capture these effects.

Similar to $\mathbb{P}_{\mathrm{rel}}^{\mathrm{sig}}$, we define $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ as the coverage probability for the single UAV case, it represents the probability that the $i$-th UAV is available and the SNR of the $i$-th link is not smaller than the threshold $\gamma_{0}$. Therefore, using Definition 2 and similar to (18) and (IV-B), we first can get

Then, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{sig}}$ is approximately obtained as shown in Theorem 4.

Since Marcum Q-function is a standard function that is easy to compute[7], using Theorem 4, we can quickly evaluate the QoS of the considered system without resorting to time-consuming simulation, especially the impact of UAV position fluctuations on the coverage probability. In addition, when an open area communication scenario is considered, the coverage probability can be obtained by setting $\mathbb{P}(C_{i})\!=\!1\!-\!\frac{\theta}{2\pi}$, the reason is consistent with the proof of Corollary 3.

Similar to $\mathbb{P}_{\mathrm{rel}}^{\mathrm{mul}}$, we define $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ as the coverage probability for the multiple UAVs case, it represents the probability that at least one UAV is available with a SNR not smaller than the threshold $\gamma_{0}$. Then, we can obtain $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ as follows:

Similar to (23), given the limited number of UAVs in practice, $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ can be rewritten as

Furthermore, a lower bound of $\mathbb{P}_{\mathrm{cov}}^{\mathrm{mul}}$ is obtained as shown in Corollary 5.