Improving IoT-over-Satellite Connectivity
using Frame Repetition Technique

Consider a satellite in a circular orbit at an altitude $h$ from the mean sea level. Ground IoT devices are assumed to be distributed by an isotropic and homogeneous Poisson Point Process (PPP) on the Earth’s surface. An important parameter that impacts the performance is the angle formed between the satellite point, the Earth center, and the observer, this angle is termed as zenith angle, denoted $\varphi$.The elevation angle $\theta$, which is the angle between satellite and horizon seen by ground user, can be obtained from the zenith angle using geometric reasoning as follows [4]

$\displaystyle\theta=\mathrm{acot}\left(\frac{\sin~{}\varphi}{\cos~{}\varphi-\alpha}\right),~{}\text{where}~{}\alpha=\frac{R}{h+R}.$

(1)

The domain of the elevation angle is $\theta\in[0,\frac{\pi}{2}]$, where $\theta=0$ represents the case when the satellite is at the horizon and $\theta=\pi/2$ is when the satellite is at the device zenith. Accordingly, $\varphi\in[\varphi_{\mathrm{h}},0]$ $\rightarrow$ $\theta\in[0,\frac{\pi}{2}]$, where $\varphi_{\mathrm{h}}$ is the maximum zenith angle formed at the horizon. The scenario is depicted in Fig. 1. It can be shown that the maximal horizon zenith angle is given by [12], $\varphi_{\mathrm{h}}=\text{acos}~{}\alpha$. As such, the serving satellite can be geometrically seen from any point in a certain spherical cap region. However, note that IoT devices that have very low elevation angle are deemed to have an extremely poor radio link performance as a result of deep path-loss. Consequently, the probability of successful transmission in such cases is low. As such, we define a minimum elevation angle, denoted as $\theta_{\mathrm{min}}$, which limits the admittance region of IoT devices. There are multiple ways to limit the admittance region practically; for example by controlling the beamwidth of the satellite or by programmaticaly preventing IoT devices from transmitting when the satellite is below the threshold elevation angle, i.e., $\theta\leq\theta_{\mathrm{min}}$. Due to this impact, the zenith angle becomes lower than $\varphi_{\mathrm{h}}$ and thus limited by the general threshold $\varphi_{\mathrm{max}}$, i.e. $\varphi\in[0,\varphi_{\mathrm{max}}]$ $\rightarrow$ $\theta\in[\frac{\pi}{2},\theta_{\mathrm{min}}]$. The relation between these two angles is given by (1) as follows

$\displaystyle\theta_{\mathrm{min}}=\mathrm{acot}\left(\frac{\sin~{}\varphi_{\mathrm{max}}}{\cos~{}\varphi_{\mathrm{max}}-\alpha}\right),~{}\text{where}~{}\alpha=\frac{R}{h+R}.$

(2)

Due to the near ground objects and foliage, i.e., also called clutter, geometric LoS is sometimes blocked by these obstacles. The probability of line-of-sight $p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}$ is an important parameter that heavily influences the path-loss and the probability of successful transmission. Therefore, we adopt the LoS model developed in [12]. However, we re-write the $p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}$ in terms of $\varphi$ as follows

$\displaystyle p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}$

$\displaystyle=\exp\left(-\beta\cot\theta\right)=\exp\left(-\beta\frac{\sin~{}\varphi}{\cos~{}\varphi-\alpha}\right),$

(3)

where $\beta$ is a controllable modeling parameter related to the underlaying propagation environment. Note that the probability of non-line-sight is given as $p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}}=1-p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}$.

Since the purpose of repetitions to improve the coverage [13], the devices with lower line-of-sight probability are expected to rely on this feature more often in order to achieve successful transmission. This intuitively indicates that devices with larger zenith angle, $\varphi$, will need more repetitions. Hence, the developed model in [14] which primarily depends on the distance towards the terrestrial base station is adopted but with changing this dependency to the zenith angle. Assuming an initial duty cycle of $D_{\mathrm{o}}$¹¹1$D_{\mathrm{o}}$ is ratio of the time-on-air of message $T_{\mathrm{m}}$ to the maximum duration till next update $T_{\mathrm{max}}$, the effective duty cycle, when, repetition is applied, is modeled as

$$D\left(\varphi\right)=D_{\mathrm{o}}+(1-D_{\mathrm{o}})g(\varphi),$$

(4)

where $g(\varphi)$ is a custom shaping function that satisfies the criterion $g\left(\varphi\right)\in\left[0,1\right]$, $D\in\left[D_{\mathrm{o}},1\right]$, since the duty cycle cannot exceed unity. A suitable function that satisfies the above requirement is a modified version of the probability of the non-line-of-sight as follows,

$$D\left(\varphi\right)=D_{\mathrm{o}}+(1-D_{\mathrm{o}})\left[1-\exp\left(-a\beta\frac{\sin~{}\varphi}{\cos~{}\varphi-\alpha}\right)\right],$$

(5)

where $\varphi\in\left(0,\varphi_{\mathrm{max}}\right)$ and $a$ $\in\left[0,1\right]$ is a controllable tuning factor. Accordingly, the number of transmissions, i.e., repetitions is give by

$$N\left(\varphi\right)=\left\lceil{\frac{D\left(\varphi\right)}{D_{\mathrm{o}}}}\right\rceil.$$

(6)

The impact of tuning factor $a$ on the duty cycle and repetitions is depicted in Fig. 2 and Fig. 3, respectively. Smaller values of $a$ correspond to lower overall repetitions and vice-versa. Furthermore, the figures also illustrate the effect of the admittance angle $\theta_{\mathrm{min}}$. It is to be noted that both $a$ and $\theta_{\mathrm{min}}$ can be tuned in order to optimize the overall success probability. Due to increased transmissions as a result of repetition, the effective device density in the admittance region becomes $\lambda(\varphi)=\lambda_{o}D\left(\varphi\right)$, where $\lambda_{\mathrm{o}}$ is the density of ground devices. That is, in case of no repetition, $D(\varphi)\to D_{\mathrm{o}}$ and $\lambda(\varphi)\to\lambda_{\mathrm{o}}$. A thorough explanation of repetition model is given in [14].

For a given satellite admittance region defined by the angle domain, $\theta\in\left[\theta_{\mathrm{min}},\frac{\pi}{2}\right)$, the users will view the serving satellite within the zenith angle domain, $\varphi\in\left(0,\varphi_{\mathrm{max}}\right)$, where $\varphi_{\mathrm{max}}\leq\varphi_{\mathrm{h}}$, as shown in Fig. 1. As introduced earlier, the elevation angle or the equivalent zenith angle of the satellite heavily impacts the repetition rate. Thus, the analysis of satellite’s zenith angle distribution is essential to obtain the probability of successful transmission. Assuming PPP, the ground devices or IoT devices are randomly placed within the admittance region. Accordingly, the probability of finding devices within a given angle $\varphi_{\mathrm{o}}$ is formulated as follows

$\displaystyle\scriptstyle F_{\varphi}(\varphi_{\mathrm{o}})=\mathbb{P}\left(\varphi\leq\varphi_{\mathrm{o}}\right)$

$\displaystyle=\frac{\text{avg. no. of active devices in $\varphi_{\mathrm{o}}$}}{\text{avg. no. of active devices in $\varphi_{\mathrm{max}}$}}~{},$

(7)

which represents the cumulative distribution function (CDF) of the satellite zenith angle. The average number of points can be found by invoking Campbell theorem [15] for a PPP. Accordingly, the average number of points in a region bounded by an angle $\left[0,\varphi\right]$ is given by

$$K(\varphi)=\int_{0}^{\varphi}2\pi R^{2}\lambda(\varphi)\sin(\varphi)~{}\mathrm{d}\varphi,$$

(8)

because the spherical strip area is given by, $2\pi R^{2}\sin(\varphi)\mathrm{d}\varphi$. Substituting in (8) and replacing $\lambda(\varphi)=\lambda_{\mathrm{o}}D(\varphi)$ we obtain

$$F_{\varphi}(\varphi_{\mathrm{o}})=\frac{K(\varphi_{\mathrm{o}})}{K(\varphi_{\mathrm{max}})}=\frac{\int_{\mathrm{0}}^{\varphi_{o}}\sin(\varphi_{\mathrm{o}})D(\varphi_{\mathrm{o}})~{}\text{d}\varphi}{\int_{0}^{\varphi_{\text{max}}}\sin(\varphi)D(\varphi)~{}\text{d}\varphi}.$$

(9)

The probability density function (PDF) is thus obtained by differentiating (9) as follows

$\displaystyle f_{\varphi}(\varphi_{\mathrm{o}})=\frac{\sin(\varphi_{\mathrm{o}})D(\varphi_{\mathrm{o}})}{\int_{0}^{\varphi_{\text{max}}}\sin(\varphi)D(\varphi)~{}\text{d}\varphi}.$

(10)

An illustration of the CDF at $\theta_{\mathrm{min}}=10^{\mathrm{o}}$ is shown in Fig. 4 along with Monte-carlo simulations to verify the analytical results. Note that the distribution changes for different tuning factor $a$.

For a given device with transmitting power $p_{\mathrm{t}}$, at an angle $\varphi_{\mathrm{o}}$, the received signal power in the uplink can be modeled as $S=p_{\mathrm{t}}l(\varphi_{\mathrm{o}})\zeta(\varphi_{\mathrm{o}})$[16], where $l(\varphi_{\mathrm{o}})$ is the reciprocal of the free space path-loss (FSPL) $\left(\frac{4\pi df}{c}\right)^{2}$, while $\zeta(\varphi_{\mathrm{o}})$ is the excess gain caused by ground clutter, and its distribution follows simple Gaussian mixture, this path-gain heavily depends on the probability of LoS $p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}$ as given in eq. (7) in [17].

For a given device located at $x_{\mathrm{o}}$, the simultaneous transmissions imposed by other devices within the admittance region cause interference with a total aggregated power, denoted $I$, obtained as follows

$\displaystyle I=\sum_{x_{\mathrm{i}}\in\Phi\text{\textbackslash}x_{\mathrm{o}}}\kappa p_{\mathrm{t}}\zeta(\varphi_{i})l(\varphi_{i}),$

(11)

where $\Phi$ is the set of all devices in the admittance region, $\kappa$ is a system parameter that captures the impact of interference mitigation caused by resource coordination in the access system. Note that $\kappa=1$ indicates the worst case scenario when no coordination is implemented. Accordingly, the spatial average of the interference is obtained by invoking Campbell’s theorem of sums and is given as

$\displaystyle\mathbb{E}[I]=2\pi R^{2}\lambda_{\mathrm{o}}\int_{o}^{\varphi_{\mathrm{max}}}\bar{\zeta}(\varphi)l(\varphi)\sin(\varphi)D(\varphi)\mathrm{d}\varphi$

(12)

where $\bar{\zeta}$ is the average excess path-gain expressed as [6]

$\displaystyle\bar{\zeta}=p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}\exp\left(-\mu_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}+\frac{\sigma^{2}_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}}{2}\right)+p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}}\exp\left(-\mu_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}}+\frac{\sigma^{2}_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}}}{2}\right),$

(13)

where $\mu$ and $\sigma$ are the mean and standard deviation parameters associated with respective LoS and non-LoS components. To validate the above analytic results, we depict in Fig. 5 a comparison between the analytic results of (12) and Monte Carlo simulations at various values of $a$ and $\theta_{\mathrm{min}}$. Note the increase in the aggregated interference power when deploying the repetition scheme. Nevertheless, it is shown in the next subsection that the probability of success can be further increased with careful tuning of $a$ and $\theta$.

To measure the probability of success under influence of frame repetition, we utilize a vital metric of communication link which is the ratio of received signal power to the interference plus noise power (SINR). The probability of a frame having an SINR greater than a given design threshold $\gamma$, is equivalent to the frame success probability. For a given IoT device viewing the satellite at a zenith angle $\varphi_{\mathrm{o}}$, the success probability of a single transmission is formulated as follows

	$\displaystyle p_{|\varphi_{\mathrm{o}}}(1)$	$\displaystyle=\mathbb{P}\left(SINR>\gamma\right)=\mathbb{P}\left(\frac{S}{\bar{I}+W}>\gamma\right)$
		$\displaystyle=\mathbb{P}\left({\zeta}>\frac{\gamma(\bar{{I}}+{W})}{\mathrm{p}_{\mathrm{t}}l(\varphi)}\right)=1-F_{\mathrm{\zeta}}\left(\frac{\gamma(\bar{{I}}+{W})}{\mathrm{p}_{\mathrm{t}}l(\varphi)}\right),$		(14)

where ${W}$ is the average thermal noise power, $\bar{I}$ is given in (12) and $F_{\mathrm{\zeta}}$ is given as [4]

$\displaystyle\scriptstyle F_{\mathrm{\zeta}}=p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}F_{\mathrm{X}}\left(x,-\rho\mu_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}},\frac{\sigma_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}^{2}\rho^{2}}{2}\right)+p_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}}F_{\mathrm{X}}\left(x,-\rho\mu_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{NLoS}$}}}},\frac{\sigma_{\vbox{\hbox{\scalebox{0.5}{$\mathrm{LoS}$}}}}^{2}\rho^{2}}{2}\right),$

(15)

where $F_{\mathrm{X}}$ is the CDF of the log-normal distribution as ${F_{\mathrm{X}}(x)=\frac{1}{2}+\frac{1}{2}\text{erf}\left(\frac{\ln x-\mu}{\sigma\sqrt{2}}\right)}$ and $\rho=\frac{\ln 10}{10}$. For a device attempting its transmission $N$ number of times, the probability of at least one of the frames to be successfully received is formulated as

$\displaystyle p_{|\varphi_{\mathrm{o}}}(N)=1-[1-p_{|\varphi_{\mathrm{o}}}(1)]^{N}.$

(16)

We illustrate the success probability in Fig. 6 with respect to a range of elevation angles for three given examples of the admittance region. We note that for smaller admittance region, i.e., larger values of $\theta_{\mathrm{min}}$, the probability of success increases. For small $\theta_{\mathrm{min}}$ the increase in admittance region causes devices to incur more repetition, hence, increasing the interference level, which overshadows the performance enhancement as shown for $\theta_{\mathrm{min}}=5^{\circ}$. Subsequently, to find the average success probability for a given admittance region bounded by $\varphi_{\mathrm{max}}$, we decondition (16) over the distribution of $\varphi$ as follows

$\displaystyle p(N)=\int_{0}^{\varphi_{\mathrm{max}}}p_{|\varphi_{\mathrm{o}}}(N)f_{\varphi}(\varphi_{\mathrm{o}})~{}\mathrm{d}\varphi.$

(17)

It is illustrated in Fig. 7 where the success probability of the repetition scenario clearly outperforms the no repetition scenario. Note that this performance is for a single satellite spot, i.e. single admittance region. The following section extends results to a constellation of satellites.

Assuming most of the satellites are positioned in a non-GEO (Geosynchronous Equatorial Orbit), which are in a relative motion with respect to the ground IoT devices. As such, when reducing the size of the admittance region, the probability of having the coverage spot of the device is reduced. Accordingly there is a trade-off between satellite availability and its success rate within the spot. To capture this trade-off, we extend the performance by including the satellite availability, which is the ratio of satellite admittance area to the total surface area of Earth’s sphere. For a small number of satellites, denoted $k$, and assuming non-overlapping spots, then the probability that a device is within the coverage is given as

$\displaystyle p_{\mathrm{spot}}$

$\displaystyle=\frac{k~{}2\pi R^{2}\left(1-\cos\varphi_{\mathrm{max}}\right)}{4\pi R^{2}}=\frac{k\left(1-\cos\varphi_{\mathrm{max}}\right)}{2},$

(18)

where $0\leq p_{\mathrm{spot}}\leq 1$. Accordingly, the probability of a successful frame delivery in a constellation of $k$ satellite with non-overlapping spots is given by

$\displaystyle p_{\mathrm{s}}=p_{\mathrm{spot}}\times p(N).$

(19)

We depict in Fig. 8 the global average success probability evaluated for one tuning factor and various minimum elevation angles. We can clearly note the trade-off induced by adjusting the admittance region, i.e. by controlling $\theta_{\mathrm{min}}$. Moreover, as the shape of the curve is impacted by the tuning factor $a$, an optimal point for a given admittance angle exists. In the following section we shed light on the joint optimization of these two parameters.