# On the Fraction of LoS Blockage Time in mmWave Systems with Mobile Users and Blockers

Dmitri Moltchanov, Aleksandr Ometov

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Dmitri Moltchanov, Aleksandr Ometov. On the Fraction of LoS Blockage Time in mmWave Systems with Mobile Users and Blockers. International Conference on Wired/Wireless Internet Communication (WWIC), Jun 2018, Boston, MA, United States. pp.183-192, 10.1007/978-3-030-02931-9_15 . hal-02269730

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# On the Fraction of LoS Blockage Time in mmWave Systems with Mobile Users and Blockers

Dmitri Moltchanov<fnref n="1" /> and Aleksandr Ometov<fnref n="2" />

Email: aleksandr.ometov@tut.fi

**Abstract.** Today, one of the emerging trends for the next generation (5G) networks is utilizing higher frequencies in closer premises. As one of the enablers, small cells appear as a cost-effective way to reliably expand network coverage and provide significantly increased capacity for end users. The ultra-high bandwidth available at millimeter (mmWave, 30–300GHz) and Terahertz (THz, 0.3–3THz) frequencies can effectively realize short-range wireless access links in small cells. Those technologies could also be utilized for direct communications for users in proximity. At the same time, the performance of mobile wireless systems operating in those frequency bands depends on the availability of line-of-sight (LoS) between communicating entities. In this paper, we estimate the fraction of LoS time for randomly chosen node moving according to different mobility models in a field of *N* moving blocking nodes for both base station and device-to-device (D2D) connectivity scenarios. We also provide an extension to the case of a random number of moving blockers. The reported results can be further used to assess the amount of traffic offloaded to other technologies having greater coverage, e.g., LTE.

## 1 Background and Motivation

Due to tremendous increase in traffic demand, researchers in both industry and academia already focus on numerous advanced networking solutions such as client-relaying [1], heterogeneous networking [2], the use of micro/pico/femto cells [3] and Device-to-Device (D2D) communications [4–6] to satisfy the requirements of fifth generation (5G) mobile wireless systems. These mechanisms alone, however, are limited in achieving the required data rates.

One of the initial steps forward is moving up in the frequencies from microwaves to millimeter waves (mmWave, 30–300GHz) [7] and further to the terahertz (THz, 0.3–3THz) band [8] is considered as a viable solution to principally increase the capacity of wireless channels and eventually satisfy the requirements of 5G-grade networks. Operating at those frequencies could be described by the following features: (i) higher throughput; (ii) shorter propagation distances due to higher attenuation; (iii) better spatial resolution due to smaller beamwidth; (iv) smaller antenna size in general; and (v) higher blockage affection.

The principle discord between mmWave/THz systems, compared to microwaves, is that there is an extreme difference in the received signal strength in line-of-sight (LoS)

<footnote n="1">¹ Tampere University of Technology, Tampere, Finland</footnote>

<footnote n="2">² National Research University Higher School of Economics, Moscow, Russia</footnote>
and nLoS conditions [9, 10]. Such systems are expected to be installed in open in-
door and outdoor environments such as conference halls, lobbies, squares, crossroads,
parks [11]. Nevertheless, as electromagnetic waves cannot “travel around” the objects
whose size is smaller than their wavelengths while the human bodies serve as perfect
absorbers [12], the crowd around the receiver blocks the LoS. To understand perfor-
mance bounds of mmWave and THz systems, it is essential to predict the probability of
having LoS at a particular instant of time as well as the fraction of time there is a LoS
between communicating entities in crowded mobile environments.

The question of the LoS propagation blockage has recently received particular at-
tention from the research and standardization community [13, 14]. As demonstrated by
field measurements in [15], the loss of LoS path in mmWave systems may result in
sharp drops of the (up to 30 ~ 40dB) in the received signal strength. The first analyti-
cal studies on this topic addressed the question of LoS blockage in environments with
static users and static blockers [16, 17]. These results have been recently extended to the
case of moving blockers and static users in [18] as well as static blockers and moving
users in [19]. In these studies, it has been shown that the mobility of both blockers and
users drastically affect the blockage statistics leading to additional uncertainly in chan-
nel state. However, to the best of the authors' knowledge, there have been no studies
addressing the case when both blockers and users are simultaneously mobile.

In this paper, we derive the probability of having nLoS at a random instant of time
as well as the fraction of LoS for a user in a group of *N* + 1 blockers moving around
in closed indoor compartment according to the random direction mobility (RDM) and
random waypoint (RWP) models. We consider mmWave access point (AP) and D2D
scenarios and also sketch the extension to the open outdoor scenarios, where a user and
blockers may freely enter and leave the service area of interest. The presented results
can further be used to assess the amount of traffic that needs to be offloaded to another
type of connection.

The rest of the manuscript is organized as follows. Section 2 provides the system model and depicts the scenarios of interest. Next, we elaborate on the applicability of the model to dynamic conditions and extend it in Section 3. The numerical examples are given in Section 4. The last section concludes the manuscript.

## 2 Indoor Closed Compartment Scenario

Two scenarios of interest are described in this section. We focus on both conventional
infrastructure-based case, where AP serves the tagged user and the remaining ones act
as potential blockers, and D2D scenario, where two users are attempting to initiate
direct link that could be blocked by other users. Both users and blockers are mobile and
move according to a RDM mobility model [20]. Recall that according to RDM a point
first randomly chooses the direction of movement uniformly in (0, 2π). Then, it moves
in the chosen direction for exponentially distributed time with constant speed *v*. The
process is restarted at the stopping point.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Considered scenarios.</figcaption></figure>

## 2.1 Access Point Scenario

In the first scenario, we assume $N + 1$ mobile users (UEs), acting as blockers, that randomly move according to RDM in the coverage area with radius $r$ of a mmWave or THz AP, which height is $h_{Tx}$ (AP is selected as transmitter (Tx)). The height of the blockers is assumed to be the constant and equal to $h_B$. The radius of a blocker is $r_B$. The height of the receiver (Rx) associated with the tagged UE is $h_{Rx} < h_B$. We are interested in the probability of nLoS at the arbitrary time instant and the fraction of nLoS. Note that these metrics are symmetric irrespective of which entity (AP or UE) acts as Tx or Rx correspondingly.

We assume that UE is located at the distance $x$ from the AP at time $t$, as shown in Fig. 1(a). The LoS to the UE could be blocked by the blockers that are located in the so-called *LoS blocking zone*, marked in gray. The length of this zone is

$$d(x) = \frac{x(h_B - h_{Rx})}{h_{Tx} - h_{Rx}}. \qquad (1)$$

Observing the top view of the scenario, Fig. 2(a), notice that the area of the LoS blocking zone is more complicated than a rectangle. To prevent overlapping there cannot be a blocker located closer than at $2r_B$ to the user. The area of the LoS blocking zone is

$$S_B = 2r_B[x - d(x)] - 2r_B^2 - \frac{1}{2}\pi r_B^2. \qquad (2)$$
<figure><img src="image_3.png" /><figcaption>Fig. 2. Top view of the LoS blocking zone.</figcaption></figure>

The existence of the zone around the UE prohibiting the presence of blockers im-
plies that the minimum distance from the AP resulting in non-zero LoS blocking zone
is $2r_B$. Recall that the limiting distribution of the RDM is uniform over the area of
interest [20]. Observing the system in stationary regime, the probability of a point uni-
formly distributed in the circle hitting an LoS blocking area is given by the ratio $S_B/\pi r^2$.
Generalizing to $N$ blockers, we arrive at

$$
p_B(N) = 1 - \left( 1 - \frac{2r_B[x-d(x)] - 2r_B^2 - \frac{1}{2}\pi r_B^2}{\pi r^2} \right)^N . \quad (3)
$$

The probability density function (pdf) of a point uniformly distributed in a circle of radius $r$ is given by $f(x) = 2x/r^2$. Thus, the probability of having nLoS at a random instant of time, $p_{nL}$, coinciding with the fraction of nLoS, $f_{nL}$, is

$$
f_{nL} = \int_{2r_B}^{r} \frac{2x}{r^2} \left( 1 - \left[ 1 - \frac{2r_B[x-d(x)] - 2r_B^2 - \frac{1}{2}\pi r_B^2}{\pi r^2} \right]^N \right) dx. \quad (4)
$$

Integrating (4) we arrived at the final result in the following closed-form

$$
f_{nL} = 1 - \frac{4r_B^2}{r^2} - \frac{\left[ \frac{4(A-1)r_B^2 + B + \pi r^2}{r^2} \right]^{N+1} [\pi r^2 - 4(A-1)(N+1)r_B^2 + B]}{2\pi^N (A-1)^2 (N+1)(N+2)r_B^2} + \quad (5)
$$

$$
+ \frac{\left[ \frac{r(2(A-1)r_B + \pi r) + B}{r^2} \right]^{N+1} [r(\pi r - 2(A-1)(N+1)r_B) + B]}{2\pi^N (A-1)^2 (N+1)(N+2)r_B^2}, \quad (6)
$$

where the shortcuts are

$$
A = \frac{(h_B - h_{Rx})}{h_{Tx} - h_{Rx}}, \quad B = 2r_B^2 - \frac{\pi r_B^2}{2}. \qquad (7)
$$
## 2.2 D2D Scenario

Since the height of Tx and Rx is assumed to be the same, $h_{Tx} = h_{Rx} < h_B$, it suffices to consider two dimensional case, as shown in Fig. 2(b). Given the distance $x$ between Tx and Rx the area of the LoS blocking zone is

$$S_B(x) = 2r_Bx - 4r_B^2 - \pi r_B^2. \qquad (8)$$

Similarly to the AP blocking model, there is no blocking when Tx and Rx are closer than $4r_B$. Recalling the stationary property of RDM model and the fact that the distance between two points uniformly distributed in the circle of radius $\pi r^2$ is given by [21]

$$f(x) = \frac{2x}{r^2} \left( \frac{2}{\pi} \arccos \left( \frac{x}{2r} \right) - \frac{x}{r\pi} \sqrt{1 - \frac{x^2}{4r^2}} \right), \quad 0 < x < 2r, \qquad (9)$$

we have the following for the fraction of nLoS

$$f_{nL} = \int_{4r_B}^{r} f(x) \left( 1 - \left( 1 - \frac{2r_Bx - 4r_B^2 - \pi r_B^2}{\pi r^2} \right)^N \right) dx, \qquad (10)$$

that also coincides with the nLoS blocking probability. Note that the integral cannot be expressed in elementary functions but can be computed numerically.

# 3 Extensions and applications

## 3.1 Outdoor environment

The scenarios considered previously are unrealistic for outdoor deployments, where blockers/user may freely enter and leave an area of interest. For AP scenario this area coincides with the service area of an AP while for the D2D case this is the area, where D2D connectivity is feasible, e.g., UE proximity characterized by the short-range radio coverage area. In this section, we address this by introducing dynamics to the model.

Let $A_0$ be the circular service area of an AP and let $A_1$ be the greater circular area where $N$ blockers are allowed to move according to RDM. We are interested in the probability that a randomly chosen node currently located in the AP service area is in nLoS state. This probability no longer coincides with the fraction of nLoS as the latter needs to be obtained knowing the AP service area residence time of a user.

Recall that in the stationary regime $N$ nodes moving in $A_1$ are uniformly distributed in $A_1$ [20]. Thus, the number of nodes in $A_0$ follow binomial distribution with parameter $A_0/A_1$ and these nodes are all uniformly distributed in $A_0$. Increasing $A_1$ and $N$ such that $\lambda = N/S_{A_1}$ is kept constant $p = A_0/A_1 \to 0$ and the binomial distribution approaches the Poisson one with parameter $\lambda$. Thus, the probability of having LoS at the distance $x$, $p_L(x)$, is given by the void probability of a Poisson process with the area estimated in (2), i.e.,

$$p_L(x) = \exp \left( -\lambda \left[ 2r_B[x - d(x)] - 2r_B^2 - \frac{1}{2}\pi r_B^2 \right] \right), \qquad (11)$$
where $d(x)$ is provided in (1).

The probability that a node is moving according to RDM and located in $A_0$ at time $t$ experiences nLoS conditions is obtained similarly to (4) with exponential blocking probability replacing the binomial one. It is provided in (12). For D2D scenario the integral for nLoS probability is similar to (10), with $1 - p_L(x)$ from (11) replacing the binomial blocking. It also cannot be solved in elementary functions but can be numerically evaluated with any given accuracy.

$$ 
\begin{aligned}
p_{nL} &= \int_{2r_B}^{r} \frac{2x}{r^2} \left( 1 - e^{-\lambda[2r_B[x-d(x)] - 2r_B^2 - \pi/2r_B^2]} \right) = \\
&= 1 - \frac{e^{\lambda r_B([\pi+4]r_B - 2r)} (2\lambda r r_B + 1) - 32\lambda^2 r_B^4}{2\lambda^2 r^2 r_B^2} + \frac{e^{(\pi-4)\lambda r_B^2} (8\lambda r_B^2 + 1)}{2\lambda^2 r^2 r_B^2}.
\end{aligned} 
\quad (12) $$

To evaluate the fraction of nLoS for outdoor deployment, it is necessary to obtain the time corresponding to the UE staying in the service area $A_0$. This is a first passage time in a circle for a random point moving according to RDM with some initial position that depends on the size of $A_0$ and parameters of RDM model. Neither pdf nor mean of this metric can be obtained analytically [22]. Nevertheless, one could always use computer simulations or Brownian motion approximation of RDM (see [22], Ch. 4) to quantify this metric.

## 3.2 Further extensions

We specifically note that the model proposed in this paper can be extended in many different ways. First of all, one can use any mobility model having well-defined stationary distribution, for example, random waypoint (RWP) mobility model whose distribution in a square with side $A$ is available in closed form [23], i.e.,

$$ f_{X,Y}(x,y) = \frac{36}{A^6} \left( x^2 - \frac{A^2}{4} \right) \left( y^2 - \frac{A^2}{4} \right). \quad (13) $$

The only additional step compared to the presented model is to determine the distance to the mmWave AP or the distance between two nodes. This can be done applying the random variables transformation technique. Notably, for the AP to UE case, the transformation of interest reads as $r(x,y) = \sqrt{x^2 + y^2}$ and the procedure is reduced to finding the Jacobian of the transformation [24].

We also would like to specifically note that LoS blockage does not always lead to the outage event, i.e., for a given propagation model the outage even may only occur when the current distance between communicating nodes is higher than the specific value. The result for a fraction of time can be extended to this case as well. Considering AP to UE scenario as an example, we first obtain the fraction of time UE is farther than a certain propagation distance $D$, then observe that the UE position conditional on being farther than $D$ is still uniformly distributed in the ring $(D, R)$ with density $f(x) = 2x/(R^2 - D^2)$ and then determine the fraction of LoS blockage time using $f(x)$.
## 4 Numerical illustrations

In this section, we illustrate the obtained numerical results. First, we consider the LoS blockage probability for AP to UE communications scenario. We then address D2D communications case. Finally, we consider the outdoor scenario with a Poisson distribution of the number of blockers.

Consider first the AP to UE indoor communications scenario. Fig. 4 illustrates the fraction of LoS blockage time as a function of systems parameters including mmWave AP height, $h_A$, and UE height, $h_U$ for a range of the number of blockers in the compartment. In these illustrations, the height of Rx is set to 1.2 meters, and the height of blockers is 1.7 meters. The radius of the coverage is 30 meters. Expectedly, as the number of blockers increases the probability of LoS blockage increases. Further, it is essential to observe that the change in AP and UE heights does not drastically affect the blockage probability. The most significant effect stems from the number of blockers in the area.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Fraction of LoS blockage time in AP to UE scenario.</figcaption></figure>

The effect of input parameters on the fraction of LoS blockage time for the D2D scenario is illustrated in Fig. 4 as the function of the number of blockers for different radii of the service area. Recall that in this scenario the height of communicating entities is the same. Logically, by keeping the number of blockers constant and increasing the service area of interest the fraction of LoS blockage time increases. Finally, by comparing the results in Fig. 4 with Fig. 3(a) and 3(b), we observe that the fraction of the LoS blockage time for the D2D scenario is significantly bigger. The rationale is that in the case of D2D communications the heights of entities are all the same implying that the area of the LoS blockage zone is significantly larger.

Finally, we compare the fraction of LoS blockage time in indoor and outdoor scenarios for AP to UE case. The comparison is facilitated by parameterizing indoor model
<figure><img src="image_5.png" /><figcaption>Fig. 4. The fraction of nLoS for D2D scenario.</figcaption></figure>

with the number of blockers corresponding to $\lambda S_A$, where $S_A$ is the service area of interest, where $\lambda$ is the density of blockers in the Poisson model. The results are demonstrated in Fig. 5. As one may observe, the outdoor model with a Poisson distribution of a number of blockers is characterized by a much higher value of the fraction of the blockage time. The difference is maximized for medium values of the blockers density in the range (0.10 – 0.25) blockers per squared meter.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Fraction of LoS blockage time in indoor and outdoor scenarios.</figcaption></figure>
# 5 Conclusions

In this manuscript, we derived the probability of nLoS and the fraction of time a user spends in nLoS state in a group of *N* blockers moving according to RDM in the indoor closed compartment for both AP and D2D connectivity cases. We then provided the extension to the case for outdoor deployment, where UEs freely enter and leave the service area. Our results apply to performance analysis of mmWave and THz wireless systems.

The central application area of the proposed model is performance analysis of traf-fic offloading in heterogeneous multi-layer wireless access networks featuring mmWave and/or terahertz APs. Notably, the fraction of LoS blockage and/or outage time charac-terize the amount of time resources of systems have larger coverage, and usually smaller capacity needs to be used.

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