A Holistic Assessment of Directional Deafness in mmWave-based Distributed 3D Networks

The adoption of abundant millimeter-wave (mmWave) spectrum offers higher capacity for short-range connectivity in various Unmanned Aerial Vehicle (UAV)-centric communications scenarios. In contrast to the conventional cellular paradigm, where the coordination of connected nodes is highly centralized, the distributed deployments, such as those operating over unlicensed frequency bands, maintain robust interactions in the absence of central control. These agile decentralized systems are being naturally created by dynamic UAV swarms that form a temporary 3D structure without reliance on remote management or pre-established network infrastructures. While much effort has been invested in the performance assessment of distributed, directional, and 3D systems individually, a combination of these three angles allows capturing more realistic UAV swarm scenarios and produces a novel research perspective. This work addresses one of the fundamental challenges in mmWave-based 3D networks – directional deafness – which is known to adversely affect the overall system performance. Particularly, we develop a mathematical framework by taking into account the peculiarities of 3D directional and distributed deployments. We provide a holistic analytical assessment of directional deafness and propose several powerful approximations that capture realistic antenna patterns.


A. Research Motivation
The fifth-generation (5G) and beyond communication technology facilitates an expanding range of applications, such as enhanced virtual and augmented reality, industrial control systems, self-driving vehicles, and, notably, autonomous robots and drones. These require increasingly intelligent and effective operations [1] as they impose unprecedented performance demands and may thus encounter radio connectivity O. Chukhno This work was also supported by the Academy of Finland (Projects RA-DIANT and IDEA-MILL) and by the JAES Foundation (Project STREAM). This paper has been supported by the RUDN University Strategic Academic Leadership Program (recipient K.S., Section I). The reported study was funded by RSF, project number 22-29-00694 (recipient Yu.G., Section VII). challenges. To this end, the next-generation wireless systems adopt communications over extremely high frequency (EHF) bands, such as millimeter-wave (mmWave) spectrum [2] that offers high capacity for short-range connectivity.
In the mmWave spectrum, the use of highly directional transmissions is essential to support high antenna gains and, thus, compensate for severe signal attenuation, large atmospheric absorption, and high vulnerability to channel fluctuations [3]. Three-dimensional (3D) beamforming (BF) techniques allow devices to concentrate the transmit power over narrower beams and maintain precise alignment [4] increasing the antenna gain and improving the radio connectivity. In 3D networks that naturally emerge in various Unmanned Aerial Vehicle (UAV)-centric communication scenarios [5], this procedure is of crucial importance due to higher altitudes and faster mobility of the involved drones.
Not limited to mmWave cellular connectivity, the 802.11 Wireless LAN Working Group has recently ratified advanced system specifications [6] that enable high data rate transmissions by expanding the legacy IEEE 802.11ad capabilities [7]. Different from the conventional cellular paradigm, where the coordination of connected nodes is highly centralized, distributed deployments, such as those operating over unlicensed frequency bands, maintain robust interactions in the absence of central control, ownership, or regulation. Here, the centralized connectivity arbitration procedures typically driven by a cellular base station become distributed [8], [9]. These decentralized systems are being naturally created by UAV swarms that form a temporary structure without reliance on remote management or pre-established network infrastructures [10].
As UAV swarms are becoming increasingly large and dynamic and require stringent performance guarantees for critical (e.g., merge and split) operations, it becomes essential to enable bandwidth-hungry distributed applications (e.g., in-swarm communications and control convergence). With 2D antenna arrays required by 3D BF techniques, the number of on-board antenna elements may be increased without exceeding the inherent form-factor limitations, which offers reduced halfpower beamwidth (HPBW) and higher antenna gains. The 3D BF allows adapting the beam pattern in both elevation and azimuth planes to exploit more degrees of freedom. Hence, analysis of directional 3D systems is demanded to capture more realistic UAV situations that become commonplace for communications across both UAV-to-UAV and UAV-toground links [11]. While much effort has been invested in the performance assessment of distributed, directional, and 3D systems individually, a combination of these three angles produces a novel perspective for the research community. The need for directional distributed 3D networking in UAV swarm communications serves as the motivation behind this work.

B. Essential Related Work
Over the recent years, the research community focused efforts on equipping flying platforms with radio communications capabilities by advancing the UAV setups in both centralized [12] and distributed systems [13]. There is already a large body of literature on mmWave-based UAV cellular networking, which includes channel [14] and antenna [15] modeling, coverage analysis [16], 3D beam management [17], positioning [18], trajectory design [19], interference assessment [20], performance optimization [21], experimental studies [22], and UAV prototype development [23].
Despite the benefits of cellular-connected UAV applications, larger UAV swarms require reliable support beyond the coverage of cellular deployments by employing hybrid 3D connectivity. In this context, distributed mmWave-based (e.g., IEEE 802.11ay) solutions have become of growing interest. They can help coordinate drones within a large and dynamic group as it travels through space and changes its configuration. However, the utilization of directional beams in distributed mmWave-based systems introduces unique peculiarities that stem from the notion of directivity [24]. A cornerstone phenomenon is directional deafness that occurs where a transmitter fails to communicate to its intended receiver because the latter is beamformed in a different direction [25]. Deafness is typically caused by a lack of state information from the neighboring nodes and may significantly degrade the system performance.
The research work on the coordination of nodes suffering from deafness has accelerated recently but remained focused on directional terrestrial networks. In [26], a channel access protocol that copes with deafness over a single channel and one radio interface was designed. An alternative multi-band approach to address the deafness effects was proposed in [27]. In [28], the use of adequate association together with relaying techniques was shown to alleviate the impact of deafness. In [29], another protocol was coined that combines directional antennas with predictive positioning to overcome deafness. Most recently, the work in [30] aimed to resolve deafnessinduced problems by resorting to preamble-based detection mechanisms.
However, a quantitative analysis of directional deafness effects has only been conducted in [31] and, more recently, in [32]. In [31], a deafness probability assessment was proposed for the nodes that are randomly located on a plane. In [32], the authors extended that work to a 3D deployment scenario while controlling the radiation pattern in one plane. A limitation of the existing knowledge is not only in reliance on 2D geometry that applies solely to legacy 2D BF techniques. Past literature also fails to cover the impact of deafness in UAV scenarios, which is increasingly timely as drone formations grow larger, become more distributed, and employ directional antennas. This work aims to bridge the indicated gap by contributing a new mathematical methodology, which brings together distributed UAV communications, directional mmWave connectivity, and 3D BF features with the aim to conduct a holistic deafness probability evaluation.

C. Main Contributions
In this work, by relying upon stochastic geometry tools, we formulate a new methodology for characterizing directional inswarm UAV communications in the unlicensed bands under realistic assumptions. We deliver a multifaceted analytical assessment of directional deafness in distributed 3D UAV networks. The essential contributions of this study are summarized in more detail as follows.
• We propose an analytically tractable model for the antenna pattern represented by a linear function of beamwidth, which mimics realistic radiation patterns and makes it possible to produce closed-form performance estimates. In particular, in this work, we obtain a lower bound for distance-dependent 3D deafness probability. • We offer an approximation for the coverage area of a symmetric antenna as an ellipse, which helps reduce the analysis complexity and improve the accuracy of the results. We assess the tightness of our approximation and establish the parameters required to accurately model the system at hand. • By further relaxing the assumption on symmetric antenna properties, we develop a more precise yet more complex ellipsoidal approximation of the coverage area for antennas with asymmetric radiation pattern. By employing this approximation, we offer an expression for the deafness probability in directional UAV systems with arbitrary antenna arrays. • We extend the methodology for deafness probability evaluation to the case of arbitrary real-world deployments of UAV swarms and provide selected numerical results to demonstrate its modeling capabilities, thus completing the proposed framework. The provided illustrative example mimics distributed UAV-to-UAV communications over the emerging mmWave radio technology by incorporating directionality, decentralized operation, and 3D structure. The remainder of this text is organized as follows. In Section II, we outline the system modeling assumptions that underpin our subsequent analysis. This is followed by an analytically tractable solution for the deafness probability estimation developed in Section III, which is based on a linear approximation of antennas. Further, our geometry-based approach to the coverage area approximation for both symmetric and asymmetric antennas is detailed in Sections IV and V, respectively. The practical application of the framework is briefly summarized in Section VI. Our numerical evaluation is reported in Section VII. The work concludes in Section VIII.
II. SYSTEM MODEL This section outlines a representative UAV scenario that captures highly directional communications in a distributed 3D setting. We summarize our modeling assumptions on the deployment, antenna radiation pattern, and signal propagation and define the target metrics of interest.

A. 3D Deployment
We consider a distributed network comprising aerial (e.g., autonomous UAV) and ground (e.g., access point, AP) nodes equipped with highly directional antennas. Particularly, we focus on an open-air scenario where the drones are elevated at different heights within a certain 3D area of interest as shown in Fig. 1.
For the sake of mathematical tractability, we divide the UAV swarm into more understandable primitives. In this work, a triangle represents the primary building block based on which more complex setups can be analyzed, as discussed in Section VI. Our target example is demonstrated in Fig. 2, where transmitter A (termed responder), which may operate as an AP, has established a connection with device B (primary initiator). We consider secondary initiator C that attempts to connect to the AP (device A) and may experience directional deafness.
Assumption 1. We assume that the location of receiver B is distributed uniformly inside a sphere of radius R d around A, as shown in Fig. 2a. Therefore, the probability density function (pdf) and the cumulative distribution function (CDF) of distance r between the connected devices A and B are given by Spatial locations of target devices A, B, and C form a triangle △ABC. We further analyze the deafness problem by considering the plane passing through these three points, as shown in Fig. 2b. For clarity, we denote the angles ∠BAC and ∠ABC as α and β, correspondingly, whereas d is the distance between A and C, d BC is the distance between B and C. Here, α and r are independent random variables, d BC and β depend on r, d, and α as follows from the triangle: (2) In general, the pdf f α (α) of angles α converges to the following expression [33]: where n is the number of space dimensions. For instance, for n = 2, f α (α) = 1/π, while in our 3D setup, the pdf f α (α) and the CDF F α (α) are given as Note that the distributions of r, d, and α can vary and rely on the shape of the considered area and the location of A as well as on the spatial distributions of B and C.

B. Realistic Antenna Pattern
Devices A and B are equipped with highly directional transceivers. We assume that a narrow radio beam between A and B is perfectly aligned as a result of employed beamforming procedures, for example, based on compressive sensing [34]. To explicitly model the antenna radiation patterns,  we make the following assumption on symmetric antenna properties.

Swarm of drones
Assumption 2. To preserve analytical tractability, we assume that the main antenna lobe is symmetric w.r.t. the antenna boresight axis, i.e., has a symmetric beam pattern in two planes.
Assumption 2 allows incorporating the randomness of antenna rotation around the boresight axis. The antenna directivity gain D comprises the maximum directivity D 0 along the boresight and a directivity function ρ(α) of the angular deviation from the boresight direction, α ∈ [0, π]. Then, the total directivity is defined by D(α) = D 0 ρ(α), where ρ(0) = 1 corresponds to the antenna boresight.
There are several ways to evaluate the coefficient ρ(α) numerically, namely, by using (i) radiation pattern measurements, (ii) simulation via CEM tools, or (iii) calculation of electromagnetic field produced by specific antenna arrays. It can also be approximated by popular theoretical models, e.g., (i) sector/two-sector antenna pattern for HPBW θ [31], (ii) multi-cosine [35], and (iii) multi-level flat-top antenna patterns [36], or (iv) as outlined in [32] and below.
Assumption 2.1 (linear function). The function ρ(α) of the relative angle α can be approximated by a piecewise linear function of misalignment of α: We emphasize that in the proposed approximation, we disregard the presence of minor antenna lobes (e.g., side-or backlobes) contrary to the popular model that consists of a sector and a circle of a short radius. A critical drawback of such models that rely on smaller circles is in their trivial cutoff solution [31], which means that they cannot be exploited for the deafness probability analysis. Assumption 2.2 (directivity gain for symmetric radiation patterns). Here, we employ an approximation for the radiation pattern in the form of a conical area. For the HPBW of θ, we may obtain the maximum value of the antenna directivity gain D 0 as a ratio between the solid angle Ω covered by the beam, Ω = 2π(1 − cos θ/2), and the solid angle of a sphere [37] To model the coverage area, we formulate the following assumptions.
Assumption 2.3 (coverage area approximation). We assume that the coverage area of a highly directional antenna has the shape of an ellipse E, E ≡< x 0 , y 0 , a, b >, where x 0 , y 0 are the coordinates of the ellipse center in the horizontal and vertical axes and a, b are the lengths of semiaxes. Then, the standard equation of an ellipse is determined in Cartesian coordinates as This approximation for the coverage area of a directional antenna by an ellipse is advantageous in terms of (i) reduction in analysis complexity and (ii) increase in simulation accuracy for symmetric antennas subject to adequate selection of ellipse parameters, which is described in Section IV. To further account for more complex antenna patterns, we adopt the following assumptions.
Assumption 3. For the case of asymmetric antenna configurations with 2D antenna elements (see Fig. 3), sufficient steering capability is provided in both horizontal and vertical planes by taking into account HPBW for the azimuth and elevation angles.
Assumption 3.1 (directivity gain for asymmetric radiation patterns). Here, we assume that antenna directivity can be determined as D 0 ≈ 4π θazθ el [37], where θ az is the HPBW in the horizontal plane and θ el is the HPBW in the vertical plane.
Assumption 3.2 (coverage area approximation). We assume that the shape of 3D coverage area of a directional antenna is represented by an ellipsoid E 3-axial ≡< x 0 , y 0 , z 0 , a, b, c> with center (x 0 , y 0 , z 0 ) and semiaxis lengths (a, b, c). The ellipsoid is characterized by We demonstrate the difference between symmetric and asymmetric antenna properties w.r.t. the deafness probability and confirm the feasibility of the above assumptions in Section VII.

C. Propagation Characterization
We assume that nodes A and B transmit in the directional mode and C listens omnidirectionally, as per IEEE 802.11ad/ay specifications w.r.t. the connection establishment [38]- [40]. Directional reception would require sweeping through all combinations of receiving and transmitting antenna codebooks and result in overheads, which may prohibit efficient UAV support.
Assumption 4. The average path loss can be calculated as L(d) = Cd κ , where d is the distance between the transmitter and the receiver, κ is the path loss exponent that depends on the environment, C = (4π/λ) 2 is the propagation constant, and λ is the wavelength. The received power can then be calculated as where P tx is the transmit power, G tx = D 0 ρ(α) and G rx = 1 are transmit and receive antenna gains. The sensitivity P thr determines the minimum power required by the control physicallayer functions [38]. Therefore, to calculate the radius of the coverage area R, we substitute d = R and P rx = P thr into (9) as R = PtxD0

D. Directional Deafness
We define directional deafness as a situation where a UAV attempts to communicate with an active device (UAV AP) and fails because it is unable to sense the medium busy during the RTS/CTS handshake procedure employed in IEEE 802.11 systems. In our three-device setup, A and B are connected, beamformed toward each other, and actively exchange data, whereas C aims to establish a connection with A. While attempting to access the medium, device C may face the following outcomes: 1) Device C receives a signal from either A or B (the received power from A or B exceeds the sensitivity threshold, i.e., P rx,A ≥ P thr or P rx,B ≥ P thr ). 2) Device C fails to receive a signal from A and B, or both received powers P rx,A and P rx,B at C are lower than the minimum threshold, i.e., P rx,A < P thr and P rx,B < P thr . In this case, device C may interpret the absence of a response from A as an indication of a collision and retransmit repeatedly, which leads to an inflated backoff time and eventually to a packet drop. We refer to this event as to directional deafness. Hereinafter, we evaluate the deafness probability, i.e., the probability that secondary initiator C fails to communicate with responder A as antenna beams of both A and primary initiator B are directed toward each other: Our system model introduces a set of assumptions to capture the essential directionality effects in 3D distributed UAV networks. The proposed framework goes beyond a particular area-of-interest shape and may capture any 3D configuration (e.g., an ellipsoid, a hemisphere, a dodecahedron, etc.) across random drone swarm layouts. Based on these assumptions, we first obtain a closed-form lower bound on the 3D distancedependent deafness probability by applying a linear antenna approximation (Section III). This approximation represents a simple-to-use function for the misplacement of the antenna directivity, which is compatible with the existing solutions (sector, etc.). Then, we approximate the coverage area of a symmetric antenna as an ellipse (Section IV). We derive an expression for the more complex asymmetric antenna model that characterizes the coverage area (ellipsoid-based), which is representative of real-world UAV systems (Section V). We elaborate on our methodology for analyzing directional deafness effects by adapting it to practical UAV deployments (Section VI).

III. DEAFNESS PROBABILITY ANALYSIS
In this section, we offer a general formulation for the deafness probability, which can be analyzed numerically. We also obtain the respective lower bound represented by a closed-form expression for the antenna patterns approximated according to (5).

A. Distance-Dependent Deafness Probability
First, we derive a general expression for the directional deafness probability in a 3D system. Proposition 1. For a fixed distance of d between responder A and secondary initiator C in a space of any dimension and given radius R d of the area of interest, the deafness probability averaged over spatial locations of initiator B can be expressed by where I(.) is an indicator function and β and d BC are given by (2).
Proof. The proof is based on the received power in the omnidirectional reception mode (9): Corollary 1. If locations of device B are distributed uniformly within a sphere of radius R d , and f r (r), f α (α) follow (1) and (4), correspondingly, the deafness probability is given by: Proof. The proof follows from Proposition 1.
Expression (13) provides an approximation that can be applied to different types of antenna radiation patterns by selecting an appropriate reduction factor ρ(α) that captures the decrease in power due to deviation α from the boresight axis.

B. Lower Bound on Deafness Probability
The use of the linear antenna pattern approximation (5) permits the derivation of a simple and analytically tractable solution in a closed form.
Proposition 2. If locations of B are distributed uniformly within a sphere of radius R d and linear function (5) approximates the antenna directivity, then a lower bound on the probability of deafness at a fixed distance of d may be obtained as follows: For shorter distances d ≤ R d sin θ, or, otherwise, for d > R d sin θ, where Proof. The proof is detailed in [32].

IV. ELLIPSE-BASED APPROACH: SYMMETRIC ANTENNAS
In this section, we outline a geometric approach to the analysis of directional deafness probability by employing an ellipse as an approximation of the coverage area.

A. General Description
We study a three-node system, which represents a minimal feasible set that can be extended to arbitrary deployments as discussed in Section VI. We consider two ellipses, E A (a, 0, a, b) and E B (r− R 2 , 0, a, b), which correspond to the coverage areas of devices A and B, and circle E C (0, 0, d, d) that denotes the locus of point C as illustrated in Fig. 4. We further define the deafness zones for device C.
Our approach is to evaluate the deafness probability as a ratio between the weighted arc length of the deafness zone and the length of the semicircle. Below, we define the locus of points in the deafness zone in Lemma 1; the formulation for the deafness probability in our 3D UAV deployment follows in Proposition 3 and Corollary 2.

B. Estimation of Directional Deafness Probability
Lemma 1. The locus of node locations in the deafness zone is given by the arcs of circle E C (0, 0, d, d) bounded by points A i , B i , where the circle intersects two ellipses E A (a, 0, a, b) and E B (r − R 2 , π, a, b). Here, angle α belongs to the area where γ 1 = arcsin Bi ) are the intersection points of E C and E A , E B and their corresponding coordinates, and d * is the maximum distance between A and C that allows C to remain inside the ellipse corresponding to B as indicated in Fig. 4. Proof. The coordinates of the intersection points of the ellipses and the circle can be established from the following systems of equations: and Solving the former system, we obtain the roots (18)- (20) that correspond to the intersection points A 1 , A 2 of the locus circle and the ellipse related to device A. Similarly, we solve the second system for the locus circle and the ellipse related to device B and obtain the coordinates of points B 1 , B 2 , B 3 , B 4 as provided in (21)- (23). The coordinates are as follows: x B1,2 = 1 2 Hence, depending on distance d, we have the following possible options: • no deafness and no equation roots if d < d * or no deafness and 2 roots if d = d * (see Fig. 4a), • 2 deafness zones and 6 roots if d * < d ≤ R−r (see Fig. 4b), • 1 deafness zone and 2 roots if R−r < d ≤ R (see Fig. 4c), • full deafness and no equation roots if d > R (see Fig. 4d). Here, d * represents the maximum distance between A and C that allows C to remain inside the ellipse corresponding to B (no deafness) and is calculated from the zero discriminant Utilizing the result of Proposition 1, we arrive at the method of calculating the deafness probability under the assumption of symmetric antennas. This means that the main antenna lobe is symmetric w.r.t. the antenna boresight axis and that the antenna pattern in both horizontal and vertical planes remains the same (i.e., has approximately cone shape).
Proposition 3. For any device C in any space dimension, the deafness probability at a fixed distance of d can be approximated as follows: where α represents ∠BAC, Ω α is interpreted as (16), f r (r) and f α (α) follow from (1) and (4).
Proof. The proof follows from the definition of the deafness probability and Lemma 1 based on the assumption on the symmetry of our model w.r.t. the horizontal axis: Corollary 2. For devices located in 3D space, the deafness probability may be derived as where y A1 is given by (20), and y B1 , y B2 correspond to (22).
To implement the proposed ellipse-based approach in UAV scenarios of an arbitrary dimension, it is essential to (i) identify the intersection points of geometrical figures that determine the coverage area of A and B, and the locus of locations of C; (ii) based on (20), (22), and (23), calculate angles γ 1 and γ 2 that correspond to the arc lengths of non-deafness zones and find the deafness zones on the semicircle; (iii) define the indicator function I(Ω α ) by taking into account d and r; and (iv) assign weights to the values of angle α according to (4).

C. Parametrization of Ellipse-based Model
Here, we address the derivation of the parameters for each ellipse in the proposed UAV scenario with three nodes. The major semiaxis a of the ellipse equals half of the maximum distance R between the transmitter and the receiver, at which a radio link can be maintained: Then, b may be produced from (7) with x = cos (θ/2) d hpbw and y = sin(θ/2) d hpbw , where d hpbw is the distance between the antenna and the half-power points (-3dB) as detailed in Fig. 5. It can be derived from the following relationship: where G max is the maximum gain and G(θ) = G max /2. By substituting the parameters a, x, and y into (7), we characterize the minor semiaxis b as: However, realistic antenna patterns are closer to the form of a drop (dotted pattern in Fig. 5) rather than an ellipse (line), and depend on the beamwidth. Hence, we can obtain b ′ from the upper triangle through hypotenuse d hpbw from (29) and angle θ/2: b ′ = sin(θ/2) R 0.5 1/κ .
We also formulate a more coarse approximation for b, which may be derived from the bottom triangle. Here, by analogy, we have angle θ/2 and the adjacent cathetus a. Then, In practice, an antenna pattern might have no definite form; therefore, we validate and analyze our results against realistic settings in Appendix.
V. ELLIPSOID-BASED APPROACH: ASYMMETRIC ANTENNAS In this section, we derive formulations for more practical but complex antenna models, i.e., those that are not symmetric w.r.t. the antenna boresight (referred to as asymmetric antennas). We characterize the shape of the coverage area for devices A and B as an ellipsoid (see Fig. 6). T Fig. 6: Ellipsoid as approximation of coverage area for asymmetric antennas.

A. General Description and Parametrization
We consider the intersection of an ellipsoid that approximates the coverage area of a device (A or B) and a sphere that reflects the possible locations of device C: where semiaxis a corresponds to (28) from the ellipse-based approach, b corresponds to (30), (31), or (32) with θ = θ az , and semiaxis c can be obtained from (30), (31), or (32) by substituting θ el into θ. In other words, the ellipsoid projections are two ellipses in azimuth and elevation planes with semiaxes a, b and a, c, correspondingly. We define the deafness probability via the surface area, which is not cut by ellipsoids (A and B) from the sphere. Importantly, we derive the general expressions for both ellipsoids A and B with the centers along the X-axis at x 0 . Note that the ellipsoid that approximates the coverage area of device A is centered at (a, 0, 0). In the case of device B, the center of the ellipsoid is (r − R/2, 0, 0) by analogy with the ellipse-based approach.

B. Deafness Probability Estimation
We estimate the deafness probability by assessing the outer area of the solid of intersection of the sphere and the ellipsoids. We determine the surface area of the spherical cap inside the ellipsoid as a non-deafness zone, while the outer pieces of the sphere represent deafness zones.
Proposition 4. Depending on distance d, the deafness probability can be obtained from where S A,right (σ), S B,right (σ), S B,left (σ) are the sought surfaces of the sphere bounded by ellipsoids A and B (from the right and the left sides), respectively; f r (r) follows (1).
Proof. Let surface σ be defined by the expression x=f (z, y) and its projection onto the plane zOy is the region D zy . Then, the area of the surface σ is determined by The proof of this proposition is structured as follows. First, we determine the plane projection D zy of surface σ. Then, we proceed with the surface σ and define the surface area of the portion of the sphere inside the ellipsoid.
Projection: Plane projection D zy of surface σ is an ellipse with semiaxes a 1 and b 1 (see Fig. 6). To obtain the intersection curve (ellipse) that defines D zy , we consider each of the horizontal E h and vertical E v ellipses of the ellipsoid separately for the sake of exposition. Therefore, for the vertical plane, we characterize the intersection of ellipse E v with the equation of a line: To establish the intersection point T 1 (x T1 (x 0 ), 0, z T1 (x 0 )) of line d with ellipse E v , we solve (36): In the case of device A, coordinates of T 1 can be transformed into 2ac 2 c 2 +a 2 tan 2 ϕel , 0, 2ac 2 tan ϕel c 2 +a 2 tan 2 ϕel . Since x 2 T1 + z 2 T1 = d 2 , we obtain ϕ el (x 0 ) as follows: where . Similarly, we solve the system of equations for ellipse E h and equation of a straight line: For brevity, we omit the derivation details and expressions for x T2 (x 0 ), y T2 (x 0 ), ϕ az (x 0 ). The projection D zy of surface σ onto plane zOy is an ellipse with semiaxes a 1 (x 0 )=z T1 (x 0 ) and b 1 (x 0 )=y T2 (x 0 ): where a 1 (x 0 ) = tan ϕ el x T1 (x 0 ) and b 1 (x 0 ) = tan ϕ az x T2 (x 0 ). Surface σ: Further, we formulate the expression for σ, which bounds the ellipsoid. Since x = d 2 − y 2 − z 2 , the partial derivatives of x with respect to z and y can be obtained from (42) The surface integral over projection σ onto plane zOy has the following form To calculate (43), we transition to the polar coordinate system by replacing the variables z = ρ cos ϕ, y = ρ sin ϕ, ρ 2 = z 2 + y 2 , and dzdy = ρdρdϕ. Hence, (43) transforms to As ellipse equation determines the limits of integration, the outer ones are 0 and 2π, while the inner integration limits can be derived from Consequently, the surface of the sphere bounded by the ellipsoid can be established from Then, the sought surface is given by the following expression: Importantly, we characterize the surface area cut by the ellipsoid given by (x−x 0 ) 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1 from the sphere x 2 +y 2 +z 2 =d 2 . We obtain surface region (47) bounded by the intersection curves, which represents the non-deafness zone. Note that ellipsoid A cuts one (or two) part(s) out of the sphere depending on distance r, while ellipsoid B and sphere C have only one intersection curve. Therefore, the deafness probability can be obtained via the numerical integral (34).

VI. PRACTICAL CONSIDERATIONS
In this section, we summarize the extensions of our methodology for practical use. Given the reference three-device setup, we consider the following cases that cover all possibilities to construct an N -node UAV system as illustrated in Fig. 7.
1) Case 1: The N -node system comprises (N − 2) devices of type C, one device A, and one device B. In this case, the deafness probability P D (d) depends only on distance d. The resulting expression is the same for any device C i , i = 1, ..., N − 2, i.e., corresponds to (10), which is averaged over all possible locations of B. Due to averaging, we further refer to the cases where there is only one device of type C in the system.
2) Case 2: The N -node system comprises one device A, which communicates with (N − 2) devices of type B, while one device C aims to connect to A. Here, there are two options: Case 2.1: Success of C depends on B i , i = 1, ..., N , which is active when C activates. Since the deafness probability is averaged over all possible locations of B i and, thus, does not depend on i, the resulting expression for this subcase corresponds to (10). This operation mode is foreseen to be the most utilized in 802.11ad/ay networks. Case 2.2: Device A is equipped with multiple beams to communicate with (N − 2) devices of type B and serves them simultaneously. This subcase implies multi-user multiple-input multiple-output (MU-MIMO) operation, which is uncommon but can be supported by, e.g., the emerging IEEE 802.11ay protocol [40]. The deafness probability, i.e., the probability that secondary initiator C fails to communicate with responder A as antenna beams of device A direct toward B i , i = 1, ..., N is given by rx,A < P thr , P rx,B1 < P thr , ..., where P (i) rx,A represents the received power (at C) from A when it transmits to B i , P rx,Bi is the received power from B i , and the maximum power budget of A is split among (N − 2) devices of type B.
3) Case 3: The N -node system comprises N −1 2 devices of type A, N −1 2 devices of type B (each A is connected to one B), and one device C that aims to join the system. Device C attempts to connect to one of devices A i at a time. The target AP A i is selected according to a certain association rule, which defines the shape of the distribution of distances d between C and A i . If the same association rule is adopted across the entire network, i.e., employed also by primary initiators B i , then it as well affects spatial locations of B i around A i and should be carefully taken into account.
Case 3.1: No coordination in the system of beams between A i and B i . The success of C depends on the pair A i and B i . We, thus, have P D = Pr{P rx,A1 < P thr , P rx,B1 < P thr , ..., The deafness probability in the case of random deployments of devices of type A is of particular interest from the network point of view and is analyzed numerically in Section VII. We note that it is also possible to find analytical solutions for particular types of deployments of A i given the distribution of distances from C to A i , i = 1, ..., N −1 2 . The simplest case of Poisson Point Process (PPP) of A i locations yields independent distances d and results in factorization of (48), while deployments with more deterministic structure, such as grid, might require introducing further approximations.
Case 3.2: There is coordination between APs A i . For example, the IEEE 802.11ad/ay amendment supports clustering of distributed APs to mitigate high interference between them [41]. Such clustering allows a group of APs to schedule their transmissions in non-overlapping time periods, thereby allowing co-channel APs to coordinate beaconing and avoid interference in dense environments. This subcase is more complex as it includes spatial correlation between the active beams and, particularly, locations of active B i . However, it could be approximated via thinning of the point process of A i locations.
Case 4: Finally, we consider a union of all the above cases as an N -mode system comprising M devices of type A, N − M −1 devices of type B, and one device C. Then, the deafness probability corresponds to the union of cases. For example, for cases 2.2 and 3.1, it can be formulated as rx,A1 < P thr , P rx,B1 < P thr , ..., where P (j) rx,Ai represents the received power (at C) from A i when it transmits to B j , P rx,Bi is the received power from B i . Presented cases cover the entire event space for directional deafness evaluation in a realistic UAV environment. We note that at a particular instant, device C may also receive high interference from AP A j , other than the target A i . In this case, we assume that C senses the channel as occupied and does not attempt to initiate a transmission. Such event is not classified as network-based deafness as there is no deafness w.r.t. A j ; however, in this case, the deafness probability would be lower than the deafness estimated in a three-node system.

VII. NUMERICAL ANALYSIS
This section collects important numerical results on analyzing the deafness probability in 3D directional and distributed UAV systems. We illustrate the capabilities of our proposed framework under assumptions of symmetric and asymmetric radiation patterns. Furthermore, we study the impact of antenna pattern rotation and system parameters on the deafness probability. The validation of our framework by comparing the results of the proposed mathematical approximations against those obtained via simulations for realistic antenna radiation patterns is detailed in Appendix.

A. Simulation Setup and System Parameters
As an illustrative example, we consider a UAV swarm elevated at different altitudes within the area of interest of radius R d (see Fig. 1, Fig. 2). Other nodes are served by a UAV AP equipped with, e.g., novel IEEE 802.11ay chipsets, which provide reliable high-rate connectivity in a noise-limited environment [42]. Following the recommendations from the available standardization documents [6], [38], [39], we assume the fixed transmit power of P tx = 23 dBm, while the receive power exceeds the threshold P thr = −78 dBm that corresponds to the lowest modulation and coding scheme (MCS0).
To accurately capture the performance of real-world UAV deployments, we employ the path loss model with κ = 2.25 as the propagation of 60 GHz signals exhibits higher losses for aerial links [9]. We assume uniform rectangular and linear arrays (URAs/ULAs) of isotropic/cosine elements; the antenna patterns are constructed using MATLAB Sensor Array Analyzer. The remaining system parameters are summarized in Table I.

B. Practical Setup Analysis
We begin by evaluating the proposed analytical framework for the N -node scenario and the antenna patterns that are symmetric w.r.t. the boresight axis. To this aim, we employ 2x2, 4x4, 8x8, 16x16, 32x32, and 64x64 URAs of isotropic antenna elements. We analyze the probability of deafness for cases 3.1, 2.2, and 4 outlined in Section VI as demonstrated in Fig. 8, Fig. 9, and Fig. 10, respectively.  First, we employ multi-connectivity degrees ranging from 1 to 5 while assuming the use of IEEE 802.11ad/ay decentralized clustering. We compare the deafness probability for the case of N = 3, 5, 7, 9, 11, i.e., 1 to 5 devices of type A, 1 to 5 devices of type B, and one device C that aims to establish a connection. In this scenario, devices A are distributed according to a PPP. In Fig. 8, the deafness probability is illustrated as a function of the number of nodes N . We observe that the increase in N leads to reduced deafness probability for this particular case with distributed coordination mechanisms. We then analyze the case with a highly directional multi-beam antenna design on the AP side. In Fig. 9, we demonstrate the probability of deafness for the case of N = 3, 4, 5, 6, 7, i.e., one device A, 1 to 5 devices of type B, and one device C. As one may observe, multi-user operation compensates deafness probability offering more options to avoid communication failures for device C.
To complement the discussion, we demonstrate the deafness probability for case 4 in Fig. 10. As a general trend, one may notice that the system performance depends on the antenna array size. For instance, 32x32 and 64x64 URAs are more vulnerable to deployment density in terms of deafness probability and demonstrate up to 95% deafness probability reduction with an increase in the number of devices. In practice, the operator may consider the choice of the beam resolution that might be related to a particular deployment and the required level of reliability. We emphasize that densification in the systems with coordination among APs should not degrade the system performance due to the interference. In contrast, densification without coordination may increase the number of collisions leading to system operation failures.

C. Effects of Antenna Pattern Rotation
One of the benefits of our ellipsoid-based approach is that it captures the properties of asymmetric antennas. In Fig. 11, we illustrate the deafness probability for ULA. We observe that the curves display behavior similar to that of URAs. Particularly, the deafness probability declines down to 50% for the ULA consisting of 4 isotropic antenna elements. We emphasize that the methods with centrally symmetric approximations of antenna patterns might not apply to linear arrays. In contrast, the ellipsoid-based approximation as per (34) allows modeling a wider range of antennas and, hence, facilitates the analysis of various mmWave-based UAV systems.   We study the impact of beam rotation on the deafness probability, as illustrated in Fig. 11. We employ different antenna patterns of linear arrays for the ellipsoid-based coverage area approximation and rotate ellipsoid B through 90 • about the X-axis. As one may observe, such antenna rotation reduces the deafness probability in the case of asymmetric antennas. For symmetric URA antennas, the deafness probability exhibits a consistent behavior irrespectively of the UAV axial movement. This suggests a conclusion that representation of many reallife scenarios requires modeling 2D antenna array in 3D and confirms that the proposed framework can effectively capture in-swarm UAV communications. We note that an imperfect alignment of antenna beams of the transmitter and the primary initiator (A and B) may have effects on the deafness probability; however, this analysis goes beyond the scope of our work.
Finally, we assess the impact of antenna settings on the deafness probability. Accordingly, for ULA of isotropic/cosine antenna elements, we apply the Hamming tapering function (also known as a window function), as shown in Fig. 11. The behavior of the resultant curves remains relatively consistent regardless of the variations in the antenna properties, while the deafness probability can be controlled to a certain extent by employing various antenna settings. For instance, the 2x2 array of isotropic elements with the Hamming tapering reduces the deafness probability by up to 45% compared to the Chebyshev tapering. Generally, tapering windows are applied for sidelobe reduction [43] and, hence, for focusing the beam energy more efficiently, which may lead to different large-scale effects depending on the system settings, spatial deployments, and antenna configurations. For example, for "more vertical" 3D links such as those in wearable networks, narrow beams lead to better spatial reuse and increased spectral efficiency. Alternatively, for "more horizontal" connections, the interference in the system may increase, resulting in a deterioration of the system performance. In our formulation, Hamming tapering ensures the smallest peak of the sidelobes, which leads to an  increase in the deafness probability. Therefore, the antenna settings that enable a reduction in the deafness probability may, in practice, yield different results for other system parameters, such as spectral efficiency, SINR, etc.

D. Effects of Radio Settings
In Fig. 12 and Fig. 13, we study the impact of the antenna pattern on the distance-averaged deafness probability. For example, Fig. 12 illustrates the changes brought by the number of antenna elements for isotropic and cosine antennas and different window functions, such as Hamming, Taylor, and Chebyshev. In general, doubling the antenna size reduces the beamwidth by half and increases the directivity by approximately 3 dB, thus yielding increased deafness probability. As seen in Fig. 12, Chebyshev window function affects the deafness probability differently than Hamming and Taylor functions for both isotropic and cosine antenna elements. Particularly, for 128x128 arrays, the curves corresponding to the Chebyshev tapering decline with a growing number of antenna elements, which can be explained by the specific configuration of the antennas. More precisely, different window functions may result in dissimilar attenuation shapes and coefficients, thus bringing variance in deafness probability values. Here, all curves display a close agreement for smaller numbers of antenna elements, while for larger arrays, Chebyshev tapering results in a lower deafness probability.
In Fig. 13, we explore the impact of the transmit power on the distance-averaged deafness probability for a URA of isotropic antennas. Notably, transmit power P tx affects the selected antenna arrays differently. For example, when P tx increases, the deafness probability is reduced significantly more for the 2x2 array than in other cases. Moreover, the narrower the beams become, the less pronounced the deafness probability decline is. Based on this evaluation, we may conclude that although the dependency of the average deafness probability on the radio-level parameters might seem intuitive, it remains, in fact, susceptible to the selected antenna settings.

VIII. CONCLUSIONS
Employing highly directional transmissions in distributed aerial networks (e.g., mmWave-based UAV swarms) significantly increases the chances of directional deafness, which may detrimentally affect the system performance. In this work, we developed a mathematical framework to analyze deafness effects in 3D directional distributed UAV systems. To provide mathematical evaluation for both symmetric and asymmetric drone antenna patterns, we proposed three approximations: (i) a linear model for the antenna pattern and an approximation of the beam coverage by (ii) an ellipse and (iii) an ellipsoid.
Employing stochastic geometry tools, we provided four approaches for evaluating the deafness probability: an integral expression for numerical computation of the distancedependent deafness probability, its closed-form lower bound, and two expressions for the deafness probability within the ellipse and ellipsoid models. The latter allows considering more complex asymmetric antenna patterns as well as estimating the effects of antenna rotation. We also investigated the impact of the essential system parameters on the 3D directional deafness, which include antenna pattern and shape, number of antenna elements, and tapering functions.
We emphasize that our mathematical methodology remains applicable beyond the considered example scenario. The proposed approximations and their extensions can be utilized for modeling a wide range of UAV-based systems as well as may aid in designing, optimizing, and evaluating radio communications algorithms in emerging aerial 3D deployments. UAV-to-AP association problem is but one example of the proposed methodology extensions. In this case, the output from our framework -estimated deafness probability -can be utilized to evaluate and optimize various association policies. Alternatively, the deafness probability can be used as a criterion to assess the performance of future radio protocol enhancements.

APPENDIX
In this section, we validate our approximations using simulations based on realistic antenna patterns. In Fig. 14(a), we plot a set of curves illustrating the deafness probability P D (d) for the following five cases: (i) our analytical model as per (13), (2), (ii) the system-level simulation results, (iii) the linear model as per (5), (iv) the closed-form lower bound as per (14), (15), and (v) the ellipse-based approximation as per (26). The linear model offers a reasonable approximation as it preserves the same trend as the realistic antenna pattern (see "Simulation" curves). The deviation stems from the fact that realistic antenna patterns have more substantial variation due to a higher number of parameters. We emphasize that our lower bound on the deafness probability, which is based on the linear model, agrees well with the simulation results and indeed represents a lower bound of the linear approximation as it captures an extended set of antenna patterns.
Further, we evaluate the results of our ellipse-based approach as per (26) under the assumption of symmetric antenna patterns. As noted in Section IV, the antenna pattern varies depending on the beamwidth. In Fig. 14(a), we employ the closest b variations and apply b ′ from (31) for wider beams (i.e., θ > 30 • ) and b ′′ from (32) for narrower beams (θ = 8 • , 18 • ). Even though the difference between the curves in Fig. 14(a) remains noticeable from the quantitative perspective, their essential behavior is similar and repeats the trend for realistic antenna arrays. Therefore, all four approaches, namely, our linear model, numerical analysis, closed-form lower bound, and geometrical ellipse-based approximation, yield adequate results suitable for the UAV systems with centrally symmetric antennas.
In Fig. 14(b), we demonstrate the performance of our ellipsoid-based model as per (34) by controlling the radiation beam pattern in both elevation and azimuth planes for 4x4, 8x8, and 16x16 URAs of isotropic elements (nearly axissymmetric antennas). As one may observe, the results for symmetric and nearly symmetric antennas show a close agreement for the considered system parameters and UAV distributions. We also provide a comparison of the ellipse-and ellipsoidbased approaches, which reveals a perfect match between the results where the antenna pattern has an entirely symmetric form (e.g., for 16x16 array, HPBWs in the horizontal and vertical planes are the same). We highlight that even for nearly symmetric antennas, the absolute deviations from symmetric ones are relatively small, which implies that the developed models accurately capture the deafness probability in the case of URAs.