In this scheme, the receiver is assumed to be served successively by its nth nearest UAV-BS. The special case of the closest selection strategy is of particular interest, which provides the maximum average received power, i.e., y c = argmin y i ∈ Ψ ‖ y i ‖ , (1) where y _c denotes the location of the closest UAV-BS and ‖ ⋅‖ denotes the Euclidean norm. For notational simplicity, we also define r ₁ = ‖y _c ‖.

In this scheme, the receiver is selected randomly with uniform distribution among all UAV-BSs. This leads to the following: y u = U n i f Ψ | n Ψ > 0 , (2) where n (⋅) denotes the number of elements in a set and Unif{⋅} denotes the uniform-selection operation. The main merit of this scheme is that the receiver performs a fast response due to not acquiring other UAV-BSs’ channel state information (CSI), and therefore, each of the UAV-BS has the same opportunity to be selected as the serving BS.

In this scheme, the receiver is simultaneously served by its nth, n > 1, and nearest UAV-BSs. In this case, the n nearest UAV-BSs are assumed to use the same system resources, and their spatial locations form the cooperation set C n .

The channel conditions are characterized by different path-loss exponents, which are denoted by α. Then, following the standard power-law path-loss model for the path between the receiver and a UAV-BS located at y _i ∈ Ψ, the random path-loss function is defined as the following: l ‖ y i ‖ = K ‖ y i ‖ − α , (3) where K = c 4 π f c 2 , with c being the speed of light and f _c the carrier frequency.

Due to the presence of obstacles between a UAV-BS and the receiver, e.g., buildings and trees, it is assumed that shadowing exists. Therefore, in the considered wireless channel model, large-scale fading (or shadowing) co-exists with small-scale fading. Subsequently, the shadowing random fluctuations are modeled by the inverse-gamma distribution, which has been recently used in UAV-assisted communication scenarios (Bithas et al., 2020). Notably, the inverse-gamma distribution can not only adequately capture the shadowing conditions in UAV communications but also maintain analytical tractability. The probability density function (PDF) of the inverse-gamma distribution is given by the following equation: f S i x = γ q Γ q x q + 1 exp − γ x , (4) for i = 1, … N, where q > 1 is the shaping parameter of the distribution related to the severity of the shadowing, i.e., lower values of q result in lighter shadowing conditions, γ denotes the scaling parameter, and Γ(⋅) is the gamma function (Gradshteyn and Ryzhik, 2007, eq. (8.310.1)). Note that the inverse-gamma distribution is suitable and has been exploited for real-world UAV communication scenarios, and at the same time, it allows for a tractable performance analysis (Bithas and Moustakas, 2023).

The channels are assumed to experience Nakagami-m fading with a different fading parameter, m. Therefore, the channel power gains {h _i }, following a gamma distribution, h _i ∼gamma m , 1 m , with the shape and scale parameters of h _i being m and 1/m, respectively. The PDF of h _i is given by the following equation: f h i x = m m x m − 1 Γ m exp − m x . (5)

Note that E [ h i ] = m 1 m = 1 .

Under the n th nearest UAV-BS association policy, the SNR at the receiver is given by the following equation: S N R = p h i S i l r n σ 2 , (6) where σ ² is the additive white Gaussian noise power. For the special case of the nearest selection association scheme, i.e., n = 1, the SNR at the receiver can be rewritten as follows: S N R = p h i S i l ‖ y c ‖ σ 2 = p h i S i K r 1 − α σ 2 . (7)

Under the random selection scheme, the SNR at the receiver is given by the following equation: S N R = p h i S i l ‖ y u ‖ σ 2 = p h i S i l d i σ 2 . (8)

Under the JT-CoMP scheme, the SNR at the receiver is given by the following equation: S N R = ∑ i = 1 n p h i S i l r i σ 2 . (9)

Remark 1. Similarly to Hou et al. (2022), this work argues that based on low-complexity channel estimation algorithms (Jiang et al., 2020), the global CSI is assumed to be perfectly known at the UAV-BSs, if required. In addition, the assumption of perfect global CSI knowledge is common in stochastic geometry-based scientific papers related to UAV-assisted communication networks (Hou et al., 2019; Hou et al., 2020), where extensive signaling overhead is assumed for CSI acquisition at the UAVs. Nevertheless, performance analysis of UAV-assisted communications under imperfect CSI is a very common optimization problem, which requires a new analytical framework to be developed that is beyond the scope of the paper.

The coverage probability is defined as the probability that the SNR at the receiver exceeds a predefined threshold γ _th , P c γ t h ≜ P S N R ⩾ γ t h = 1 − F S N R γ t h , (10) and the average achievable rate of the receiver is given by Armeniakos and Kanatas (2022) as the following: R = E B w ⁡ log 2 1 + S N R = − B w ∫ 0 ∞ ⁡ log 2 1 + γ t h d 1 − F S N R γ t h = B w ln ⁡ 2 ∫ 0 ∞ 1 − F S N R γ t h 1 + γ t h d γ t h = B w ln ⁡ 2 ∫ 0 ∞ P c γ t h 1 + γ t h d γ t h = ( a ) B w ln ⁡ 2 ∫ 0 ∞ P c e t − 1 d t , (11) where (a) follows through the change in the variable γ _th + 1 = e ^t and B _w is the bandwidth available to the link of interest. The average energy efficiency is defined as the ratio of the average rate and the total power consumption, i.e., E E a v ≜ R P tot , (12) where P _tot is the total power consumption given by Bithas and Moustakas (2023) as the following: P tot = P cRx N + n ω + 1 p + P cTx , (13) where P _cRx and P _cTx denote the UAVs’ transmitter and receiver circuit powers required for signaling purposes, respectively. Note that Eq. 13 indicates that all UAV-BSs collect signaling information required for communication purposes, whereas only n UAV-BSs transmit information at a given time. Moreover, in Eq. 13, ω denotes the amplifier power efficiency given by ω = 2 2 R − 1 η 2 R + 1 , with η denoting the drain efficiency.

Lemma 1. The PDF of the distance r _n is given by the following equation: f r n r = N ! r r 2 − h 2 n − 2 R − r 2 − h 2 N − n R N Γ N − n + 1 Γ n , h ≤ r ≤ h 2 + R 2 . (14)

Proof. Let u _i denote the horizontal distance between the receiver and the projection of a UAV on the ground. Then, the unordered set of distances {u _i } comprises independent and identically distributed (i.i.d.) random variables, with the cumulative distribution function (CDF) of each element given by F u i ( u i ) = u i R , u _i ∈ [0, R]. Through a simple transformation on F u i ( u i ) , the CDF of the unordered set of distances {d _i } comprises i.i.d., with the CDF of each element given by F d i ( d i ) = d i 2 − h 2 R . Then, the PDF f d i ( d i ) can simply be obtained as f d i ( d i ) = d F d i ( d i ) d d i , and therefore, it is given by the following: f d i d i = d i R d i 2 − h 2 , d i ∈ h , h 2 + R 2 . (15)

By order statistics (Ahsanullah et al., 2013), the PDF of r _n can be obtained as the following: f r n r = N ! N − n ! n − 1 ! F d i d i n − 1 1 − F d i d i N − n f d i d i . (16)

After applying some simplifications, Lemma 1 yields.

Lemma 2. The PDF of the path-loss term l (r _n ) is given by the following equation: f l r n y = N ! R − n α K n − 1 ! N − n ! y K − 1 − 2 α y K − 2 α − h 2 n 2 − 1 R − y K − 2 α − h 2 R N − n , (17) for y ∈ [ l ( h 2 + R 2 ) , l ( h ) ] .

Proof. The PDF of r n − α can first be obtained through the corresponding CDF obtained as the following: P r n − α ≤ y = P r n ≥ y − 1 α = 1 − P r n ≤ y − 1 α = 1 − F r n y − 1 α . (18)

After applying the binomial expansion to f r n ( r ) , we can rewrite f r n ( r ) as f r n r = ∑ k = 0 N − n − 1 k N ! R − n − k r n − 1 ! N − n ! k N − n r 2 − h 2 n + k − 2 . (19)

Therefore, P [ r n − α ≤ y ] is obtained in terms of f r n ( r ) as P r n − α ≤ y = 1 − ∫ 0 y − 1 α f r n r d r = 1 − ∑ k = 0 N − n − 1 k N ! R − n − k n − 1 ! N − n ! k N − n y − 2 α − h 2 n + k 2 n + k . (20)

Through the change of variables n + k = i and with the aid of the definition of the Gauss hypergeometric function ₂ F ₁ (⋅, ⋅, ⋅, x), as defined by Gradshteyn and Ryzhik (2007); (Eqs 9, 10) through power series, the following formula is exploited: ₂ F ₁ ( − m , b , c , x ) = ∑ n = 0 m ( − 1 ) n m n ( b ) n ( c ) n z n , where (⋅) _n denotes the Pochhammer symbol, and therefore, the sum converges to a Gauss hypergeometric function ₂ F ₁ (⋅, ⋅, ⋅, x). Then, f r n − α ( y ) can be obtained as f r n − α ( y ) = d F r n − α ( y ) d y . Next, f l ( r n ) ( y ) can be obtained as f l ( r n ) ( y ) = 1 K f r n − α ( y K ) . After applying some algebraic manipulations, Lemma 2 yields.

Theorem 1. The coverage probability of the receiver under the nth nearest UAV-BS association policy is given by the following equation: P c γ t h = 1 − ∫ 0 ∞ ∫ l h 2 + R 2 l h γ m s , m s c 0 γ t h w Γ m s y q ⁡ exp γ y − 1 w f S i w f l r n y d y d w , (21) where c 0 = σ 2 p and γ(⋅, ⋅) denotes the lower incomplete gamma function defined by Gradshteyn and Ryzhik (2007); (eq. (8.350.1)).

Proof. First, let W = S _i l (r _n ). PDF f _W (w) can directly be obtained through the well-known formula for the product of two independent random variables as f W w = ∫ l h 2 + R 2 l h 1 y f l r n y f S i w y d y . (22)

Next, let Z = h _i W. The PDF f _Z (z) is similarly given by f Z z = ∫ 0 ∞ 1 w f W w f h i z w d w . (23)

Finally, the CDF of SNR F _SNR (γ _th ) can be obtained through f _Z (z) as F S N R γ t h = ∫ 0 γ t h c 0 f Z c 0 z d z . (24)

Finally, by applying Gradshteyn and Ryzhik (2007) (eq. (3.351.1) for calculating F _SNR (γ _th ), and after some manipulations, P c ( γ t h ) is obtained in terms of the complementary CDF (CCDF), i.e., P c ( γ t h ) = 1 − F S N R ( γ t h ) , and this completes the proof.

Lemma 3. The PDF of the path-loss term l (d _i ) is given by the following equation: f l d i x = x − 1 − 2 α a R K − 2 α x K − α 2 − h 2 , (25) for x ∈ [ l ( h 2 + R 2 ) , l ( h ) ] .

Proof. The PDF of d i − α can first be obtained through the corresponding CDF obtained as P d i − α ≤ x = P r n ≥ x − 1 α = 1 − P d i ≤ x − 1 α = 1 − F d i x − 1 α , (26) where F d i ( d i ) = d i 2 − h 2 R . Then, the proof follows the same lines as the proof presented in Lemma 1.

Theorem 2. The coverage probability of the receiver under the random selection scheme is given by the following equation: P c γ t h = 1 − ∫ 0 ∞ ∫ l h 2 + R 2 l h x q ⁡ exp γ x − 1 w γ m s , m s c 0 γ t h w Γ m s f S i w f l d i x d x d w . (27)

Proof. The proof follows similar lines as the proof presented for theorem 2, and thus, it is omitted here.

Lemma 4. The joint PDF f r 1 , … , r n ( r 1 , … , r n ) of the n nearest distances between the receiver and its n nearest UAVs is given by the following equation: f r 1 , … , r n r 1 , … , r n = N ! N − n ! 1 − F d i r n N − n ∏ i = 1 n f d i r i , h ≤ r 1 ≤ … ≤ r n ≤ h 2 + R 2 . (28)

Proof. By order statistics, the joint PDF of all order distances { r n } n = 1 : N is given with respect to f d i ( d i ) by f r 1 , … , r N ( r 1 , … , r N ) = N ! ∏ i = 1 N f d i ( r i ) . After integrating out the N − n distances { r i } i = n + 1 : N from f r 1 , … , r N ( r 1 , … , r N ) , the joint PDF f r 1 , … , r n ( r 1 , … , r n ) can be obtained by exploiting order statistics (Ahsanullah et al., 2013), as in Lemma 4.

Lemma 5. The Laplace transform of the SNR at the receiver under the JT-CoMP scheme assuming the n nearest UAV-BSs in C n is given by the following equation: L S N R s = ∫ 0 ∞ … ∫ 0 ∞ ⏟ n ∫ h h 2 + R 2 … ∫ r n − 1 h 2 + R 2 ⏟ n ∏ i = 1 n 1 + s p x i K r i − α m s − m s f S i x i × f r 1 , … , r n r 1 , … , r n d r n … d r 1 d x 1 … d x n . (29)

Proof. The Laplace transform of the SNR can be obtained as follows: L S N R s = E S N R exp − s S N R = E x i , r i ∏ i = 1 n E h i exp − s p x i h i l r i σ 2 = ( a ) E x i , r i ∏ i = 1 n 1 + s p x i K r i − α m s σ 2 − m s = ( b ) ∫ 0 ∞ … ∫ 0 ∞ ⏟ n E r i ∏ i = 1 n 1 + s p x i K r i − α m s σ 2 − m s f S i x i d x 1 … d x n = ( c ) ∫ 0 ∞ … ∫ 0 ∞ ∫ h h 2 + R 2 … ∫ r n − 1 h 2 + R 2 ∏ i = 1 n 1 + s p x i K r i − α m s − m s f S i x i × f r 1 , … , r n r 1 , … , r n d r n … d r 1 d x 1 … d x n , (30) where (a) follows from independence between x _i , h _i , and r _i , and after applying the moment generating function (MGF) of the gamma random variable h _i , (b) follows after deconditioning over the i.i.d. random variable x _i with the PDF f S i ( x i ) ; (c) follows after deconditioning over r _i with the joint PDF f r 1 , … , r n ( r 1 , … , r n ) given by Lemma 4, and this completes the proof.

Theorem 3. The coverage probability of the receiver under the JT-CoMP scheme assuming the n nearest UAV-BSs in C n is given by the following equation: P c γ t h = 1 − L S N R − 1 1 s L S N R s s s = γ t h . (31)

Proof. Having obtained L S N R ( s ) , the CDF of SNR is first obtained through the well-known formula for the inverse Laplace transform of L S N R ( s ) , i.e., F S N R ( γ t h ) = L − 1 { 1 s E S N R [ exp ( − s S N R ) ] } ( γ t h ) . Finally, P c ( γ t h ) is given through the CCDF of SNR, i.e., P c ( γ t h ) = 1 − F S N R ( γ t h ) , and this completes the proof.