Let Φ = ∑ n = 1 ∞ δ X n denote a homogeneous PPP on R d , which works as the parent process, and let Ψ n = ∑ k = 1 κ n δ Y n , k , n ∈ N , denote a family of finite (therefore inhomogeneous) and mutually independent PPPs on R d , which are also independent of Φ and work as the offspring processes. Then, PPCP Ψ = ∑ m = 1 ∞ δ Y m is given as Ψ = ∑ n = 1 ∞ S − X n Ψ n = ∑ n = 1 ∞ ∑ k = 1 κ n δ X n + Y n , k . (3) The PPCP Ψ constructed as above is stationary since the parent process Φ is stationary and the offspring processes Ψ _n , n ∈ N , are independent and identically distributed [see, e.g., (Baccelli et al., 2020, Example 2.3.18)]. We assume that Φ has a positive and finite intensity λ _Φ, and Ψ _n , n ∈ N , have an identical intensity measure Λ _o = μ Q, where μ is a positive constant and Q is a probability distribution on ( R d , B ( R d ) ) . Thus, the number of points in each offspring process follows a Poisson distribution with mean μ, so that the intensity of Ψ is equal to λ _Ψ = λ _Φ μ, and offspring points are scattered on R d according to Q independently of each other. We further assume that Q is diffuse; that is, Q({x}) = 0 for any x ∈ R d , to make Ψ simple. We refer to S − X n Ψ n in (Eq. 3) as the cluster associated with X _n for n ∈ N . Two main examples of the PPCPs are the (modified) Thomas point process and the Matérn cluster process [see, e.g., (Chiu et al., 2013, Example 5.5)]. When Q is an isotropic normal distribution, then the obtained PPCP is called the Thomas point process. On the other hand, when Q is the uniform distribution on a fixed ball centered at the origin, then the result is called the Matérn cluster process. Note that the PPCP Ψ and its parent process Φ fulfill the conditions of Theorem 1.

For a stationary point process Ψ, let Ψ^!≔Ψ − δ ₀ on the event {Ψ({0}) = 1}, which is referred to as the reduced Palm version of Ψ.

Lemma 1. For the stationary PPCP Ψ described in Section 3.1, the intensity measure of the reduced Palm version Ψ^! (with respect to the Palm distribution) is given by Λ Ψ 0 B ≔ E Ψ 0 Ψ ! B = λ Φ μ | B | + μ ∫ R d Q B − y Q − d y , B ∈ B R d , (4) where |⋅| denotes the Lebesgue measure on ( R d , B ( R d ) ) and Q ⁻(B) = Q(−B) with − B = { − x: x ∈ B} for B ∈ B ( R d ) .

Proof

Since the offspring processes Ψ _n , n ∈ N , are PPPs, the PPCP Ψ is a Cox point process; that is, once the parent process Φ = ∑ n = 1 ∞ δ X n is given, Ψ is conditionally an inhomogeneous PPP with a conditional intensity measure μ ∑ n = 1 ∞ S − X n Q [see, e.g., (Baccelli et al., 2020, Example 2.3.13)]. Since the reduced Palm version of a PPP is identical in distribution to its original version (not conditioned on {Ψ({0}) = 1}) by Slivnyak’s theorem [see, e.g., (Daley and Vere-Jones, 2008, Proposition 13.1.VII) or (Baccelli et al., 2020, Theorem 3.2.4)], we have E Ψ 0 Ψ ! B ∣ Φ = E Ψ B ∣ Φ = μ ∑ n = 1 ∞ Q B − X n , B ∈ B R d . Taking the expectation with respect to P Ψ 0 and then applying Theorem 1, we obtain E Ψ 0 Ψ ! B = μ E Ψ 0 ∑ n = 1 ∞ Q B − X n = E Φ 0 ∫ R d ∑ n = 1 ∞ Q B − X n ◦ θ y Ψ 0 d y = E Φ 0 ∫ R d ∑ n = 0 ∞ Q B − X n + y Ψ 0 d y = μ ∫ R d Q B + y + E Φ 0 ∑ n = 1 ∞ Q B − X n + y Q d y , where λ _Ψ = λ _Φ μ is used in the second equality, the third equality follows because, for any n ∈ N and y ∈ R d , there exists an n ′ ∈ N ∪ { 0 } such that X _n ◦θ _y = X _n’ − y on {Φ({0}) = 1}, and in the last equality, we apply Campbell’s formula [see, e.g., (Last and Penrose, 2017, Proposition 2.7) or (Baccelli et al., 2020, Theorem 1.2.5)] for Ψ₀. For the expectation in the last expression above, Slivnyak’s theorem, Campbell’s formula for Φ and then Fubini’s theorem yield E Φ 0 ∑ n = 1 ∞ Q B − X n + y = λ Φ ∫ R d Q B − x d x = λ Φ ∫ R d ∫ B − x Q d z d x = λ Φ ∫ R d ∫ B − z d x Q d z = λ Φ | B | , which completes the proof.

Remark 2. The second term on the right-hand side of (Eq. 4) is of course equal to μ∫Q(B + y) Q(dy). We adopt the form in Lemma 1 due to its interpretability. Since Q is the distribution for the position of an offspring point viewed from its parent, Q⁻ represents the distribution for the location of the parent of the offspring point at the origin on the event {Ψ({0}) = 1}. On the other hand, μ Q(B − y) gives the expected number of offspring points falling in B ∈ B ( R d ) among a cluster whose parent is shifted to y ∈ R d . In other words, the second term on the right-hand side of (Eq. 4) represents the expected number of offspring points falling in B among the cluster which is given to have one point at the origin. Since the first term on the right-hand side of (Eq. 4) is equal to Λ Ψ ( B ) = E [ Ψ ( B ) ] , Lemma 1 states that the intensity measure for the reduced Palm version of a stationary PPCP is given as the sum of the intensity measure of the stationary version and that of a cluster which has one point at the origin. Lemma 1 is also a slight generalization of the result in (Tanaka et al., 2008, Section 2.2).

The observation in Remark 2 is further enhanced by the following proposition.

Proposition 1. For the stationary PPCP Ψ = ∑ m = 1 ∞ δ Y m described in Section 3.1, the generating functional of the reduced Palm version Ψ^! (with respect to the Palm distribution) is given by G Ψ 0 h ≔ E Ψ 0 ∏ m = 1 ∞ h Y m = G Ψ h ∫ R d h ̃ z Q − d z , (5) for any measurable function h : R d → [ 0,1 ] , where G Ψ is the generating functional of the stationary version of Ψ given as G Ψ h ≔ E ∑ m = 1 ∞ h Y m = G Φ h ̃ = exp − λ Φ ∫ R d 1 − h ̃ x d x , (6) and h ̃ ( x ) denotes the generating functional of an offspring process Ψ₁ whose parent is shifted to x ∈ R d ; h ̃ x = G Ψ 1 S x h = exp − μ ∫ R d 1 − h x + y Q d y . (7)

Note that in Proposition 1 above, G Φ is the generating functional of the parent process Φ. The relation G Ψ ( h ) = G Φ ( h ̃ ) with h ̃ ( x ) = G Ψ 1 ( S x h ) in Eqs 6, 7 is known to hold for more general cluster point processes [see, e.g., (Daley and Vere-Jones, 2003, Example 6.3(a)] or [Baccelli et al., 2020, Proposition 2.3.12 and Lemma 2.3.20)], whereas the last equalities in Eqs 6, 7 follow because Φ and Ψ₁ are PPPs, respectively (see, e.g., (Last and Penrose, 2017, Exercise 3.6), or [Baccelli et al., 2020, Corollary 2.1.5)]. The relation (Eq. 5) is derived in (Ganti and Haenggi, 2009, Lemma 1), to which we give another proof using the exchange formula in Theorem 1.

Proof

As stated in the proof of Lemma 1, once the parent process Φ = ∑ n = 1 ∞ δ X n is given, the PPCP Ψ is conditionally an inhomogeneous PPP with the conditional intensity measure μ ∑ n = 1 ∞ S − X n Q . Since the reduced Palm version of a PPP is identical in distribution to its original (not conditioned) version, we have E Ψ 0 ∏ m = 1 ∞ h Y m Φ = E ∏ m = 1 ∞ h Y m Φ = exp − μ ∑ n = 1 ∞ ∫ R d 1 − h y Q d y − X n = ∏ n = 1 ∞ h ̃ X n , where the generating functional of a PPP is applied in the second equality. Taking the expectation with respect to P Ψ 0 and then applying Theorem 1, we obtain G Ψ 0 h = E Ψ 0 ∏ n = 1 ∞ h ̃ X n = 1 μ E Φ 0 ∫ R d ∏ n = 0 ∞ h ̃ X n − z Ψ 0 d z = ∫ R d h ̃ − z E Φ 0 ∏ n = 1 ∞ h ̃ X n − z Q d z , where Campbell’s formula for Ψ₀ is applied in the last equality. By Slivnyak’s theorem and the stationarity for Φ, we have E Φ 0 ∏ n = 1 ∞ h ̃ ( X n − z ) = E ∏ n = 1 ∞ h ̃ ( X n ) = G Φ ( h ̃ ) , which completes the proof.

Remark 3. The right-hand side of (Eq. 5) is given as the generating functional G Ψ ( h ) of the stationary version of Ψ multiplied by the integral term ∫ h ̃ ( z ) Q − ( d z ) . Since h ̃ ( z ) represents the generating functional of an offspring process whose parent is shifted to z ∈ R d and Q ⁻ is the distribution of the location of the parent point of the offspring at the origin on the event {Ψ({0}) = 1}, this integral term represents the generating functional of the cluster which is given to have a point at the origin. In other words, Proposition 1 implies that, for a stationary PPCP, its Palm version is obtained by the superposition of the original stationary version and an additional independent offspring process whose parent is placed such that it has an offspring point at the origin. This observation is already found in, e.g., (Saha et al., 2019) and is also interpreted such that a point z ∈ R d is first sampled from the distribution Q ⁻ and the Palm version of Φ at z is then obtained as Φ + δ _z by Slivnyak’s theorem, which works as a parent process of the Palm version of Ψ. Proposition 1 indeed supports this interpretation.

For a stationary point process Ψ on R d , let Ψ ! = ∑ m = 1 ∞ δ Y m be its reduced Palm version on {Ψ({0}) = 1} and let Y _* denote the nearest point of Ψ^! from the origin. Then, the nearest-neighbor distance distribution for Ψ is defined as the probability distribution for ‖Y _*‖ with respect to P Ψ 0 , where ‖ ⋅‖ denotes the Euclidean distance. We show below that the nearest-neighbor distance distribution for a stationary PPCP is obtained in a similar way to Proposition 1.

Proposition 2. For the stationary PPCP Ψ described in Section 3.1, the complementary nearest-neighbor distance distribution is given by P Ψ 0 ‖ Y ∗ ‖ > r = G Φ h r * ∫ h r ∗ t Q − d t , r ≥ 0 , (8) where h r ∗ ( x ) = e − μ Q ( b 0 ( r ) − x ) and b ₀(r) denotes a d-dimensional ball centered at the origin with radius r .

Proof

As with the proof of Proposition 1, we consider the conditional probability given the parent process Φ = ∑ n = 1 ∞ δ X n and obtain P Ψ 0 ‖ Y * ‖ > r ∣ Φ = P Ψ 0 Ψ ! b 0 r = 0 ∣ Φ = P Ψ b 0 r = 0 ∣ Φ = ∏ n = 1 ∞ e − μ Q b 0 r − X n = ∏ n = 1 ∞ h r ∗ X n , (9) where the second equality follows from Slivnyak’s theorem and the third does because Ψ^! is conditionally an inhomogeneous PPP with the intensity measure μ ∑ n = 1 ∞ S − X n Q when Φ is given. The rest of the proof is similar to that of Proposition 1.

Remark 4. In (Eq. 8), the term G Φ ( h r * ) is the complementary contact distance distribution and that is obtained by taking the expectation of (Eq. 9) with respect to P , instead of P Ψ 0 [see, e.g., (Miyoshi, 2019)]. The result of Proposition 2 is consistent with the existing ones in, e.g., (Baudin, 1981; Afshang et al., 2017a,b; Pandey et al., 2020) and gives a unified approach to derive the nearest-neighbor distance distributions for stationary PPCPs.

Wireless devices are distributed on R d according to a stationary point process Ψ = ∑ m = 1 ∞ δ Y m . At each time slot, each device is in transmission mode with probability p ∈ (0, 1) or in receiving mode with probability 1 − p independently of the others (half duplex with random access). Devices in the transmission mode transmit signals but can not receive ones, whereas the devices in the receiving mode can receive signals but can not transmit ones. We assume that all transmitting devices transmit signals with identical transmission power (normalized to one) and share a common frequency spectrum. The path-loss function representing attenuation of signals with distance is given by ℓ satisfying ℓ(r) ≥ 0, r > 0, and ∫ ϵ ∞ ℓ ( r ) r d − 1 d r < ∞ for ϵ > 0. We further assume that all wireless links receive Rayleigh fading effects while we ignore shadowing effects. We focus on the device at the origin, referred to as the typical device, under the condition of {Ψ({0}) = 1} and examine whether the typical device can decode messages from other transmitting devices. Let Ψ T x = ∑ m = 1 ∞ δ Y m ′ denote the sub-process of Ψ representing the locations of devices in the transmission mode and for each m ∈ N , let H _m denote a random variable representing the fading effect on signals transmitted from the device at Y m ′ , where H _m , m ∈ N , are mutually independent, independent of Ψ T x and exponentially distributed with unit mean due to the Rayleigh fading. With this setup, the received signal power by the typical device amounts to H m ℓ ‖ Y ′ _ m ‖ when it receives signals from the device at Y m ′ . Hence, if the typical device is in the receiving mode and communicates with the transmitting device at Y m ′ , the signal-to-interference-plus-noise ratio (SINR) is given as S I N R m = H m ℓ ‖ Y m ′ ‖ ∑ j = 1 j ≠ m ∞ H j ℓ ‖ Y j ′ ‖ + N , (10) where N denotes a constant representing noise at the origin. We suppose that the typical device can successfully decode a message from the device at Y m ′ if the typical device is in the receiving mode and S I N R m in (Eq. 10) exceeds a predefined threshold θ > 0.

We here suppose that a device in the receiving mode communicates with the nearest device in transmission mode. The probability that the typical device can successfully decode a message from its partner is called the coverage probability and is given by C P θ = 1 − p ∑ m = 1 ∞ P Ψ 0 S I N R m > θ , ‖ Y m ′ ‖ ≤ ‖ Y j ′ ‖ , j ∈ N , (11) where 1 − p on the right-hand side indicates that the typical device must be in the receiving mode and the sum over m ∈ N represents the probability that the SINR from the nearest transmitting device exceeds the threshold θ. We now suppose that the point process Ψ representing the locations of devices is given as a stationary PPCP studied in Section 3. Then, Ψ T x = ∑ m = 1 ∞ δ Y m ′ representing the locations of devices in the transmission mode is also a stationary PPCP, where the parent process remains the same as the homogeneous PPP Φ with intensity λ _Φ, whereas the offspring processes Ψ n ′ = ∑ k = 1 κ n ′ δ Y n , k ′ , n ∈ N , are finite PPPs with the intensity measure pμQ.

Theorem 2. For the model of a D2D network described in Section 4.1 with the devices deployed according to a stationary PPCP in Section 3.1, the coverage probability is given by C P θ = 1 − p p μ ∫ R d I 1 , θ t + I 2 , θ t Q − d t , (12) where Q ⁻ is given in Lemma 1 and I 1 , θ t = ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , t E θ y Q d y − t , I 2 , θ t = λ Φ ∫ R d ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , t C θ y , x E θ y Q d y − x d x , E θ y = exp − λ Φ ∫ R d 1 − C θ y , w d w , C θ y , x = exp − p μ 1 − ∫ ‖ z ‖ > ‖ y ‖ 1 + θ ℓ ‖ z ‖ ℓ ‖ y ‖ − 1 Q d z − x . (13)

Before proceeding on the proof of Theorem 2, we give an intuitive interpretation to the result of it. First, as stated in the preceding section, Q ⁻ denotes the distribution for the location of the parent point of the typical device at the origin. Thus, pμI _1,θ (t) and pμI _2,θ (t) in (Eq. 12) represent the cases where the typical device, whose parent is located at t ∈ R d , communicates with the transmitting device in the same cluster and in a different cluster, respectively; that is, the location of the communication partner is sampled from a finite PPP with the intensity measure pμQ(dy − t) in I _1,θ (t) and is from one with pμQ(dy − x) in I _2,θ (t), where x is also sampled from a homogeneous PPP with intensity λ _Φ. Moreover, E _θ (y) represents the effect from other clusters which are neither the one having the typical device nor the one having its communication partner at y. Finally, C _θ (y, x) represents the effect of the cluster with the parent point at x ∈ R d when the typical device communicates with the transmitting device at y.

Proof

Similar to the proof of Proposition 1, once the parent process Φ = ∑ n = 1 ∞ δ X n is given, the point process Ψ T x representing the locations of devices in the transmission mode is conditionally an inhomogeneous PPP with the conditional intensity measure p μ ∑ n = 1 ∞ S − X n Q . Thus, we can use the corresponding approach to that obtaining the coverage probability for a cellular network with BSs deployed according to a PPP [see, e.g., (Andrews et al., 2011) or (Błaszczyszyn et al., 2018, Section 5.2)]. Since H _m , m ∈ N , are mutually independent, exponentially distributed, and also independent of Φ, we have from (Eq. 11), P Ψ 0 S I N R m > θ , ‖ Y m ′ ‖ ≤ ‖ Y j ′ ‖ , j ∈ N Φ = P Ψ 0 H m > θ ℓ ‖ Y m ′ ‖ ∑ j = 1 j ≠ m ∞ H j ℓ ‖ Y j ′ ‖ + N , ‖ Y m ′ ‖ ≤ ‖ Y j ′ ‖ , j ∈ N Φ = E Ψ 0 e − θ N / ℓ ‖ Y m ′ ‖ ∏ j = 1 j ≠ m ∞ 1 + θ ℓ ‖ Y j ′ ‖ ℓ ‖ Y m ′ ‖ − 1 1 ‖ Y j ′ ‖ > ‖ Y m ′ ‖ Φ , where 1 _A denotes the indicator function for set A and we use P ( H m > a ) = e − a for a ≥ 0 and E [ e − s H j ] = ( 1 + s ) − 1 in the last equality. Summing the above expression over m ∈ N , we have from Slivnyak’s Theorem for Ψ conditioned on Φ and the refined Campbell formula [see, e.g., (Daley and Vere-Jones, 2008, Theorem 13.2. III), (Last and Penrose, 2017, Theorem 9.1) or (Baccelli et al., 2020, Theorem 3.1.9)], ∑ m = 1 ∞ P Ψ 0 S I N R m > θ , ‖ Y m ′ ‖ ≤ ‖ Y j ′ ‖ , j ∈ N Φ = p μ ∑ n = 1 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ E ∏ j = 1 ∞ 1 + θ ℓ ‖ Y j ′ ‖ ℓ ‖ y ‖ − 1 1 ‖ Y j ′ ‖ > ‖ y ‖ Φ Q d y − X n . (14) Furthermore, the generating functional of a PPP applying to the above expectation yields E ∏ j = 1 ∞ 1 + θ ℓ ‖ Y j ′ ‖ ℓ ‖ y ‖ − 1 1 ‖ Y j ′ ‖ > ‖ y ‖ Φ = exp − p μ ∑ i = 1 ∞ ∫ R d 1 − 1 + θ ℓ ‖ z ‖ ℓ ‖ y ‖ − 1 1 ‖ z ‖ > ‖ y ‖ Q d z − X i = ∏ i = 1 ∞ exp − p μ 1 − ∫ ‖ z ‖ > ‖ y ‖ 1 + θ ℓ ‖ z ‖ ℓ ‖ y ‖ − 1 Q d z − X i = ∏ i = 1 ∞ C θ y , X i . Plugging this into (Eq. 14), taking the expectation with respect to P Ψ 0 and then applying Neveu’s exchange formula in Theorem 1, we have C P θ = 1 − p p μ E Ψ 0 ∑ n = 1 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ ∏ i = 1 ∞ C θ y , X i Q d y − X n = 1 − p p E Φ 0 ∫ R d ∑ n = 0 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ ∏ i = 0 ∞ C θ y , X i − t Q d y − X n + t Ψ 0 d t = 1 − p p μ ∫ R d E Φ 0 ∑ n = 0 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ ∏ i = 0 ∞ C θ y , X i − t Q d y − X n + t Q d t , (15) where we note the existence of X ₀ = 0 on {Φ({0}) = 1} in the second equality and apply Campbell’s formula in the third equality. Noting that X ₀ = 0 on {Φ({0}) = 1}, we separate the expectation in (Eq. 15) into E Φ 0 ∑ n = 0 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ ∏ i = 0 ∞ C θ y , X i − t Q d y − X n + t = ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , − t E Φ 0 ∏ i = 1 ∞ C θ y , X i − t Q d y + t + E Φ 0 ∑ n = 1 ∞ ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , − t C θ y , X n − t ∏ i = 1 i ≠ n ∞ C θ y , X i − t Q d y − X n + t , (16) and consider the two terms on the right-hand side of (Eq. 16) one by one. For the first term, the generating functional of a PPP yields (1st term of (Eq. 16)) = ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , − t exp − λ Φ ∫ R d 1 − C θ y , w d w Q d y + t = I 1 , θ − t . (17) On the other hand, applying Campbell’s formula and the generating functional for Φ to the second term on the right-hand side of (Eq. 16), we have (2nd term of (Eq. 16)) = λ Φ ∫ R d ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , − t C θ y , x E Φ 0 ∏ i = 1 ∞ C θ y , X i − t Q d y − x d x = λ Φ ∫ R d ∫ R d e − θ N / ℓ ‖ y ‖ C θ y , − t C θ y , x exp − λ Φ ∫ R d 1 − C θ y , w d w Q d y − x d x = I 2 , θ − t . (18) Finally, plugging (Eqs 17, 18) into (Eq. 16), and then into (Eq. 15), we have (Eq. 12) and the proof is completed.

When d = 2 and the distribution Q for the locations of offspring points depends only on the distance; that is, Q(dy) = f _o(‖y‖) dy for y ∈ R 2 , we obtain a numerically computable form of the coverage probability.

Corollary 1. When d = 2 and Q(dy) = f _o(‖y‖) dy , y ∈ R 2 , the coverage probability in Theorem 2 is reduced to C P θ = 2 π 1 − p p μ ∫ 0 ∞ ∫ 0 ∞ e − θ N / ℓ s E ̂ θ s C ̂ θ s , u I ̂ θ s , u d s f o u u d u , (19) where I ̂ θ s , u = g s ∣ u + 2 π λ Φ ∫ 0 ∞ C ̂ θ s , r g s ∣ r r d r , E ̂ θ s = exp − 2 π λ Φ ∫ 0 ∞ 1 − C ̂ θ s , v v d v , C ̂ θ s , r = exp − p μ 1 − ∫ s ∞ 1 + θ ℓ q ℓ s − 1 g q ∣ r d q , g s ∣ r = 2 s ∫ 0 π f o s 2 + r 2 − 2 s r ⁡ cos ⁡ φ d φ . (20)

Proof

Since the distribution Q depends only on the distance, it holds that Q ⁻(dt) = Q(dt) = f _o(‖t‖) dt, t ∈ R 2 , and (Eq. 12) is reduced to C P θ = 1 − p p μ ∫ R 2 I 1 , θ t + I 2 , θ t f o ‖ t ‖ d t = 2 π 1 − p p μ ∫ 0 ∞ I ̂ 1 , θ u + I ̂ 2 , θ u f o u u d u , (21) where the polar coordinate conversion is applied in the second equality and I ̂ 1 , θ u = ∫ 0 ∞ e − θ N / ℓ s C ̂ θ s , u E ̂ θ s g s ∣ u d s , I ̂ 2 , θ u = 2 π λ Φ ∫ 0 ∞ ∫ 0 ∞ e − θ N / ℓ s C ̂ θ s , u C ̂ θ s , r E ̂ θ s g s ∣ r d s r d r . Therefore, we have I ̂ 1 , θ u + I ̂ 2 , θ u = ∫ 0 ∞ e − θ N / ℓ s C ̂ θ s , u E ̂ θ s I ̂ θ s , u d s . Plugging this into (Eq. 21), we have (Eq. 19) and the proof is completed.

We next consider the problem of device discovery. Devices in the transmission mode transmit broadcast messages, whereas a device in the receiving mode can discover the transmitters if it can successfully decode the broadcast messages. When a device in the receiving mode receives the signal from one transmitting device, the signals from all other transmitting devices work as interference. Then, the expected number of transmitting devices discovered by the typical device is represented by N θ = 1 − p E Ψ 0 ∑ m = 1 ∞ 1 S I N R m > θ . (22)

Proposition 3. Consider the D2D network model described in Section 4.1 with the devices deployed according to a stationary PPCP given in Section 3.1. Then, the expected number N ( θ ) of transmitting devices discovered by the typical device is obtained by (Eq. 12) in Theorem 2 replacing the integral range ‖z‖ > ‖y‖ in (Eq. 13) by R d . Moreover, when d = 2 and Q(dy) = f _o(‖y‖) dy for y ∈ R 2 , N ( θ ) is reduced to (Eq. 19) in Corollary 1 replacing the integral range (s, ∞) in (Eq. 20) by (0, ∞).

Proof

Since E Ψ 0 ∑ m = 1 ∞ 1 { S I N R m > θ } = ∑ m = 1 ∞ P Ψ 0 ( S I N R m > θ ) , the difference between (Eq. 11) and (Eq. 22) is only the event { ‖ Y m ′ ‖ ≤ ‖ Y j ′ ‖ , j ∈ N } . This leads to the difference of the integral ranges in C _θ (y, x) in (Eq. 13) and in C ̂ θ ( s , r ) in (Eq. 20). Remark 5. Since P Ψ 0 ⋃ m = 1 ∞ { S I N R m > θ } ≤ ∑ m = 1 ∞ P Ψ 0 ( S I N R m > θ ) = N ( θ ) , Proposition 3 also gives an upper bound for the coverage probability with the max-SINR association policy, where a device in the receiving mode receives a message with the strongest SINR. This upper bound is known to be exact for θ > 1 since ∑ m = 1 ∞ 1 { S I N R m > θ } ≤ 1 + θ − 1 almost surely [see (Dhillon et al., 2012) or (Błaszczyszyn et al., 2018, Lemma 5.1.2)].