SIR Analysis in 3D UAV Networks: A Stochastic Geometry Approach

In this work, a mathematical framework to evaluate the performance of a finite, three dimensional (3D) unmanned aerial vehicle (UAV) network in the presence of interference is developed. The framework builds upon stochastic geometry tools and specifically the binomial point process (BPP) for the spatial distribution of the UAVs. A UAV base station (UAV-BS) reference receiver is located at the center of a sphere and communicates with the nearest transmitting UAV node, whose distance from the UAV-BS reference receiver is either fixed or random. The reverse link suffers from the presence of a single dominant interferer, whose location is random within the sphere. Closed-form expressions are derived for statistical metrics of the signal-to-interference ratio (SIR) for two scenarios, namely for fixed or random location of the desired transmitting node. Then, the impact of the location of the transmitting node on the coverage probability (CP) is studied while the average error probability and the ergodic capacity have been also analytically investigated. Finally, the theoretical results are numerically evaluated and compared to simulation to reveal some useful insights.


I. INTRODUCTION A. MOTIVATIONS AND RELATED WORK
Unmanned aerial vehicles (UAVs) have emerged as key enablers of seamless wireless connectivity in diverse scenarios such as large-scale temporary events, military operations, or emergency situations, and capacity enhancement in occasional demands [1], [2]. The spatial modeling of UAV networks is especially important for their design and performance analysis. In realistic networks with multitude of nodes, the space spanned by the nodes is three-dimensional (3D). This is quite obvious when UAVs are spread over a large area and ground-based network nodes are also located on the streets or in high buildings. Stochastic geometry has already been used extensively as a key tool for the spatial modeling and analysis of cellular networks. However, the spatial modeling and performance analysis of a cellular The associate editor coordinating the review of this manuscript and approving it for publication was Maurizio Magarini . two dimensional (2D) network differs significantly from a 3D one. Subsequently, only a limited number of works have exploited stochastic geometry's capabilities for the analysis of UAV networks. In this work, a 3D model for the reverse link analysis of a finite UAV network, i.e., a network with finite number of nodes, is considered for the first time, borrowing tools from stochastic geometry.
The homogeneous infinite Poisson point process (HPPP) is a very popular choice for the distribution of the spatial locations of terrestrial transceivings due to its simplicity and tractability [3]- [5], whereas recently has been adopted for the spatial modeling of UAV networks [6]- [8]. In particular, in [6], a coverage and rate analysis is conducted by assuming that UAVs are deployed in a plane above the ground forming a PPP. In [7], a detailed analysis of a UAV network is performed in terms of outage probability (OP), where the locations of UAVs are modeled as an HPPP inside a sphere. In [8], PPP is incorporated in the coverage probability (CP) analysis of a UAV network. The authors in [9] studied the performance of VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ a multi-tier network, including UAVs, whose locations modeled as an HPPP on the Euclidean plane. However, the HPPP cannot be used in modeling finite UAV networks with a given number of UAVs. In realistic scenarios, a predefined number of UAVs is deployed to cover a finite region, e.g., in a post disaster scenario. In such cases, the PPP is clearly not an adequate model for UAV networks since different realizations constitute of random number of points. For such scenarios, a simple yet proper model for the spatial distribution of a finite number of UAVs is the binomial point process (BPP) [10], [11]. The performance analysis of a UAV network modeled by a BPP is significantly more challenging as compared to an infinite PPP. This is the main reason why BPP has not been widely adopted for spatial modeling and analysis of UAV networks. In addition, the number of nodes in disjoint areas, within the network area, are not independent [12], which increases the complexity of the BPP. The authors in [11] presented a detailed performance analysis of a finite UAV network, where the UAVs were distributed inside a disk above the ground forming a 2D BPP. In [13], a similar model to the one presented in [11] was adopted, with the UAVs moved in stochastic trajectories. While 2D BPP can be used in spatial modeling of current cellular networks, a 3D UAV network cannot be modeled in the same way as shown in subsection V.A. As UAV-enabled networks have received significant attention, the research interest focuses on the design and modeling of such networks in order to incorporate UAVs user equipments (UAVs-UEs). For this reason, research works have employed 3D models for UAVs [14]- [16]. However, none of them has considered the performance analysis in a realistic 3D network model to fully capture the effect of 3D coverage space. In this work, for the first time, to the best of the authors' knowledge, a 3D BPP is considered to model the spatial locations of a realistic UAV network, where a finite number of UAVs is assumed to be deployed inside a sphere.
While the choice of proper spatial modeling is considered crucial in the performance analysis of UAV networks, it strongly depends on the statistical characterization of the distance distributions. The authors in [12] studied the distance distributions for a finite number of uniformly distributed nodes inside an arbitrary compact set. However, the derivation of distance distributions in a BPP, strongly depends on the shape of the set where the nodes are distributed. If a UAV is served by its nearest transmitting node, which is a common assumption [11], [17], the distribution of the point process after removing this nearest serving node, is not the same as that of the original BPP. Thus, the statistical characterization of distance distributions becomes more challenging. To highlight the importance of the locations where UAVs are deployed within the network, many works are investigating the optimization of trajectories and locations of UAVs [18]- [20].
From the performance analysis point of view, the adoption of the bit error rate (BER) as a performance indicator was investigated in [21], [22]. In particular, in [21], among several research targets, a closed-form expression for the average BER was derived, while in [22], analytical expressions for the average BER of a reconfigurable intelligent surface (RIS)assisted UAV scheme, were obtained. Other recent works that focused on the study of BER can be found in [22]- [24]. Moreover, various studies have investigated the impact of interference on the performance analysis of a UAV network. More specifically, in [11], the impact of total network interference on the CP was investigated. In [9], a study in terms of CP was presented by considering the HPPP and the Poisson cluster process in spatial system modeling. However, in all the above works, the analysis resulted in complicated expressions with integral representations. As far as the ergodic capacity is concerned, the authors in [11] first studied the ergodic capacity in a UAV network modeled as a BPP, while in [25] the ergodic capacity was considered for a UAV network modeled as a sphere, where UAVs moved in stochastic trajectories.

B. CONTRIBUTIONS
In this paper, a 3D UAV network with a finite number of nodes is assumed and important statistical characteristics of the received signal-to-interference ratio (SIR) are analytically investigated by exploiting the tool of stochastic geometry. More specifically, the main contributions of this paper are threefold:

1) MODELING OF FINITE UAV NETWORK IN A SPHERE
A finite number of UAV nodes uniformly distributed in a sphere is considered. To the best of the authors' knowledge, such an approach has never been reported in the past.

2) STOCHASTIC GEOMETRY ANALYSIS
Based on the BPP, a new stochastic analysis framework has been developed. This framework is used to calculate the SIR for realistic network scenarios.

3) CLOSED-FORM EXPRESSIONS
In contrast to previous work in the past, closed-form expressions have been derived for key performance indicators, namely the CP, BER, average output SIR, and ergodic capacity.

C. STRUCTURE
The remainder of the paper is organized as follows. In Section II, the system and channel models are described. In Section III, stochastic geometry analysis for studying the behavior of SIR, is conducted. In Section IV, performance analysis is performed in terms of CP and BER, average output SIR, and ergodic capacity. In Section V, numerical evaluation is performed and analytical results are verified through simulations. Finally, in Section VI, concluding remarks are provided.

II. SYSTEM AND CHANNEL MODELS
Assume that a 3D UAV network comprised of one fixed and a finite number N of transceiving nodes is deployed in a 204964 VOLUME 8, 2020 sphere. The locations of the N transceiving nodes are modeled as a uniform BPP, where the nodes are independently and uniformly distributed in the sphere. In stochastic geometry terms, it is implied that the transceiving nodes are uniformly distributed in a finite compact set A ⊂ R 3 . For the needs of the 3D network modeling, let A = b 3 (o, R) denote a 3D ball of radius R that is centered at the origin o ≡ (0, 0, 0). In contrast to prior geometrical models assumed, e.g., [11], the sphere is a simple yet realistic 3D model to describe the UAV network coverage space. To the best of the authors' knowledge, no previous work has ever captured the effect of a spherical model in the performance analysis of a finite UAV network. A UAV -base station (UAV-BS) reference receiver is assumed to be located at the origin o and communicates with the nearest node out of the N transceiving nodes. This work focuses on the reverse link analysis and therefore, the nearest transceiving node is referred to as the desired transmitting node and the reverse link as the reference link. The distance of the transmitting node to the reference receiver is denoted by R s . The thermal noise is assumed to be negligible as compared to the interference experienced by the receiver [11]. Once the reference link has been established, the remaining transceiving nodes are uniformly distributed in b 3 In principle, the reference link may suffer from undesired signals from many interfering nodes. However, for mathematical tractability purposes, this work considers only one dominant interfering node, which is the nearest transmitting node outside the area b 3 (o, s), i.e., the second nearest node to the UAV-BS reference receiver. Such an assumption is quite common in the open technical literature, e.g., [17], [26]. Indeed, in [17], only one ground-based interferer is considered in the performance analysis, while in [26], it is stated that the CP has similar trend when multiple interfering nodes or one dominant interfering node are considered. The link between the dominant interfering node and the UAV-BS reference receiver is referred to as the interfering link and the distance of the interfering node to the reference receiver is denoted by R I . After establishing the reference link, the distance distribution of R I is no longer independent of the distance R s . In this case, the PDF of the distance R I conditioned by the distance of the reference link R s = y, is given by [12] (1) The system model described is depicted in Fig. 1. The received SIR is defined as where (·) −a models power path-loss with factor a, and P s , P I represent the transmit powers of the transmitting and dominant interfering nodes, respectively. Moreover, the random variable (RV) h ∈ {h s , h I }, models Nakagami-m channel fading for the reference and interfering link, respectively, with probability density function (PDF) given by where m ∈ {m s , m I } is the shape parameter, θ ∈ {θ s , θ I } is the scale parameter [27] of the distribution, and (·) denotes the Gamma function [28, eq. (8.310.1)]. Note that Nakagami-m fading is a generalized model that mimics various fading environments and simultaneously retains analytical tractability [11]. While the modeling assumptions and the analysis are motivated by the study of UAV networks, the proposed generic framework is designed to incorporate also terrestrial nodes. Towards this direction, two scenarios for the spatial location of the transmitting nodes, are investigated: (i) the transmitting node is assumed to be located at a deterministic distance from the UAV-BS, and (ii) the transmitting node is randomly located. The two scenarios considered, are presented in the following.

A. SCENARIO 1: FIXED LOCATION OF DESIRED TRANSMITTING NODE
For this system setup, the transmitting node is assumed to be located at a fixed distance s from the center o of A. In this case, (2) can be rewritten as An example link for this scenario is the one between a UAV and a ground-based node. Moreover, a typical use-case includes a fixed ground-based node or an aerial one acting as a relay that serves other ground-based nodes. In the latter case, the UAV transmitting node is considered to be fixed in an optimum location for maximum reliability of the link.

B. SCENARIO 2: RANDOM LOCATION OF DESIRED TRANSMITTING NODE
The UAV-BS reference receiver once again communicates with the nearest transmitting node. However, in this setup, VOLUME 8, 2020 the distance R s , between the transmitting node and the UAV-BS reference receiver, is a RV following a PDF given by For this system setup, the SIR experienced by a UAV-BS reference receiver located at the center o of A is defined as in (2). This scenario is a generic one, since all nodes' locations are random, i.e., it is applicable for UAVs and ground-based nodes placed randomly in the considered space.

III. STOCHASTIC GEOMETRY ANALYSIS
In this section, the stochastic analysis is conducted in terms of the instantaneous SIR or simply SIR for brevity. Closed-form expressions for the cumulative distribution function (CDF) of the SIR for the two scenarios studied, are derived. Before proceeding into the analysis, the distribution of the required distances is first calculated.
A. SCENARIO 1 As the distance of the reference link is deterministic, with R s = s, the interest turns on defining the distribution of the distance R I of the interfering link. However, as mentioned in Section II, after establishing the reference link, the distance distribution of R I is no longer independent of the distance R s . In this case, for R I ∈ [s, R], the PDF of the distance R I conditioned by the distance of the reference link R s , results from (1), after substituting y = s.
Let I = P I h I R I −a represent the RV of the denominator of (4), directly proportional to the received power of the interfering signal. The conditional PDF of the RV I , given the interfering link distance r I , is given by The unconditional distribution of the RV I can now be obtained by averaging (6) over (1).
Proof: See Appendix A. The RV of the nominator of (4), S = P s h s R s −a , is directly proportional to the received power experienced by the UAV-BS reference receiver. The PDF of the RV S results immediately from [27] with a simple change of variable and has similar form with (6). The distribution of the SIR can now be derived in the following Lemma.
Proof: See Appendix B. The corresponding CDF is now derived from (8) in the following Lemma.
Lemma 3: The CDF F fix SIR (γ |R s = s) of a UAV-BS reference receiver served by its nearest, fixed-located node in the presence of a dominant interferer, is given in closed Note that in this setup, it is convenient to take the product of RVs resulting in the desired SIR. Towards this direction, the distribution of the ratio V of random distances R s /R I is first obtained. Notice that the two RVs are dependent. In this context, let V = R s /R I represent the ratio of the distances. Through a change of variables, R I can be expressed as R I = R s /V . The corresponding CDF and PDF of the RV V are provided in the following Lemma. Lemma 4: The CDF and PDF of V are given by Proof: The CDF of V yields from double integration of the joint distance PDF with respect to the limits of r s , r I and consequently where v ∈ [0, 1]. By substituting the distance distributions in the above integral and applying the Binomial expansion, is then obtained by simply taking the derivative of F V (v) and that completes the proof.
Interestingly, it is noticed that the distribution of ratio V confirms i) a square dependence between distances and ii) independence from the number of nodes N and the radius R, i.e., from the area of the spherical region of the network. Based on (12) and (13), the CDF F rand SIR (γ ) of SIR can now be obtained as where E[·] denotes the expectation operator and P(·) denotes probability operator.
The above equation results in the following Lemma. Lemma 5: The CDF F rand SIR (γ ) and the corresponding PDF f rand SIR (γ ) are given by (16), (17), as shown at the bottom of the page, respectively.
Proof: See Appendix D.

IV. PERFORMANCE ANALYSIS
In this Section, using the closed-form expressions for the CDF derived in the previous Section, a performance analysis in terms of CP and BER, is conducted. Subsequently, the analysis focuses on the derivation of closed-form expressions for the average output SIR and the ergodic capacity for the generic case where the distance of the reference link is random.

A. COVERAGE PROBABILITY
In this subsection, the analysis focuses on studying the network performance in terms of CP, which is defined as the probability that the received SIR at the UAV-BS exceeds a threshold value γ th . Mathematically, it is expressed as The CP has been evaluated for the two scenarios studied through the following Propositions.
Proposition 1: Under the assumptions made for Scenario 1, the CP is expressed as P c = 1 − F fix SIR (γ th |R s = s). Proposition 2: Under the assumptions made for Scenario 2, the CP is expressed as P c = 1 − F rand SIR (γ th ).

B. BER
In this subsection, the performance of the system setups proposed in Section II, is evaluated in terms of the average BER for binary modulation schemes. The average BER is the metric that is most revealing about the nature of the system behavior and the one most often illustrated in works containing system performance evaluation studies [22], [23], [30].
Using the CDF-based approach proposed in [31], the average BER is obtained by integrating the derivative of conditional error probability (CEP) P e (γ ), over the respective CDF of the SIR of each scenario. The average BER is then given bȳ where Q(·) is the area under the tail of the Gaussian PDF defined as [32, eq. (2-1-97)] and p ∈ {fix, rand}. Moreover, P e (γ ) = Q( √ 2γ ) for binary phase-shift keying (BPSK) and P e (γ ) = 1 2 e − γ 2 for differential BPSK (DBPSK).

1) SCENARIO 1
The average BER is first derived for Scenario 1 in the following Lemma. Lemma 6: Given the fixed distance s of the reference link, the average BER for the BPSK and DBPSK modulation schemes are given by (20), (21), as shown at the bottom of the page, respectively, where G m,n p,q (x) is the Meijer G-function defined in [28, eq. (9.301)].
Remark 1: Meijer G-function is a built-in function in many mathematical software packages, e.g., Mathematica, Maple, and thus it can be directly evaluated.
Proof: If any constant value is neglected, by substituting (10) and the derivative of P e (γ ) in (19),P BPSK e,fix results in Through a simple change of variables c 1 = P I θ I P s θ s γ , c 2 = By using [28, eq. (7.813.1)], the integrals in (23) can now be easily derived in closed form as in (20). The same procedure is also followed for the derivation of the average BER for the DBPSK scheme.

2) SCENARIO 2
The results obtained for the average BER are presented in the following Lemma. Lemma 7: Given a random location R s of the transmitting node, the average BER for binary modulation schemes is given byP Proof: By following a similar procedure as the one presented in the previous subsection, the previously presented closed-form expressions for the BER are deduced.

C. AVERAGE OUTPUT SIR
In this subsection, the analysis focuses on studying the network performance in terms of the average output SIR for the generic scenario where the distance of the reference link is random. The average output SIR can be calculated bȳ The above equation results in the following Lemma. Lemma 8: Given a random location R s of the transmitting node, the average output SIR is given bȳ .
Note that (27) holds for a < 3, which is a valid and logical restriction for UAV-enabled networks, in which the path loss factor takes values 1-3 for line-of-sight (LoS) scenarios, i.e., [11], [15].

D. ERGODIC CAPACITY
The ergodic channel capacity quantifies the maximum achievable rate that can be supported by the subjected channel. In this subsection, the ergodic capacity is leveraged to measure the achievable rate of the network for the generic scenario where the distance from the defined UAV-BS node is random. It is defined as follows where BW denotes the channel's bandwidth. The ergodic capacity can now be evaluated through the following Lemma. Lemma 9: Given a random location R s of the transmitting node, the ergodic channel capacity is given bȳ Proof: After expressing the log 2 (·) and the hypergeometric functions of (28) Now, (30) is given directly in closed form by using [29, eq. (07.34.21.0011.01)]. After applying some simplifications, (29) is deduced.

V. NUMERICAL RESULTS
In this section, representative numerical examples have been prepared using the expressions derived in the previous sections, while simulation results were also used for verification purposes. First, the CP is numerically evaluated under different operational scenarios. Next, the performance of the network in terms of the average BER, is demonstrated. Finally, the maximum achievable rate implied by the ergodic capacity, is evaluated. The numerical results of the proposed analysis are also discussed to reveal useful insight.

A. COVERAGE PROBABILITY EVALUATION
For all system setups θ s = θ I = 1, in order to compare the same Nakagami fading conditions and P s = P I = 0dBm, in order to preserve equal transmit powers, unless otherwise stated. Other simulation parameters are a = 3 and m s = m I = 2.75 to consider the same severity of fading. Note that these values are typically considered for UAV-enabled communication scenarios and approach LoS propagation conditions [7], [15], [25]. For Scenario 1, a finite network of UAVs is simulated with N = 10, inside a sphere of radius R = 100 m, and the CP is evaluated for various simulation parameters. In Fig. 2, the CP is plotted as a function of R for different values of γ th . One can observe that for small values of R, i.e., for a small-sized networks, the CP degrades rapidly while moving towards more demanding values of γ th . In contrast to this, for large values of R, i.e., for a large-sized network, the probability that acceptable communication is established is very high even for demanding values of γ th .  Indeed, an increase of the size of the network area increases the probability that the distance of the interfering link is longer than the one of the reference link. In Fig. 3, the CP of the UAV-BS reference receiver is plotted as a function of the distance s. Note that s is expressed as a percentage of R. It can be observed that the more demanding the values of γ th are, the more rapid is the degradation of CP as the transmitting node moves closer to the network edge. For the rest of the plots, it is assumed that s = 20 m. Fig. 4 is the representative plot chosen to compare the two scenarios for different values of s and N . It is observed that when s = 20m, the CP of Scenario 1 is far better than the CP of Scenario 2, even for N = 10. However, with the increase of s, i.e., while moving the transmitting node towards the edge of the network, it is more likely that the distance values of the interfering and the reference link are approximately equal for Scenario 1. In this case, the CP of Scenario 2 outperforms the CP of Scenario 1 even for N = 5. Moreover, with the increase of N , the CP is worsened as expected. Another important conclusion from Fig. 4 is the necessity for 3D modeling of UAV networks. Indeed, comparing the 2D and 3D model curves for Scenario 2, it is observed that the 2D BPP model provides optimistic CP compared to the realistic one provided by 3D BPP. This fact, further motivates the study of 3D network models.
For Scenario 2, the CP is independent of the system parameters R, N . In Fig. 5, the CP is plotted as a function of γ th for different fading conditions and path loss factor values. In this figure, it is shown that the better the channel conditions are, the worse the CP is achieved. This observation includes, both the mean received power dominated by the path loss factor and the fading conditions, implied by the parameters m s , m I . This is because the interfering link gets stronger and this is more profound for larger values of γ th . The second observation is that there is a cross point for small values of γ th where the behaviour of CP is inverted, i.e., the CP is slightly larger for better channel conditions. It should be pointed out that higher values of Nakagami-m parameter (m s = m I = 4) model sufficiently LoS propagation environment [15].

B. AVERAGE BER EVALUATION
For the sake of BER performance evaluation of Scenario 1, a finite network of UAVs is simulated with N = 10, in a sphere of radius R = 200m and the average BER as a function of the transmit power P s is computed. The utilized simulation parameters are a = 3 and m s = m I = 2.75. In Fig. 6, the average BER is plotted for different values of the serving distance s for the BPSK modulation scheme. Once again, the distance s is given as a function of the sphere radius R. It is observed that as s increases, the performance of the average BER degrades but the degradation is gradually smaller.
For Scenario 2, the average BER of binary modulation schemes is compared for different values of the channel fading parameters m s , m I . In Fig. 7, the average BER is plotted as a function of the transmit power P s for the BPSK modulation scheme. It can be observed that as P s increases, the average BER becomes significantly better when moving from severe fading (m s = m I = 1) to less-severe fading (m s = m I = 4) conditions. In other words, when severe fading is present, non-LoS (NLoS) propagation conditions  exist and the performance of the average BER is significantly degraded.
In Fig. 8, the average BER is plotted as a function of the transmit power P s for both scenarios, for the DBPSK, and for different values of s. It is observed that the performance of Scenario 2 approximates the performance of Scenario 1, when s = 0.4R. When s = 0.2R, the average BER of Scenario 1 is far better than the corresponding of Scenario 2. However, for s > 0.4R, i.e., when the transmitting node starts moving towards the edge of the network, the performance of Scenario 2 is clearly better.
The demonstrated results enable the system designer to quantify the effects of the distance s, transmit power, and Nakagamim fading parameter on the error performance of a UAV-BS reference receiver.

C. ERGODIC CAPACITY EVALUATION
In this subsection, the performance of the network in terms of the achievable rate is evaluated. As shown from the analytical expression, the ergodic capacity is independent of the system parameters R, N . The utilized simulation parameters are a = 2, θ s = 1 m s and θ I = 1 m I , respectively, so that the expectation of Nakagami fading equals to unity and P I = 0dBm.  In Fig. 9, the normalized ergodic capacityC/BW is plotted as a function ofγ for different fading conditions. For the needs of the numerical evaluation, the simulation parameter P s is set such that a desired output SIR is achieved every time. It can be observed that with the increase of average output SIR, the achievable rate increases as expected. As the fading conditions become more favorable, capacity increases. However, it can be observed that in the low average SIR region, the increment is rather small.

VI. CONCLUSION
In this paper, a comprehensive mathematical framework for the performance analysis of finite wireless 3D UAV networks is derived. The model assumes a UAV-BS reference receiver communicating with its nearest transmitting node, while suffering from interference from a dominant interfering node. Modeling the locations of nodes as a uniform BPP, two scenarios are examined: 1) the transmitting node is at a fixed and known distance from the UAV-BS reference receiver and 2) the transmitting node is randomly located in the spherical-shaped network area. The presented framework is utilized to study the statistics of the received SIR, the coverage probability, and the average BER for binary modulation VOLUME 8, 2020 schemes for the two scenarios. Finally, the calculation of the average output SIR is leveraged to study the ergodic capacity of the network. Indeed, analytical results have been derived and compared to simulation ones providing perfect agreement. The work in this paper allows several extensions in the future work. An interesting future direction is to extend the proposed framework to incorporate a relay selection scheme scenario. Another promising direction is to investigate the performance of the network in the presence of multiple interfering nodes and compare the results with the proposed analysis. By expressing the exponential function and the exponential integral function through the Meijer G-function according to [29,  Disregarding any constant value for brevity, the CDF is expressed as a difference of integrals = γ m s m s 3 F 2 ·, ·, ·; ·, ·; − P I θ I P s θ s γ − γ m s m s 3 F 2 ·, ·, ·; ·, ·; − P I R −a θ I P s s −a θ s γ . (C.1) In (C.1), (α) holds due to [29, eq. (07.23.21.0002.01)]. After substituting (C.1) in the initial expression and applying some simplifications, (10) yields immediately.

APPENDIX D PROOF OF LEMMA 5
Begin from the CDF F rand SIR (γ ) = P(SIR ≤ γ ) = E V {F H (γ V a )} =  (17) is now given directly by simply taking the derivative of (16).