1 The Gauss-Poisson Process for Wireless Networks and the Benefits of Cooperation Anjin Guo, Student Member, IEEE, Yi Zhong, Student Member, IEEE, Wenyi Zhang, Senior Member, IEEE, and Martin Haenggi, Fellow, IEEE

Abstract—Gauss-Poisson processes (GPPs) are a class of clus- In some circumstances, the transmitters form clusters, due to tered processes, which include the Poisson geographical factors (e.g., access points inside a building and as a special case and have a simpler structure than general groups of nodes moving in a coordinated fashion), or popula- Poisson cluster . A key property of the GPP is that it is completely defined by its first- and second-order tion factors (e.g., base stations in urban regions). Besides, the . In this paper, we first show properties of the GPP clustering phenomenon of transmitters can also be artificially and provide an approach to fit the GPP to a given point set. induced by certain MAC protocols1. Thus, clustered point A fitting example is presented. We then propose the GPP as processes are suitable to model these transmitters’ locations. a model for wireless networks that exhibit clustering behavior A few prior studies have used models of clustered point and derive the signal-to-interference-ratio (SIR) distributions for different system models: (1) the basic model where the desired processes, most notably, the Neyman-Scott process [6] [7]. transmitter is independent of the GPP and all nodes in the GPP In those works, the system performance indicators, such as are interferers; (2) the non-cooperative model where the desired success and mean achievable rate, are usually in transmitter is one of the nodes in the GPP; (3) the cooperative complex form involving multiple integrals. model, where the nodes in a GPP cluster transmit cooperatively. In this paper, we focus on the Gauss-Poisson process (GPP), The results indicate that a gain of 5-6 dB can be achieved with cooperation. which is a relatively simple clustered point process that has either one or two points in a cluster. As such, it retains a good level of tractability and constitutes a definitive improvement I.INTRODUCTION over the PPP in cases where “attraction” exists between node A. Motivation locations, and it achieves a good trade-off between modeling tools have been widely used to analyze accuracy and tractability. the performance of wireless networks [2]. A main application The motivation for introducing the GPP mainly comes from of stochastic geometry in wireless communication is to model three aspects. the node locations using spatial point processes. With the • The GPP belongs to the family of the Poisson cluster point process models, metrics like success probability, processes, with the number of points in each cluster achievable rate, local delay and so on can be derived by restricted to one or two. It well describes the scenario quantitative analysis. When applying the point processes to where each cluster has one or a pair of nodes. In model wireless networks, two key problems should be taken reality, scenarios where there are 1 or 2 transmitters close into consideration: first, whether a point process can accurately together are commonly seen including: 1) the scenario model the actual network which may employ medium access where one person uses a phone and a tablet to access control; second, whether the point process model is tractable the network concurrently, or two persons work together for analyzing the system performance. through their own devices; 2) some indoor deployments Most of the works in the literature model wireless networks of access points where there are one or two access points use the (PPP) [3]. This is because the of the cellular networks or Wi-Fi access points in one PPP model has a number of convenient features, such as the room; 3) full-duplex networks where each transceiver independence between different points and the simple form of pair has a fixed/small inter-node distance [8]; 4) multi- the probability generating functional (PGFL), which make the antenna systems where each access point is equipped with analysis tractable. But the PPP model may be inadequate for two antennas; 5) military ad hoc networks, where each those scenarios where the node locations exhibit correlations. node represents a soldier and soldiers move in pairs for their missions. Not limited to model those scenarios, the A. Guo and M. Haenggi are with the Dept. of Electrical Engineering, Uni- versity of Notre Dame, IN 46556 USA (e-mail: {aguo, mhaenggi}@nd.edu). (generalized) GPP is also suitable for many other types Y. Zhong and W. Zhang are with the Dept. of Electronic Engineering and of clusters as will be pointed out in Section II. Information Science, University of Science and Technology of China, Hefei • The generalized GPP, where the inter-point distance in 230027, China (e-mail: [email protected]; [email protected]). Part of this work [1] was presented at the 2014 IEEE International each cluster is a instead of a determinis- Symposium on Information Theory (ISIT). tic quantity, can be used in great generality to model the The research of A. Guo and M. Haenggi has been supported by the spatial distribution of transmitters based on first-order and U.S. NSF (grants CCF-1216407 and CCF-1525904), and the research of Y. Zhong and W. Zhang has been supported by the National Basic Research Program of China (973 Program) through grant 2012CB316004 and the 1Conversely, some MAC protocols may also lead to repulsion between National Natural Science Foundation of China through grant 61379003. transmitters [4], [5]. 2

second-order statistics if the deployment of the transmit- station cooperation of both non-coherent JT and coherent JT in ters appears clustered. It has been shown in [9] that this the downlink of heterogeneous cellular networks and derived property makes the GPP the point process analog of the the coverage/success probability. normal distribution, completely defined by its first- and second-order characteristics. This implies that a suitable C. Contributions GPP can be used to generate point distributions with any given intensity and two-point correlation function that This work makes the following contributions: satisfy certain constraints. In other words, given the first- • We propose an approach to using the GPP to model and second-order statistics of an actual wireless network, spatial distribution of wireless networks, for which the we can use the GPP to model this network. basic idea is to equalize both the first- and second-order • The GPP constitutes a simple network model to analyze statistics of the GPP and the point sets. A fitting case wireless networks that apply cooperative techniques (see study is provided. Sections III-V). • We derive the SIR distributions for the following four models: (a) Basic model: the desired transmitter is in- dependent of the GPP. (b) Non-cooperative model: the B. Related Work desired transmitter is a point of the GPP, and the other Both clustered point processes (e.g., the Neyman-Scott point in the same cluster (if any) acts as an interferer. (c) process) and repulsive point processes (e.g., the Ginibre point Cooperative model 1: the desired transmitter is a point of process and the Matern´ hard-core processes) have been used the GPP, the other point in the same cluster (if any) acts to model wireless networks, including cellular networks. In as a cooperator, and in the SIR expression at the receiver, [6], the authors derived the distributional properties of the the desired signals are combined by accumulating their interference and provide upper and lower bounds for its powers. (d) Cooperative model 2: it is similar to the distribution, assuming the node locations form a Poisson cooperative model 1, except that in the SIR expression cluster process on the . In [7], the Neyman-Scott process at the receiver, the desired signal power is that of the was used to model the distribution of femto-cells, and the sum of the amplitudes of the desired signals. distributions of the SINR and mean achievable rates of both • We investigate the benefits of cooperative communica- nonsubscribers and subscribers were derived. In [10], the tions in a GPP network, which gives some fundamental Ginibre point process was applied to cellular networks to insights into the benefits of cooperation techniques in model the locations of the base stations, and the interference large networks, where the interference from all transmit- and the coverage/success probability were analyzed. In [11], ting nodes is properly accounted for. we used the Strauss process, the Poisson hard-core process and the perturbed triangular lattice to model actual base station II.THE GAUSS-POISSON PROCESS locations in the UK. The GPP has been well studied in the mathematical litera- A. Definition ture. In [12], the authors determined necessary and sufficient Definition 1 (Generalized Gauss-Poisson process [21]). The conditions on the first- and second-order measures for the generalized GPP is a Poisson cluster process with homo- resulting GPP point process to be well defined, and studied geneous independent clustering. The intensity of the parent stationarity, , and infinite divisibility of the GPP. In process is denoted by λp. Each cluster has one or two points, [9], the author proposed a simple method for the simulation of with 1−p and p, respectively. If a cluster has one the GPP, given the intensity and the pair correlation function. point, it is located at the position of the parent. If a cluster Moreover, like the Poisson cluster processes, the GPP has also has two points, one of them is at the position of the parent, been used to model wireless networks. In [13], the GPP was and the other is randomly distributed around the parent with proposed to fit a cognitive radio network based on the first- and some PDF g. second-order statistics using the method of minimum contrast, Note that in Def. 1, the inter-point distance u in the two- but no analysis was carried out using this model. point cluster is random. If u is deterministic, which means that Recently, cooperative transmission techniques in cellular in two-point clusters, one point is at the position of the parent, networks, known as coordinated multi-point transmission and the other is uniformly distributed on the circle with radius (CoMP), have been widely studied [14], [15]. For the down- u centered at the location of the parent, then the GPP is called link, CoMP can be divided into two categories: coordinated a standard GPP as defined in [2, Example 3.8]. The standard scheduling/beamforming [16], [17], which reduces inter-cell GPP is motion-invariant (isotropic and stationary) [2, Ch. 2]. interference, and joint transmission (JT) [18]–[20]. In [18], Fig. 1 shows a realization of a standard GPP with λ = 0.02, the authors studied coherent JT with power-splitting in cellular p u = 1 and p = 0.5. networks where base stations cooperate in a pairwise manner. Denote the parent point process of the generalized GPP by In [19], the authors proposed a tractable model for analyzing Φ = {x , x , ...} with intensity λ . Let {Φ , x ∈ Φ } be the the base station cooperation of non-coherent JT, where the base p 1 2 p x p clusters of the GPP, denoted by stations follow a stationary PPP and at the receiver a received power boost is yielded by the non-coherent superposition of  {x} w.p. 1 − p Φx = (1) the useful signals. In [20], the authors considered the base {x, x + zx} w.p. p, 3

Since the generalized GPP is motion-invariant, ρ(2)(x, y) (2) + + only depends on kx − yk. So we define ρmi : R → R , such (2) (2) 2 that ρmi (kx − yk) ≡ ρ (x, y), for all x, y ∈ R . Without (2) ambiguity, we also call ρmi the second moment density. For (2) the GPP, an expression for ρmi is given in the following lemma. (2) y Lemma 1. The second moment density ρmi (r) of the gener- alized GPP is

(2) 2 ρmi (r) = λ + 2pλpfu(r). (2)

Proof: For any compact A, B ⊂ R2, the second is 0 10 20 30 40 50 60 Z Z Z Z ρ(2)(x, y)dxdy = λ2 + pλ g(y − x) 0 20 40 60 80 p x A B A B  + pλpg(x − y) dxdy (3) Fig. 1. A realization of the standard GPP on [0, 80]×[0, 60] with λp = 0.02, Z Z 2  u = 1 and p = 0.5. There are in total 159 points. The expected number of = λ + 2pλpg(y − x) dxdy. points is 4800 × 0.02 × 1.5 = 144. A B (4)

2 The intensity of the GPP, denoted by λ, is thus λp(1 + p). For For the right hand side of (3), inside the integral, λ is due the cluster Φx, we call x the cluster center. The GPP is the to pairs of points in different clusters, pλpg(y − x) is due to union of the clusters: the pair (x, y) of points in the same cluster, where x ∈ A is [ the cluster center and y ∈ B is the other point of the cluster, Φ = Φx, and pλpg(x − y) is from the pair (x, y) of points in the same x∈Φp cluster, where y is the cluster center and x is the other point. 2 where the translations {zx} ∈ R , x ∈ Φp, are i.i.d. with PDF Since A, B are arbitrary compact sets, we have that for any g. x, y ∈ R2, In this paper, we only consider motion-invariant (isotropic (2) 2 and stationary) point processes, so g(x) only depends on kxk, ρ (x, y) = λ + 2pλpg(y − x), (5) + + and we define the radial PDF fu : R → R by fu(kxk) (2) , r ∈ + ρ (r) = λ2 +2pλ f (r) g(x). and it follows that for any R , mi p u . The generalized GPP was first introduced by Newman (2) in [21]. Newman named the point process “Gauss-Poisson” According to Lemma 1, ρmi (r) “inherits” the properties of (2) 2 due to its property that it is completely characterized by fu(r), since ρmi (r) − λ ∝ fu(r). Indeed, defining the two- its first- and second-order statistics (i.e., intensity and two- point correlation function as point correlation function). In contrast, the stationary PPP is ρ(2)(r) characterized only by its intensity. As such, the generalized mi ξ(r) , 2 − 1, (6) GPP is a natural generalization of the PPP that retains some λ of its simplicity. In the following subsection, we discuss some we have of its pertinent properties. 2p ξ(r) = f (r). (7) λ(1 + p) u B. Properties 2p R (2) Let D . Since 2 fu(kxk)dx = 1, we have The second moment measure α [2, Def. 6.4] , λ(1+p) R (2) R 2 ξ(kxk)dx = D. Given the intensity λ and the two-point and the second moment density ρ [2, Def. 6.5] of a point R process are related by correlation function ξ(r), we obtain for the generalized GPP 6=   Z ξ(r) (2) X (2) fu(r) = , (8) α (A×B) , E 1A(x)1B(y) = ρ (x, y)dxdy, D x,y∈Φ λD A×B p = , (9) where A, B are two compact subsets of 2 and the 6= symbol 2 − λD R  λD  indicates that the sum is taken only over distinct point pairs. λp = λ 1 − . (10) The second moment density is an important statistic that 2 describes the pairwise correlation of a point process. For Thus, the generalized GPP is completely defined by its first- (2) 2 (2) the PPP, ρ (x, y) = λ , since points are independent. If order statistic λ and its second-order statistic ξ(r) (or ρmi (r)). ρ(2)(x, y) > λ2, it is likely that Φ has a point at y, if there is Note that (8), (9) and (10) have also been derived in [9] using a point of Φ located at x. a different method. 4

+ + + ++ + + + + + + ++ + ++ + + ++ + + + ++ + + +++ + + + + + + + ^ + + + ++ + +++ + + ++ ξ r of the point set + ++ + + + + + + + + ( ) + + + +++ ++ + ++ + + + ++ ++ + ξ (r) + + + ++ ++ +++ +++ + Pois + ++ + + + ++ + + + + + ++ + + +++ + + + ++ + ++ + + + + + + + + + ++ + + + + + + + + + + + + + + + + ++ ++ + ++ ++ ++ + ++++ ++ + ++ ++ + ++ + ++ + + + + + + + + + + + + + + + + +++ + + + ++ + + + + + + ++ + + + + + + + + + + + + ) + + + r

y + + + + + + + + + + + ( ^ + + ξ + ++ + + + + + + + ++ + + + + + ++ + + + + + ++ + + ++++++++ + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + ++ + + + + + + +++ + + + + + +++ + + + ++ + ++ + + + +++ + + + + +++++ + + + +++ + ++ + + ++ + + + + + + + + + + + + + + + + ++ ++ + + + + + + ++ + + ++ + + ++ + + + + + + + + + +++ 0 5 10+ 15 20 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

0 5 10 15 20 0.0 0.5 1.0 1.5 2.0

x r

Fig. 2. The illustration of the point set we consider for the GPP fitting. It Fig. 3. The estimated two-point correlation function ξˆ(r) of the point set is a realization of the PHP with λ1 = 0.8825, λ2 = 2 and r = 0.5. and the two-point correlation function ξPois(r) of the PPP.

+ R Since fu(r) ≥ 0 for all r ∈ R and 2 fu(kxk)dx = 1, S R of intensities λ1 and λ2. Further let Ξr , {b(x, r): x ∈ Φ1} by (7), we have two constraints for ξ(r), namely be the unions of all disks of radius r centered at a point of + ξ(r) ≥ 0, ∀r ∈ R , (11) Φ1. The PHP is Φ = {x ∈ Φ2 : x∈ / Ξr} = Φ2 \ Ξr, i.e., each Z 2p point in Φ1 carves out a hole of radius r from Φ2. λ ξ(kxk)dx = ≤ 1. (12) As is illustrated in Fig. 2, the point set we use for fitting 2 1 + p R is generated on a 20 × 20 window from the PHP with λ = Herein, (11) indicates that the GPP is a clustered point 1 0.8825, λ = 2 and r = 0.5. The intensity of the PHP is λ = process, and (12) implies that next to a point of the process, 2 λ exp(−λ πr2) = 1. The estimated intensity of the point set at most one other point in excess of the Poisson distributed 2 1 is λˆ = 1.06. To estimate the two-point correlation function, we points can be present. Given λ and ξ(r) that statisfy the two use the function ‘pcf’ of the package “spatstat” in the software constraints, the generalized GPP is well defined. “R”. Fig. 3 shows the estimated two-point correlation function ξˆ(r). We observe that ξˆ(r) vanishes for r > 1, i.e., ξˆ(r) ≈ 0, C. GPP Model Fitting p for r > 1/ λˆ, which means that if the distance between two Given λ and ξ(r), which satisfy (11) and (12), there is a points is larger than twice the mean nearest-neighbor distance generalized GPP. In this subsection, we introduce the GPP of the PPP with the same intensity2, the two points are likely model fitting given a point set. The basic idea is that, given ˆ to be uncorrelated. To apply the generalized GPP for modeling a point set, we first estimate its intensity λ and the two-point fitting, we let λ = λˆ and make the approximation of ξ(r) that correlation function ξˆ(r) and then fit the generalized GPP with ˆ ˆ (  p i λ and ξ(r) to the point set by letting λ = λ and ξ(r) = ξ(r). max{0, ξˆ(r)}, for r ∈ 0, 1/ λˆ , The goodness-of-fit is evaluated by comparing the nearest- ξ(r) = (13) 0, otherwise. neighbor distance function G(r) [2, Def. 2.39]. The point set we use for fitting in this subsection is drawn Denote the PDF of the inter-point distance in the two-point ˆ ˆ ˆ from a realization of the Poisson hole process (PHP). Recently, cluster as fu. We have fu(r) = 2πrfu(r). By (8), fu of the as a class of clustered point process, the PHP has drawn fitted GPP is obtained and shown in Fig. 4. By (9) and (10), much attention for its application to model some emerging we have p = 0.460 and λp = 0.726. types of networks. For example, in [22], the PHP is used to The two-point correlation functions of the point set and the model active secondary users in a cognitive network, who are fitted generalized GPP are shown in Fig. 5. The estimated outside the primary user exclusion regions; in [23], the PHP is two-point correlation functions of 99 realizations of the fitted used to model the device-to-device (D2D) transmitters in D2D generalized GPP are also illustrated. We observe that there networks where exclusion zones are introduced by interference is only a small gap between their average and ξˆ(r) of the management. A shortcoming of the PHP is that it is not very point set for r < 0.3, and their envelope fully contains ξˆ(r) tractable; as a result, it would benefit from being approximated using a simpler point process, such as the GPP. 2The nearest-neighbor distance distribution G(r) of the PPP of intensity 2 The PHP is a that has been defined in [2, λ is G(r) = 1 − e−λπr . Thus the mean nearest-neighbor distance NN = ∞ 2 2 R 2λπr2e−λπr dr = √1 . Example 3.7]. Let Φ1, Φ2 ⊂ R be independent uniform PPPs 0 2 λ 5 ) ) r r ( ( u f G ^

Superposition of realizations Envelope of realizations Average of realizations Point set PPP 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5

r r

Fig. 4. The PDF fˆu(r) of the distance between the two points in the same Fig. 6. The nearest-neighbor distance distribution functions G(r) of the data cluster for the fitted generalized GPP. (solid curve) and 99 realizations of the fitted generalized GPP (grey curves). The dotted curve are the pointwise maximum and minimum of G(r) of the 99 realizations. The dash-dotted curve is the average value of G(r) of the 99 realizations. The dashed curve is the theoretical curve of the PPP.

Superposition of realizations Envelope of realizations Average of realizations III.SYSTEM MODELS Point set PPP We model the locations of the transmitters as a generalized GPP on R2. As a special case, the modeling using the standard GPP has been considered in our prior work [1]. Our analysis )

r is focused on the typical receiver located at the origin o with ( ξ desired transmitter at x0 = (b, 0) with b 6= 0. We adopt a path loss model `(x) = kxk−α, where x ∈ R2 and α > 2, and assume the power fading coefficients to be spatially independent with of unit mean (i.e., Rayleigh fading). Denote by hx the power fading coefficient between the transmitter x and the receiver at o. We set all transmit powers to unity and focus on the interference- −0.5 0.0 0.5 1.0 1.5 2.0 2.5 limited regime, thus omitting the thermal noise. 0.0 0.2 0.4 0.6 0.8 1.0

r A. Basic Model Fig. 5. The two-point correlation functions ξ(r) of the data (solid curve) and The desired transmitter x is separate from the GPP and 99 realizations of the fitted generalized GPP (grey curves). The dotted curve 0 are the pointwise maximum and minimum of ξ(r) of the 99 realizations. The all points in the GPP are interferers. The SIR at the receiver dash-dotted curve is the average value of ξ(r) of the 99 realizations. The located at the origin o is dashed curve is the theoretical curve of the PPP. −α h0b SIR = P −α , (14) hxkxk of the point set, which implies that the approximation by (13) x∈Φ h p i provides a very good match within the r range 0, 1/ λˆ . where h0 is the power fading coefficient between the desired We next compare the G function between the point set and transmitter and the receiver. the fitted generalized GPP. The functions of the point set and 99 realizations of the fitted generalized GPP are given in Fig. B. Non-cooperative Model 6. We conclude that the PPP is not suitable to describe the point set, while the fitted generalized GPP is a good model In this case, the desired transmitter x0 is taken from the for the point set, since the functions of the point set fall into GPP and b = kx0k. All other points in the GPP are interferers. the range of the functions of the fitted generalized GPP’s 99 Therefore, there is an interferer at distance u of the desired 2p realizations. transmitter with probability 1+p , since the probability that x0 6

(1−p)λp 1−p belongs to a one-point cluster is = , and the SIR where {Φy} are the clusters of the GPP and y is the cluster λp+pλp 1+p at the receiver is center of Φy. h b−α Therefore, the SIR at the receiver o is SIR = x0 . (15) P −α 2 hxkxk x∈Φ\{x0} P ˜ −α/2 hxkxk x∈Φ0 SIR = . (20) 2 C. Cooperative Model 1 P P ˜ −α/2 hxkxk y∈Φp\Φ0 x∈Φy In this case, the desired transmitter x0 is taken from a cluster Φ0 in the GPP, and if there is another point in Φ0, it acts as a cooperator. Unlike the cooperation model in [19], where the cooperative base stations are selected based on the received IV. SIR DISTRIBUTION signal strength measurements, in our model, we assume that We assume that the receiver can decode successfully if its the cooperative pair consists of the transmitters in the same SIR exceeds a threshold θ. The interference is denoted by I, cluster and that they share the information including the data which is the denominator in (14), (15), (16) and (20). In this that needs to be transmitted. We assume that if there is a section, we derive the complementary cumulative distribution cooperator, the transmitters adopt non-coherent JT and at the function (CCDF) of the SIR distributions (or the transmission receiver, the useful signals are combined by accumulating success probabilities) for the four models. their powers, a scheme known as cyclic delay diversity for single-frequency networks using orthogonal frequency division A. Basic Model multiplexing (OFDM) as is shown in [19]. In this way, the receiver is served by both transmitters in a cluster, and all Lemma 2. Let v : R2 7→ [0, 1] be a measurable function such points from other clusters of the GPP act as interferers. The that 1 − v has bounded support. The probability generating SIR at the receiver is functional (PGFL) of the GPP is P −α hxkxk  Z SIR = x∈Φ0 . (16) h P h kxk−α G[v] = exp λp (1 − p)v(x) x∈Φ\Φ x 2 0 R Z i  D. Cooperative Model 2 + pv(x) v(x + z)fu(kzk)dz − 1 dx . (21) 2 R In this case, same as the cooperative model 1, the desired Proof: The PGFL of a Poisson cluster process is transmitter x0 is taken from a cluster Φ0 in the GPP, and Z if Φ0 has two points, the other point acts as a cooperator. PCP  [x]  G [v] = exp λp (G0 [v] − 1)dx , (22) But different from the cooperative model 1, if there exists a 2 R cooperator, the signals from the two cooperative transmitters [x] [x] are not combined at the receiver by accumulating their powers where G0 [v] is the PGFL of the cluster Φ that is centered [x] Q  but they add up in amplitude. The received signal at the typical at x, given by G0 [v] = E y∈Φ[x] v(y) [2, Cor. 4.12]. receiver is expressed as According to the definition of the generalized GPP, we have X ˜ −α/2 X ˜ −α/2 Z Yro = hxkxk Xto + hzkzk Xz, (17) [x] G0 [v] = (1 − p)v(x) + p v(x)v(x + z)fu(kzk)dz. (23) x∈Φ 2 0 z∈Φ\Φ0 R ˜ where hx are the fading coefficients and are i.i.d. complex Substituting (23) into (22), we obtain (21). Gaussian distributed random variables with mean 0 and vari- Theorem 1. In the basic model, the SIR distribution is the ance 1, Xto is the signal transmitted by the desired (coop- Laplace transform of the interference I at θbα, i.e., erative) transmitters of the typical receiver, and Xz is the α signal transmitted by the transmitter at z. Xto and Xz are Ps(θ, b, λp, α, p) = LI (θb ), (24) independent random variables with mean 0 and variance 1. The first term is the desired signal, while the second term is where the interfering signal. The desired received signal power at o  Z ∞  1 − p p LI (s) = exp 2πλp −α + −α is 0 1 + sr 1 + sr 2 Z ∞ Z 2π X ˜ −α/2 1 Pr = hxkxk . (18) · dψf (τ)τdτ 2 2 −α/2 u x∈Φ 0 0 1+s(r +τ +2rτ cos ψ) 0   So P is exponentially distributed with mean P kxk−α r x∈Φ0 −1 rdr . (25) (i.e., Rayleigh fading). Note that in the interference signal, if two transmitters belong to one cluster, they transmit the same signal. So the interference power at o is given by Proof: The SIR distribution Ps is 2 X X ˜ −α/2  −α  Io = hxkxk . (19) h0b (a) α Ps(θ, b, λp, α, p) = P > θ = LI (θb ), (26) y∈Φp\Φ0 x∈Φy I 7

α where (a) follows because h0 ∼ exp(1). The Laplace trans- where s = θb , form of I is derived from the PGFL as follows:  πλ sδ    p   LI0 (s) = exp − (1 + pδ) , (32) X −α sinc δ LI (s) = EΦ,{h} exp − shxkxk x∈Φ and δ 2 .   , α Y −α  = EΦ Eh exp(−shkxk ) Proof: For the asymptotic regime u → 0, the two points x∈Φ in any two-point cluster tend to be located at the same position.  Z h Let Φ1 ⊂ Φp be the set of parent points of the clusters with = exp λp (1 − p)v(x) + pv(x) 2 R only one point in the GPP and Φ2 = Φp\Φ1 be the set of Z i  parent points of the clusters with co-located two points in the · v(x + z)fu(kzk)dz − 1 dx , (27) P −α P 2 GPP. Let I1 = hxkxk and I2 = (hx,1 + R x∈Φ1 x∈Φ2 h )kxk−α Φ Φ where x,2 be the interference from 1 and 2 respectively, where h and h are the power fading coefficients between 1 x,1 x,2 v(x) = exp(−shkxk−α) = . (28) the two transmitters co-located at x and the typical receiver. Eh 1 + skxk−α The Laplace transform of the interference is then given by Substituting (28) into (27), we obtain (25).   The SIR distribution can be bounded in closed form for the  LI0 (s)= E exp − sI1 − sI2 standard GPP with u = 1 and α = 4.   Y −α  Corollary 1. For the standard GPP with u = 1 and α = 4, =EΦ1 Eh exp(−shkxk ) the SIR distribution has upper and lower bounds in closed x∈Φ1   form, as follows: Y −α  ·EΦ2 Ehx,1,hx,2 exp(−s(hx,1 +hx,2)kxk )  2 √  π x∈Φ2 Ps(θ, b, λp, 4, p) ≤ exp − λp(1 − p) s + λppWu(s) , 2  Y 1   Y 1  =EΦ1 EΦ2 and 1+skxk−α (1+skxk−α)2 x∈Φ1 x∈Φ2  2  π √  Z (1 + p + skxk−α)skxk−α  Ps(θ, b, λp, 4, p) ≥ exp − λp(1 − p) s + λppWl(s) , =exp − λ dx 2 p −α 2 R2 (1 + skxk ) 4  2 δ  where s = θb , δπ λps √ = exp − (1 + pδ) . (33)  sin (πδ) π s 3 s 2 W (s) = 8s 2 ln + (−3s + 24s u 4(9s2 + 40s + 16) s + 4 √ From (26), we get the SIR distribution for the GPP in the 2 21  π s limit of u → 0. + 16) arctan √ −π( s2 + 48s + 24) + s 2 2(4s + 1)   3 s 1 B. Non-cooperative Model · s 2 ln + (3s + 1)(arctan √ − π) , (29) s + 1 s Lemma 3. Conditioned on a point of the generalized GPP and being located at y, the conditional PGFL of the GPP excluding √ 2√ 1 √ 1 y is π s πs 4  √ √ 1 + s − 2s 4 Wl(s) = − − 2 2π − 2 ln √ √ 1  Z  4 8 1 + s + 2s 4 ! 1 − p 2p √ Gy[v] = G[v] + v(y + z)fu(kzk)dz . 1 1 + p 1 + p 2 √ 1 − 2 − s 4 √ 1 R − 2( 2 + 2s 4 ) arctan − 2( 2 − 2s 4 ) 1 Proof: Denote the points in the cluster which contains the √ s 4 1 3 √ desired transmitter y as Φ and all points in other clusters as − 2 + s 4  2πs 2 + π s 1 0 · arctan 1 + arctan √ Φc =Φ\Φ0. From Slivnyak’s theorem [2], conditioning on Φ0 s 4 2(4s + 1) s does not change the distribution of the other clusters, and the 2 2 3 2√ πs s 2π s 2 + π s + ln − . (30) distribution of the points excluding Φ0 remains the same as 4s + 1 s + 1 2(4s + 1) the original GPP Φ. So, the conditional PGFL excluding y is   Proof: See Appendix A. ! Y Gy[v] = E v(x) As b → 0, we have Wu → 0 and Wl → 0, and thus the gap S x∈(Φc Φ0)\{y} between Wu and Wl vanishes.  Y   Y  The following corollary gives the SIR distribution in the = E v(x) E v(x) asymptotic regime u → 0 for the standard GPP. x∈Φ x∈Φ0\{y}  Z  Corollary 2. In the basic model, for the standard GPP, the (a) 1 − p 2p = G[v] + v(y + z)fu(kzk)dz , 1 + p 1 + p 2 SIR distribution in the asymptotic regime u → 0 is equal to R α the Laplace transform of the interference I0 at θb , i.e., where (a) follows since the probability that y belongs to a one-point cluster is (1−p)λp = 1−p . Ps(θ, b, λp, α, p) = LI0 (s), (31) λp+pλp 1+p 8

Theorem 2. In the non-cooperative model, the SIR distribu- C. Cooperative Model 1 α tion is the Laplace transform of the interference I at θb In this cooperative model, the cooperator transmits the same information as the desired transmitter simultaneously. In this Ps(θ, b, λp, α, p) = L¯I (s), (34) case, the received power, denoted by Psum, is the sum of the where s = θbα and received signal powers from the desired transmitter and the cooperator, i.e.,  1 − p 2p ( −α 1−p L¯I (s) = LI (s) · + hb w.p. 1+p , 1 + p 1 + p Psum = −α −α 2p (36) h1b + h2c w.p. 1+p , Z ∞ Z 2π 1  · dψfu(τ)τdτ . 1 + s(b2 + τ 2 + 2bτ cos ψ)−α/2 where h, h1, h2 ∼ exp(1) are mutually independent, c = kx0+ 0 0 2 zck, and zc ∈ R is a random point with PDF fu. The exponential distribution has the property that if h ∼ Proof: The proof is similar to that of Theorem 1, with exp(1), then lh ∼ exp(1/l), for l > 0. Thus, conditioned on ! −α α the conditional PGFL Gy[v] instead of G[v]. c, for the case where Psum = h1b , Psum ∼ exp(b ); for the −α −α Similar to the basic model, Ps(θ, b, λp, α, p) is not in closed case where Psum = h1b + h2c , if b 6= c, Psum follows α α form. For the standard GPP with u = 1 and α = 4, however, the hypoexponential distribution Hypo(b , c ) and the PDF bαcα α α  closed-form lower and upper bounds are available. of Psum is fP (x) = bα−cα exp(−c x) − exp(−b x) , oth- α 2α α erwise, Psum ∼ (2, b ) and fP (x) = b x exp(−b x). Corollary 3. For the standard GPP with u = 1 and α = 4, !Φ0 Conditioned on y ∈ Φ0, the conditional PGFL Gy of the the SIR distribution has upper and lower bounds in closed GPP, excluding Φ0, is form, as follows: !Φ0 Gy [v] = G[v]. (37)  π2 √  P (θ, b, λ , 4, p) ≤ exp − λ (1 − p) s + λ pW (s) This follows from Slivnyak’s theorem. s p 2 p p u 1−p p  1 1  Theorem 3. In the cooperative model 1, the SIR distribution · + + , is 1+p 1+p 1 + s(b2 + 1 + 2b)−2 1 + s(b2 + 1)−2 1−p 2p P (θ, b, λ , α, p) = L (θbα)+ H(kx + z k) and s p 1+p I 1+pEzc 0 c (38)  π2 √  P (θ, b, λ , 4, p) ≥ exp − λ (1 − p) s + λ pW (s) where s p 2 p p l bα cα 1−p p  1 1  H(c) L (θcα) − L (θbα) (39) · + + , , bα − cα I bα − cα I 1+p 1+p 1 + s(b2 + 1 − 2b)−2 1 + s(b2 + 1)−2 and the PDF of zc is fu. where s = θb4. Proof: The SIR at the receiver is Psum/I. We have Proof: See Appendix B. P  P (θ, b, λ , α, p) = sum > θ The SIR distribution in the asymptotic regime u → 0 for s p EPsum P I the standard GPP is given by the following corollary. 1 − p hb−α  = P > θ Corollary 4. In the non-cooperative model, for the standard 1 + p I 2p h b−α + h c−α  GPP, the SIR distribution as u → 0 is equal to the Laplace + 1 2 > θ α P transform of the interference I0 at θb , i.e., 1 + p I 1 − p α 2p ¯ = LI (θb ) + Q, (40) Ps(θ, b, λp, α, p) = LI0 (s), (35) 1 + p 1 + p

−α −α α h1b +h2c where s = θb and where Q , P( I > θ). Since the case of c = b has a vanishing probability thus contributing zero to Q, we α  δ  α α (1 + p)b + (1 − p)s πλps b α c α  L¯ (s) = exp − (1 + pδ) . have Q = EI,c bα−cα exp(−θc I) − bα−cα exp(−θb I) = I0 (1 + p)(bα + s) sinc δ  Ec H(c) . 2 As c = kx0 + zck, where zc ∈ R is a random point with Proof: The proof follows the same reasoning as that of  PDF fu, we have Q = Ezc H(kx0 + zck) . Corollary 2 except for the conditional PGFL of the GPP as For the standard GPP with u = 1 and α = 4, upper and u → 0. lower bounds of the SIR distribution can be derived. Compared with the corresponding result for the basic model, Corollary 5. For the standard GPP with u = 1 and α = 4 the result for the non-cooperative model has an extra factor α and b 6= 1/2, the SIR distribution has upper and lower bounds, (1+p)b +(1−p)s . This factor is a result of the extra factor of (1+p)(bα+s) as follows: the conditional PGFL of the GPP, compared with the PGFL 4 4 of the GPP. Ps(θ, b, λp,4, p) ≤ C1LI (θb ) + C2LI (θ|b − 1| ), (41) 9 and Based on the expression of the Laplace transform of the in- terference power, we give the SIR distribution in the following 4 4 Ps(θ, b, λp,4, p) ≥ C3LI (θb ) + C4LI (θ(b + 1) ), (42) theorem. 1−p 2p|b−1|4 2pb4 Theorem 4. In the cooperative model 2, the SIR distribution where C1 = 1+p − (1+p)(b4−|b−1|4) , C2 = (1+p)(b4−|b−1|4) , 1−p 2p(b+1)4 2pb4 is C3 = − 4 4 and C4 = 4 4 . 1+p (1+p)(b −(b+1) ) (1+p)(b −(b+1) ) 1 − p P (θ, b, λ , α, p) = L˜ (θbα) Proof: See Appendix C. s p 1 + p Io To get the bounds in closed form, we may apply (24) and Z 2p ˜  θ  Corollary 1 to (41) and (42). In (41), for each of the two terms + LIo −α −α fu(kzk)dz, (45) 1 + p 2 b + kx + zk 4 4 R 0 C1LI (θb ) and C2LI (θ|b − 1| ), we apply the upper bound ˜ of LI (·), if the corresponding factor (C1 or C2) is larger than where LIo (s) is given in (44). 0 L (·) , and apply the lower bound of I otherwise. While in Proof: According to the SIR expression given in (20), we (42), for each of the two terms, we apply the lower bound of have L (·) if the corresponding factor is larger than 0, and apply I 2 P ˜ −α/2 the upper bound of LI (·) otherwise. Note that the upper bound hxkxk  x∈Φ0  becomes loose as b → 0.5, as will be observed in Section V. P (θ, b, λ , α, p) = > θ s p P I The SIR distribution in the asymptotic regime u → 0 for o ˜ −α/2 2 1−p |hx b |  the standard GPP is given by the following corollary. = P 0 >θ 1+p Io Corollary 6. In the cooperative model 1, as u → 0 for the ˜ −α/2 ˜ −α/2 2 2p |h1b +h2c |  standard GPP, the SIR distribution is + P >θ 1+p Io Z ∞ (1 + 3p)θbα − (1 + p)2jπω 1 − p 2p h i P (θ, b, λ , α, p) = ˜ α ˜ −α −α −1 s p α 2 = LIo (θb ) + Ec LIo (θ(b + c ) ) −∞ (1 + p)(θb − 2jπω) 1 + p 1 + p  δ  1−p 2p Z  θ  πλp(2jπω) ˜ α ˜ · exp − (1 + pδ) dω. = LIo (θb )+ LIo −α −α fu(kzk)dz, 1+p 1+p 2 b +kx +zk sinc δ R 0 (43) (46) 2 where c = kx0 + zck, zc ∈ R is a random point with PDF f , and h˜ , h˜ , h˜ ∼ CN (0, 1) are independent. Proof: See Appendix D. u x0 1 2 The SIR distribution in the asymptotic regime u → 0 for the standard GPP is given by the following corollary. D. Cooperative Model 2 Corollary 7. In the cooperative model 2, as u → 0, the SIR In this cooperative model, the interference signals from distribution is the same cluster can be treated as one signal, since in any  δ 2 δ  1−p πλpθ b (1−p+2 p) cluster Φy with y ∈ Φp \ Φ0, the interference power is Ps(θ, b, λp, α, p)= exp − 2 1+p sinc δ P h˜ kxk−α/2 , which follows the exponential distri- x∈Φy x  δ 2 −δ −δ  2p πλpθ b (2 −2 p+p) bution with mean P kxk−α. In this way, each cluster + exp − . (47) x∈Φy 1+p sinc δ can be treated as one interferer, which follows the PPP with intensity λp. Thus, the Laplace transform of the interference power Io is given by Proof: See Appendix E. It is worth noting that if p = 0, the GPP reduces to the   X X 2 ˜ ˜ −α/2 PPP with intensity λ = λp. Substituting p = 0 into either one LIo (s)= EΦ,{h˜} exp − s hxkxk of Theorems 1-4, we obtain the well-known result P (θ) = y∈Φp\Φ0 x∈Φy s δ 2   2  exp − πλpθ b Γ(1 + δ)Γ(1 − δ) , see, e.g., [2, Ch. 5.2]. Y X ˜ −α/2  = Φ ˜ exp(−s hxkxk ) E EΦy ,{h} y∈Φp\Φ0 x∈Φy V. NUMERICAL RESULTS  Y  1 We consider all system models we analyzed and show the = (1 − p) EΦp 1 + skxk−α numerical results. The numerical results are obtained from the x∈Φ p analytical results we have derived. Z 1  + p −α −α fu(kzk)dz 2 1 + s(kxk + kx + zk ) R A. Deterministic u (Standard GPP)  Z h 1 = exp λp (1 − p) −α Fig. 7 shows the SIR distribution and the closed-form 2 1 + skxk R Z  bounds of the basic model as a function of the distance 1 i between the receiver and the desired transmitter. We observe +p −α −α fu(kzk)dz−1 dx . 2 1+s(kxk +kx+zk ) R that the upper bounds are satisfactorily tight for the basic (44) model. 10

1 1 Numerical result Numerical result of CM1 Bounds Bounds of CM1 0.9 0.9 Numerical result of CM2 b = 1.5 0.8 0.8 b = 0.2 0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

Success Probability b = 0.5

0.3 Success Probability 0.3 b = 3 b = 1 0.2 0.2

0.1 0.1 b = 3 0 0 -20 -10 0 10 20 30 -20 -10 0 10 20 30 40 SIR Threshold (dB) SIR Threshold (dB)

Fig. 7. The SIR distribution and the closed-form bounds with different Fig. 9. The SIR distribution and the closed-form bounds with different distances between the receiver and the desired transmitter for the basic model distances between the receiver and the desired transmitter for the cooperative (λp = 0.1, p = 0.5, α = 4, u = 1). model 1 (CM1) and the SIR distribution for the cooperative model 2 (CM2). (λp = 0.1, p = 0.5, α = 4, u = 1.)

1 Numerical result 0.9 Bounds 1 Basic model 0.8 Non-cooperative model 0.9 Cooperative model 1 0.7 Cooperative model 2 0.8 0.6 0.7 0.5 0.6 0.4 0.5 Success Probability 0.3 b = 0.5 0.4 0.2 b = 3

Success Probability 0.3 0.1 b = 1 0.2 0 -20 -10 0 10 20 30 0.1 SIR Threshold (dB) 0 -25 -20 -15 -10 -5 0 5 10 15 20 Fig. 8. The SIR distribution and the closed-form bounds with different SIR Threshold (dB) distances between the receiver and the desired transmitter for the non- cooperative model (λp = 0.1, p = 0.5, α = 4, u = 1). Fig. 10. The SIR distributions of the basic model, the non-cooperative model, the cooperative model 1 and the cooperative model 2 (λp = 0.1, p = 0.5, b = 1.5, α = 4, u = 1). Fig. 8 shows the SIR distribution and the closed-form bounds of the non-cooperative model as a function of the distance between the receiver and the desired transmitter. We observe that the upper bounds are tight for both large and assumptions, the two cooperative techniques have the same small values of b. However, when b approaches u, the bounds performance gain over the non-cooperative model w.r.t. the become loose, since if the cluster containing the desired success probability. transmitter has another transmitter z0, we use√ the bounds for Fig. 10 compares the SIR distribution curves of the four 2 2 the distance dzr from z0 to the receiver that b √+ u < dzr < models. As expected, we observe that the performance of 2 2 b + u with probability 0.5 and |b − u| < dzr < b + u with the non-cooperative model is the worst while those of the probability 0.5. cooperative models are the best, with a horizontal gap of 5−6 Fig. 9 shows the SIR distribution and the closed-form dB. The performance of the two cooperative models is nearly bounds of the cooperative model 1 as a function of the distance identical. The benefit of cooperation is significant, since by between the receiver and the desired transmitter. We also cooperation, some nearby interferers turn into cooperators. observe that the bounds are tight for both large and small Comparing the non-cooperative model and the basic model, values of b, while they become loose when b is close to u. we conclude that a horizontal gap of 2−3 dB can be obtained The SIR distributions for the cooperative model 2 are also by silencing some nearby interferers. And by the comparison shown in Fig. 9. Interestingly, they are approximately the same of the cooperative models and basic model, a horizontal gap of as those for the cooperative model 1. So under our system about 3 dB can be gained by adding some nearby cooperators. 11

1 We also showed that by cooperation, the SIR distribution is u ∼ gamma(0.5,2) u ∼ gamma(2,0.5) improved significantly—it has a horizontal gain of more than 0.9 u = 1 5 dB compared with the non-cooperative schemes. Since the 0.8 two cooperative schemes have similar performance in terms of 0.7 the SIR distribution, the cooperative scheme without the cyclic

0.6 delay diversity (scheme 2) is preferred, due to the lower system Basic model complexity. 0.5 Non-cooperative model 0.4 Cooperative model 1 APPENDIX A Success Probability 0.3 PROOFOF COROLLARY 1 0.2 Proof: For convenience, let 0.1  Z   0 B1 , exp λp(1 − p) v(x) − 1 dx (48) -20 -15 -10 -5 0 5 10 15 20 25 2 SIR Threshold (dB) R and Fig. 11. The SIR distributions of the basic model, the non-cooperative  Z  Z 2π 1   model, and the cooperative model 1, when u ∼ gamma(0.5, 2), u ∼ B exp λ p v(x) v(x + w(ψ)) dψ − 1 dx . gamma(2, 0.5) u = 1 λ = 0.1, p = 0.5, b = 1, α = 4 2 , p and .( p ). The 2 0 2π results of the cooperative model 2 are omitted, since they are very similar to R (49) those of the cooperative model 1. It follows from Theorem 1 that

Ps(θ, b, λp, α, p) = B1B2. (50) B. Random u (Generalized GPP) We investigate the generalized GPP with u gamma dis- We have tributed. In Fig. 11, we consider the cases of u ∼  Z ∞ 1  B = exp λ (1 − p)2π ( − 1)rdr gamma(0.5, 2) and u ∼ gamma(2, 0.5), where [u] = 1, 1 p −4 E 0 1 + sr and show the SIR distributions of the basic model, the non-  π2 √  cooperative model, and the cooperative model 1. For com- = exp − λp(1 − p) s (51) 2 parison, the case of u = 1 is also drawn. The results of the cooperative model 2 are close to those of the cooperative and model 1, so they are omitted.   For the basic model, the SIR distribution is invariant under B2 = exp λppW , (52) different settings of u, since all points of the GPP are inter- ferers and the interference changes little when the intensity of R ∞ 1 R 2π 1 1 where W = 2π 0 1+sr−4 0 1+s(r2+1+2r cos ψ)−2 2π dψ − the GPP remains unchanged with different distribution of u. 1dr. For the non-cooperative model, the success probability is On the one hand, improved for random u compared with the deterministic u in Z ∞  the high-reliability regime. The case of u ∼ gamma(0.5, 2) 1 1 1 W < 2π −4 2 −2 performs better than the case of u ∼ gamma(2, 0.5), which 0 1 + sr 2 1 + s(r + 1 + 2r) means that as the variance of u becomes larger, the success 1 1   + − 1 rdr probability becomes better. 2 1 + s(r2 + 1)−2 For the cooperative model 1, the success probability for Z ∞  1 1 1 random u is worse than that for the deterministic u in the < 2π 1 + sr−4 2 1 + s(2(r2 + 1))−2 low-reliability regime, and when the variance of u becomes 0 1 1   larger, the benefit from the cooperation is weakened. + − 1 rdr 2 1 + s(r2 + 1)−2

VI.CONCLUSION = Wu(s). (53) In this paper, we first investigated the GPP and used it as a On the other hand, model for point set fitting. The fitted GPP has approximately Z ∞  the same intensity and two-point correlation function as the 1 1 1 W > 2π −4 2 −2 given point set. Although the GPP cannot model all clustered 0 1 + sr 2 1 + s(r + 1 − 2r) point processes/sets, it can well model a wide class of them. 1 1   + − 1 rdr We then proposed the application of the GPP in several dif- 2 1 + s(r2 + 1)−2 ferent wireless network models with and without cooperation Z ∞ 1 1 > 2π − 1rdr and derived the SIR distributions and their bounds for the con- 1 + sr−4 1 + s(r2 + 1 − 2r)−2 sidered models. The results indicate that the bounds, especially 0 Z ∞ Z ∞ the upper bounds, provide useful approximations that well 1 1 = π − s 4 rdr − s 4 rdr fit the actual SIR distribution for different operating regimes. 0 r + s 0 (r − 1) + s 12

! Z ∞ 1 2 where LI0 (s) and LPsum (s) are the Laplace transform of the + s 4 4 rdr 0 (r + s)((r − 1) + s) interference and the desired signal power respectively. The

Z ∞ Laplace transform of the interference LI0 (s) is given by (32). 1 1  + π −4 2 −2 − 1 rdr The Laplace transform of the desired power is given by 0 1 + sr 1 + s(r + 1) ∞ ∞ Z 1 Z 1 LPsum (s) = EPsum (exp(−sPsum)) > π − s rdr − s rdr r4 + s (r − 1)4 + s 1 − p −α 0 0 = Eh(exp(−shb )) ! 1 + p Z ∞ 1 + s2 rdr 2p −α −α 4 2 2 + Eh1,h2 (exp(−sh1b − sh2b )) 0 (r + s)((r + 1) + s) 1 + p Z ∞ 1 1 1 − p 2p + π − 1rdr = + −4 2 −2 (1 + p)(1 + sb−α) (1 + p)(1 + sb−α)2 0 1 + sr 1 + s(r + 1) (1 + p) + (1 − p)sb−α = Wl(s). (54) = . (59) (1 + p)(1 + sb−α)2 Combining (50), (51), (52), (53), and (54), we obtain (29) and (29). Plugging LI0 (s) and LPsum (s) into (58), we get the SIR distribution in Corollary 6. APPENDIX B PROOFOF COROLLARY 3 APPENDIX E Proof: Let PROOFOF COROLLARY 7 1 − p p Z 2π 1 Proof: For the cooperative model 2, as u → 0, by (45), B = + dψ. 3 2 −2 we have 1 + p π(1 + p) 0 1 + s(b + 1 + 2b cos ψ)  α  On the one hand, 1 − p ˜ α 2p ˜ θb Ps(θ, b, λp, α, p) → LIo (θb ) + LIo . 1−p p  1 1  1 + p 1 + p 2 B3 < + + . (60) 1+p 1+p 1+s(b2 +1+2b)−2 1+s(b2 +1)−2 (55) Since according to (44), as u → 0, On the other hand,  Z h 1−p p i  1−p p  1 1  ˜ LIo (s) → exp λp −α + −α −1 dx B3 > + + . 2 1+skxk 1+2skxk 1+p 1+p 1+s(b2 +1−2b)−2 1+s(b2 +1)−2 R (56)  Z  1  = exp λp(1 − p) −α − 1 dx Applying Corollary 1, we obtain the results. 2 1 + skxk R Z  1   APPENDIX C + λpp −α − 1 dx 2 1 + 2skxk PROOFOF COROLLARY 5  R  πλp −α −α −α −α = exp − (1 − p)sδ + p(2s)δ , (61) Proof: As h1b + h2c ≤ h1b + h2|b − 1| , sinc δ h b−α + h |b − 1|−α  Q < 1 2 > θ substituting (61) into (60), we obtain (47). P I bα |b − 1|α = L (θ|b − 1|α) − L (θbα) REFERENCES bα − |b − 1|α I bα − |b − 1|α I [1] A. Guo, Y. Zhong, M. Haenggi, and W. Zhang, “Success probabilities in = H(|b − 1|), (57) Gauss-Poisson networks with and without cooperation,” in 2014 IEEE −α −α International Symposium on Information Theory (ISIT), Jun. 2014, pp. where H(·) is defined in (39). Similarly, as h1b +h2c ≥ 1752–1756. −α −α h1b + h2(b + 1) , Q > H(b + 1). [2] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge Substituting the bounds of Q into (40), we obtain (41) and University Press, 2012. [3] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and (42). M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” IEEE Journal on APPENDIX D Selected in Communications, vol. 27, no. 7, pp. 1029–1046, Sep. 2009. ROOFOF OROLLARY P C 6 [4] H. Nguyen, F. Baccelli, and D. Kofman, “A stochastic geometry analysis Proof: The proof for the cooperative model 1 as u → 0 of dense IEEE 802.11 networks,” in IEEE INFOCOM 2007, May 2007, pp. 1199–1207. is different from that in Theorem 3 since when the typical [5] Y. Zhong, W. Zhang, and M. Haenggi, “Stochastic analysis of the mean receiver is served by two co-located transmitters cooperatively, interference for the RTS/CTS mechanism,” in 2014 IEEE International the desired signal power is not a hypoexponential distribution Conference on Communications (ICC), Jun. 2014, pp. 1996–2001. [6] R. K. Ganti and M. Haenggi, “Interference and outage in clustered but an . We turn to use the following wireless ad hoc networks,” IEEE Transactions on Information Theory, equation to derive the SIR distribution [24]. vol. 55, no. 9, pp. 4067–4086, Sep. 2009. Z ∞ L (−2jπω/θ) − 1 [7] Y. Zhong and W. Zhang, “Multi-channel hybrid access femtocells: A Psum stochastic geometric analysis,” IEEE Transactions on Communications, Ps(θ) = LI0 (2jπω) dω, (58) −∞ 2jπω vol. 61, no. 7, pp. 3016–3026, Jul. 2013. 13

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