{'fraction_non_alphanumeric': 0.038715504890051704, 'fraction_numerical': 0.012988226499719679, 'mean_word_length': 4.231709304220303, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Spectral Efficiency of Dynamic Coordinated Beamforming: A Stochastic Geometry Approach

This paper characterizes the performance of coordinated beamforming with dynamic clustering. A downlink model based on stochastic geometry is put forth to analyze the performance of such base station (BS) coordination strategy. Analytical expressions for the complementary cumulative distribution function (CCDF) of the instantaneous signal-to-interference ratio (SIR) are derived in terms of relevant system parameters, chiefly the number of BSs forming the coordination clusters, the number of antennas per BS, and the pathloss exponent. Utilizing this CCDF, with pilot overheads further incorporated into the analysis, we formulate the optimization of the BS coordination clusters for a given fading coherence. Our results indicate that (i) coordinated beamforming is most beneficial to users that are in the outer part of their cells yet in the inner part of their coordination cluster, and that (ii) the optimal cluster cardinality for the typical user is small and it scales with the fading coherence. Simulation results verify the exactness of the SIR distributions derived for stochastic geometries, which are further compared with the corresponding distributions for deterministic grid networks.


A. Background
Base station (BS) coordination is regarded as an effective approach to mitigate intercell interference [2]- [4]. The idea is to allow multiple BSs to coordinate their transmit/receive strategies (e.g., beamforming, power control, and scheduling) by utilizing channel state information (CSI). The performance would increase monotonically with the number of coordinated BSs if such CSI could be acquired at no cost and thus, ideally, entire systems should be coordinated [5]- [9]. In practice though, coordination of an entire (large) network is not only computationally unfeasible, but undesirable once the ensuing overheads are taken into account [10]. A central concept in the implementation of BS coordination is then that of a cluster, defined as the set of BSs that a given user coordinates with. From the vantage of a user then, only those BSs outside the cluster are sources of interference. Intuitively, a larger cluster reduces intercell interference but it also increases the overheads required to acquire the necessary CSI at the BSs.
It follows that determining the optimal cluster cardinality is a key step to assess the true benefits of BS coordination. This paper tackles such optimization for a particular coordination strategy.

B. Related Work
In toy setups where all the BSs can participate in the coordination, centralized schemes have been shown to yield sum spectral efficiencies that increase unboundedly with the transmit powers [5]- [8]. However, in large networks where the CSI-acquisition overheads and the channel uncertainty caused by fading selectivity prevent large cluster cardinalities, outof-cluster interference is inevitable and the spectral efficiency has been shown to be fundamentally bounded no matter how sophisticated the cooperation [9]- [13]. Nevertheless, small cooperation clusters not incurring too much overhead do provide performance improvements with respect to a noncooperative baseline.
Since, despite their regularity, deterministic grid models are remarkably unfriendly to analysis, most of the results on BS coordination for grid networks are simulation-based. Analytical results are available only for the simplest embodiments thereof, in particular for the so-called Wyner model where BSs and mobile users are located along a one-dimensional universe [13], [14].
Approaches based on stochastic geometry are rapidly gaining momentum because of their superior analytical tractability and because they happen to match well the heterogeneous nature of emerging networks [15]. Within this framework, the performance of BS coordination schemes with fixed cluster structures established a-priori has been studied [16]- [18].
Dynamic BS clustering is a way of forming coordinated BS sets based on users' locations and channel quality. Given the evidence (e.g., [19], [20], [22]) that dynamic clusters based on user locations and channel propagation features perform far better than their fixed counterparts, there is clear interest in extending the existing stochastic geometry analyses to such dynamic cooperation structures.

C. Contribution
We consider the downlink of a network whose topology is modeled through stochastic geometry. BS locations are modeled as a homogeneous Poisson point process (PPP) with the corresponding cells determined from a tessellation of the plane into Voroni regions. Users in each cell are randomly located and each one then defines its own cluster, i.e., the set of BSs it coordinates with, on the basis of such location. Under this  Fig. 1(a) shows an instance of the first-order Voronoi tessellation on a two-dimensional plane. Each BS location d i for i ∈ {1, 2, . . . , 11} is the center of a cell V 1 (d i ). Fig. 1(b) illustrates the second-order Voronoi tessellation associated with the same BS locations in the left figure. Users in the region V 2 (d 1 , d 2 ) choose to connect with BS 1 and BS 2.
dynamic BS clustering policy, the set of users that share the same BS cluster are served through coordinated beamforming, a method that seeks to have each user communicate with one of the BSs in its cluster with no interference from all other in-cluster BSs. Our contribution is to characterize the performance of such dynamic coordinated beamforming. We derive analytical expressions for the signal-tointerference ratio (SIR) distributions and the ergodic spectral efficiency in terms of system parameters, chiefly the pathloss exponent, the number of antennas per BS, the cluster cardinality, and the pilot overhead. We obtain these analytical results for users with specific in-cluster relative locations and, by marginalizing over such locations, for the typical user. Utilizing this latter result, we then characterize the benefits of coordination in terms of the net ergodic spectral efficiency, incorporating the pilot overheads required for coordinated beamforming. From this, we obtain the optimal cluster cardinality for the typical user. Our finding is that coordinated beamforming is most beneficial to users that are in the outer part of their cells yet in the inner part of their coordination cluster, and that the optimal cluster cardinality for the typical user is small and scales with the fading coherence Through simulation, the accuracy of the derived SIR distributions is verified.
The remainder of the paper is organized as follows. Section II describes the proposed models as well as the performance metrics for the considered coordinated beamforming scheme. In Section III, analytical expressions for the SIR distribution are derived for specific relative in-cluster locations. The SIR distribution for the typical user is derived in Section IV, and then utilized in Section V to analyze the optimal cluster cardinality. Section VI concludes the paper.

A. Network Model
We consider a cellular network consisting of BSs, each equipped with N t antennas. The locations of these BS are established according to a homogeneous PPP, Φ = {d k , k ∈ N}, on the plane R 2 . This PPP has density λ. By tessellating the plane into Voronoi regions around each BS, we can define cells in the traditional manner. We further consider a population of single-antenna users distributed according to another homogeneous PPP, Φ U = {u k , k ∈ N}, which has density λ U and is independent of Φ.

B. Dynamic Clustering Model
The dynamic clustering technique that we analyze relies on the policy of having each user coordinate with the K closest BSs, where K ≤ N t is the cardinality of the coordination clusters. With this clustering policy, the set of users' locations that can be served by their K nearest BSs is formally defined by the notion of Kth-order Voronoi tessellation. Denoted by V K (d 1 , . . . , d K ), the Kth-order Voronoi cell associated with K distinct points d 1 , . . . , d K is the set of points closer to d 1 , . . . , d K than to any other point of Φ, i.e., where ∀d j ∈ Φ/{d 1 , . . . , d K }. Taking K = 2 as an example, Fig. 1 depicts the second-order Voroni tessellation with the corresponding Voroni cells for the proposed dynamic clustering method. Users in the region V 2 (d 1 , d 2 ) connect to the cluster formed by BSs at d 1 and d 2 . Further, the area of the Kth-order Voronoi cell is nonzero with probability one. Then, provided that the user density is much higher than the BSs density, i.e., λ U λ, there will be (with high probability) at least K users in V K (d 1 , . . . , d K ) choosing to connect with the K BSs at d 1 , . . . , d K . Each BS serves one user per timefrequency resource, for a total of K users per cluster, with the remaining users in the network accommodated in different signaling resources.
For later use, we also introduce a geometric parameter δ 1 = d1 2 d K 2 defined as the distance to the closest BS normalized by the distance to the furthest BS in the cluster. This parameter plays an important role in interpreting the SIR distribution and the ergodic spectral efficiency for any specific in-cluster geometry, i.e, for any specific user locations within the cluster. Specifically, a smaller δ 1 implies a larger protection area and vice versa, with this area being the minimum that is sure to be free of out-of-cluster interfering BSs. For example, when two BSs at d 1 and d 2 serve two users in V 2 (d 1 , d 2 ) through coordinated beamforming, the user associated with the BS in d 1 has a protection area A = π( d 2

C. Signal Model
Under the premise of separate encoding at each BS, the kth BS sends an information symbol s k (intended for the kth user) through a linear beamforming vector v k = [v 1 k , v 2 k , . . . , v Nt k ] T with unit norm, v k 2 = 1, k ∈ {1, 2, . . . , K}. Without loss of generality, let us focus on a user located at the origin. The observation at this user is where h 1,k = [h 1 1,k , h 2 1,k , . . . , h Nt 1,k ] ∈ C 1×Nt represents the downlink channel between the kth BS and the user, with entries that are independent and identically distributed (IID) complex Gaussian random variables having zero mean and unit variance, i.e., CN (0, 1). The channels vary over time in an IID block-faded fashion. Further, β represents the pathloss exponent and z 1 denotes the additive Gaussian noise, z 1 ∼ CN (0, σ 2 ). The transmit power at each BS satisfies E |s k | 2 ≤ P .
Each user learns the downlink channels from the K BSs within its cluster by means of orthogonal pilot symbols and then conveys this information back to the BSs via error-free feedback links. From this CSI, the coordinated beamforming scheme constructs the beamforming vectors v k , k ∈ {1, . . . , K}, that nullify intra-cluster interference while maximizing the desired signal strength for the K users in the cluster. Thus, the k-th BS selects beamforming vector v k solving which always exists when N t ≥ K. The corresponding instantaneous SIR for the user 1 at the origin is where is the aggregate out-of-cluster interference power. This instantaneous SIR in (4) involves multiple levels of randomness: 1) The randomness associated with the user location relative to its serving BS; this is incorporated through d 1 .
Equivalently, and more conveniently to the analysis that follows later, it can be incorporated through d K and 2) The randomness associated with the user location relative to the interfering BSs; this is incorporated through d k , k > K.

3) The randomness associated with the desired link fading;
this is incorporated through h 1,1 . 4) The randomness associated with the interference fading; this is incorporated through h 1,k , k > K.

D. Performance Metrics
The CCDF of the instantaneous SIR is characterized at two different levels, with the absolute dimensions of the network abstracted out.
1) Specific Relative Cluster Geometry: First, we characterize the CCDF of the SIR for some given relative distances {δ 1 , . . . , δ K }, but with the absolute distances {d 1 , . . . , d k } and the out-of-cluster interference I K marginalized over. Since the signals received from BSs k = 2, . . . , K do not contribute interference by virtue of (3), it suffices to condition on δ 1 and the ensuing conditional CCDF is where the expectation over d K and I K , characterized in Section III, effect the marginalization. This conditional CCDF does not correspond to the distribution of the SIR experience by any actual user in the system, but it is representative of the average behavior in all possible cluster geometries that share a particular δ 1 .
2) Average Cluster Geometry: By further marginalizing over δ 1 , we obtain the SIR distribution averaged over all possible geometries, which is less informative than the one in (7). In particular, this fully marginalized distribution does not allow discriminating between situations that are either favorable or adverse to coordinated beamforming, but it does serve as a stepping stone towards the computation of average quantities. The fully marginalized CCDF of the SIR is where the expectation is now also over δ 1 in addition to d K and I K . This distribution will be used to characterize the performance of the typical user in Secs. IV and V.

III. SPECIFIC RELATIVE CLUSTER GEOMETRY
In this section, we characterize the conditional CCDF in (7) in terms of K, N t and β.

A. General Characterization
We begin by providing a general characterization in integral form.
Theorem 1. For a given δ 1 , sĨr denotes the Laplace transform ofĨ r , which is given in (57) in Appendix B, and the expectation is over d K = r, distributed as per Lemma 3.
Proof: See Appendix A. Although general and exact, the form given in Theorem 1 is rather unwieldy, motivating the interest in more compact characterizations. Still in full generality, we next provide closed-form upper and lower bounds to the distribution.
Proof: See Appendix B. The upper and lower bounds coincide when N t = K, implying that for this most important case we obtain the exact CCDF.

B. Special Cases
To shed further light on the significance of the expressions in Thm. 2 and Cor. 1, it is instructive to consider certain special cases.

1) Noncoordinated Network:
The most basic special case is the one where there is no coordinated beamforming, i.e., where K = 1 (and δ 1 = 1 with the conditioning thereupon immaterial). By setting N t = 1 we then recover the CCDF of the SIR given in [15], namely which Thm. 2 therefore generalizes. For this special case, the derived expressions are useful to characterize the benefits of having N t antennas. As illustrated in Fig. 2, the upper bound tightly matches the exact CCDF over the entire range of SIRs of interest and for distinct values of N t , i.e., With the coordinated beamforming activated and N t = K, the behavior is characterized by the simple expression in Cor. 1. Particularized to β = 4, which is a standard value in terrestrial outdoor wireless systems, D(·, ·) reduces to which, plugged into (15), yields This simple CCDF facilitates gauging different scenarios as indicated earlier: δ 1 1 corresponds to users well isolated from out-of-cluster interference (i.e., with a large protection area) whilst δ 1 ≈ 1 corresponds to users susceptible to strong out-of-cluster interference (small protection area). Specifically, as illustrated in Fig. 3, when K = N t = 2 and β = 4, a user with δ 1 = 0.5 has a better SIR distribution-and thus higher benefits from coordinated beamforming-than a user with δ 1 = 0.8. A similar observation can be made for δ 1 = 1/4  and δ 1 = 1/3 when K = N t = β = 4. Further, we compare the SIR distributions derived for the given specific in-cluster geometry δ 1 ∈ {0.5, 0.8} against those without coordination, i.e., for K = 1 and N t = 2 (and hence for δ 1 = 1). As illustrated in Fig. 3, on the one hand the user with a small protection area (δ 1 = 0.8) has a worse SIR distribution than that without BS coordination for all range of γ and, on the other hand, the user with a larger protection area (δ = 0.5) exhibits better SIR distributional results up to γ = 16 dB. Agreeing with intuition, if a small protection area is created by coordinated beamforming, it might be better to use the two antennas for maximizing the desired signal power instead of canceling the nearest interferer. Conversely, in situations with a large protection area, canceling the nearest interference yields more benefits than maximizing the desired signal power.
The following example further illustrates the benefits of coordinated beamforming in terms of ergodic spectral efficiency for different relative locations, δ 1 .
Example 1: For N t = K = 2 and β = 4, the ergodic spectral efficiency (in bits/s/Hz) is where in (20) integration by parts was applied. The values for different relative locations δ 1 , listed in Table I, reveal that the most substantial gains of coordinated beamforming are obtained for small δ 1 . This is because users with large δ 1 remain subject to strong interference from out-of-cluster BSs.

IV. AVERAGE CLUSTER GEOMETRY
This section is devoted to the characterization of the CCDF in (9). Specifically, tight lower and upper bounds are derived as a function of K, N t and β.

A. Upper and Lower Bounds
The following Lemma provides the probability density function (PDF) of δ 1 = d1 d K induced by the underlying PPP. Lemma 1 (PPP distance ratio). Let d 1 and d K denote the distances from the origin to the first and the Kth BSs. The PDF of δ 1 = d1 d K is given by Proof: See Appendix C. Applying the above lemma to (12) and (13) we can readily write for which analytical approximations are derived next.

B. Bound Approximations
As shown in App. D, the integrals within (23) and (24) which, for the most relevant case where β = 4 and N t = K, coincide yielding The simple expression in (28) clearly shows how coordination with the K nearest BSs improves the CCDF of the SIR with K because arccot(x) is a decreasing function of x. Furthermore, this approximation recovers the exact CCDF of the SIR for K = 1.
To validate (26) and (27), we compare them with simulation results in Fig. 4. The agreement is excellent for the various  values of K considered and the entire range of SIRs of interest. As K increases, the SIR improves because the out-of-cluster interference abates, but that would come at the expense of further overheads. This motivates the optimization of K with the overhead incorporated, a problem that is tackled in Section V.

C. Low-SIR Analysis
To wrap up our characterization of the SIR distribution under coordinated beamforming, we specialize it to the low-SIR regime, which is of particular relevance for the purposes of establishing communication outages in quasi-static communication settings [29]. Proposition 1. The unconditioned CCDF of the SIR for K = N t expands as Proof: See Appendix E. For β = 4, the expansion in (29) further simplifies into which evidences that, by allowing coordinated beamforming from the K nearest BSs, the outage probability in a quasi-static setting would decrease linearly with K.

V. OPTIMAL CLUSTER CARDINALITY
Having characterized the SIR distributions of a user at a specific relative location and of the typical user, in this section we establish the corresponding optimal (in terms of ergodic spectral efficiency) cluster cardinalities. To compute the effective ergodic spectral efficiency, we incorporate pilot overheads into the formulation so as to account for the cost of acquiring the CSI required for the coordinated beamforming.
Although we have ignored background noise in the analysis of SIR distributions, we take it into account for the purpose of pilot transmissions with out-of-cluster interference, as these take place in an orthogonal fashion among users within a cluster. For a given fading coherence L b (in symbols), the pilot overhead is where L p (K, N t , SINR) denotes the number of symbols reserved for pilots. This number varies with K and N t , and also with the pilot-transmission SINR, which can be obtained from a parametric estimation method. We henceforth model the number of pilot symbols as L p (K, N t , SINR) = ηKN t where η ≥ 1 is an SINR-dependent parameter that signifies the number of pilots per transmit antenna and fading coherence interval. For instance, utilizing the result in [28], we can explicitly express η as a function of the channel estimation minimum mean square error (MMSE) and the average SINR of the pilot signals, namely Hence, the number of pliots η approaches 1 for SINR → ∞ .
With the SIR distributions obtained in Secs. III and IV, we compute the optimal K for different situations.

A. Optimal Cluster Cardinality for a Specific Relative Cluster Geometry
Assuming that perfect CSI is gathered from the pilot observations at the receiver and complex Gaussian codebooks are used, the ergodic spectral efficiency (in bits/s/Hz) for a specific relative cluster geometry is from which the cluster cardinality K that optimizes the effective spectral efficiency is obtained as the solution of the integer optimization Note that (35) provides the optimal cluster size for a given in-cluster geometry, with such geometry determined by δ 1 , which is itself a function of K. The dependence of δ 1 with K can be seen from its distribution, shown in Lemma 1. From this, we can compute the mean value of δ 1 as K , which decreases as K increases. In other words, adding more BSs to the cluster implies that d K grows and, consequently, that δ 1 diminishes. This dependence is incorporated in the maximization problem by setting δ 1 = (c/K) 1/2 with c being a constant value which determines the in-cluster geometry independently of K. Having this dependence in mind, Fig. 5 shows the ergodic spectral efficiency as a function of the cluster size when the user's relative location δ 1 is consistently scaled with K as we control the initial size of the protection area, which subsequently scales with K. Therefore, as depicted in Fig.  5, if the user has initially a large protection area (small c), the benefit of adding more BSs to the cluster does not overcome the overheads and the optimal cluster size remains small.

B. Optimal Cluster Cardinality for an Average Cluster Geometry
The effective spectral efficiency averaged over δ 1 is from which the optimal cluster cardinality K for an average cluster geometry is obtained as the solution of the integer optimization whose solution can be obtained by a numerical line search technique. Fig. 6 depicts the effective spectral efficiency as a function of K for different ratios L b /η, with β = 4 and N t = K. Notice how the optimum cluster cardinality increases with the channel coherence relative to the pilot cost, ranging from K = 2 when L b /η = 20 to K = 5 when L b /η = 200.
For N t = K = 4, the effective average spectral efficiencies for different fading coherences are summarized in Table II. Recall that these spectral efficiencies are obtained for the coordinated beamforming method in (3), which cancels the K − 1 nearest interference signals while maximizing the desired signal power with the remaining N t − K degrees of freedom. As shown in the first row of Table II, even without the Cluster size (K) Ergodic Spectral Efficiency Optimal Cluster Size pilot overheads accounted for, the ergodic spectral efficiency does not necessarily increase with the cluster size K because, once the main interferers have been canceled, beamforming power gain can be more beneficial than the cancellation of additional sources of interference. When the pilot overhead is considered, the optimal cluster cardinality with a short fading coherence (L b /η = 20) is K = 1, implying that the cost of the coordination is in this case higher than the return. Alternatively, for a relatively long coherence (L b /η = 200) the optimal cluster cardinality is K = 2 with a 23.8% gain in average spectral efficiency relative to the K = 1 baseline. Altogether, except for very short fading coherence, the optimal ergodic spectral efficiency for N t = 4 is attained with clusters of size K = 2 so that the strongest interference signal is canceled and the remaining degrees of freedom (provided by the 2 extra antennas) are put towards maximizing the desired signal strength.

VI. COMPARISONS
In this section, we compare the SIR distributions derived under our stochastic geometry model with the corresponding results-obtained through simulation-for a deterministic grid model. Since grid and PPP stochastic geometry models correspond to optimistic and pessimistic scenarios of real BS deployments as argued in [15], this comparison can be seen to convey upper and lower bounds to the actual benefits of dynamic coordinated beamforming.
In a deterministic square network, the BSs are arranged into a periodic square lattice on a plane. For simulation purposes (cf. Fig. 7), we consider 36 BSs located at regular grid points and drop a user uniformly within the highlighted square. Since we are considering dynamic clustering, in each realization the user selects its K nearest BSs for coordination and the rest of the BSs constitute sources of interference. Without loss of generality, we can again index the BSs in increasing distance  Fig. 7. A snapshot of the grid model for K = 2. A user is uniformly located within the square region highlighted in the center, and it selects its two nearest BSs for coordination. In this snapshot, these BSs are the ones located at (0, 0) and (0, 500).
from the user and express the SIR as where H k and d k denote fading coefficient and distance from the kth nearest BS to the user, respectively. In particular, H 1 has a chi-squared distribution with 2(N t − K + 1) degrees of freedom. Meanwhile, the fading coefficients of all the interfering links, H K+1 , H K+2 , . . . , H 36 , are exponential with unit mean. Then, the CCDF of the SIR is where the expectation is over d 1 and { d k , H k } for k ∈ {K + 1, . . . , 36}. Fig. 8 shows the CCDFs for the two different models in which the CCDF of the grid model is obtained by Montecarlo simulation. As one would expect, the CCDF of the SIR in the deterministic grid model is somewhat more favorable than in the stochastic geometry model because, under PPP, the nearest interferer's location can be arbitrarily close to the in-cluster BSs. Nevertheless, the shapes of the CCDFs in the two models are analogous, and they allow us to gauge the potential gains by coordinated beamforming in actual deployments.

VII. CONCLUSION
In this paper, we have characterized the performance of coordinated beamforming with dynamic BS clusters. Capitalizing on the tools of stochastic geometry, we have derived SIR distributions in terms of the number of BSs per cluster, the number of antennas per BS and the pathloss exponent. From these distributions, we have obtained analytical expressions for the effective ergodic spectral efficiency and optimized the cluster cardinality as function of the fading coherence. Our key finding is that coordinated beamforming is most beneficial to users in the inner part of the coordination clusters as opposed to users near the edges, where the mitigation of in-cluster interference makes less of a difference because of the strong out-of cluster interference component. Further, we have found that the optimal cluster cardinality for the typical user is small and that it scales with the fading coherence.

APPENDIX A PROOF OF THEOREM 1
The proof relies on two Lemmas, reproduced next for the sake of completeness.

Lemma 3.
[PPP distance distribution] Given a PPP in a plane with intensity λ, the distribution of the distance d K between a typical user and its Kth serving BS is the generalized Gamma distribution where Γ(K) = ∞ 0 e −x x K−1 dx is the Gamma function. Proof: See [23].

Let us denote by
the aggregate interference power from all the out-of-cluster BSs, conditioned on the Kth in-cluster BS location satisfying d K = r where B(0, r) denotes a circle centered at the origin with radius r. Then, the conditional CCDF of the SIR is given by whereĨ r = δ β 1 γI r and the expectation is over the distribution of r. From Lemma 2, since where the inner expectation is over the distribution ofĨ r . From the derivative property of the Laplace transform, which is E X m e −sX = (−1) m d m L X (s) ds m , we finally obtain which completes the proof.

APPENDIX B PROOF OF THEOREM 2
To prove this result, the following lemma is needed. [26], [27]] If H 1 is chi-squared with 2M degrees of freedom, then the CDF F H1 (γ) = P[H 1 < γ] is upper and lower bounded by

Lemma 4. [Alzer's Inequalty
where F H1 (γ) = Now we are ready to prove Theorem 2. We focus on proving the upper bound therein because the lower bound is directly obtained from the former by setting κ = 1.
Conditioned on the Kth BS being located at distance r from the user, the conditional CCDF of the SIR can be written as Applying Lemma 4 and the binomial expansion, (49) from which the conditional CCDF of the SIR in (48) is upper bounded as where the expectation is over the distribution of I r and M = N t − K + 1.
Unconditioning with respect to the location of the Kth BS, with inner and outer expectations over the distributions of I r and r, respectively. To evaluate these expectations, we first compute the conditional Laplace transform of I r . Conditioned on d K = r, such Laplace transform is where (55) follows from the independence of d k and H k , (56) holds because H k is exponentially distributed and unit mean for k ∈ {K + 1, K + 2, . . .}, and (57) follows from the probability generating functional of the PPP. Evaluating this conditional Laplace transform at s = κ δ β 1 γr β , where (58) follows from the variable change while (59) holds because To uncondition the foregoing Laplace transform, we marginalize it with respect to r using the distribution in Lemma 3.
With that, the Laplace transform of the aggregate out-of-cluster interference power emerges as where (63) follows from the variable change whereas (64) follows from the definition of the Gamma function. By plugging (64) into (52), we finally obtain and, by setting κ = 1, we further have the lower bound in (13), which completes the proof.
APPENDIX C PROOF OF LEMMA 1 We start by computing the joint PDF of d 1 and d K . Consider the four nonoverlapping areas A 1 = B(0, r 1 ), A 2 = B(0, r 1 + dr 1 )/B(0, r 1 ), A 3 = B(0, r K )/B(0, r 1 + dr 1 ), and A 4 = B(0, r K + dr K )/B(0, r K ). By definition of the PPP, the joint probability that d 1 and d K belong to the two (thin ring) areas A 2 and A 4 , respectively, is given by the product of the four independent probability events as P [ d 1 ∈ A 2 , d K ∈ A 4 ] = P 1 P 2 P 3 P 4 if r 1 ≤ r K 0 otherwise (67) where P 1 = P[No points in A 1 ] = e −λπr 2 1 P 2 = P[One point in A 2 ] = λπ2r 1 dr 1 e −λπ2r1dr1 From the limits of the joint probability in (67), the joint PDF of d 1 and d K emerges as Utilizing the joint PDF of d 1 and d K , we derive the CDF of δ 1 = d1 d K as where 0 ≤ x ≤ 1. Therefore, the PDF of δ 1 is given by APPENDIX D PROOF OF EQ. (25) Recall that D(γδ β 1 , β) =γ 2/β δ 2 1 ∞ 1/(γ 2/β δ 2 1 ) where δ 1 is distributed as per Lemma 1. We approximate the integral above as a constant value that captures the effect of the randomness induced by δ 1 , where the expectation is over δ 1 . From (76), D(γδ β 1 , β) Plugging (77) into the left side of (25) and marginalizing with respect to δ 1 , Recall that the upper bound in Thm. 2 is exact for K = N t . The outage probability P out (K, K, β, δ 1 ; γ) = 1 − F c SIR (K, K, β, δ 1 ; γ) expands at γ = 0 as P out (K, K, β, δ 1 ; γ) = 1 − F c SIR (K, K, β, δ 1 ; γ) (79) where we have invoked the series expansion of the Gauss hypergeometric function at γ = 0, as well as By dropping the second order error term and marginalizing with respect to δ 1 , we obtain the average outage probability 1 − F c SIR (K, K, β; γ) which completes the proof.