Laplace Functional Ordering of Point Processes in Large-Scale Wireless Networks

Hindawi Wireless Communications and Mobile Computing Volume 2018, Article ID 4307136, 14 pages https://doi.org/10.1155/2018/4307136

Research Article Laplace Functional Ordering of Point Processes in Large-Scale Wireless Networks

Junghoon Lee 1 and Cihan TepedelenlioLlu 2

1 Electronics and Telecommunications Research Institute (ETRI), Daejeon, Republic of Korea 2School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA

Correspondence should be addressed to Cihan Tepedelenlio˘glu; [email protected]

Received 11 June 2018; Revised 26 September 2018; Accepted 18 October 2018; Published 1 November 2018

Academic Editor: Yu Chen

Copyright © 2018 Junghoon Lee and Cihan Tepedelenlio˘glu. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Stochastic orders on point processes are partial orders which capture notions like being larger or more variable. Laplace functional ordering of point processes is a useful stochastic order for comparing spatial deployments of wireless networks. It is shown that the ordering of point processes is preserved under independent operations such as marking, thinning, clustering, superposition, and random translation. Laplace functional ordering can be used to establish comparisons of several performance metrics such as coverage probability, achievable rate, and resource allocation even when closed form expressions of such metrics are unavailable. Applications in several network scenarios are also provided where tradeofs between coverage and interference as well as fairness and peakyness are studied. Monte-Carlo simulations are used to supplement our analytical results.

1. Introduction Recently stochastic ordering theory has been used for performance comparison in wireless networks which are Point processes have been used to describe spatial distri- modeled as point processes [16–20]. Directionally convex bution of nodes in wireless networks. Examples include (DCX) ordering of point processes and its integral shot randomly distributed nodes in wireless sensor networks noise felds have been studied in [16]. Te work has been or ad hoc networks [1–4] and the spatial distributions for extended to the clustering comparison of point processes base stations and mobile users in cellular networks [5– with various weaker tools, including void probabilities and 10]. In the case of cognitive radio networks, locations of moment measures, than DCX ordering in [17]. In [18], usual primary and secondary users have been modeled as point stochastic ordering of random variables capturing carrier-to- processes [11–14]. Random translations of point processes interference ratio has been established in cellular systems. have been used for modeling of mobility of networks Ordering results for coverage probability and per user rate in [15]. Stationary Poisson processes provide a tractable have been shown in multiantenna heterogeneous cellular net- framework, but sufer from notorious modeling issues in works [19]. In [20], Laplace functional (LF) ordering of point matching real network distributions. Stochastic ordering of processes has been introduced and used to study interference point processes provides an ideal framework for compar- distributions in wireless networks. Several examples of the LF ing two deployment/usage scenarios even in cases where ordering of specifc point processes have been also introduced the performance metrics cannot be computed in closed in [20], including stationary Poisson, mixed Poisson, Poisson form. Tese partial orders capture intuitive notions like one cluster, and binomial point processes. point process being more dense or more variable. Existing While limited examples of the LF ordering have been works on point process modeling for wireless networks provided in [20], in this paper we apply the LF ordering have paid little attention to how two intractable scenarios concept to several general classes of point processes such can be nevertheless compared to aid in system optimiza- as Cox, homogeneous independent cluster, perturbed lattice, tion. and mixed binomial point processes which have been used 2 Wireless Communications and Mobile Computing to describe distributed nodes of wireless systems in the liter- Ten � is said to be smaller than � in the Laplace transform ature. We also investigate the preservation properties of the (LT) order (denoted by �≤Lt �). For example, when � and � LF ordering of point processes with respect to independent are the instantaneous SNR distributions of a fading channel, operations such as marking, thinning, random translation, (2) can be interpreted as a comparison of average bit error and superposition. We prove that the LF ordering of original rates for exponentially decaying instantaneous error rates point processes still holds afer applying these operations on (as in the case for diferential-PSK (DPSK) modulation and the point processes. To the best of our knowledge, there is Chernof bounds for other modulations) [23]. Te LT order no study on LF ordering of general classes of point processes �≤Lt � is equivalent to and their preservation properties in the literature. Trough the preservation properties of LF ordering with respect to E [� (�)] ≥ E [� (�)] , (3) independent operations on a point process such as marking, thinning, random translation, and superposition which is for all completely monotonic (c.m.) functions �(⋅) [22, pp. 96]. done in our manuscript, we can consider several efects of real By defnition, the derivatives of a c.m. function �(�) alternate � � � systems such as propagation efects over wireless channels, in sign: (−1) d �(�)/d� ≥0,for� = 0,1,2,...,and�≥ multiple access schemes, heterogeneous network scenarios, 0. An equivalent defnition is that c.m. functions are positive and mobile networks and compare performances without mixtures of decaying exponentials [22]. A similar result to (3) having to obtain closed form results for a wide range of with a reversal in the inequality states that performance metrics such as coverage probability, achievable rate, and resource allocation under these real system efects. �≤ �⇐⇒E [� (�)] ≤ E [� (�)] , In addition to the performance comparison, the stochastic Lt (4) ordering of point processes provides insights into the system design and evaluation such as network deployment and user for all �(⋅) that have a completely monotonic derivative selection schemes as we summarize in this paper. (c.m.d.). Finally, note that �≤st ��⇒�≤Lt �.Tiscan Te paper is organized as follows: In Section 2, we be shown by invoking the fact that �≤st � is equivalent to introduce mathematical preliminaries. Section 3 introduces E[�(�)] ≤ E[�(�)] whenever �(⋅) is an increasing function [22] ordering of point processes. In Section 4, we show the and that c.m.d. functions in (4) are increasing. preservation properties of LF ordering. Sections 5.1 and 5.2 introduce applications of stochastic ordering of point pro- 2.2. Point Processes and Random Measures. Point processes cesses in wireless networks. We also elaborate the qualitative have been used to model large-scale networks [1, 2, 13, 24– insights of the stochastic ordering of point processes on 29]. Since wireless nodes are usually not colocated, our focus the system design and evaluation in Section 5.3. Section 6 is on simple point processes, where only one point can exist at presents simulations to corroborate our claims. Finally, the a given location. In addition, we assume the point processes paper is summarized in Section 7. are locally fnite; i.e., there are fnitely many points in any bounded set. Unlike [20], stationary and isotropic properties 2. Mathematical Preliminaries are not necessary in this paper. In what follows, we introduce some fundamental notions that will be useful. 2.1. Stochastic Ordering of Random Variables. Before intro- ducing ordering of point processes, we briefy review some common stochastic orders between random variables, which 2.2.1. Campbell’sTeorem. It is ofen necessary to evaluate the can be found in [21, 22]. expected sum of a function evaluated at the point process Φ. Campbell’s theorem helps in evaluating such expectations. For any nonnegative measurable function � which runs over 2.1.1. Usual Stochastic Ordering. Let � and � be two random � the set U of all nonnegative functions on R , variables (RVs) such that � (�>�) ≤�(�>�) ,−∞<�<∞. (1) E [∑� (�)]=∫ � (�) Λ (d�) . (5) � � � Ten is said to be smaller than in the usual stochastic �∈Φ R order (denoted by �≤st �). Roughly speaking, (1) says that � is less likely than � to take on large values. To see the inter- Te intensity measure Λ of Φ in (5) is a characteristic pretation of this in the context of wireless communications, � � analogous to the mean of a real-valued random variable and when and are distributions of instantaneous SNRs due to Λ(�) = E[Φ(�)] �⊂R� fading, (1) is a comparison of outage probabilities. Since �, � defned as for bounded subsets . + Λ(�) � Φ are positive in this case, �∈R is sufcient in (1). Tus, is the mean number of points in .If is stationary, then the intensity measure simplifes as Λ(�) = �|�| � � � for some nonnegative constant , which is called the 2.1.2. Laplace Transform Ordering. Let and be two intensity of Φ,where|�| denotes the � dimensional volume nonnegative RVs such that of �. For stationary point processes, the right side in (5) is L (�) = E [ (−��)]≥E [ (−��)]=L (�) �∫ �(�) � � exp exp � equal to R� d . Terefore, any two stationary point (2) processes with the same intensity lead to equal average sum �>0. for of a function (when the mean value exists). Wireless Communications and Mobile Computing 3

ArandommeasureΨ is a function from Borel sets in 2.2.3. Voronoi Cell and Tessellation. Te Voronoi cell �(�) of � + � R to random variables in R .TeLaplace functional � of apoint� of a general point process Φ⊂R consists of those � random measure Ψ is defned by the following formula locations of R whose distance to � is not greater than their Φ −∫ � �(�)Ψ(d�) distance to any other point in ; i.e., �Ψ (�) ﬂ E [� R ]. (6) � � � � � � (�) ﬂ {� ∈ R : ��−�� ≤ ��−�� ∀� ∈ Φ \ �} . (10) Te Laplace functional completely characterizes the distri- � � � � bution of the random measure [25]. A point process Φ is Ψ Te Voronoi tessellation (or Voronoi diagram) is decompo- a special case of a random measure where the measure sition of the space into the Voronoi cells of a general point takes on values in the nonnegative integer random variables. process. InthecaseoftheLaplacefunctionalofapointprocess, ∫ �(�)Φ( �) ∑ �(�) R� d can be written as �∈Φ in (6). As an important example, the Laplace functional � of Poisson point 3. Ordering of General Classes of process of intensity measure Λ is Point Processes

�Φ (�) = exp {− ∫ [1−exp (−� (�))] Λ (d�)} . (7) Te examples for LF orderings of some specifc point pro- R� cesses have been provided in [20]. In this section, we intro- If the Poisson point process is stationary, the Laplace func- duce the LF ordering of general classes of point processes. tional simplifes with Λ(d�) = �d�. 3.1. Cox Processes. A generalization of the Poisson process is 2.2.2. Laplace Functional Ordering. In this section, we intro- to allow for the intensity measure itself being random. Te duce the Laplace functional stochastic order between random resulting process is then Poisson conditional on the intensity measures which can also be used to order point processes. measure. Such processes are called doubly stochastic Poisson processes or Cox processes. Consider a random measure Ψ on � Defnition 1. Let Ψ1 and Ψ2 be two random measures such R . Assume that, for each realization Ψ=Λ, an independent that Poisson point process Φ of intensity measure Λ is given. Te −∫ �(�)Ψ (d�) −∫ �(�)Ψ (d�) Ψ � (�) = E [� R� 1 ]≥E [� R� 2 ] random measure is called the driving measure for a Cox Ψ1 (8) process. Te LF ordering of Cox processes depends on their =� (�) driving random measures. Ψ2 Teorem 2. Φ Φ where �(⋅) runs over the set U of all nonnegative functions Let 1 and 2 be two Cox processes with driving � random measures Ψ1 and Ψ2, respectively. If Ψ1 ≤Lf Ψ2,then on R .TenΨ1 is said to be smaller than Ψ2 in the Laplace Φ1 ≤Lf Φ2. functional (LF) order (denoted by Ψ1 ≤Lf Ψ2).

In this paper, we focus on the LF order of point processes Proof. Te proof is given in Appendix A. unless otherwise specifed. Note that the LT ordering in (2) is Te mixed Poisson process is a simple instance of a Cox for RVs, whereas the LF ordering in (8) is for point processes process, where the random measure Ψ is described by a or random measures. Tey can be connected in the following positive random constant � so that Ψ(�) = �|�|.Since way: the Laplace functional of the mixed Poisson process can be expressed as �Φ(�) = E�[exp{−� ∫ � [1 − exp(−�(�))]d�}], Φ1≤Lf Φ2 ⇐⇒ ∑ � (�) ≤Lt ∑ � (�) ,∀�∈U. R �∈Φ �∈Φ (9) ∫ [1 − (−�(�))] �≥0 1 2 using (7), and because R� exp d and the c.m. property of exp(−��), � ≥ 0, the LF ordering of mixed Hence, it is possible to think of LF ordering of point processes Poisson processes has the following relationship: if �1 ≤Lt �2, as the LT ordering of their aggregate processes. Intuitively, then Φ1 ≤Lf Φ2. the LF ordering of point processes can be interpreted as the LT ordering of their aggregate interferences. Te Laplace transform of interference is particularly convenient for the 3.2. Homogeneous Independent Cluster Processes. A general determination of outage probabilities under certain fading cluster process is generated by taking a parent point process distributions (e.g., Rayleigh) [30] and is also useful in and daughter point processes, one per parent, and translating performance comparison. Te LF ordering of point pro- the daughter processes to the position of their parent. Te cesses also can be translated into the ordering of coverage cluster process is then the union of all the daughter points. Φ ={�,� ,...} probabilities and spatial coverage which will be discussed Denote the parent point process by p 1 2 ,and �∈N ∪{∞} in detail in Section 5. Te probability generating functional let be the number of parent points. Further let {Φ }, � ∈ N is another mathematical tool for studying point processes � , be a family of fnite points sets, the untranslated [1]. Te probability generating functional will provide the clusters or daughter processes. Te cluster process is then the same results in our scenarios. However, we will focus on the union of the translated clusters: Laplace functional of point process in our study since it is � easy to relate the Laplace functional to interference metrics Φ ﬂ ⋃ Φ� +��. (11) as mentioned above. �=1 4 Wireless Communications and Mobile Computing

If the parent process is a lattice, the process is called a lattice From Teorem 4 and (13), the uniformly perturbed lattice cluster process.Analogously,iftheparentprocessisaPoisson processes with replicating points with nonnegative integer point process, the resulting process is a Poisson cluster process. valued distribution � and � will be the smallest and biggest If the parent process is stationary and the daughter LF ordered point processes, respectively, among uniformly processes {Φ�}� are fnite point sets which are independent perturbed lattice processes with the same average number of of each other, are independent of Φp,andhavethesame points. Te smallest LF ordered uniformly perturbed lattice distribution, the procedure is called homogeneous indepen- process exhibits clustering since some Voronoi cells have 2 dent clustering. In this case, only the statistics of one cluster points but other cells do not have any point. Tis observation need to be specifed, which is usually done by referring to is in line with the intuition that clustering diminishes point the representative cluster, denoted by Φ0 which is distributed processes in the LF order. the same as any Φ�,�∈N. In this class of point processes, Φ the LF ordering depends on the parent process p and the 3.4. Mixed Binomial Point Processes. In binomial point Φ representative process 0 as follows. processes, there are a total of fxed � points uniformly � distributed in a bounded set �∈R . Te density of the Teorem 3. Let Φ1 and Φ2 be two homogeneous independent process is given by �=�/|�|where |�| isthevolumeof�.If cluster processes having representative clusters Φ0 and Φ0 , 1 2 the number of points � is random, the point process is called respectively. Also, let Φp and Φp be the parent point processes 1 2 a mixed binomial point process. As an example, with Poisson of two homogeneous independent cluster processes Φ1 and Φ2, distributed �, the point process is called a fnite Poisson respectively. If Φp ≤Lf Φp and Φ0 ≤Lf Φ0 ,thenΦ1 ≤Lf Φ2. 1 2 1 2 point process. Te intensity measure of mixed binomial point Λ(�) = �|�| Proof. Te proof is given in Appendix B. processes is . In these point processes, one can show the following.

3.3. Perturbed Lattice Processes with Replicating Points. Lat- Teorem 5. Let Φ1 and Φ2 be two mixed binomial point tices are deterministic point processes defned as process with nonnegative integer valued random distribution � �1 and �2, respectively. If �1 ≤Lt �2,thenΦ1 ≤Lf Φ2. L ﬂ {� ∈ Z : G�} , (12)

�×� Proof. Te proof is given in Appendix D. where G ∈ R is a matrix with det G =0̸ , the so-called generator matrix. Te volume of each Voronoi cell is �= Similar to Teorem 4, Teorem 5 enables LF ordering of |det G| and the intensity of the lattice is �=1/�[31]. Te two point processes whenever an associated discrete random perturbed lattice process is a lattice cluster process. Denote variable is LT ordered. the lattice point process by L ={�1,�2,...}, and let �∈ N ∪{∞} {Φ }, � ∈ be the number of lattice points. Further let � 4. Preservation of Stochastic Ordering of N, be untranslated clusters. In each cluster, the number of daughter points are a random variable �, independent of Point Processes each other and identically distributed. Moreover, these points In what follows, we will show that the LF ordering between are distributed by some given spatial distribution. Te entire two point processes is preserved afer applying independent process is then the union of the translated clusters as in operations on point processes such as marking, thinning, (11). If the replicating points are uniformly distributed in the random translation, and superposition of point processes. Voronoi cell of the original lattice, the resulting point process is a stationary point process and called a uniformly perturbed � 4.1. Marking. Consider the � dimensional Euclidean space lattice process. If, moreover, the numbers of replicas are � R , �≥1, as the state space of the point process. Consider Poisson random variables, the resulting process is a stationary ℓ Poissonpointprocess[32].Now,wecandefnethefollowing asecondspaceR , called the space of marks. A marked point ̃ � ℓ � ℓ LF ordering of such point processes. process Φ on R ×R (with points in R and marks in R )isa � locally fnite, random set of points on R ,withsomerandom Teorem 4. Φ Φ ℓ Let 1 and 2 be two uniformly perturbed lattice vector in R attached to each point. A marked point process processes with numbers of replicas being nonnegative integer is said to be independently marked if, given the locations of � � valued random variables 1 and 2, respectively, and with the the points in Φ, the marks are mutually independent random E[� ]=E[� ]=1 � ≤ � Φ ≤ Φ ℓ same mean 1 2 .If 1 Lt 2,then 1 Lf 2. vectors in R , and if the conditional distribution of the mark �� of a point �∈Φdepends only on the location of the point Proof. Te proof is given in Appendix C. � it is attached to.

Based on Teorem 5.A.21 in [21], the smallest and biggest � � Lemma 6. Let Φ1 and Φ2 be two point processes in R .Also LT ordered random variables can be defned as follows: Let ̃ ̃ be a random variable such that �{�=0}=1−�{�=2}=1/2 let Φ1 and Φ2 be independently marked point processes with ℓ and let � be a random variable degenerate at 1.Let� be a marks �� with identical distribution in R .IfΦ1 ≤Lf Φ2,then ̃ ̃ nonnegative random variable with mean 1.Ten Φ1 ≤Lf Φ2. �≤ �≤ �. Lt Lt (13) Proof. Te proof is given in Appendix E. Wireless Communications and Mobile Computing 5

4.2. Tinning. A thinning operation uses a rule to delete to each point, the random translation can be considered as points of a basic process Φ, thus yielding the thinned point the independent marking operation on a point process. process Φth, which can be considered as a subset of Φ. Te simplest thinning is �-thinning: each point of Φ has Φ Φ 1−� 4.4. Superposition. Let 1 and 2 be two point processes. probability of sufering deletion, and its deletion is Consider the union independent of locations and possible deletions of any other points of Φ. A natural generalization allows the retention Φ=Φ1 ∪Φ2. (14) probability � to depend on the location � of the point. A � deterministic function �(�) is given on R ,with0≤�(�)≤ Φ Φ 1 � Φ 1−�(�) Suppose that with probability one the point sets 1 and 2 do .Apoint in is deleted with probability and again not overlap. Te set-theoretic union then coincides with the its deletion is independent of locations and possible deletions superposition operation of general point process theory. Te of any other points. Te generalized operation is called �(�)- � superposition preserves the LF ordering of point processes as thinning. In a further generalization the function is itself follows. random. Formally, a random feld � ={0≤�(�)≤1: �∈R�} Φ is given which is independent of . A realization Lemma 9. Let Φ1,� and Φ2,�, � = 1,...,� be mutually � �th of the thinned process Φth is constructed by taking a independent point processes and Φ1 =⋃�=1 Φ1,� and Φ2 = realization � of Φ and applying �(�)-thinning to �,using � � ⋃�=1 Φ2,� be the superposition of point processes. If Φ1,� ≤Lf Φ2,� for �(�) asample{�(�) : � ∈ R } of the random feld �. for �=1,...,�,thenΦ1 ≤Lf Φ2. Given �(�) = �(�) and given Φ=�,theprobabilityof� in Φ Φ �(�) also belonging to th is .Aslongasanindependent Proof. Te proof is given in Appendix H. thinning operation regardless of �, �(�),and�(�) is applied on point processes, the LF ordering of the original pair of point processes is retained.

� 5. Applications to Wireless Networks Lemma 7. Let Φ1 and Φ2 be two point processes in R ,and let Φth,1 and Φth,2 be independently thinned point processes In the following discussion, we will consider the applications both with any identical independent thinning operation which of stochastic orders to wireless network systems. could be either �, �(�),or�(�)-thinning on both Φ1 and Φ2. Φ ≤ Φ Φ ≤ Φ If 1 Lf 2,then th,1 Lf th,2. 5.1. Cellular Networks. In this section, the comparisons of performance metrics will be derived based on the LF ordering Proof. Te proof is given in Appendix F. of point processes for spatial deployments of base stations (BSs) and mobile stations (MSs). Since the thinned point process Φth is a locally fnite � random set of points on R , with a binary random variable + 5.1.1. System Model. We consider the downlink cellular net- in R attached to each point, independent thinning can be work model consisting of BSs arranged according to some considered as the independent marking operation on a point point process ΦB in the Euclidean plane. For the deployment processasdiscussedintheprevioussection. of BSs, a deterministic network such as lattice points or stochastic network such as a Poisson point process may 4.3. Random Translation. In this section, the stochastic oper- be considered. Consider an independent collection of MSs, � ation that we consider is random translation. Each point located according to some point process ΦM which is inde- Φ in the realization of some initial point process is shifed pendent of ΦB. Figure 1 shows an example of cellular network independently of its neighbors through a random vector �� consisting of stationary Poisson point processes with diferent in R where {��}� are independent of each other and the intensities for BSs and MSs, respectively. For a traditional conditional distribution of a random vector �� of a point �∈ cellular network, assume that each user associates with the Φ depends only on the location of the point �.Teresulting closest BS, which would sufer the least path-loss during process is Φrt ﬂ {� + �� :�∈Φ}.Terandomtranslation wireless transmission. It is also assumed that the association preserves the LF ordering of point process as follows. between a BS and a MS is carried out in a large time scale

� compared to the coherence time of the channel. Te cell Lemma 8. Let Φ1 and Φ2 be two point processes in R , boundaries are defned through the Voronoi tessellation of and let Φrt,1 and Φrt,2 be the translated point processes with the BS process. Our goal is to compare performance metrics common distribution for the translation ��.IfΦ1 ≤Lf Φ2,then such as total cell coverage probability through stochastic Φrt,1 ≤Lf Φrt,2. ordering tools. Te spatial coverage of cellular networks is also compared based on the LF order of the BS point Proof. Te proof is given in Appendix G. processes. In order for the total cell coverage probability to be Similar to the independent thinning operation, since the compared, the signal to interference plus noise ratio (SINR) of Φ random translated point process rt is a locally fnite random auserat�∈ΦM should be quantifed. Te efective channel � � set of points on R ,withsomerandomvectorinR attached power between a user � and its associated typical BS �0 is 6 Wireless Communications and Mobile Computing

(＂,－) = (0.1,1) 10

−2

−4

−6

−8

−10 −10 −5 0 5 10

Figure 1: Illustration of a cellular network. Te triangles represent base stations which form a stationary Poisson point process ΦB with intensity �B =0.1and the dots represent users which also form a stationary Poisson point process ΦM with intensity �M =1.0.

(�) ℎS , which is a nonnegative RV.Te SINR with additive noise in the cell C0 which is associated with the BS �0.Bystudying 2 power � is given by spatial character of networks and investigating the spatial distributions of mobile users, we can compare system per- ℎ(�)� (‖�‖) formances using stochastic ordering approach without actual �� (�) = S , (15) �2 +�(�) system performance evaluation. In addition, the preservation properties of the LF order in Lemmas 6–9 guarantee that the + + where �(⋅) : R �→ R is the path-loss function which is a performance comparison results based on the LF ordering continuous, positive, nonincreasing function of the Euclidean of point processes are not changed with respect to any Φ Φ distance ‖�‖ from the user located at � to the typical BS �0. identical independent random operation on M or B such Te following is an example of a path-loss model [1, 26, 33, as marking, thinning, translation, and superposition. 34]: (a) Total Cell Coverage Probability. In multicast/broadcast −1 � (‖�‖) = (�+�‖�‖�) (16) scenarios, multiple users receive a common signal from their associated BS. Terefore, the probability that SINRs of all � for some �>0, �>�and �∈{0,1},where� is served users are greater than a minimum threshold is an called the path-loss exponent, � determines whether the path- important measure to ensure the signal reception quality of loss model belongs to a singular path-loss model (�=0) every user and it is called total cell coverage probability.Tis or a nonsingular path-loss model (�=1), and � is a metric can be ordered, if the underlying point processes are compensation parameter to keep the total received power LF ordered. normalized regardless of the values of path-loss exponent. �(�) Teorem 10. Let ΦB be an arbitrary fxed BS deployment. In (15), is the accumulated interference power at a user Φ Φ � Also, let M1 and M2 be two point processes for MS deploy- located at given by 1 1 2 2 ments, and let �1,...,�� ∈ΦM ∩ C0 and �1,...,�� ∈ (�) � � 1 1 2 � (�) =∑ℎ �(��−��) ΦM ∩ C0 be two user location sets belonging to the typical I � � (17) 2 �∈ΦB\{�0} Voronoi cell C0. �1 and �2 are random variables that are Φ Φ {ℎ(�)}, � ∈ (Φ ∪ functions of M1 and M2 , respectively. Let S M1 where ΦB denotes the set of all BSs which is modeled as a Φ )∩C (�) M2 0 in (15) be exponentially distributed independent RVs, point process and ℎI is a positive random variable capturing (�) and let {ℎ }, � ∈ ΦB in (17) be independent RVs that are also � �th I the (power) fading coefcient between a user and the {ℎ(�)} Φ Φ Φ ≤ Φ (�) independent of S , M1 ,and M2 .If M1 Lf M2 ,thenfor interfering BS. Moreover, ℎI are i.i.d. random variables and �>0 Φ Φ independent of M and B. 1 1 �(��(�1)≥�,...,��(�� )≥�) 1 (18) 5.1.2. Ordering of Performance Metrics in a Cellular Network. ≥�(��(�2)≥�,...,��(�2 )≥�) 1 �2 In the following discussion, we will introduce performance �(⋅) ℎ(�) ℎ(�) Φ Φ metrics involving the stochastic ordering of aggregate process where is over all RVs, S , I , M1 ,and M2 . Wireless Communications and Mobile Computing 7

Proof. Te proof is given in Appendix I.

Note that Teorem 10 holds for any arbitrary ΦB as long Primary as it is identical under both scenarios. Equation (18) shows transmitter(user) that LF ordering of user point processes implies ordering of the multivariate complementary cumulative distribution functions (CCDFs) of the SINRs.

(b) Network Spatial Coverage. Te network spatial coverage is Primary an important performance metric to design BS deployment receiver in cellular networks. We assume that BSs are distributed by a point process ΦB and each BS has a fxed radius of coverage �. Denote the random number of BSs covering a fxed location � by

�(�)= ∑ 1 {� ∈ � (�)}, � (19) �∈ΦB Active SUs Desired signal De-activated SUs Interference where ��(�) is a �-dimensional ball of radius � centered at the point �. Denote the probability generating function of Figure 2: Illustration of a cognitive network. the random number of BSs covering location � by �(�) = �(�) � E[� ], 0 ≤ � ≤ 1.Since0≤�≤1, � is a c.m. function with �.Notethat1−�(0)represents the probability whether the developing efcient methods for spectrum management and location � is covered by at least one BS from the defnition sharing. of the probability generation function. Tus, if ΦB ≤Lf ΦB , � (�) ≥ � (�) 1 2 then 1 2 from the property of LT ordering in (3) 5.2.1. System Model. Let us consider an underlay cognitive and consequently 1−�1(0) ≤ 1 − �2(0).Tismeansthe � radio network which contains a primary user (PU) and many probability that any arbitrary point � in R is covered by secondary users (SUs) with an average interference power Φ at least one BS with the cell deployment by B1 is always constraint ΓI. Te PU is located at the origin. Te � SUs are Φ � less than the probability with B2 . Due to random efects uniformly randomly located in a certain area �⊂R [35– of real systems such as shadowing, diferent transmission 37]. It is assumed that there is a BS to coordinate the SUs’ � power per each BS, and obstacles, the range of coverage can transmission. In order to satisfy the interference constraint, be a nonnegative random variable. Since the random range the BS selects only � active number of users out of � total � of coverage can be considered as independent marking users through user selection schemes and the selected active �=� in Section 4.1, the ordering of spatial coverage SUs are allowed to transmit their signals to the BS. User probabilities still holds from Lemma 6 under the assumption selection creates a thinned point process ΦSU,th for the active � that the random range of coverage is independent of the BS SUs which has uniformly distributed � points over the area Φ deployment B. Te ordering of spatial coverage probabilities �. Te resulting thinned point process can be considered also holds with random ellipse or square instead of a ball as a mixed binomial point process as in Section 3.4. Te � (�) � from Lemma 6. system model of the cognitive radio network is illustrated With performance metrics such as network spatial cover- in Figure 2. Under the system model, based on Teorem 5, age and network interference, our study can provide design one can design user selection schemes which guarantee the guidelines for network deployment to increase spatial cover- same average interference power from the active SUs to age of networks or provide less interference from networks. the PU and the same average sum rate for the active SUs. As an example, consider the efects of clustering as in Sec- On the other hand, the distribution of the instantaneous tion 3.2. In the case that the daughter clusters are assumed to interferences from the active SUs to the PU is such that they be Poisson, the clustering of nodes diminishes a point process are LT ordered and cause ordered performance metrics such in the LF order. From an interference point of view, the as a coverage probability and an achievable rate of the PU clustering of interfering nodes causes less interference in the according to the user selection schemes. In other words, the LT order between interference distributions. Tis translates smaller LT ordered interference distribution guarantees the into an increased coverage probability and improved capacity better coverage probability and achievable rate of the PU. We for the system. However, the clustering of nodes also causes now discuss these in detail. less spatial coverage. Terefore, the proper point process for network deployment should be studied for balancing between (a) Average Interference Power Constraint.Teinstantaneous interference and spatial coverage. aggregate interference from the active SUs to the PU can be expressed as 5.2. Cognitive Networks. In the following discussion, we will � =∑ℎ(�)� (‖�‖) , consider the applications of stochastic orders to cognitive SU I �∈Φ (20) network systems where there is an increasing interest in SU,th 8 Wireless Communications and Mobile Computing

where ΦSU,th is a point process for the active SUs and a predetermined failure numbers � with a probability � (�) ℎ is a positive random variable capturing the (power) occuring, then the number of selected active SUs follows I � � fading coefcient between an active SU located at � and a negative binomial distribution with parameter and � � ≤ � Φ ≤ Φ the PU. From Campbell’s theorem, the average of aggregate denoted as NB.Since NB Lt B [38], NB Lf B from interference from the active SUs, E[�SU],isthesameaslong Teorem 5. Terefore, the aggregate interferences from the � ≤ � E[� ]=E[� ] as the average number of the active SUs is fxed to �=E[�] active SUs are LT ordered NB Lt B,while NB B E[� ]=E[� ]≤Γ regardless of the distribution for � (for a proof, please see and NB B I. For these operations, the BS for SUs Appendix J). Terefore, we need to select � in order to satisfy only needs to know the number of total SUs � and the average � the average interference power constraint E[�SU]≤ΓI. number of active SUs . From the discussion in Sections 5.2.1(a) and 5.2.1(b), it (b) Average Sum Rate. On the other hand, if there is is noted that the smaller LT ordered random number of no interference between the active SUs by adopting code- active SUs � causes less interference to the PU, while the division multiple access (CDMA), the instantaneous sum of average interference E[�SU] and average sum rate E[�SU] are achievable rates of the active SUs, �SU,isalsorandomandcan the same. From Teorem 5.A.21 in [21] and the discussion in be expressed as Section 3.3, the smallest LT ordered random number �min is �{�=0}=1−�{�=2�}=1/2and the biggest (�) ℎ � (‖�−�‖) �max is the fxed �=�. Even though the �min causes � (�) =∑ (1 + S ), SU log 2 (21) the smallest LT ordered interference to the PU, when �= �∈Φ � +�P (�) SU,th 2�, it causes large instantaneous interference to the PU at � ℎ(�) this moment. Otherwise, the max provides balanced trafc where S is an efective fading channel between an active SU from the active SUs since the fxed number of random set of � � � (�) = ℎ(�)�(‖�‖) located at andtheBSforSUslocatedat , P I users � is always selected. However, it causes the bigger LT is the interference power from the PU to the SU located 2 ordered interference to the PU than any other distributions at �,and� is an additive noise power. From Campbell’s for active random SUs �. Terefore, the proper distribution theorem, the average of sum of achievable rates of active SUs, for random number of active SUs should be studied for E[� (�)] SU , is the same regardless of the point processes for balancing peakyness of instantaneous interference power and the active SUs as long as the average number of the active fairness of active SUs’ trafc. SUs is equal to �=E[�] (proof is the same as the average interference case in Appendix J with a diferent function). 5.3. Discussion and Tradeofs 5.2.2. Randomized User Selection Scheme for Secondary Users. 5.3.1. Coverage versus Interference. Asshownintheprevious We now discuss diferent ways to perform randomized user sections, the stochastic ordering of point processes can be selection which corresponds to diferent ways of thinning applied to performance comparison and system design. Since the SU point process. One can apply the stochastic ordering the performance metrics depend on the stochastic ordering tool to design user selection schemes based on Teorem 5 of point processes for spatial distribution of nodes, we can which produce mixed binomial point processes for active study spatial character of networks and investigate the spatial SUs. Given �=E[�] which satisfes the interference distributions of transmitting nodes in order to evaluate sys- constraint E[�SU]≤ΓI, the smaller LT ordered random tem performances using stochastic ordering approach. Ten number of active SUs � provides the smaller LF ordered point our investigations can be applied to the comparison of perfor- process ΦSU,th;thatis,�1 ≤Lt �2 �⇒ Φ SU,th ≤Lf ΦSU,th from 1 2 mance metrics such as number of served users and total cell Teorem 5. Consequently, the resulting mixed binomial point coverage probability without having closed form expressions processes cause LT ordered aggregate interferences to the for the metrics. With performance metrics such as network PU, �SU ≤Lt �SU . Te LT ordered aggregate interferences from 1 2 spatial coverage and network interference [20], our study can the active SUs to the PU yield ordered performance metrics providedesignguidelinesfor networkdeploymenttoincrease for the PU such as a coverage probability and an achievable spatial coverage of networks or provide less interference from rate due to the LT ordered interferences and the coverage networks. As an example, consider the efects of clustering. probability and the achievable rate having c.m. property with Te clustering of nodes reduces a point process in the LF respect to the interferences [20]. On the other hand, the order. From an interference point of view, the clustering of average sum of achievable rates of the active SUs and the interfering nodes causes less interference in the LT order average of interference power remain the same, E[�SU ]= 1 between interference distributions. Tis translates into an E[�SU ] and E[�SU ]=E[�SU ] as discussed in Sections 2 1 2 increased coverage probability and improved capacity for the 5.2.1(a) and 5.2.1(b). system [20]. However, the clustering of nodes also causes We now give examples of a user selection scheme using less spatial coverage. Terefore, the proper point process for the stochastic ordering approach. If the active SUs are chosen network deployment should be studied for balancing between among � total SUs with a probability � independently, the interference and spatial coverage. number of active SUs is a binomial random variable, �B. Te number of active SUs can follow discrete distributions other than binomial if diferent modes of operation are 5.3.2. Burstiness versus Fairness. Te stochastic ordering adopted. For another example, if the SUs are selected before approach can also be applied to design user selection schemes Wireless Communications and Mobile Computing 9

(＂３,－３) = (0.1, 3) 100

− 10 1

− 10 2

Total Cell Prob. Coverage Total − 10 3

− 10 4 −30 −25 −20 −15 −10 −5 015 0 SINR Treshold (dB)

PPP MS & PPP BS MPP MS & PPP BS Figure 3: Total cell coverage probabilities. for SUs in cognitive networks which satisfy the constraints of node is moving. If the initial point processes of two mobile the system. Under the system setup described in Section 5.2, networks are LF ordered, then interferences raised from the the smaller LT ordered active SUs � causes less interference mobile networks are LT ordered. Consequently the ordered tothePU,whiletheaverageinterferenceE[�SU] and average interferences imply ordered performance metric results such sum rate E[�SU] are the same. From Teorem 5.A.21 in [21] as coverage probability and achievable rate. Te ordered and the discussion in Section 3.3, the smallest and biggest LT performance metric results are not changed regardless of the ordered � can be obtained as follows: diferent mobility models such as constrained i.i.d. mobility, ≤ �≤ � , random walk, and Brownian motion. Since these random Nmin Lt Lt min (22) efects do not afect the performance comparison results and where � is any nonnegative integer random variable, �min design guidelines using stochastic ordering approach, we is �{�=0}=1−�{�=2�}=1/2,and�max is the need to focus the stochastic ordering of spatial deployments fxed �=�. Even though the smallest LT ordered random of nodes. number � causes the smallest LT ordered interference to the PU, it causes more bursty trafc from the active SUs and more instantaneous interference when �=2�.Otherwise, 6. Numerical Results the biggest LT ordered � provides balanced trafc from the In this section, we now corroborate our theoretical results active SUs since the fxed number of random set of users � through numerical examples. Since the number of BSs is is always selected. However, it causes the bigger LT ordered smaller than the number of MSs in general, we choose the interference to the PU than any other distributions for active smaller value of �B for BSs than �M for MSs. random SUs �. Terefore, the proper distribution for active random SUs should be studied for balancing burstiness of interference and fairness of active SUs’ trafc. 6.1. Cellular Networks. We show in Figure 3 the total cell coverage probabilities. It is assumed that the BS distribution 5.3.3. Random Efects. Te system performances do depend is a stationary Poisson point process ΦB,andthecompared on not only spatial deployment of a network, but also random user distributions follow also a Poisson point process ΦPPP efects of real systems such as random fading efects, random and mixed Poisson process ΦMPP with the same intensity �M. moving of mobile stations, coexisting of heterogeneous net- Since the user distributions are LF ordered as ΦMPP ≤Lf ΦPPP, works, and shadowing. From the preservation properties of from Teorem 10, the total cell coverage probability of the LF ordering of point processes, it is noted that the random users distributed by ΦMPP is greater than that of ΦPPP in the efects do not change the LF ordering of original point process entirerangeoftheSINRthresholdasshowninFigure3.Using when independent identical random efects are considered the stochastic ordering approach, one can compare the cov- in two diferent spatial network deployment scenarios. As erage probabilities by investigating the spatial distributions of an example, consider two mobile networks where every mobile users without actual system evaluation. 10 Wireless Communications and Mobile Computing

CDF of SINR (SNR=20dB) 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 5 10 15 20 25 30 35 40 45 50 SINR (dB)

Binomial Poisson Negative Binomial

Figure 4: Coverage probabilities of primary user.

Avg. of sum rates for secondary users 180

160

140

120

100

20 0 5 10 15 20 SNR (dB)

Binomial Poisson Negative Binomial

Figure 5: Average sum rates of secondary users.

6.2. Cognitive Networks. In Figure 4, the CDFs of the SINR the ordering ΦNB ≤Lf ΦP ≤Lf ΦB of the corresponding mixed of the primary user are shown where the spatial distributions binomial processes, from Teorem 5. Since the aggregate of the active SUs are diferent mixed binomial processes. Te interferences from LF ordered point processes for spatial LT ordered number of the active SUs �NB ≤Lt �P ≤Lt �B of distributions of the active SUs are LT ordered, we observe the negative binomial, Poisson, and binomial RVs ensures ��NB ≥st ��P ≥st ��B which are the ratio between the PU’s Wireless Communications and Mobile Computing 11 efective fading channel and the interference from active SUs. from the defnition of Laplace functional of random measure TisisbecausewhenthePU’sefectivefadingchannelis in (8). Ten, the proof follows from Defnition 1. exponentially distributed and interferences are LT ordered, the SIRs are reverse ordered in usual stochastic order [20]. Te B. Proof of Theorem 3 aforementioned relationship between SIR and interference (�) distributions holds as long as the CCDF of the PU’s efective Let �0 denote the Laplace functional of Φ0 +�, i.e., the fading channel is a c.m. function by the property of LT Laplace functional of the representative cluster translated by ordering in (3). An example is the exponential distribution. �. It can be expressed as follows: Despite the ordering of the SIRs in Figure 4, the average of the sum of achievable rates of the active SUs in (21) is the same �(�) (�) ﬂ E [ ∏ (−� (�))] regardless of LF ordered point processes as shown in Figure 5. 0 Φ0 exp �∈Φ +� In order to guarantee the best coverage probability for the [ 0 ] PU, one can use the stochastic ordering approach to design (B.1) the user selection schemes which cause less interference from [ ] = EΦ ∏ (−� (� + �)) . active SUs to the PU. Since the negative binomial distribution 0 exp �∈Φ for the number of active SUs causes the smallest LT ordered [ 0 ] interference, one can design the user selection scheme with From (B.1), the Laplace functional of the homogeneous the minimum information such as the total number of SUs � Φ Φ � independent cluster process with parent process p can be and the average number of active SUs which provides the expressed as follows [25]: negative binomial distribution as discussed in Section 5.2.

[ (�) ] �Φ (�) = EΦ ∏ � (�) ,�∈U. 7. Summary p 0 (B.2) �∈Φ [ p ] In this paper, Laplace functional ordering of broad classes of Φp ≤Lf Φp point processes was investigated. We showed the preservation From Defnition 1 and (B.2), if 1 2 and the same �(�)(�) Φ ≤ Φ Φ ≤ Φ of LF ordering of point process with respect to several 0 in both, then 1 Lf 2. Similarly, if 01 Lf 02 and operations, such as independent marking, thinning, random the same Φp in both, then Φ1 ≤Lf Φ2. Terefore, if Φp ≤Lf Φp Φ ≤ Φ Φ ≤ Φ 1 2 translation, and superposition. We introduced the applica- and 01 Lf 02 ,then 1 Lf 2. tions of LF ordering of point processes to wireless networks such as cellular networks and cognitive networks, which C. Proof of Theorem 4 provided guidelines for design of user selection schemes and transmission strategy for wireless networks. Tradeofs Te uniformly perturbed lattice process can be considered between coverage and interference as well as fairness and as the superpositions of independent fnite point processes peakyness were also discussed. Te power of this approach corresponding to each Voronoi cell. Te fnite point process is that network performance comparisons can be made even in one of Voronoi cells consists of uniformly distributed in cases where a closed form expression for the performances random number of points �. Te Laplace functional of the is not analytically tractable. We verifed our results through fnite point process can be expressed as follows: Monte-Carlo simulations. 1 � �Φ (�) = E� [( ∫ exp (−� (�)) �d�) ], (C.1) Appendix � |�| � where � is a region of the Voronoi cell. By denoting �= A. Proof of Theorem 2 1/(�|�|) ∫ (−�(�))� � �� exp d , is a c.m. function of since 1/(�|�|) ∫ (−�(�))� �≤1 � ≤ � A Cox process is a Poisson point process conditional on the � exp d . Terefore, if 1 Lt 2,then realization of the intensity measure. Terefore, the Laplace we have the following relation: functional of the Cox process Φ with driving random mea- � � sure Ψ can be expressed as follows [25]: E [� 1 ]≥E [� 2 ]. �1 �2 (C.2)

�Φ (�) =∫ �Λ (�) Ψ (dΛ) (A.1) From (C.2) the following LF ordering is obtained: M � 1 1 E� [( ∫ exp (−� (�)) �d�) ] =∫ exp (− ∫ [1 − exp (−� (�))]Λ(d�))Ψ(dΛ) (A.2) 1 � |�| � M R� (C.3) � 1 2 =�Ψ (1 − exp (−�)), (A.3) ≥ E� [( ∫ exp (−� (�)) �d�) ]. 2 � |�| � where �Λ is the Laplace functional of the Poisson process of intensity measure Λ and M is the set of intensity measures Λ Since the fnite point processes are LF ordered, their super- in (A.1). Equation (A.2) follows from the defnition of Laplace positions are also LF ordered from Lemma 9. Te proof of functional of Poisson point process in (6), and (A.3) follows Teorem 4 is completed. 12 Wireless Communications and Mobile Computing

D. Proof of Theorem 5 is subset of �(�)-thinning. Analogous formula for �(�)- thinning follows by averaging with respect to the distribu- Similar to Appendix C, the Laplace functional of a mixed tion of the random process �.Sincetheinequalityholds � binomial point process with random number of points can under every realization �(�), their expectations also hold the be expressed as follows: inequality. 1 � �Φ (�) = E� [( ∫ exp (−� (�)) �d�) ], (D.1) � |�| � G. Proof of Lemma 8 �(⋅) � Let denote the common distribution for the translations where is the bounded set of the point process. By denoting � �∈U � = 1/(�|�|) ∫ (−�(�))� � �� .For ,theLaplacefunctionalaferrandomtranslation � exp d , is a c.m. function of since takes the form 1/(�|�|) ∫ exp(−�(�))�d�≤1. Terefore, if �1 ≤Lt �2,then � � (�) =� (� ) , we have the following relation: Φrt Φ t (G.1)

�1 �2 E [� ]≥E [� ]. where �t(�) = −log(∫ � exp(−�(� + �))�(d�|�)).From �1 �2 (D.2) R Defnition 1, if Φ1 ≤Lf Φ2,thenΦrt,1 ≤Lf Φrt,2. From (D.2) the following LF ordering is obtained:

� 1 1 H. Proof of Lemma 9 E [( ∫ (−� (�)) � �) ] �1 exp d � |�| � Let Φ1,� and Φ2,�, � = 1,...,� be mutually independent (D.3) � � � point processes and Φ1 =⋃�=1 Φ1,� and Φ2 =⋃�=1 Φ2,� be 1 2 ≥ E� [( ∫ exp (−� (�)) �d�) ]. the superposition of point processes. Te Laplace function of 2 � |�| � superposition of mutually independent point processes can be expressed as follows: Te proof of Teorem 5 is completed. � � (�) = ∏� (�) . Φ Φ� (H.1) E. Proof of Lemma 6 �=1

Te Laplace functional of an independently marked point �Φ(�) converges if and only if the infnite sum of point ̃ � ℓ + � process Φ with a nonnegative function �:R × R ��→ R processesisfniteonboundedarea�∈R . Terefore, from can be expressed as follows [39]: (H.1) and the defnition of LF ordering in (8), if Φ1,� ≤Lf Φ2,� for �=1,...,�,thenΦ1 ≤Lf Φ2.

�Φ̃ (�) = EΦ [∏ ∫ exp (−� (�, �)) � (d�|�)] ℓ (E.1) �∈Φ R I. Proof of Theorem 10 Φ = EΦ [∏exp (− (−log ∫ exp (−� (�, �)) � (d�|�)))] To prove Teorem 10, we frst express for a generic M ℓ (E.2) �∈Φ R �(��(�1)≥�,...,��(��)≥�) (I.1)

= EΦ [∏exp (−�̃ (�))]=EΦ [exp (− ∑ �̃ (�))] . (E.3) = EΦ [� (�� (�1)≥�,...,��(��)≥�)| �∈Φ �∈Φ M (I.2) Φ ] �(�)̃ = − ∫ (−�(�, �))�( �|�) 0≤ M Here log Rℓ exp d .Since ∫ (−�(�, �))�( �|�)≤1�(�)̃ Rℓ exp d , is a nonnegative function of �.Ten,theLaplacefunctionalofmarkedpoint = E [ ∏ � (�� (�) ≥�) |Φ ] ΦM M (I.3) Φ̃ �∈� ∩Φ process follows from [ 0 M ]

̃ 2 �Φ̃ (�) =�Φ (�) . (E.4) �(� +�(�)) [ (�) ] = EΦ ∏ �(ℎ ≥ )|ΦM (I.4) M S � � Terefore, from (E.4) and the defnition of LF ordering in (8), �∈� ∩Φ (‖ ‖) ̃ ̃ [ 0 M ] if Φ1 ≤Lf Φ2,thenΦ1 ≤Lf Φ2. �(�2 +�(�)) = EΦ [exp (− ∑ 1 {� ∈ �0}) | F. Proof of Lemma 7 M � (‖�‖) �∈ΦM (I.5) If �Φ is the Laplace functional of Φ, then that of Φth is Φ ]. � (�) =� (� ) �∈U, M Φth Φ p for (F.1) where �p(�) = −log(exp(−�(�))�(�) + 1 − �(�)).From Equation (I.3) follows from the assumption that ℎS and ℎI in Defnition 1, if Φ1 ≤Lf Φ2,thenΦth,1 ≤Lf Φth,2.Te�-thinning (15) and (17) are independent under the given realizations Wireless Communications and Mobile Computing 13

(�) of ΦM. From the assumption that ℎS is exponentially Harvesting Using Ginibre Point Processes,” IEEE Transactions distributed (Rayleigh fading), (I.5) follows. In (I.5), �(�) = on Wireless Communications,vol.16,no.6,pp.3700–3713,2017. 2 (�(� + �(�))/�(‖�‖))1{� ∈ �0} is a nonnegative function of [5] J. G. Andrews, F.Baccelli, and R. K. Ganti, “Atractable approach � Φ ≤ Φ to coverage and rate in cellular networks,” IEEE Transactions on . Terefore, if M1 Lf M2 , then (18) follows by (3) because exp(−�) is a c.m. function. Communications,vol.59,no.11,pp.3122–3134,2011. [6] S. Mukherjee, “Distribution of downlink SINR in heteroge- E[� ] neous cellular networks,” IEEE Journal on Selected Areas in J. Proof of �� Communications,vol.30,no.3,pp.575–585,2012. From Campbell’s theorem in (5) and the intensity measure of [7]T.Zhang,K.Lu,S.Fu,Y.Qian,W.Liu,andJ.Wang,“Improving the mixed binomial point processes Λ(�) = �|�|,theaverage the capacity of large-scale wireless networks with network- assisted coding schemes,” IEEE Transactions on Wireless Com- of interference can be expresses as follows: munications,vol.11,no.1,pp.88–96,2012. [8] Q. Ye, B. Rong, Y. Chen, M. Al-Shalash, C. Caramanis, and J. G. E [� ]=E [ ∑ ℎ(�)� (‖�‖) |Φ ,ℎ ] Andrews, “User association for load balancing in heterogeneous SU ΦSU,th,ℎI I SU,th I (J.1) �∈Φ cellular networks,” IEEE Transactions on Wireless Communica- [ SU,th ] tions,vol.12,no.6,pp.2706–2716,2013. [9] M. Afshang and H. S. Dhillon, “Fundamentals of Modeling [ (�) ] Finite Wireless Networks Using Binomial Point Process,” IEEE = Eℎ [�∫ ℎI � (‖�‖) d� |ℎI] . (J.2) I ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟� Transactions on Wireless Communications,vol.16,no.5,pp. [ � ] 3355–3370, 2017. [10] D. Jiang and Y. Cui, “Partition-Based Caching in Large-Scale Conditioned on ℎI, � is always equal. Terefore, the expec- ℎ SIC-Enabled Wireless Networks,” IEEE Transactions on Wireless tation of (J.2) with respect to I is still equal. Te proof is Communications,vol.17,no.3,pp.1660–1675,2018. completed. [11]A.Rabbachin,T.Q.S.Quek,H.Shin,andM.Z.Win,“Cog- nitive network interference,” IEEE Journal on Selected Areas in Data Availability Communications,vol.29,no.2,pp.480–493,2011. [12] Y. Wen, S. Loyka, and A. Yongacoglu, “Asymptotic analysis Te simulation data used to support the fndings of this study of interference in cognitive radio networks,” IEEE Journal on are available from the frst author upon request. Selected Areas in Communications,vol.30,no.10,pp.2040– 2052, 2012. Conflicts of Interest [13] C.-H. Lee and M. 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