Wireless Communications and Mobile Computing

Volume 2018, Article ID 4307136, 14 pages

https://doi.org/10.1155/2018/4307136

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

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

Correspondence should be addressed to Cihan Tepedelenlioğlu; ude.usa@nahic

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 Tepedelenlioğlu. This 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.

#### Abstract

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 tradeoffs between coverage and interference as well as fairness and peakyness are studied. Monte-Carlo simulations are used to supplement our analytical results.

#### 1. Introduction

Point processes have been used to describe spatial distribution of nodes in wireless networks. Examples include randomly distributed nodes in wireless sensor networks or ad hoc networks [1–4] and the spatial distributions for base stations and mobile users in cellular networks [5–10]. In the case of cognitive radio networks, locations of primary and secondary users have been modeled as point processes [11–14]. Random translations of point processes have been used for modeling of mobility of networks in [15]. Stationary Poisson processes provide a tractable framework, but suffer from notorious modeling issues in matching real network distributions. Stochastic ordering of point processes provides an ideal framework for comparing two deployment/usage scenarios even in cases where the performance metrics cannot be computed in closed form. These partial orders capture intuitive notions like one point process being more dense or more variable. Existing works on point process modeling for wireless networks have paid little attention to how two intractable scenarios can be nevertheless compared to aid in system optimization.

Recently stochastic ordering theory has been used for performance comparison in wireless networks which are modeled as point processes [16–20]. Directionally convex (DCX) ordering of point processes and its integral shot noise fields have been studied in [16]. The work has been extended to the clustering comparison of point processes with various weaker tools, including void probabilities and moment measures, than DCX ordering in [17]. In [18], usual stochastic ordering of random variables capturing carrier-to-interference ratio has been established in cellular systems. Ordering results for coverage probability and per user rate have been shown in multiantenna heterogeneous cellular networks [19]. In [20], Laplace functional (LF) ordering of point processes has been introduced and used to study interference distributions in wireless networks. Several examples of the LF ordering of specific point processes have been also introduced in [20], including stationary Poisson, mixed Poisson, Poisson cluster, and binomial point processes.

While limited examples of the LF ordering have been provided in [20], in this paper we apply the LF ordering concept to several general classes of point processes such as Cox, homogeneous independent cluster, perturbed lattice, and mixed binomial point processes which have been used to describe distributed nodes of wireless systems in the literature. We also investigate the preservation properties of the LF ordering of point processes with respect to independent operations such as marking, thinning, random translation, and superposition. We prove that the LF ordering of original point processes still holds after applying these operations on the point processes. To the best of our knowledge, there is no study on LF ordering of general classes of point processes and their preservation properties in the literature. Through the preservation properties of LF ordering with respect to independent operations on a point process such as marking, thinning, random translation, and superposition which is done in our manuscript, we can consider several effects of real systems such as propagation effects over wireless channels, multiple access schemes, heterogeneous network scenarios, and mobile networks and compare performances without having to obtain closed form results for a wide range of performance metrics such as coverage probability, achievable rate, and resource allocation under these real system effects. In addition to the performance comparison, the stochastic ordering of point processes provides insights into the system design and evaluation such as network deployment and user selection schemes as we summarize in this paper.

The paper is organized as follows: In Section 2, we introduce mathematical preliminaries. Section 3 introduces ordering of point processes. In Section 4, we show the preservation properties of LF ordering. Sections 5.1 and 5.2 introduce applications of stochastic ordering of point processes in wireless networks. We also elaborate the qualitative insights of the stochastic ordering of point processes on the system design and evaluation in Section 5.3. Section 6 presents simulations to corroborate our claims. Finally, the paper is summarized in Section 7.

#### 2. Mathematical Preliminaries

##### 2.1. Stochastic Ordering of Random Variables

Before introducing ordering of point processes, we briefly review some common stochastic orders between random variables, which can be found in [21, 22].

###### 2.1.1. Usual Stochastic Ordering

Let and be two random variables (RVs) such thatThen is said to be smaller than in the* usual stochastic order* (denoted by ). Roughly speaking, (1) says that is less likely than to take on large values. To see the interpretation of this in the context of wireless communications, when and are distributions of instantaneous SNRs due to fading, (1) is a comparison of outage probabilities. Since are positive in this case, is sufficient in (1).

###### 2.1.2. Laplace Transform Ordering

Let and be two nonnegative RVs such thatThen is said to be smaller than in the* Laplace transform* (LT)* order* (denoted by ). For example, when and are the instantaneous SNR distributions of a fading channel, (2) can be interpreted as a comparison of average bit error rates for exponentially decaying instantaneous error rates (as in the case for differential-PSK (DPSK) modulation and Chernoff bounds for other modulations) [23]. The LT order is equivalent tofor all* completely monotonic* (c.m.) functions [22, pp. 96]. By definition, the derivatives of a c.m. function alternate in sign: , for , and . An equivalent definition is that c.m. functions are positive mixtures of decaying exponentials [22]. A similar result to (3) with a reversal in the inequality states thatfor all that have a completely monotonic derivative (c.m.d.). Finally, note that . This can be shown by invoking the fact that is equivalent to whenever is an increasing function [22] and that c.m.d. functions in (4) are increasing.

##### 2.2. Point Processes and Random Measures

Point processes have been used to model large-scale networks [1, 2, 13, 24–29]. Since wireless nodes are usually not colocated, our focus is on* simple* point processes, where only one point can exist at a given location. In addition, we assume the point processes are locally finite; i.e., there are finitely many points in any bounded set. Unlike [20], stationary and isotropic properties are not necessary in this paper. In what follows, we introduce some fundamental notions that will be useful.

###### 2.2.1. Campbell’s Theorem

It is often necessary to evaluate the expected sum of a function evaluated at the point process . Campbell’s theorem helps in evaluating such expectations. For any nonnegative measurable function which runs over the set of all nonnegative functions on ,The intensity measure of in (5) is a characteristic analogous to the mean of a real-valued random variable and defined as for bounded subsets . Thus, is the mean number of points in . If is stationary, then the intensity measure simplifies as for some nonnegative constant , which is called the intensity of , where denotes the dimensional volume of . For stationary point processes, the right side in (5) is equal to . Therefore, any two stationary point processes with the same intensity lead to equal average sum of a function (when the mean value exists).

A random measure is a function from Borel sets in to random variables in . The* Laplace functional * of random measure is defined by the following formulaThe Laplace functional completely characterizes the distribution of the random measure [25]. A point process is a special case of a random measure where the measure takes on values in the nonnegative integer random variables. In the case of the Laplace functional of a point process, can be written as in (6). As an important example, the Laplace functional of Poisson point process of intensity measure isIf the Poisson point process is stationary, the Laplace functional simplifies with .

###### 2.2.2. Laplace Functional Ordering

In this section, we introduce the Laplace functional stochastic order between random measures which can also be used to order point processes.

*Definition 1. *Let and be two random measures such thatwhere runs over the set of all nonnegative functions on . Then is said to be smaller than in the Laplace functional (LF) order (denoted by ).

In this paper, we focus on the LF order of point processes unless otherwise specified. Note that the LT ordering in (2) is for RVs, whereas the LF ordering in (8) is for point processes or random measures. They can be connected in the following way:Hence, it is possible to think of LF ordering of point processes as the LT ordering of their aggregate processes. Intuitively, the LF ordering of point processes can be interpreted as the LT ordering of their aggregate interferences. The Laplace transform of interference is particularly convenient for the determination of outage probabilities under certain fading distributions (e.g., Rayleigh) [30] and is also useful in performance comparison. The LF ordering of point processes also can be translated into the ordering of coverage probabilities and spatial coverage which will be discussed in detail in Section 5. The probability generating functional is another mathematical tool for studying point processes [1]. The probability generating functional will provide the same results in our scenarios. However, we will focus on the Laplace functional of point process in our study since it is easy to relate the Laplace functional to interference metrics as mentioned above.

###### 2.2.3. Voronoi Cell and Tessellation

The Voronoi cell of a point of a general point process consists of those locations of whose distance to is not greater than their distance to any other point in ; i.e.,The Voronoi tessellation (or Voronoi diagram) is decomposition of the space into the Voronoi cells of a general point process.

#### 3. Ordering of General Classes of Point Processes

The examples for LF orderings of some specific point processes have been provided in [20]. In this section, we introduce the LF ordering of general classes of point processes.

##### 3.1. Cox Processes

A generalization of the Poisson process is to allow for the intensity measure itself being random. The resulting process is then Poisson conditional on the intensity measure. Such processes are called* doubly stochastic Poisson processes* or* Cox processes*. Consider a random measure on . Assume that, for each realization , an independent Poisson point process of intensity measure is given. The random measure is called the driving measure for a Cox process. The LF ordering of Cox processes depends on their driving random measures.

Theorem 2. *Let and be two Cox processes with driving random measures and , respectively. If , then .*

*Proof. *The proof is given in Appendix A.

The mixed Poisson process is a simple instance of a Cox process, where the random measure is described by a positive random constant so that . Since the Laplace functional of the mixed Poisson process can be expressed as , using (7), and because and the c.m. property of , the LF ordering of mixed Poisson processes has the following relationship: if , then .

##### 3.2. Homogeneous Independent Cluster Processes

A general cluster process is generated by taking a parent point process and daughter point processes, one per parent, and translating the daughter processes to the position of their parent. The cluster process is then the union of all the daughter points. Denote the parent point process by , and let be the number of parent points. Further let , be a family of finite points sets, the untranslated clusters or daughter processes. The cluster process is then the union of the translated clusters:If the parent process is a lattice, the process is called a* lattice cluster process*. Analogously, if the parent process is a Poisson point process, the resulting process is a* Poisson cluster process*.

If the parent process is stationary and the daughter processes are finite point sets which are independent of each other, are independent of , and have the same distribution, the procedure is called* homogeneous independent clustering*. In this case, only the statistics of one cluster need to be specified, which is usually done by referring to the* representative cluster*, denoted by which is distributed the same as any . In this class of point processes, the LF ordering depends on the parent process and the representative process as follows.

Theorem 3. *Let and be two homogeneous independent cluster processes having representative clusters and , respectively. Also, let and be the parent point processes of two homogeneous independent cluster processes and , respectively. If and , then .*

*Proof. *The proof is given in Appendix B.

##### 3.3. Perturbed Lattice Processes with Replicating Points

Lattices are deterministic point processes defined aswhere is a matrix with , the so-called generator matrix. The volume of each Voronoi cell is and the intensity of the lattice is [31]. The perturbed lattice process is a lattice cluster process. Denote the lattice point process by , and let be the number of lattice points. Further let , be untranslated clusters. In each cluster, the number of daughter points are a random variable , independent of each other and identically distributed. Moreover, these points are distributed by some given spatial distribution. The entire process is then the union of the translated clusters as in (11). If the replicating points are uniformly distributed in the Voronoi cell of the original lattice, the resulting point process is a stationary point process and called a* uniformly perturbed lattice process*. If, moreover, the numbers of replicas are Poisson random variables, the resulting process is a stationary Poisson point process [32]. Now, we can define the following LF ordering of such point processes.

Theorem 4. *Let and be two uniformly perturbed lattice processes with numbers of replicas being nonnegative integer valued random variables and , respectively, and with the same mean . If , then .*

*Proof. *The proof is given in Appendix C.

Based on Theorem 5.A.21 in [21], the* smallest* and* biggest* LT ordered random variables can be defined as follows: Let be a random variable such that and let be a random variable degenerate at . Let be a nonnegative random variable with mean . ThenFrom Theorem 4 and (13), the uniformly perturbed lattice processes with replicating points with nonnegative integer valued distribution and will be the* smallest* and* biggest* LF ordered point processes, respectively, among uniformly perturbed lattice processes with the same average number of points. The smallest LF ordered uniformly perturbed lattice process exhibits clustering since some Voronoi cells have 2 points but other cells do not have any point. This observation is in line with the intuition that clustering diminishes point processes in the LF order.

##### 3.4. Mixed Binomial Point Processes

In binomial point processes, there are a total of fixed points uniformly distributed in a bounded set . The density of the process is given by where is the volume of . If the number of points is random, the point process is called a mixed binomial point process. As an example, with Poisson distributed , the point process is called a finite Poisson point process. The intensity measure of mixed binomial point processes is . In these point processes, one can show the following.

Theorem 5. *Let and be two mixed binomial point process with nonnegative integer valued random distribution and , respectively. If , then .*

*Proof. *The proof is given in Appendix D.

Similar to Theorem 4, Theorem 5 enables LF ordering of two point processes whenever an associated discrete random variable is LT ordered.

#### 4. Preservation of Stochastic Ordering of Point Processes

In what follows, we will show that the LF ordering between two point processes is preserved after applying independent operations on point processes such as marking, thinning, random translation, and superposition of point processes.

##### 4.1. Marking

Consider the dimensional Euclidean space , , as the state space of the point process. Consider a second space , called the space of marks. A marked point process on (with points in and marks in ) is a locally finite, random set of points on , with some random vector in attached to each point. A marked point process is said to be independently marked if, given the locations of the points in , the marks are mutually independent random vectors in , and if the conditional distribution of the mark of a point depends only on the location of the point it is attached to.

Lemma 6. *Let and be two point processes in . Also let and be independently marked point processes with marks with identical distribution in . If , then .*

*Proof. *The proof is given in Appendix E.

##### 4.2. Thinning

A thinning operation uses a rule to delete points of a basic process , thus yielding the* thinned point process *, which can be considered as a subset of . The simplest thinning is *-thinning*: each point of has probability of suffering deletion, and its deletion is independent of locations and possible deletions of any other points of . A natural generalization allows the retention probability to depend on the location of the point. A deterministic function is given on , with . A point in is deleted with probability and again its deletion is independent of locations and possible deletions of any other points. The generalized operation is called *-thinning*. In a further generalization the function is itself random. Formally, a random field is given which is independent of . A realization of the thinned process is constructed by taking a realization of and applying -thinning to , using for a sample of the random field . Given and given , the probability of in also belonging to is . As long as an independent thinning operation regardless of , and is applied on point processes, the LF ordering of the original pair of point processes is retained.

Lemma 7. *Let and be two point processes in , and let and be independently thinned point processes both with any identical independent thinning operation which could be either , or -thinning on both and . If , then .*

*Proof. *The proof is given in Appendix F.

Since the thinned point process is a locally finite random set of points on , with a binary random variable in attached to each point, independent thinning can be considered as the independent marking operation on a point process as discussed in the previous section.

##### 4.3. Random Translation

In this section, the stochastic operation that we consider is random translation. Each point in the realization of some initial point process is shifted independently of its neighbors through a random vector in where are independent of each other and the conditional distribution of a random vector of a point depends only on the location of the point . The resulting process is . The random translation preserves the LF ordering of point process as follows.

Lemma 8. *Let and be two point processes in , and let and be the translated point processes with common distribution for the translation . If , then .*

*Proof. *The proof is given in Appendix G.

Similar to the independent thinning operation, since the random translated point process is a locally finite random set of points on , with some random vector in attached to each point, the random translation can be considered as the independent marking operation on a point process.

##### 4.4. Superposition

Let and be two point processes. Consider the unionSuppose that with probability one the point sets and do not overlap. The set-theoretic union then coincides with the superposition operation of general point process theory. The superposition preserves the LF ordering of point processes as follows.

Lemma 9. *Let and be mutually independent point processes and and be the superposition of point processes. If for , then .*

*Proof. *The proof is given in Appendix H.

#### 5. Applications to Wireless Networks

In the following discussion, we will consider the applications of stochastic orders to wireless network systems.

##### 5.1. Cellular Networks

In this section, the comparisons of performance metrics will be derived based on the LF ordering of point processes for spatial deployments of base stations (BSs) and mobile stations (MSs).

###### 5.1.1. System Model

We consider the downlink cellular network model consisting of BSs arranged according to some point process in the Euclidean plane. For the deployment of BSs, a deterministic network such as lattice points or stochastic network such as a Poisson point process may be considered. Consider an independent collection of MSs, located according to some point process which is independent of . Figure 1 shows an example of cellular network consisting of stationary Poisson point processes with different intensities for BSs and MSs, respectively. For a traditional cellular network, assume that each user associates with the closest BS, which would suffer the least path-loss during wireless transmission. It is also assumed that the association between a BS and a MS is carried out in a large time scale compared to the coherence time of the channel. The cell boundaries are defined through the Voronoi tessellation of the BS process. Our goal is to compare performance metrics such as total cell coverage probability through stochastic ordering tools. The spatial coverage of cellular networks is also compared based on the LF order of the BS point processes.