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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
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