Stochastic Geometry Analysis of Spatial-Temporal Performance in Wireless Networks: A Tutorial
Xiao Lu, Mohammad Salehi, Martin Haenggi, Ekram Hossain, and Hai Jiang
11 Stochastic Geometry Analysis of Spatial-TemporalPerformance in Wireless Networks: A Tutorial
Xiao Lu,
Member, IEEE , Mohammad Salehi, Martin Haenggi,
Fellow, IEEE ,Ekram Hossain,
Fellow, IEEE , and Hai Jiang,
Senior Member, IEEE
Abstract —The performance of wireless networks is fundamen-tally limited by the aggregate interference, which depends onthe spatial distributions of the interferers, channel conditions,and user traffic patterns (or queueing dynamics). These factorsusually exhibit spatial and temporal correlations and thus makethe performance of large-scale networks environment-dependent(i.e., dependent on network topology, locations of the blockages,etc.). The correlation can be exploited in protocol designs (e.g.,spectrum-, load-, location-, energy-aware resource allocations) toprovide efficient wireless services. For this, accurate system-levelperformance characterization and evaluation with spatio-temporalcorrelation are required. In this context, stochastic geometrymodels and random graph techniques have been used to developanalytical frameworks to capture the spatio-temporal interferencecorrelation in large-scale wireless networks. The objective of thisarticle is to provide a tutorial on the stochastic geometry analysisof large-scale wireless networks that captures the spatio-temporalinterference correlation (and hence the signal-to-interference ratio(SIR) correlation). We first discuss the importance of spatio-temporal performance analysis, different parameters affecting thespatio-temporal correlation in the SIR, and the different perfor-mance metrics for spatio-temporal analysis. Then we describe themethodologies to characterize spatio-temporal SIR correlations fordifferent network configurations (independent, attractive, repul-sive configurations), shadowing scenarios, user locations, queueingbehavior, relaying, retransmission, and mobility. We concludeby outlining future research directions in the context of spatio-temporal analysis of emerging wireless communications scenarios.
Index Terms —Large-scale wireless access networks, signal-to-interference ratio (SIR), spatio-temporal correlation, point processmodeling, stochastic geometry.
I. I
NTRODUCTION
A. Background and Objective
Wireless communications systems are evolving toward aheterogeneous architecture (e.g., multi-tier and cell-free) withthe dense deployment of different types of access points (e.g.,smallcells and hotspots) to enable pervasive wireless Inter-net access [1]. The evolving wireless networks are expectedto provide seamless connectivity to ubiquitous and/or high-mobility devices and users with millisecond delay and gigabitsper second data rate [2]. The ever-increasing demand for low-latency high-reliability services from pervasive terminals willlead to an explosive increase in mobile traffic. To accommodate
X. Lu and H. Jiang are with the Department of Electrical and ComputerEngineering, University of Alberta, Canada. M. Salehi and E. Hossain are withthe Department of Electrical and Computer Engineering, University of Man-itoba, Canada. M. Haenggi is with the Department of Electrical Engineering,University of Notre Dame, Notre Dame, USA. the massive traffic volume, high network densification andaggressive spatial frequency reuse will be required, which willresult in high levels of interference in the network.Signal propagation over a radio link is impaired by small-scale fading, large-scale path loss and shadowing, as well asco-channel interference from concurrent transmissions. Sinceall of these effects are heavily location-dependent, the net-work spatial configurations become a dominant factor thatdetermines the system-level performance. Hence, developingtractable approaches for modeling large-scale wireless systemsand analyzing their statistical performance taking into accountthe randomness (due to the above-mentioned factors) havebecome compelling. In this context, stochastic geometry [3](also referred to as geometric probability), a probabilisticanalytical approach to study (random) point configurations,has become a necessary theoretical tool for the analysis andcharacterization of large-scale wireless systems including het-erogeneous cellular networks [4]–[6], dynamic spectrum accesssystems [7], [8], wireless ad hoc networks [9]–[12], dronenetworks [13]–[15], vehicular networks [16]–[18], and lowearth orbit satellite networks [19], [20]. Spatio-temporal aspectsof mobile communications, including the spatial distributionof network nodes, wireless channels and traffic patterns haveto be considered for system development, resource allocation,performance evaluation and optimization. The purpose of thispaper is to provide a tutorial on how to quantitatively analyzethe effects of spatial and temporal fluctuations of interference(resulting from the factors above) on the system-level networkperformance.
B. Importance of Characterization of Signal-to-Interference-plus-Noise Ratio Correlation
Wireless communications systems need to preserve the qual-ity of radio links in time-varying environments. The signal-to-interference-plus-noise ratio (SINR) statistics is commonlyused as the main measure of the quality of links [21], andmost of the performance metrics for system-level evaluation(to be introduced in Section II-C) are based on the SINR. Thevariation of the wireless environment (and hence the SINR)is mainly attributed to two causes. On the one hand, thechanges of the relative positions of communication devicesand surrounding obstructions affect the multipath propagationand thus the received power of both desirable and interfering a r X i v : . [ c s . N I] F e b Fig. 1:
Sources of SINR correlation. signals. Also, variations in traffic patterns cause fluctuations inthe interference and hence the SINR.Owing to the spatio-temporal fluctuations of network distri-butions, wireless channels and traffic patterns, the interferencesand hence SINRs (at different locations and time instants)are correlated (Fig. 1). Although rapid channel fluctuationsdue to small-scale fading can result in reduced interferencecorrelation [22], the correlation still exist due to large-scalefading, i.e., path loss and shadowing [23]. For example, in astatic network, as shown in Fig. 2(a), if the ambient trans-mitters have data packets to send during two time intervals,the interferences at the receiver are temporally correlated.Besides, in a full-duplex communication system as shown inFig. 2(b), the interferences at the transceiver pair are spatiallycorrelated at any time instance. In a mobile network, as shownin Fig. 2(c), the interference at the receiver is spatially andtemporally correlated. The causes of the temporal and/or spatialinterference correlation in the above examples all arise fromthe fact that the interference comes from the same group oftransmitters. The interference correlation results in the spatio-temporal correlation of transmission outage/success [24]–[26],throughput [27], [28], mean local delay [29], [30], etc., thusneeds to be treated carefully in the designs of mobile systems.Since SINR correlation affects the performance at different transmission attempts (e.g., when using an error recoverymethod) and/or locations (e.g., in a relay-based system), anaccurate characterization of it is essential to the understandingof wireless network performance. Information about SINRcorrelation can be exploited to optimize the performance anddesign of the system accordingly.
C. Related Work
Several survey and tutorial papers have focused on thestochastic geometry analysis of wireless communication net-works. In particular, reference [31] provides a survey of pointprocess models and stochastic geometry tools that have beenused to analyze static wired, wireless, ad hoc and cellularnetworks prior to 2009. Reference [32] overviews the impact ofspatial modeling on the SINR-based performance metrics, i.e.,connectivity, coverage area, and capacity of different types ofsystems, including ad hoc, cellular and cognitive networks. Thesurvey in [33] comprehensively reviews the works on stochasticgeometry analysis of multi-tier and cognitive cellular systemsprior to 2013.In addition to the above survey papers, tutorial papers onthe mathematical tools used for stochastic geometry analysisof large-scale systems have also been written. Reference [34]is the first tutorial on point process theory, random geometry (a) Example 1: Temporal interference correlation in a static network.(b) Example 2: Spatial interference correlation ina full-duplex communication system.(c) Example 3: Spatio-temporal interference correlation in a mobile network.
Fig. 2:
Examples of interference correlation. graphs, and percolation theory for interference characteriza-tions in ad hoc networks. Targeting cellular networks, [35]provides a tutorial on stochastic geometry analysis of bothdownlink and uplink networks based on Poisson point process(PPP) modeling. The focus is on characterizing interferencein different scenarios by exploiting the properties of the PPPunder Rayleigh fading assumption. As PPP modeling fails tocapture the spatial correlation among the random points, theauthors in [36] emphasize the use of repulsive point processesto model cellular networks, where the base station locationsare usually systematically planned with a moderate degree ofirregularity due to different development issues. To this end,the authors present a tutorial on the SINR distribution analysisof downlink cellular networks based on the β -GPP (Ginibrepoint process), which is a fairly tractable model for randompoints with spatial repulsion. The tutorial in [37] focuses ona unifying analysis of bit/symbol error probability, coverageoutage probability, and ergodic capacity in cellular networks. However, none of the existing survey and tutorial papersfocus on the stochastic geometry techniques to characterizethe spatial and temporal correlations in their consideredsystems . Moreover, this is the first article to include the refined-grained analysis tool of the (SIR) meta distribution.
D. Contributions and Organization
This tutorial aims to concisely present the analytical ap-proaches for performance evaluation of large-scale wirelessnetworks taking into account various correlation causes andeffects, such as shadowing, traffic queueing, and spatial dis-tribution of network nodes. We focus on the characterizationof the interference distribution and signal-to-interference ratio(SIR)-based performance metrics , e.g., success probability , joint success probability , and moments of conditional successprobability (CSP) given the point process . For spatial pointprocess models, we additionally derive the asymptotic SIR gain ,which is the horizontal gap between a target SIR distributionand a reference SIR distribution. This metric directly reflects thevariation of SIR due to the changes in the network model withrespect to (w.r.t.) the reference model. Furthermore, this metriccan be utilized to simplify the analysis of non-Poisson networksbased on the PPP [39] (to be introduced in Section III-B).This tutorial considers the following correlation effects inwireless networks. • The spatial correlation (i.e., attraction and repulsion)among the locations of the transmitters. • The spatially-correlated shadowing experienced by thelinks that traverse common obstacles (e.g., buildings). • The spatially and temporally correlated queue statusamong the transmitters due to the cross-interference im-posed on each other over space and time. Since in large-scale cellular networks, the impact of aggregate interferencetypically dominates that of noise [6], [38], this tutorial focuses on analyzingthe interference-limited cases with the noise ignored. However, without loss ofgenerality, the same analytical approaches can also be applied to characterizeSINR-based performance metrics. • The spatial correlation experienced by users located in thecell-center and cell-boundary regions. • The temporal interference correlation between multipletransmission attempts due to the correlation in the loca-tions of the interferers over time. • The spatio-temporal interference correlation among relay-ing nodes in a multihop network due to the correlation inthe locations of the interferers over space and time. • The spatio-temporal interference correlation in mobile sys-tems (i.e., where the users and/or base stations (BSs) aremobile) due to the correlation among interferers’ locationsover space and time.This tutorial presents the methodologies of analyzing theinterference (and SIR) correlation and the effects of spatio-temporal SIR correlations. For a thorough exposition of theanalytical techniques, we demonstrate the step-by-step deriva-tions as well as numerical results. The main differences betweenour tutorial and the state-of-the-art discussed in the previoussubsection are summarized in Table I . Herein, Poisson spatialmodels refer to the PPP and binomial point process, while non-Poisson spatial models refer to any point process whose pointsare not independently distributed.The organization of this tutorial and the relations amongdifferent sections are shown in Fig. 3. We restrict the pointprocesses to the Euclidean spaces R and R . However, thesame methodologies can be applied to analyze point processesin higher dimensions without loss of generality. Section I intro-duces the role of stochastic geometry analysis and highlightsthe importance of spatio-temporal correlation characterization.Section II explains the correlation effects of different networkfactors that may affect the SIR-based network performance.Sections III-VI present exact methodologies to analyze networkperformance with correlations in node distribution, link distancedistribution, shadowing, and queueing, respectively. Moreover,Sections VII and VIII present a performance characterizationwith multiple transmission attempts (i.e., retransmission) andmultihop relaying, respectively, for correlated and indepen-dent interference. Section IX characterizes the spatio-temporalperformance with mobility. Future directions and researchchallenges are then discussed in Section X followed by theconclusion in Section XI. Additionally, for convenience, welist the abbreviations used in Table II. Notations : The notations defined in Table III are usedthroughout this tutorial.II. O
VERVIEW OF S PATIO -T EMPORAL P ERFORMANCE C ORRELATION
A. Parameters Impacting SIR Correlation
Let o denote the origin of R d . The SIR at a target receiverlocated at o expressed asSIR = P y h y S y (cid:96) ( (cid:107) y (cid:107) ) (cid:80) x ∈ Φ ! ι x P x h x S x (cid:96) ( (cid:107) x (cid:107) ) , (1) Herein, success probability refers to the complementary cumulative distri-bution function (CDF) of the SINR/SIR where Φ ! denotes the set of all the interfering transmitters, y thelocation of the transmitter associated with the target receiver, P x the transmit power of transmitter at x , h x ( S x ) the fading(shadowing) coefficient between the transmitter located at x andreceiver located at o , (cid:96) the path loss function, and ι x the stateindicator of the transmitter located at x which equals 1 and 0when the transmitter is on and off, respectively. The physicalimplications of the SIR components are shown in Table IV.The SIR varies due to the fading and shadowing effects andsystem factors such as timing-varying traffic loads and positionsof transmitters and receivers. Their impacts are discussed below.
1) Spatial Distribution:
The spatial distribution of the net-work nodes can be categorized into three types: independent,repulsive and attractive. • With independent distribution, the locations of the trans-mitters are independent of each other. Spatial point processmodels to characterize independent distributions includePPP and binomial point process (BPP) [40] models. • With repulsive distribution, the wireless transmitters trans-mitting simultaneously are not too close to each other. Therepulsion may arise from planned deployment, physical re-strictions (e.g., geographic exclusion and terrain occlusion)and channel access control (e.g., carrier sense multipleaccess (CSMA) [41]–[44] and licensed-user activity [7],[45]) as shown in Fig. 4. Lattice processes [46], Ginibrepoint processes [36], [47], [48], and Mat´ern hardcorepoint processes [49], [50] are some well-known examplesof point process models to characterize this repulsivebehavior. • Attractive distributions can be observed when wirelesstransmitters are only clustered in certain regions, i.e., notidentically distributed over the entire plane, as illustratedin Fig. 5. This can be caused by base station (BS)-centricuser gathering [51], (e.g., around open-access WiFi spots)or due to user-centric BS deployment [52]. The Mat´erncluster process [53], [54], Thomas cluster process [51],Gauss-Poisson process [55], [56], Cox process [16], [57],[58], and the Poisson hole process [7], [45] are somerepresentative point processes to model attractive spatialdistributions.Additionally, a point process can be a mix of the above threetypes. Whether a point process is of any type may depend onthe distance between the locations considered. For example, thetype I user point process introduced in [59] (for modeling userdistribution in cellular networks) is repulsive at short distances,attractive at intermediate distances and eventually approachesindependence at larger distances.For network performance analysis, a significant body ofliterature adopts the independent distribution assumption forsimplicity and tractability (see [33] and references therein).However, this assumption is too idealized to hold in practice asmost of the cellular networks are deployed under system-levelplanning [60]. Due to factors such as geographical restriction,access control and resource allocation, active network nodescan exhibit a spatial pattern with a degree of correlation (i.e.,
TABLE I:
Main differences between other survey and tutorial papersReference(year) Type ofReview Examined systems Spatial models Target performance metrics Spatio-temporalcorrelationanalysisAd hoc Cellular Multihop Mobile Poisson Non-Poisson[31](2009) Survey (cid:88) (cid:88) (cid:88)
Success probability, paging,handover No[34](2009) Tutorial (cid:88) (cid:88)
Interference characterization,outage probability, capacity,and area spectral efficiency No[32](2010) Survey (cid:88) (cid:88) (cid:88)
Success probability, coveragearea, capacity No[33](2013) Survey (cid:88) (cid:88)
Success probability, capacity No[35](2016) Tutorial (cid:88) (cid:88)
Success probability No[36](2016) Tutorial (cid:88) (cid:88)
Success probability No[37](2017) Tutorial (cid:88) (cid:88) (cid:88)
Interference characterization,error probability, error rate,outage probability, capacity,and handover NoThis work Tutorial (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
Success probability, joint suc-cess probabilities, conditionalsuccess probabilities, momentsof conditional success prob-ability given the point pro-cess, SIR meta distribution,SIR gain, interference correla-tion coefficient Yes
TABLE II:
List of abbreviations
Abbreviation DescriptionPPP Poisson point processPLP Poisson line processBPP Binomial point processMCP Mat´ern cluster processGPP Ginibre point processRDP Relative distance processSINR Signal-to-interference-plus-noise ratioSIR Signal-to-interference ratioCSP Conditional success probabilityJSP Joint success probabilityMISR Mean interference-to-signal ratioLSU Location-specific userASAPPP Approximate SIR analysis based on the PPPMIMO Multiple-input and multiple-outputPDF Probability density functionCDF Cumulative distribution functionPCF Pair correlation functionPGFL Probability generating functionalMAC Medium access controlCSMA Carrier sense multiple accessBS Base stationDF Decode-and-forwardHARQ Hybrid automatic repeat requestQSI Quasi-static interferenceFVI Fast-varying interferenceIoT Internet of thingsIoNT Internet of nanothingsIoST Internet of space thingsML Machine learning repulsion or attraction) which cannot be ignored in performancecharacterization.
2) Locations of Receivers:
In cellular networks, the trans-mission performance of a mobile user is highly dependent onits location w.r.t. the serving BS and interfering BSs [61].Specifically, for a cell-edge user, the desired signal is weakerand the interference signal is stronger compared to those for acell-center user (Fig. 6). Different techniques have been devel-oped to strengthen the desirable signals and/or to mitigate theinterference of cell-edge users, e.g., through coordinated beam-forming [62] and intercell interference coordination [63], [64].With these techniques, resource allocation strongly dependson the spatial variation of mobile users. Therefore, location-dependent modeling of user performance is fundamental to theunderstanding of spatio-temporal performance of these systems.
3) Small-Scale Fading:
Owing to small-scale fading, overonly a fraction of a wavelength, the signal power variation mayreach up to 40 dB [65]. The correlation of the signal strengthcan be both temporal and spatial and is frequency depen-dent [21]. The temporal correlation occurs due to movement ofenvironmental scatters. The spatial correlation is caused by cor-related multipath components due to the common propagationenvironment. However, small-scale fading is only correlatedover a short time duration and a small distance (e.g., severalwavelengths) in environments with moving scattering objectsthat change the multipath propagation. Hence it is reasonableto adopt independent and identically distributed (i.i.d.) small-scale fading models for receiving antennas with wavelengthseparation. This assumption is widely used in the existing
Fig. 3:
Organization of this paper. literature (see [33], [37] and references therein).
4) Shadowing:
Compared to small-scale fading, shadowingis correlated at a much larger time and space scale [66].For example, in an urban environment, shadowing causedby obstacles in communication paths can be geographicallycorrelated on a scale from 50 to 200 meters [21]. Moreover,temporal shadow fades with variations below 1 dB (i.e., highlycorrelated) over an IEEE 802.11ad channel can be commonlymeasured in urban streets [67]. Fig. 7 demonstrates an exampleof correlated shadowing. As the two links between the BSand users experience a similar propagation environment, theirshadowing attenuations tend to be correlated. As evidenced byexperimental studies in [68] and [69], path attenuations in a region are correlated due to shadowing effects. Therefore, cor-related shadowing should be carefully treated when assessingsystem performance that is heavily impacted by environmentalblockages.
5) Transmission Buffer Status:
The buffer status (i.e., queue-ing status) of each transmitter depends on the packet arrivaland service processes, and it impacts the activation of thetransmitters, and therefore, the mutual interference. In view ofthis, the buffer statuses of the transmitters are interdependent,leading to interacting queues . The queues are spatially coupledsince the mutual interference directly affects the transmissionsuccess probability (i.e., service processes of the queues). Also,the queues are temporally coupled since the current buffer status
TABLE III:
Notations
Symbol Definition = √− The imaginary unit ≡ Equivalence relation N , R , R + Natural numbers, real numbers and positive real numbers, respectively R d d -dimensional Euclidean space o The origin of R d B ( x, R ) Disk of radius R centered at x Φ Point process representing the nodes in the network Φ ! The set of interferers Φ( A ) The number of elements in Φ ∩ A (cid:44) The definition operator ( d ) = Equivalence in distribution {·} Indicator function which equals 1 and 0 the statement {·} is true and false, respectively E [ · ] Expectation operator E ! x [ · ] Expectation operator w.r.t. the reduced Palm measure of x P [ · ] Probability measure P ! x [ · ] Reduced Palm probability measure at x V [ · ] Variance | · |
Modulus operator (cid:107) · (cid:107)
Euclidean norm Γ( · ) Gamma function γ ( s, x ) Lower incomplete gamma function, i.e., γ ( s, x ) = (cid:82) x u s − e − u d u E ( a ) Exponential distribution with rate parameter a P ( a ) Poisson distribution with rate parameter a G ( a, b ) Gamma distribution with shape parameter a and scale parameter b Det
Determinant operator F ( ., . ; . ; z ) The Gauss hypergeometric function f X ( · ) , F X ( · ) , L X ( · ) The probability density function (PDF), the cumulative distribution function, and Laplace transform ofrandom variable X . W ( · ) The Lambert- W function, i.e., the inverse function of f ( x ) = xe x TABLE IV:
SIR components
Component Representation Characterization Φ Spatial distribution of interferers Spatial point process models (in Section III) r = (cid:107) x (cid:107) Contact distance distribution Location-dependent analysis of cellular models (in Section IV) S x Blockage Shadowing models (in Section V) ι x Buffer status Queueing models (in Section VI) is affected by the previous departure process. Due to the randomnature of channel fading and aggregate interference, the serviceprocesses can be very dynamic.Fig. 8 depicts an example of the correlation between thequeues at two transmitters. Transmitter T1, which serves re-ceiver R1, has a longer transmission link and a shorter in-terference link compared to those of transmitter T2, whichserves receiver R2. On the one hand, the channel disparitybetween the two communication links results in different packetservice rates. Compared to T2, T1 suffers from more severepath loss and interference. Correspondingly, given a similartraffic load T1 tends to vacate its queue more slowly and thusremain active more frequently than T2. On the other hand,the queue lengths of T1 and T2 determine the activation ofT1 and T2. In particular, if both transmitters are busy, theirtransmissions will cause mutual interference, which slows downthe departure process. If one of the transmitters has an emptyqueue, the other enjoys a speedy departure process. Due tothe interacting queues, transmissions in a large-scale systeminherently experience spatial and temporal correlation.
B. Spatio-Temporal SIR Correlation and Network Scenarios
Even with independent fading, aggregated interference is spa-tially correlated when it comes from the same group of interfer-ers. Similarly, interference becomes temporally correlated whencommon interferers exist across time. Spatially and temporally-correlated interference causes spatio-temporal SIR correlation.When different transmission attempts are affected by the samegroup of static interferers (e.g., the same point process), it givesrise to a quasi-static interference (QSI) scenario. By contrast,a fast-varying interference (FVI) [70] scenario results whendifferent transmission attempts are affected by different sets ofinterferers (e.g., independent point processes).Spatio-temporal SIR correlation affects the performance ofmany practical systems, where the transmission performancedepends on the SIR measured over different space and timespots. Typical examples of these systems include the following: • Multihop relaying : The end-to-end performance of a mul-tihop relaying system depends on the SIRs at the receivers The QSI and FVI assumptions can be used for modeling static and highly-mobile network scenarios, where the interferers across different transmissionsremain the same and become completely different, respectively.
Fig. 4:
Examples of wireless systems with spatially repulsive node locations.
Fig. 5:
Illustration of a wireless system with spatially attractivedeployment.
Fig. 6:
Illustration of location-specific users. at different hops. The SIRs at different hops can becorrelated due to spatio-temporal interference correlation(e.g., common interferers exist from one transmission toanother). • Retransmission : The reliability of a retransmission scheme
Fig. 7:
Illustration of correlated shadowing.
Fig. 8:
Illustration of the interacting queues. or a multi-packet transmission scheme depends on the SIRat the receiver during multiple transmissions. The SIRsfrom different transmissions can are correlated if theyare subject to the interference from the same group oftransmitters. • Mobile networks : For a mobile user, transmissions couldoccur at different spatial locations and times. The cor-relation between the SIRs at the receiver for differenttransmissions are dependent on the mobility.The SIR correlation affects the performance metrics suchas temporal joint success probability (JSP) [71], end-to-endsuccess probability [72], the local delay [29], and joint SIRmeta distribution [73].
C. Metrics for Spatio-Temporal Performance Analysis
Definition 1. (Success Probability): The (average) successprobability of a user can be defined as the probability that thereceived SIR is greater than a target SIR threshold θ , i.e., P s = ¯ F SIR ( θ ) (cid:44) P [SIR > θ ] , θ ∈ R + . (2)where ¯ F SIR represents the complementary CDF of SIR.
Definition 2. (Moments of Conditional Success Probabilitygiven the Point Process): Let P s | Φ ( θ ) = P (cid:2) SIR > θ | Φ (cid:3) represent the CSP given the point process Φ (abbreviated asCSP Φ ), which is averaged over the fading of all the links and the random channel access (if applicable). The b -th moment of P s | Φ is defined as M P s ( b ) (cid:44) E (cid:104)(cid:0) P s | Φ ( θ ) (cid:1) b (cid:105) , b ∈ C . (3)Note that the first moment of the CSP Φ is the average successprobability defined in (2), i.e., M P s (1) ≡ P s , and the varianceis V ( P s ) = M P s (2) − M P s (1) . Besides, the mean local delay,defined as the average number of transmission attempts toaccomplish a success [74], is given by M P s ( − . It is worthmentioning that for static random networks and b ∈ N , the b -thmoment of the CSP Φ is equivalent to the JSP of b transmissionsof the same link [75] and JSP that the b antennas of a multiple-antenna receiver all succeed in reception (i.e., SIR exceeds θ )[76]. Definition 3. (SIR Meta Distribution): The SIR meta distribu-tion is the complementary CDF of the CSP Φ , defined as [77] ¯ F P s ( θ, s ) (cid:44) P (cid:2) P s | Φ ( θ ) > s (cid:3) , s ∈ [0 , , (4)where s represents the target success probability.In ergodic point processes, the SIR meta distribution indi-cates the fraction of links that can achieve successful transmis-sion with probabilities greater than x in any realization of Φ .Compared to the average success probability defined in (2), theSIR meta distribution also characterizes the variability of linksuccess probabilities. For example, the inverse function of theSIR meta distribution ¯ F − P s ( p ) , p ∈ [0 , , yields the successprobability x that − p fraction of the links can achieve whilethe rest do not. Definition 4. (SIR Gain): The SIR gain is the horizontal gapbetween the complementary CDF of two SIR distributions.Evaluated at the target success probability P t , the SIR gainis defined as [78, eqn. (1)] G ( P t ) (cid:44) ¯ F − tm ( P t )¯ F − rm ( P t ) , P t ∈ (0 , , (5)where ¯ F − represents the inverse function of the complemen-tary CDF of the SIR, and SIR tm and SIR rm denote the SIRsof a target model and a reference model, respectively.The SIR gain can be used to quantify the impact of a targetmodel on the SIR distribution w.r.t. that of a reference model.An SIR gain greater or less than 1 indicates that the targetmodel can achieve the same success probability with a largeror smaller SIR threshold, respectively, than the reference model.In wireless networks, the SIR gain is usually not sensitiveto the target success probability to be evaluated [78], [79].Therefore, the SIR gain can be approximated by the asymptoticSIR gain evaluated in the high-reliability regime, i.e., P t → or θ → , defined as G (cid:44) lim P t → G ( P t ) , (6)whenever the limit exists. Note that a necessary and sufficient condition for the asymp-totic SIR gain to exist is that the slopes of the two CDFs ofthe SIR are asymptotically the same as θ → [78]. Definition 5. (Joint Success Probability): Let x = { x k } k ∈{ ,...,K } ∈ ( R d ) K and t = { t k } k ∈{ ,...,K } ∈ N K denote deterministic vectors of locations and time instances,respectively, and SIR k denote the SIR measured at location x k and time t k . The JSP is defined as the probability thatSIR i is greater than the corresponding SIR threshold θ k forall k ∈ { , . . . , K } . Given the locations x , times t , and thetarget SIR thresholds θ = { θ k } k ∈{ , ,...,K } ∈ ( R + ) K , the JSPis defined as J K ( θ , x , t ) (cid:44) P [SIR > θ , SIR > θ , . . . , SIR K > θ K ] , (7)where SIR k denotes the SIR of the k -th transmission.The SIR corresponding to the different transmissions canexhibit spatial and/or temporal correlation, the effect of whichwill be reflected in the JSP. The JSP can be for temporal, spatial,or spatio-temporal transmission events. In particular, we have • Temporal JSP of multiple transmission attempts occurringat the same location but different time instances (e.g.,multiple transmissions [75] and retransmissions [80] bya transmitter); • Spatial JSP of multiple transmission attempts occurringat the same time but different locations (e.g., joint uplinkand downlink transmissions [81], [82]); • Spatio-temporal JSP of multiple transmission events oc-curring at different space and time intervals (e.g., for thesame mobile user [71]).For a multihop communication scenario, the probability thatthe transmissions at each hop are successful for a given packetis referred to as the end-to-end success probability . Definition 6. (Conditional Success Probability): The CSP isthe probability of achieving a successful transmission given that K − ( K ≥ ) such events have occurred. Given the locations x , times t , and the target SIR thresholds θ , the CSP is givenby C K ( θ , x , t ) = J K ( θ , x , t ) J K − ( θ , x , t ) , (8)where J K ( θ , x , t ) is defined in (7).The CSP reveals the dependence between two success-ful transmission events. If the two events are positivelycorrelated (spatially or temporally), one has C ( θ , x , t ) > J ( θ , x , t ) [75]. Moreover, if the events are independent, onehas C ( θ , x , t ) = J ( θ , x , t ) = (cid:112) J ( θ , x , t ) . Similar to thedefinition of the JSP, the CSP of spatial events, temporal events,and spatial-temporal events are referred to as spatial CSP , temporal CSP , and spatial-temporal CSP , respectively. Definition 7. (Product SIR Meta Distribution): Let J K | Φ ( θ , x , t ) (cid:44) P [SIR > θ , . . . , SIR K > θ K | Φ] denote the JSP of K transmissions given the point process. Theproduct SIR meta distribution is defined as the complementaryCDF of the JSP at locations x and times t given the pointprocess Φ [73], i.e., ¯ F J K ( θ , x , t , s ) (cid:44) P (cid:104) J K | Φ ( θ , x , t ) > s (cid:105) , s ∈ [0 , . (9) Definition 8. (Joint SIR Meta Distribution): Let P ( k )s | Φ ( θ k ) = P (cid:2) SIR k > θ k | Φ (cid:3) represent the CSP Φ at x k . The joint SIRmeta distribution is defined as the joint distribution of CSP Φ atlocations x and times t [73], i.e., ¯ F ( K ) P s ( θ , x , t , s ) (cid:44) P (cid:20) K (cid:92) k =1 (cid:8) P ( k ) s | Φ ( θ k ) > s k (cid:9)(cid:21) , (10)where s = { s , s , . . . , s K } ∈ [0 , K is the vector of targetsuccess probabilities corresponding to the locations x . Definition 9. (Interference Correlation Coefficient): Let I ( t ) x denotes the aggregated interference received at location x attime t . The correlation degree of interference at two locationsand times can be quantified by the Pearson correlation coeffi-cient defined as ζ t ,t ( u , u ) (cid:44) E (cid:2) I ( t ) u I ( t ) u (cid:3) − E (cid:2) I ( t ) u (cid:3) E (cid:2) I ( t ) u (cid:3)(cid:113) E (cid:2)(cid:0) I ( t ) u (cid:1) (cid:3) − E (cid:2) I ( t ) u (cid:3) (cid:113) E (cid:2)(cid:0) I ( t ) u (cid:1) (cid:3) − E (cid:2) I ( t ) u (cid:3) , (11)where the numerator computes the covariance of I ( t ) u and I ( t ) u and the denominator is the product of the standard deviationsof I ( t ) u and I ( t ) u .The interference correlation coefficient is a statistical mea-sure that quantifies the extent to which the interferences at u and u are associated.Note that when the interferers are motion-invariant, one has I ( t ) o ( d ) = I ( t ) u . In this case, ζ ( (cid:107) u (cid:107) ) can be simplified to [22] ζ t ,t ( (cid:107) u − u (cid:107) ) (cid:44) E (cid:104) I ( t ) u I ( t ) u (cid:105) − E (cid:104) I ( t ) u (cid:105) E (cid:104)(cid:0) I ( t ) u (cid:1) (cid:105) − E (cid:104) I ( t ) u (cid:105) . (12) Definition 10. (Interference Coherence Time): The interferencecoherence time is defined as the minimum time lag such thatthe interference correlation coefficient is below a threshold ζ th [83], i.e., τ ct (cid:44) min (cid:8) τ = t − t ∈ N | ζ t ,t ≤ ζ th (cid:9) , ζ th ∈ R + . (13)Fig. 9 summarizes the above performance metrics for spatio-temporal performance analysis. Fig. 9:
Metrics for spatio-temporal performance analysis.
III. S
PATIAL P OINT P ROCESS M ODELS
This section formally defines some common point processes,also referred to as random point fields, and illustrates how thespatial distribution of the random points affects the networkperformance. To evaluate the impacts of independent, attractiveand repulsive spatial configurations, we choose three represen-tative point processes, namely the Poisson point process (PPP),the Mat´ern cluster process (MCP), and the β -Ginibre pointprocess (GPP), due to their tractability. • In a PPP, each point is located independently from theothers. • In an MCP, the locations of the random points have apropensity to be clustering. • In a β -GPP, the random points are scattered with repulsion.This section considers both ad hoc and downlink cellular net-works modeled based on the above point processes and analyzesthe interference correlation coefficient, success probability, SIRmeta distribution and SIR gain. A. System Models1) System Configurations:
We focus on analyzing the per-formance of a target in both ad hoc and downlink cellularnetworks. The target receiver is considered to be at the origin o ∈ R and attempts to decode the transmitted signals from theassociated transmitter subject to the aggregate interference. • Ad hoc networks : The target receiver at o is served bythe transmitter located at x t ∈ R , where (cid:107) x t (cid:107) = r t . Thetarget link is impaired by the interference from a randomfield of interferers modeled by a stationary point process Φ of intensity λ . Φ does not contain the serving transmitterat x t . • Downlink cellular networks : The transmitters (i.e., BSs)form a stationary point process Φ with intensity λ . Theusers are located according to a stationary point processindependent of the BS process. Each user is served by thenearest BS in Φ .In the networks considered, the transmitters stay active withunit transmit power, i.e., there is no MAC scheme. All thetransmitters and receivers are equipped with one antenna. Thesystem employs universal frequency reuse. The channels of thelinks experience i.i.d. block Rayleigh fading and power-lawpath loss, i.e., d − α , where d and α > are the link distanceand the path-loss exponent, respectively.For the convenience of notation, the points in Φ are assumedto be ordered from nearest to farthest to the origin, i.e., (cid:107) x j (cid:107) < (cid:107) x j +1 (cid:107) . Subsequently, the set of interferers in the ad hocnetworks and downlink cellular networks is Φ ! = Φ = { x j } j ∈ N and Φ ! = Φ \{ x } , respectively. The distance from the j -thnearest point in Φ to the origin is denoted as r j = (cid:107) x j (cid:107) .The SIR at the target receiver in an ad hoc and downlinkcellular network is given by, respectively, as η = h t (cid:107) x t (cid:107) − α (cid:80) j ∈ N h j (cid:107) x j (cid:107) − α (14)and η = h (cid:107) x (cid:107) − α (cid:80) ∞ j =2 h j (cid:107) x j (cid:107) − α , (15)where h t and h j denote the power fading coefficients betweenthe transmitters at x t , x j and the target receiver, which are ex-ponential random variables with unit mean, i.e., h t , h j ∼ E (1) .It is noted that (14) represents the SIR of a specific receiverinstead of the typical receiver in ad hoc networks, while (15)is the SIR of a user at the origin that, upon averaging overthe base station and user point processes, becomes the typicaluser in the downlink cellular network, for any stationary pointprocess of users that is independent of the base station pointprocess.
2) Spatial Configurations:
We consider the homogeneousPPP, MCP and β -GPP, formally defined as follows. Definition 11. (Homogeneous Poisson point process): A pointprocess
Φ = { x j } j ∈ N ⊂ R d is a homogeneous PPP ifit possesses the following two properties: i) the number ofpoints in any compact set B ⊂ R d is Poisson-distributed withmean λ |B| ; and ii) the numbers of points in disjoint sets areindependent. Properties: Density : For a homogeneous PPP, the first moment density(or first-order density) and second moment density (or second-order product density) are given, respectively, as [84] ρ (1) ( x ) = λ, (16)and ρ (2) ( x, y ) = (cid:0) ρ (1) ( x ) (cid:1) = λ . (17)The n -th moment density is the density pertaining to the n -thorder factorial moment measure, which, for n > , indicatethe spatial correlation. For example, ρ (1) ( x )d x measures theprobability of having a point at x in some infinitesimal region d x .2) Contact Distance Distribution : Contact distance refersto the distance between a reference location and the nearestpoint in Φ [85]. The probability density function (PDF) andCDF of the contact distance in a homogeneous PPP are given,respectively, as [86] f r ( r ) = 2 πλ exp( − λπr ) (18)and F r ( r ) = exp( − λπr ) . (19)3) Joint Distance Distribution : The joint PDF of thedistances to n nearest points is [87, Eqn. (30)] f r ,r ,...,r n ( x , x , . . ., x n ) = e − λπx n (2 λπ ) n x x . . . x n . (20)4) Distance Ratio Distribution : Let (cid:37) j = r r j , j ∈ N , denotethe distance ratio of the nearest point to the j -th nearest point.The CDF and PDF of (cid:37) j are given, respectively, by [88, Lemma3] F (cid:37) j ( (cid:37) ) = 1 − (1 − (cid:37) ) j − , (cid:37) ∈ [0 , (21)and f (cid:37) j ( (cid:37) ) = 2( j − (cid:37) (1 − (cid:37) ) j − , (cid:37) ∈ [0 , . (22)5) Probability Generating Functional (Product Func-tional) : For υ ( x ) ∈ [0 , and (cid:82) R (1 − υ ( x ))d x < ∞ , theprobability generating functional (PGFL) for the PPP is givenby E (cid:20) (cid:89) j ∈ N υ ( x j ) (cid:21) = exp (cid:18) − λ (cid:90) R d (cid:0) − υ ( x ) (cid:1) d x (cid:19) . (23)According to Slivnyak’s theorem [3, Theorem 8.10], thereduced Palm distribution of the PPP coincides with its ordinarydistribution. Therefore, the conditional PGFL for the PPP canalso be expressed as (23).Let Φ R = (cid:8) x j ∈ Φ \{ x } : (cid:107) x (cid:107) / (cid:107) x j (cid:107) (cid:9) = (cid:8) j ∈ N \{ } : (cid:37) j (cid:9) ⊂ (0 , denote the relative distance process (RDP) of aPPP Φ . The PGFL of Φ R is given by [79, Lemma 1] E (cid:20) (cid:89) j ∈ N \{ } υ ( (cid:37) j ) (cid:21) = 11 + 2 (cid:82) (cid:0) − υ ( (cid:37) ) (cid:1) (cid:37) − d (cid:37) . (24)for functions υ ( x ) ∈ [0 , such that (cid:82) R (1 − υ ( x ))d x is finite. 6) Sum Functional : For any measurable function g ≥ on R d , the sum functional for a stationary point process is givenby Campbell’s Theorem [3, Theorem 4.1] as E (cid:20) (cid:88) j ∈ N g ( x j ) (cid:21) = λ (cid:90) R d g ( x )d x. (25)7) Sum-Product Functional : Let p (cid:44) ( p , p , . . . , p q ) ∈ N q , (cid:107) p (cid:107) (cid:44) (cid:80) qk =1 p k and M = { j ∈ N : M j } denote a set of i.i.d.random marks associated with the points in Φ = { x j } j ∈ N . If g k ( x ) ∈ [0 , ∞ ) , ≤ k ≤ q , and υ ( x ) ∈ [0 , are measurablefunctions for all x ∈ R d , the sum-product functional for thePPP with (cid:107) p (cid:107) > is given by [89, Theorem 1] E (cid:34) q (cid:89) k =1 (cid:18) (cid:88) j ∈ N g ( x j , M j ) (cid:19) p i (cid:89) j ∈ N υ ( x j , M j ) (cid:35) = exp (cid:18) − λ (cid:90) R d (cid:16) − E M j (cid:104) υ ( x, M j ) (cid:105)(cid:17) d x (cid:19) × (cid:107) p (cid:107) (cid:88) l =1 (cid:88) N ∈N pl D N l ! l (cid:89) k =1 λ (cid:90) R d E M (cid:20) υ ( x, M j ) × q (cid:89) i =1 f n ki i ( x, M j ) (cid:21) d x, (26)where D N (cid:44) (cid:81) qk =1 p k ! (cid:81) li =1 m ik ! and N p l ⊂ N q × l is the classof all q × l matrices with the columns (cid:107) n · i (cid:107) > , ∀ i ∈{ , , . . . , l } and the rows (cid:107) n k · (cid:107) = p k , ∀ k ∈ { , . . . , q } .In the special case q = 1 and p = 1 , the sum-product func-tional for the PPP is given by the Campbell-Mecke Theorem [3,Theorem 8.2] as E (cid:34) (cid:88) j ∈ N g ( x j , M j ) (cid:89) j ∈ N υ ( x j , M j ) (cid:35) = exp (cid:18) − λ (cid:90) R d (cid:16) − E M j (cid:104) υ ( x, M j ) (cid:105)(cid:17) d x (cid:19) × λ (cid:90) R d E M j (cid:104) g ( x, M j ) υ ( x, M j ) (cid:105) d x. (27) Definition 12. (Mat´ern cluster process): The MCP Φ M isa doubly Poisson cluster process constructed from a parentPPP Φ p = { x j } j ∈ N with intensity λ p with each point of Φ p substituted by a daughter cluster consisting of a PPP with anaverage number of points ¯ c within a disc of radius R d centeredat that point. Properties: Density : For an MCP, the first moment and secondmoment densities are given, respectively, by ρ (1) ( x ) = λ = λ p ¯ c, (28) ρ (2) ( x, y ) = λ ¯ c + λ p ¯ c π R A R d ( (cid:107) x − y (cid:107) ) , (29) A R d represents the area of the intersection of two disks withradius R d at a distance r > given as A R d ( r ) = R arccos( r/ R d ) − r (cid:113) R − r / , ≤ r ≤ R d , otherwise. (30)2) Daughter Point Distribution : Each daughter point islocated uniformly within a disk of radius R around the originwith the PDF given by f M ( y ) = (cid:40) πR , if (cid:107) y (cid:107) ≤ R d , otherwise . (31)3) Contact Distance Distribution : The PDF and CDF ofthe contact distance in the MCP are given as (32) and (33),respectively [85].4)
Probability Generating Functional : The PGFL for anMCP is given as follows [9]: E (cid:20) (cid:89) j ∈ N υ ( x j ) (cid:21) = exp (cid:32) − λ (cid:90) R (cid:20) − M (cid:18) (cid:90) R υ ( x + y ) f M ( y )d y (cid:19)(cid:21) d x (cid:33) d y, (34)where M ( t ) = e − ¯ c (1 − t ) is the moment generating function ofthe representative cluster in an MCP.Let E ! o [ · ] denote the expectation operation based on thereduced Palm measure [84] which takes the expectation for apoint process conditioned at a point of the process at o withoutincluding the point. The conditional PGFL of an MCP is [9,Lemma 1] E ! o (cid:20) (cid:89) j ∈ N υ ( x j ) (cid:21) = exp (cid:32) − λ p (cid:90) R (cid:20) − M (cid:18) (cid:90) R υ ( x + y ) f ( y )d y (cid:19)(cid:21) d x (cid:33) × (cid:90) R G Md (cid:16) υ ( x − y ) (cid:17) f M ( y )d y, (35)where G Md is the PGFL for the representative cluster given by G Md ( υ ) = M (cid:18) (cid:90) R υ ( x ) f M ( x )d x (cid:19) . (36) Definition 13. ( β -Ginibre point process): A β -GPP Φ G β = { x j } j ∈ N is a determinantal point process with the kernel givenby [47] K β,λ ( x, y ) = λe − πλ | x − y | β , x, y ∈ C , (37)w.r.t. the Lebesgue measure on C [90]. Properties: The kernel represents the interaction force among the spatial points of theprocess. Density : For a β -GPP, the first moment density and thesecond moment density of the β -GPP are given, respectively,by ρ (1) ( x ) = det[ K β,λ ( x, x )] = λ, (38) ρ (2) ( x, y ) = det (cid:20) K β,λ ( x, ¯ x ) K β,λ ( x, ¯ y ) K β,λ (¯ x, y ) K β,λ ( y, ¯ y ) (cid:21) = λ (cid:18) − exp (cid:18) − πλ | x − y | β (cid:19)(cid:19) . (39)2) Link Distance Property : Let λ represents the intensityof Φ G β . Let { Q j } j ∈ N be a set of independent gamma randomvariables with PDF f Q j ( q ) = q j − e − πλβ q (cid:16) βπλ (cid:17) j Γ( j ) , (40)i.e., Q j ∼ G ( j, β/πλ ) . Then the set {(cid:107) x j (cid:107) } j ∈ N is equivalent indistribution with the set constructed by retaining each elementfrom { Q j } j ∈ N independently with probability β [91, Theorem4.7.1].Fig. 10(a), (b), (c), respectively, illustrate realizations ofMCP, PPP, and β -GPP. As shown by the realizations of MCPand β -GPP, the point sets are attractive and repulsive, respec-tively.The correlation between the spatial points can be measuredby the pair correlation function (PCF). For a point process Φ ⊂ R d , the PCF is defined as g ( x, y ) (cid:44) ρ (2) ( x,y ) ρ (1) ( x ) ρ (2) ( y ) . If Φ is motion-invariant, the first moment density is the constantintensity and the second moment density ρ (2) ( x, y ) only de-pends on the difference r = (cid:107) x − y (cid:107) . Hence, the PCF can beexpressed as g ( r ) = ρ (2) ( r ) λ .Let Φ( A ) denote the number of points in Φ ∩ A . ThePCF quantifies the degree of correlation between the randomvariables Φ( A ) and Φ( B ) in a non-centered way. If g ( x, y ) ≡ then for disjoint A and B the covariance of Φ( A ) and Φ( B ) is zero, which means the two variables are uncorrelated. Forspatial point processes, the PCF describes how a point issurrounded by others. The PCF equals one if the points areuncorrelated (e.g., as in the PPP), and is greater (smaller) than1 if the points are attractive (repulsive).The PCFs of the MCP, PPP and β -GPP are given, respec-tively, as [3, Page 153], [47] g ( r ) = cA R d ( r ) λπ R , MCP , PPP − exp( − r /β ) πλ , β -GPP , (41)where A R d ( r ) is given in (30). Properties:
1) For r ≥ R d , for an MCP, the PCF is the same as that ofthe PPP, because two points with a distance greater than R d must belong to different clusters and thus are independent. f M r ( r | x ) = rR , if ≤ r ≤ R d − x for (cid:107) x (cid:107) < R d rπR cos − (cid:16) r + x − R rx (cid:17) , if R d − (cid:107) x (cid:107) ≤ r ≤ R d − (cid:107) x (cid:107) for (cid:107) x (cid:107) < R d , and if (cid:107) x (cid:107) > R d , (32) F M r ( r ) = 1 − exp (cid:32) − λ p (cid:18) (cid:90) B (0 ,R d ) (cid:20) − exp (cid:18) − ¯ c (cid:18) (cid:90) min( r,R d − x )0 yR d y + (cid:90) min( r,R d + x )min( r,R d − x ) yπR cos − (cid:16) y + x − R yx (cid:17) d y (cid:19)(cid:19)(cid:21) d x + (cid:90) R \ B (0 ,R d ) (cid:20) − exp (cid:18) − ¯ c (cid:90) min( r,R d + x )min( r,R d − x ) yπR cos − (cid:16) y + x − R yx d y (cid:17)(cid:19)(cid:21) d x (cid:19)(cid:33) , (33) -6 -4 -2 0 2 4 6-6-4-20246 (a) Realization of an MCP ( λ p = 0 . , ¯ c = 5 and R d = 1 ). -6 -4 -2 0 2 4 6-6-4-20246 (b) Realization of a homogeneous PPP ( λ = 0 . ). -6 -4 -2 0 2 4 6-6-4-20246 (c) Realization of a β -GPP ( λ = 0 . , β = 0 . ). r P a i r C o rr e l a t i on F un c t i on g (r) MCPPPP-GPP (d) Pair correlation functions ( λ = 1 , R d = 1 ). Fig. 10:
Illustration of spatial point processes.
2) Given the intensity λ , the PCF of an MCP approaches thatof the PPP as ¯ c → , since the points are less likely to belongto the same cluster.3) For a PPP, the PCF is not affected by the intensity sincethe points are independently distributed.4) For a β -GPP, the PCF approaches that of the PPP as β → or λ → ∞ .Fig. 10(d) shows the PCFs of the spatial point processes anddemonstrates the properties discussed above. B. Performance Analysis1) Spatio-Temporal Interference Correlation in Ad Hoc Net-works:
We first study the correlation of interference observed attwo locations o and u in two time slots t and t , respectively,with the interferers distributed as the three types of pointprocesses considered. As PPP, MCP and β -GPP are motion-invariant, the spatio-temporal interference correlation can bemeasured by the Pearson correlation coefficient defined in (12).As can be seen from (12), the expectation and the secondmoments of I ( t ) o and the mean product of I ( t ) o and I ( t ) u areneeded to quantify the interference correlation. However, thesetwo quantities do not exist because of the singularity of thepath-loss function (cid:96) ( x ) . To cope with this issue, we follow theapproach in [9] by defining (cid:96) (cid:15) ( x ) = (cid:15) + (cid:107) x (cid:107) α , α > , (cid:15) > ,such that (cid:96) ( x ) = lim (cid:15) → (cid:96) (cid:15) ( x ) .The expectation of I ( t ) o can be derived as E (cid:2) I ( t ) o (cid:3) ( a ) = E (cid:2) I o (cid:3) = E (cid:20) (cid:88) j ∈ N h j (cid:96) (cid:15) ( x j ) (cid:21) = E (cid:2) h (cid:3) (cid:90) R (cid:96) (cid:15) ( x ) ρ (1) ( x )d x ( b ) = 2 πλ (cid:90) ∞ x(cid:15) + x α d x ( c ) = δπ λ(cid:15) δ − csc( δπ ) , (42)where ( a ) follows since MCP, PPP, and β -GPP are all motion-invariant and the superscript ( t ) is dropped for conciseness, ( b ) applies the conversion from Cartesian to polar coordinates,and ( c ) substitutes α with δ .If the densities of the interferers following a PPP, MCP, and β -GPP as given in (16), (28), and (38), respectively, are thesame, we can observe from (42) that the three point processescause the same mean interference at an arbitrary location. Thisindicates that spatial attraction and repulsion do not affect thefirst moment of the interference.We then continue to derive the second moment of I ( t ) o as E (cid:104)(cid:16) I ( t ) o (cid:17) (cid:105) = E (cid:34)(cid:18) (cid:88) j ∈ N h j (cid:96) (cid:15) ( x j ) (cid:19) (cid:35) = E (cid:34) (cid:88) j ∈ N h j (cid:96) (cid:15) ( x j ) + j (cid:54) = i (cid:88) j,i ∈ N h j h i (cid:96) (cid:15) ( x j ) (cid:96) (cid:15) ( x i ) (cid:35) = E [ h ] (cid:90) R (cid:96) (cid:15) ( x ) ρ (1) ( x )d x + E [ h ] (cid:90) R (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) ρ (2) ( x, y )d x d y = 2 δλπ (1 − δ ) (cid:15) δ − csc( δπ )+ (cid:90) R (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) ρ (2) ( x, y )d x d y. (43)By plugging in the second moment density ρ (2) ( x, y ) for MCP,PPP, and β -GPP given, respectively, in (29), (17), (39), weobtain the second moment of the interference in (44), where ¯ I PPP is ¯ I PPP = 2 δλπ (1 − δ ) (cid:15) δ − csc( δπ )+ δ π λ (cid:15) δ − csc( δπ ) . (45)Table V shows the variance of the interference with differentfields of interferers. It can be observed that the MCP andthe β -GPP cause larger and smaller interference variance thanthe PPP, respectively, as illustrated in Fig. 11. Moreover, itcan be found that, for an MCP, given the interference density λ = λ p ¯ c , the variance increases when the points are moredensely clustered (i.e., with smaller cluster density λ p and alarger average number of points ¯ c within each cluster).Similarly, we have the mean product of I ( t ) o and I ( t ) u , t (cid:54) = t , as E (cid:104) I ( t ) o I ( t ) u (cid:105) = E (cid:34) (cid:88) j ∈ N h ( t ) j (cid:96) (cid:15) ( x j − o ) (cid:88) i ∈ N h ( t ) i (cid:96) (cid:15) ( x i − u ) (cid:35) = E (cid:34) (cid:88) j ∈ N h ( t ) j h ( t ) j (cid:96) (cid:15) ( x j ) (cid:96) (cid:15) ( x j − u )+ j (cid:54) = i (cid:88) j,i ∈ N h ( t ) j h ( t ) i (cid:96) (cid:15) ( x j ) (cid:96) (cid:15) ( x i − u ) (cid:35) = E [ h ] (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u ) ρ (1) ( x )d x + E [ h ] (cid:90) R (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) ρ (2) ( x, y )d x d y, (46)which is an integral function of the first and second momentdensities. Subsequently, (46) can be obtained by following thederivation of the second moment of interference as in (47).Finally, by inserting the expectation, second moment, andmean interference product given in (42), (43), and (46), re-spectively, into (12), we have the spatio-temporal interferencecorrelation coefficient in the following theorem. Theorem 1.
The spatio-temporal correlation coefficient withinterferers distributed as MCP, PPP and β -GPP and path-lossfunction (cid:96) (cid:15) ( x ) = (cid:15) + x α is given by (48), where (cid:36) MCP ( R d , ¯ c ) and (cid:36) β -GPP ( λ ) are given, respectively, by (cid:36) MCP ( R d , ¯ c )= ¯ cπ R (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) A R d ( (cid:107) x − y (cid:107) )d x (cid:19) d y, E (cid:20)(cid:16) I ( t ) o (cid:17) (cid:21) ( a ) = ¯ I PPP + λ ¯ cπ R (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) A R d ( (cid:107) x − y (cid:107) )d x (cid:19) d y, MCP ¯ I PPP , PPP ¯ I PPP − λ (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) (cid:18) exp (cid:18) − πλ (cid:107) x − y (cid:107) β (cid:19)(cid:19) d x (cid:19) d y, β -GPP . (44) TABLE V:
Variance of InterferencePointProcess E [ I o ] − E [ I o ] MCP δπ λ (1 − δ ) (cid:15) δ − csc( δπ ) + λ ¯ cπ R (cid:82) R (cid:0) (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) A R d ( (cid:107) x − y (cid:107) )d x (cid:1) d y PPP δπ λ (1 − δ ) (cid:15) δ − csc( δπ ) β -GPP δπ λ (1 − δ ) (cid:15) δ − csc( δπ ) − λ (cid:82) R (cid:16) (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) exp (cid:0) − πλ (cid:107) x − y (cid:107) /β (cid:1) d x (cid:17) d y -2 -1 Density of Interferers V a r i an c e MCPPPP1-GPP
Fig. 11:
Interference variance in ad hoc networks with different fieldsof interferers ( (cid:96) ( x ) = x α and R d = 1 ). and (cid:36) β -GPP ( λ )= λ (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) exp (cid:18) − πλ (cid:107) x − y (cid:107) β (cid:19) d x (cid:19) d y. Remark 1 : The interference correlation coefficient for the MCPis greater than for the PPP, since ζ MCP ( (cid:107) u (cid:107) ) − ζ PPP ( (cid:107) u (cid:107) ) = ( C − C ) (cid:36) MCP ( R d , ¯ c ) C ( C + (cid:36) MCP ( R d , ¯ c )) ( a ) > , where ( a ) holds as (cid:36) MCP ( R d , ¯ c ) > and C − C ≥ / [9], and C = (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x and C = 2 δπ (1 − δ ) (cid:15) δ − csc( δπ ) . Remark 2 : The interference correlation coefficient for the β -GPP is smaller than for the PPP, since ζ β -GPP ( (cid:107) u (cid:107) ) − ζ PPP ( (cid:107) u (cid:107) ) = ( C − C ) (cid:36) β -GPP ( λ ) C ( C − (cid:36) β -GPP ( λ )) ( a ) < , where ( a ) holds as C < C and C > (cid:36) β -GPP (since exp (cid:0) − πλ (cid:107) x − y (cid:107) /β (cid:1) < ). Fig. 12 shows the spatio-temporal correlation coefficient fordifferent fields of interferers, which illustrates the propertiesdiscussed above.
2) Moments of the CSP Φ in Ad Hoc Networks: Next, wederive the moments of the CSP Φ in three different random fieldsof interferers. For this, • We start by obtaining the CSP Φ by averaging out therandomness of channel gains for a given point process Φ and a link distance. • We then derive the moments of the CSP Φ by averagingover the spatial distributions of the interferers in thedifferent random fields. Specifically, – after averaging over the channel gains of the contactlink and interfering links based on their distributions,the moments of the CSP Φ can be represented by theexpectation of the product of a function w.r.t. thelocations of the interferers, i.e., E (cid:2) (cid:81) x ∈ Φ f ( x ) (cid:3) . – the expectation for Mat´ern cluster and Poisson fieldsof interferers can be derived based on the PGFLs ofMCP and PPP given in (34) and (23), respectively,and that for the β -Ginibre field of interferers can bederived based on the distributions of the distances ofthe interfering links given in (40). Theorem 2.
The moments of the CSP Φ for a Mat´ern clusterfield, Poisson field and β -Ginibre fields of interferers are givenby (49), where V b ( x, θ ) = (cid:90) B (0 ,R d ) (cid:18) θr α t (cid:107) x − y (cid:107) − α (cid:19) b d y. (50) Proof.
See
Appendix A .Fig. 13 shows the average success probabilities (i.e., M P s (1) ) in ad hoc networks with different fields of interferers.In order to reveal the entire SIR distribution, we plot thesuccess probabilities in the M¨obius homeomorphic (MH) scale.The conversion from linear scale to MH scale is given by thefunction x MH = x − x [92], which maps the one-sided infinitesupport [0 , ∞ ) to the unit interval [0 , . It can be observedthat, in ad hoc networks, spatial repulsion and attraction among E (cid:104) I ( t ) o I ( t ) u (cid:105) = λ (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x + δ π λ (cid:15) δ − csc( δπ ) + λ ¯ cπ R × (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) A R d ( (cid:107) x − y (cid:107) )d x (cid:19) d y, MCP λ (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x + δ π λ (cid:15) δ − csc( δπ ) , PPP λ (cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x + δ π λ (cid:15) δ − csc( δπ ) − λ (cid:90) R (cid:18)(cid:90) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( y ) exp (cid:18) − πλ (cid:107) x − y (cid:107) β (cid:19) d x (cid:19) d y, β -GPP . (47) ζ ( (cid:107) u (cid:107) ) = (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x + (cid:36) MCP ( R d , ¯ c )2 δπ (1 − δ ) (cid:15) δ − csc( δπ ) + (cid:36) MCP ( R d , ¯ c ) , MCP (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x δπ (1 − δ ) (cid:15) δ − csc( δπ ) , PPP (cid:82) R (cid:96) (cid:15) ( x ) (cid:96) (cid:15) ( x − u )d x − (cid:36) β -GPP ( λ )2 δπ (1 − δ ) (cid:15) δ − csc( δπ ) − (cid:36) β -GPP ( λ ) , β -GPP , (48) C o rr e l a t i on C oe ff i c i en t MCPPPP-GPP
Fig. 12:
Correlation coefficient in ad hoc networks with different fieldsof interferers ( λ = 0 . , (cid:15) = 1 and R d = 1 ). the interferers result in lower and higher success probabilities,respectively, compared to the independently located interferers.This can be intuitively understood from the fact that strongerspatial attraction (repulsion) increases the chance that theinterferers are located further away (closer to) the target receiver(as shown in Fig. 10).Subsequently, we investigate the temporal dependence ofsuccessful transmissions by evaluating the temporal CSP, i.e., M P s (2) / M P s (1) , in Fig. 14. It can be found that the CSPincreases when the interferers are more clustered (i.e., withsmaller R d or larger ¯ c ) and decreases when the interferers aremore scattered (i.e., with larger β ). The reason is that spatial attraction and repulsion cause a lower and higher intensityof aggregated interference at the target receiver, and thus atransmission attempt is more likely to succeed given a previoussuccessful transmission.Furthermore, we investigate the SIR meta distribution for atarget link in ad hoc networks. According to the Gil-Pelaez the-orem [93], the exact SIR meta distribution can be representedas an integral function of the moments of the CSP Φ as ¯ F ( θ, x ) = 12 + 1 π (cid:90) ∞ (cid:61) (cid:0) e − ux M u ( θ ) (cid:1) u d u, (51)herein (cid:61) ( z ) is the imaginary part of z and = √− denotesthe imaginary unit.Fig. 15 shows the SIR meta distribution in ad hoc networkswith different fields of interferers. Compared with the successprobability in Fig. 13, the SIR meta distribution gives the entiredistribution of the CSP Φ . For example, for a . MH SIRthreshold, although the average success probability for PPP isslightly greater than that of β -GPP, the percentage of the linksachieving reliability in PPP is higher than twice of that in β -GPP.
3) Moments of the CSP Φ in Downlink Cellular Networks: With Rayleigh fading, the exact moments of the CSP given Φ M and Φ G β can be obtained using the following steps: • deriving the CSP Φ by averaging over the randomness ofthe channel gains of the contact link and interfering linksbased on the PDF of the exponential distribution due toRayleigh fading; • deriving the moments of the CSP Φ by deconditioning onthe spatial distributions of the interferers in the differentrandom fields and contact distance. Specifically, – the spatial randomness of the interferers in Mat´erncluster and Poisson downlink networks is averaged M P s ( b ) = exp (cid:32) − λ p (cid:90) R (cid:18) − exp (cid:18) − ¯ c + ¯ cV b ( x, θ ) πR (cid:19)(cid:19) d x (cid:33) , MCP exp (cid:32) − πλθ δ r Γ(1 − δ )Γ( b + δ )Γ( b ) (cid:33) , PPP (cid:90) ∞ e − πλq/β (cid:18) β θr α t q − α/ + 1 − β (cid:19) b (cid:89) j ≥ ( πλ/β ) j Γ( j ) q j − d q, β -GPP (49) SIR Threshold (MH) S u cc e ss P r obab ili t y MCPPPP-GPP
Fig. 13:
Success probability in different ad hoc networks ( λ = 0 . , r t = 1 , and R d = 1 . The curves and the markers correspond to theanalytical and simulation results, respectively.). out based on the reduced PGFL of MCP and PPP,respectively, given in (35) and (23), and that inGinibre downlink networks is averaged out based onthe PDF of the distances of the interfering links givenin (40); – based on the nearest-BS associated rule, the contactdistances in Mat´ern cluster, Poisson, and Ginibredownlink networks are averaged out based on theirPDFs given in (32), (18), and (40), respectively.For non-Poisson networks, since deriving the exact downlinksuccess probability is tedious (if not impossible) and the re-sulting expressions are cumbersome, we use an approximationmethod, referred to as Approximate SIR analysis based on thePPP (ASAPPP) method [39], [94], [97], to simplify the evalua-tion of the SIR distribution. ASAPPP provides an approximateSIR distribution in a non-Poisson network, obtained from theSIR distribution in a Poisson network along with a horizontalshift (in the dB scale). This method is based on the insight thatwhen the disparity between the target system model and thePoisson model purely lies in the spatial configuration of thepoints, shifting the SIR threshold θ of the Poisson model by acoefficient G results in a close approximation of the successprobability and the meta distribution of the target system model.The subscript in G indicates that the shift is calculated when θ approaches . In other words, the ASAPPP method becomesexact as θ → . As shown in [79], ASAPPP yields a very goodapproximation across the whole SIR distribution and is barelysusceptible to the fading and path-loss models.The asymptotic gain G can be obtained by taking the ratioof the mean interference-to-signal ratios (MISRs) of the pointprocess considered and that of the PPP as [78] G = MISR
PPP
MISR = 2 α − MISR , (52)where MISR is defined asMISR (cid:44) E (cid:34) (cid:80) ∞ k =2 (cid:107) x k (cid:107) − α (cid:107) x (cid:107) − α (cid:35) , (53)and MISR PPP is the MISR in Poisson downlink networks, whichcan be obtained by utilizing the distance ratio distribution asfollows [78]:MISR
PPP = ∞ (cid:88) j =2 E (cid:20)(cid:18) r r j (cid:19) α (cid:21) = ∞ (cid:88) j =2 E (cid:2) (cid:37) αj (cid:3) ( a ) = ∞ (cid:88) j =2 (cid:90) (cid:37) α j − (cid:37) (1 − (cid:37) ) j − d (cid:37) = ∞ (cid:88) j =2 Γ(1 + α/ j )Γ( j + α/ α − , (54)where ( a ) follows the PDF of (cid:37) j given in (22).The numerical value of G can be obtained easily fromMonte Carlo simulations for a given network geometry, i.e., λ p , ¯ c , and R d for the MCP and λ and β for the β -GPP. Notethat, as found in [4], the MISR-based gain of the β -GPP canbe accurately approximated as G ≈ β/ , (55)which is insensitive to the network density and path-lossexponent [79]. Therefore, the success probability of the typicaluser in a β -GPP network is approximately the same as thatin a Poisson network with the SIR threshold scaled from θ to θ/ (1 + β/ . C ond i t i ona l S u cc e ss P r obab ili t y (a) MCP ( λ = 0 . ) C ond i t i ona l S u cc e ss P r obab ili t y (b) β -GPP ( λ = 0 . ) Fig. 14:
Temporal CSP in ad hoc networks with non-Poisson fields of interferers ( α = 4 ). Fig. 15:
SIR meta distribution in ad hoc networks with different fieldsof interferers. ( θ = 0 . MH, r t = 1 , λ = 0 . , and R d = 1 . The curvesand the markers correspond to the analytical results and the simulationresults, respectively.) Fig. 16 and Fig. 17, respectively, confirm the effectivenessof the ASAPPP method for approximating the SIR distributionand the SIR meta distribution in MCP and β -GPP downlinknetworks. As expected, for both types of networks, the ASAPPPmethod yields a more accurate approximation of SIR distribu-tion as the SIR threshold becomes small. For the MCP model,we can observe that the ASAPPP method is more accurate when Φ M is less clustered (e.g., with smaller ¯ c given λ ). When thenetwork is more clustered (e.g., when ¯ c = 10 ), there exists anobservable gap between the approximation and the simulationresults for both success probability and SIR meta distribution.The reason is that a higher degree of clustering results in aslowly growing rate of the asymptotics [97]. Moreover, for the β -GPP, the insignificant disparity between the simulation andthe approximation results can be ascribed to the approximationof the MISR-based gain given in (55).Next, in Fig. 18(a) and Fig. 18(b), we evaluate the temporalCSPs for MCP and β -GPP downlink networks, respectively.We observe that the CSP decreases slightly when the networkbecomes more clustered (i.e., with larger ¯ c and/or smaller R d with fixed λ ) and increases slightly when the networkbecomes more repulsive (i.e., with larger β ). However, thecurves are almost flat, i.e., the temporal CSPs in Mat´ern clusterand Ginibre downlink networks are generally non-sensitive toattractiveness and repulsiveness of the spatial points. C. Summary and Discussion
We have analyzed the impact of the spatial distributionof transmitters on the interference correlation coefficient andsuccess probability. Specifically, we have presented the deriva-tions of spatio-temporal interference correlation coefficient andsuccess probability and SIR meta distribution of a target linkin MCP, PPP, and β -GPP fields of interferers. We have alsointroduced an approximation of the success probabilities andSIR meta distributions in non-Poisson networks based on theanalysis of a Poisson network by scaling the SIR threshold.The main observations are as follows. • With non-Poisson fields of interferers, the spatio-temporalinterference is more and less correlated when the inter-ferers are distributed more attractively and repulsively,respectively. • Compared with a Poisson field of interferers, the Mat´erncluster and β -Ginibre field of interferers increases anddecreases the success probability of a target link, respec-tively. Moreover, the successful transmission events aremore and less temporally-correlated when the locations ofthe interferers are more clustered and scattered, respec-tively. SIR Threshold (MH) S u cc e ss P r obab ili t y Fig. 16:
The ASAPPP approximation of non-Poisson cellular net-works ( α = 4 , λ = 0 . , R d = 4 ). -GPP, SimulationASAPPP for -GPPPPP, SimulationPPP, AnalysisMCP, SimulationASAPPP for MCP Fig. 17:
The ASAPPP approximation of SIR meta distribution innon-Poisson cellular networks ( θ = 0 . MH, α = 4 , λ = 0 . , R d = 5 ). C ond i t i ona l S u cc e ss P r obab ili t y AnalysisSimulation (a) MCP C ond i t i ona l C o v e r age P r obab ili t y (b) β -GPP Fig. 18:
Temporal CSP in non-Poisson cellular networks ( α = 4 , λ = 0 . ). • Compared with a Poisson downlink network, the Mat´erncluster and Ginibre downlink networks render a lower anda higher success probability, respectively. Additionally,the successful transmission events are less and moretemporally-correlated when the interferers are more attrac-tive and repulsive, respectively.
Open Technical Issues : The point process models presentedin Section III have been adopted to characterize the generalspatial features (e.g., independence, repulsion and clustering)of wireless networks. In addition to these generic models, somepoint process models have been established for some special-ized network scenarios. For example, reference [95] proposesan energized point process to model the spatial distribution ofwireless-powered devices. The model exhibits spatial correla- tion of the RF-powered nodes that can harvest sufficient energyfrom a Poisson field of RF sources. Besides, reference [96]introduces a
Poisson rain process to model cellular networkswith spatio-temporal traffic. Specifically, the model employsa space-time PPP to model traffic arrives which are thenassigned to PPP-distributed BSs based on different allocationschemes. Future efforts should be dedicated to developing pointprocesses that incorporate spatio-temporal features (e.g., fading,shadowing, blockage, and renewable energy arrivals) for morespecific system models.IV. L
OCATION -D EPENDENT A NALYSIS OF C ELLULAR M ODELS
The locations of the users play a crucial part in theirperformance. Conditioning the regions (e.g., cell center and boundary) of the user of interest allows a location-dependentanalysis [99], [100] for a location-specific user (LSU) in cellularnetworks. The objective of this section is to demonstrate theeffect of the location of the target user on the SIR distribution.For this, we derive the moments of the CSP Φ and the SIR gainfor an LSU in different regions. A. System Models
For a stationary point process Φ ⊂ R of BSs, let x j ( u ) ∈ Φ represent the location of the j -th nearest BS to the LSU at u .Under the nearest-BS association principle, x ( u ) and x ( u ) are the locations of the associated (serving) BS and the nearestinterfering BS to the LSU, respectively. The region of a user’slocation can be defined by the relationship between the linkdistances to the associated BS and to the nearest interferingBS.For ρ ∈ [0 , , the regions of cell-center user and cell-boundary user are defined, respectively, as R c (cid:44) { u ∈ R : (cid:107) x ( u ) − u (cid:107) ≤ ρ (cid:107) x ( u ) − u (cid:107)} , R b (cid:44) { u ∈ R : (cid:107) x ( u ) − u (cid:107) > ρ (cid:107) x ( u ) − u (cid:107)} . The area fraction of a region can be determined by theprobability that an arbitrary location, say o , falls into thatregion, which solely depends on ρ . For the analysis, we focuson the case where Φ is a PPP.The area fractions of R c and R b can be obtained based onthe CDF of (cid:37) (given in (21)) as P [ o ∈ R c ] = P [ (cid:37) ≤ ρ ] = F (cid:37) ( ρ ) = ρ and P [ o ∈ R b ] = 1 − P [ o ∈ R c ] = 1 − ρ . Moreover, in the special cases when a cell-boundary userhas two or three equidistant closest BSs, it is referred to as an edge user or a vertex user , respectively. The regions of the edgeusers (one-dimensional) and vertex users (zero-dimensional) aredefined, respectively, as R e = { u ∈ R : (cid:107) x ( u ) − u (cid:107) = (cid:107) x ( u ) − u (cid:107)} , R v = { u ∈ R : (cid:107) x ( u ) − u (cid:107) = (cid:107) x ( u ) − u (cid:107) = (cid:107) x ( u ) − u (cid:107)} . (56)Note that in a two-dimensional homogeneous PPP there existsno location with more than three nearest BSs almost surely [84].In the following, we explore and the performance at fivedifferent types of typical users: (i) The standard typical user,whose performance correspond to the average of all users; itis referred to in this section as the typical general user; (ii)the typical cell-center user, whose performance correspondsto the average of all cell-center users; (iii) the typical cell-boundary user, whose performance corresponds to the averageof all cell-boundary users; (iv) the typical cell-edge user whoseperformance corresponds to the average of all locations on thecell edges; and (v) the typical vertex user, whose performancecorresponds to the average of all vertex points of the cells. In Poisson cellular networks, the PDF of the contact distanceof the typical vertex user is given by [101], [102] f v r ( r ) = 2( λπ ) r exp( − λπr ) . (57)The success probability of the typical cell-center and cell-boundary users can be defined, respectively, as ¯ F c η ( θ ) = P [ η > θ | o ∈ R c ] , and ¯ F b η ( θ ) = P [ η > θ | o ∈ R b ] . (58)Fig. 19 shows a realization of network partitions of a Poissonnetwork with unit intensity, where the locations of the BSsare represented by the black circles. Blank and cyan regionsrepresent the cell-center regions R c and cell-boundary regions R b , respectively. The blue triangle markers represent the regionof the vertex users and blue lines excluding the verticesrepresent the region of the edge users. -2 -1 0 1 2-2-1.5-1-0.500.511.52 Fig. 19:
Illustration of network partitions in a two-dimensional Poissonnetwork for ρ = 0 . . B. Performance Analysis1) Cell-Center and Cell-Boundary User:
We start by show-ing how to obtain the b -th moments of the CSP Φ for cell-centerand cell-boundary users, denoted as M c P s ( b ) and M b P s ( b ) ,respectively. The methodology of obtaining the former is toderive the SIR meta distribution for the typical general user(e.g., as in [77]) by conditioning its location to be in the cell-center and cell-boundary regions. Such conditioning restrictsthe distribution regions of interferers which will be reflected inthe lower bounds of the integrals after applying the PGFL ofPPP.Based on this approach, we obtain M c P s ( b ) in the followingtheorem. Theorem 3.
The moments of the CSP Φ for the typical cell-center user, i.e., the typical general user conditioned on R c , ina Poisson downlink network with Rayleigh fading are M c P s ( b ) = 1 F ( b, − δ ; 1 − δ ; − ρ α θ ) , (59)where δ = 2 /α . Proof.
See
Appendix B . Remark 3 : The SIR gain of the typical cell-center usercompared to the typical general user is ρ − α , which is easilyobtained by comparing (59) with the moments of the CSP Φ forthe typical general user in a Poisson downlink network givenas [77]: M P s ( b ) = 1 F ( b, − δ ; 1 − δ ; − θ ) . (60) Remark 4 : A fraction x = ρ of users who have the lowestdistance ratio of the associated BS to the interfering BS has anSIR gain of − α log x dB.Applying the law of total probability, the moments of theCSP Φ for the typical general user can be expressed as follows: M P s ( b ) = E (cid:104) P (cid:2) η > θ (cid:3) b (cid:105) = E (cid:104) P (cid:2) η > θ (cid:3) b (cid:12)(cid:12) o ∈ R c (cid:105) P (cid:2) o ∈ R c (cid:3) + E (cid:104) P (cid:2) η > θ (cid:3) b (cid:12)(cid:12) o ∈ R b (cid:105) P (cid:2) o ∈ R b (cid:3) = M c P s ( b ) P (cid:2) o ∈ R c (cid:3) + M b P s ( b ) (cid:16) − P (cid:2) o ∈ R c (cid:3)(cid:17) . Then, the moments of the CSP Φ for the typical cell-boundaryuser are given by M b P s ( b ) = M P s ( b ) − M c b ( θ ) P [ o ∈ R c ]1 − P [ o ∈ R c ]= M P s ( b ) − ρ M c P s ( b )1 − ρ . (61)We then have the following corollary by using (60) and (59)in (62). Corollary 1.
The moments of the CSP Φ of the typical cell-boundary user, i.e., o ∈ R b , in a Poisson downlink networkwith Rayleigh fading are M b P s ( b ) = 1(1 − ρ ) F ( b, − δ ; 1 − δ ; − θ ) − ρ (1 − ρ ) F ( b, − δ ; 1 − δ ; − ρ α θ ) , (62)where δ = 2 /α .
2) Edge User:
An edge user can be considered as anasymptotic case of a cell-boundary user when ρ → , i.e., P [ o ∈ R b ] → . In this case, the cell-boundary user isequidistant from the serving BS and the nearest interferer, i.e.,on the cell edge. Therefore, the performance of the typical cell-edge user can be obtained from the asymptotics of the typicalcell-boundary user as ρ → . By taking the limit ρ → of M b P s ( b ) in (62), we have themoments of the CSP Φ for the typical edge user as follows: M e P s ( b ) ( a ) = lim ρ → M b P s ( b )= 1 F ( b, − δ ; 1 − δ ; − θ ) − bθ F ( b + 1 , − δ ; 2 − δ ; − θ )(1 − δ ) F ( − b, − δ ; 1 − δ ; − θ ) , (63)where ( a ) follows from L’Hˆopital’s rule.After some mathematical manipulations, we have M e P s ( b ) inthe following corollary. Corollary 2.
With Rayleigh fading, the moments of the CSP Φ for the typical edge user in a Poisson downlink network are M e P s ( b ) = 1(1 + θ ) b F ( b, − δ ; 1 − δ ; − θ ) , b ∈ C , (64)where δ = 2 /α . Remark 5 : Compared with M P s ( b ) in (59), M e P s ( b ) in (64)can be expressed as M e P s ( b ) = M P s ( b ) (1+ θ ) b , ∀ b ∈ R + . M e P s ( b ) issmaller than M P s ( b ) and their gap depends only on θ and b .
3) Vertex User:
Finally, we compute the performance ofthe typical vertex user, which represents the worst-case per-formance. To obtain the moments of the CSP Φ , denoted as M v P s ( b ) , we follow the derivation steps similar to those forthe typical general user with the condition that the user isequidistant from the serving BS and the two nearest interferingBSs. Under this condition, the contact distance is averaged outbased on its PDF given in (57). With this methodology, wehave M v P s ( b ) given in the following theorem. Theorem 4.
With Rayleigh fading, the moments of the CSP Φ for the typical vertex user in a Poisson downlink network are M P s ( b ) = 1(1 + θ ) b F ( b, − δ ; 1 − δ ; − θ ) , (65)where δ = 2 /α . Proof.
See
Appendix C . Remark 6 : Compared with (116), the success probability ofthe typical vertex user in (65) can be expressed as M v P s ( b ) = M e P s ( b )(1+ θ ) b . M v P s ( b ) is smaller than M e P s ( b ) due to the twoequidistant closest interfering BSs.Fig. 20 depicts the average success probabilities (i.e., M P s (1) ) of the LSUs. It is straightforward that the successprobability monotonically decreases with ρ . The weighted meanof the success probabilities for the typical cell-center user andthe typical cell-boundary user with the same value of ρ areequal to that of the typical general user.
4) Temporal Effect of Location Dependence:
Next, we inves-tigate how location dependence affects the temporal correlationamong different successful transmission events by evaluatingthe CSP M P s (2) M P s (1) . Fig. 21 shows how the CSP varies with ρ . Itis found that the CSP for the typical cell-center user is greaterthan the typical cell-boundary user. Moreover, for both typical SIR Threshold (MH) S u cc e ss P r obab ili t y Cell-Center UserCell-Boundary UserGeneral User =0.8,0.6,0.4,0.2=0.8,0.6,0.4,0.2 (a) Typical general user and cell-center/boundary user
SIR Threshold (MH) S u cc e ss P r obab ili t y Edge User, =4Vertex User, =4Edge User, =3Vertex User, =3 (b) Typical edge user and vertex user
Fig. 20:
Success probability of LSUs in Poisson downlink networks ( λ = 1 , α = 4 ). C ond i t i ona l S u cc e ss P r obab ili t y Cell-center userCell-boundary user
Fig. 21:
The temporal CSP of LSUs. cell-center and cell-boundary users, their CSPs decrease with ρ . The reason is that the successful transmission events aremore temporally correlated when the received signal dominatesthe interference. Such temporal correlation decreases when theinterference becomes stronger.
5) SIR Gain:
Next, we study the asymptotic SIR gain toillustrate the SIR improvement over the typical general userdue to location dependence. The MISR of the typical cell-centeruser can be derived asMISR R c = ∞ (cid:88) i =2 E (cid:20)(cid:18) r r i (cid:19) α | o ∈ R c (cid:21) ( b ) = E (cid:20)(cid:18) r r (cid:19) α | o ∈ R c (cid:21) ∞ (cid:88) i =2 E (cid:20)(cid:18) r r i (cid:19) α (cid:21) , = E (cid:104)(cid:16) r r (cid:17) α , o ∈ R c (cid:105) ρ ∞ (cid:88) i =2 E (cid:20)(cid:18) r r i (cid:19) α (cid:21) , (66)where ( b ) holds as the condition that the typical general user isin the cell-center region, i.e., o ∈ R c , only affects the relativedistance between r and r .The first expectation in (66) can be derived based on the jointdistribution of r and r as E (cid:104)(cid:16) r r (cid:17) α , o ∈ R c (cid:105) = (cid:90) ρ f v ( v ) v α d v = 2 ρ α α . (67)The second expectation in (66) can be obtained by exploitingthe distribution of the distance ratios of a PPP as ∞ (cid:88) j =2 E (cid:20)(cid:18) r r i (cid:19) α (cid:21) = (cid:80) ∞ j =2 E (cid:104)(cid:16) r r i (cid:17) α (cid:105) E (cid:104)(cid:16) r r (cid:17) α (cid:105) ( c ) = 2 + α ∞ (cid:88) j =2 (cid:90) v αi f i ( v )d v = 2 + α (cid:90) v α v − d v = α + 2 α − , (68)where ( c ) follows the distribution of (cid:37) .Plugging (67) and (68) into (66) yieldsMISR R c = 2 ρ α α − . (69)Subsequently, with MISR PPP and MISR R c given in (54) and(69), respectively, the SIR gain of the typical cell-center usercan be obtained as G R c = MISR
PPP
MISR R c = ρ − α . (70) Similarly, the SIR gains of different LSUs can be obtained asin Table VI.
Remark 7 : Since α > and ρ ∈ [0 , , the SIR gain of thetypical cell-center user is greater than or equal to one and theequality holds when ρ = 1 . Moreover, the SIR gain of thetypical cell-boundary user is smaller than or equal to one andthe equality holds when ρ = 0 . It can be found that the influenceof ρ on the SIR gain of the typical cell-center user is muchmore significant than that of the typical cell-boundary user.Moreover, the gap between the SIR gains of the typical edgeuser and vertex user increases with α . C. Summary and Discussion
We have discussed the impact of the relative distance of arandom user on its performance in Poisson downlink networks.Based on the relationship between the contact distance and thedistance to the two nearest interferers, a user can be categorizedas cell-center, cell-boundary, cell-edge, and vertex user. Wepresent the derivations of the moments of the CSP Φ for thetypical cell-center, cell-boundary, cell-edge and vertex usersin two-dimensional Poisson downlink networks. To show thedirect impact of the location on the SIR, we also derive theSIR gain of the four types of users. The major findings are asfollows. • The gaps among the b -th moments of the CSP Φ for thetypical cell-center, cell-edge and vertex users depend onlyon b and the SIR threshold θ , i.e., M c P s ( b ) = (1 + θ ) b M e P s ( b ) = (1 + θ ) b M v P s ( b ) , ∀ b ∈ R + . • The successful transmission events are more temporallycorrelated for the typical cell-center user than the typicalcell-boundary user. • The SIR gains for the typical cell-center and cell-boundaryuser depend on the path-loss exponent and location depen-dence coefficient ρ while those for the typical cell-edgeand vertex user only depend on the path-loss exponent. Open Technical Issues : In the literature, a location-dependentanalysis has been carried out [100], and BS cooperationschemes have been designed based on the location-dependentmodeling [99]. However, the location-dependent performancefor Poisson uplink networks remains unexplored. Characteriz-ing the location-dependent uplink user is more challenging asthe uplink model is not fully tractable [59]. Another interestingdirection is to investigate the performance of LSUs in non-Poisson cellular networks, e.g., when the transmitters exhibitrepulsive or attractive distribution.Additionally, cell-free communication systems wheredensely deployed BSs could simultaneously serve a number ofusers have emerged as a practical solution for future-generationcommunication systems. In such an infrastructure, the BScooperation schemes need to take into account the userlocations. For example, it is more meaningful to assign moreresources for cell-boundary users to improve the their successprobabilities. Thus, location-dependent analysis is the key tothe design of BS cooperation schemes in cell-free massiveMIMO systems. V. S
PATIALLY C ORRELATED AND I NDEPENDENT S HADOWING
The main objective of this section is to investigate the effectof correlated and independent shadowing in large-scale systemsbased on stochastic geometry analysis.
A. System Model
To model the effect of shadowing, we consider that the wholeplane is partitioned into a set of deterministic shadowing cells S = { S k ⊂ R } k ∈ N in which the transmitters experiencesimilar shadowing. We consider an ad hoc network with aPoisson field of interferers Φ introduced in Section III-A asan example system model for the analysis in this section.The channel gain from x j ∈ Φ to o is h j S j (cid:96) ( (cid:107) x j (cid:107) ) , where h j and S j are the fading coefficient and shadowing coefficient,respectively, from x j to o , and (cid:96) ( (cid:107) x j (cid:107) ) = (cid:107) x j (cid:107) − α is the pathloss function. With a Poisson field of interferers, the aggregatedinterference at the receiver of interest is I o = (cid:88) j ∈ N h j S j (cid:96) ( (cid:107) x j (cid:107) ) , (71)and the SIR follows as η = h t S t (cid:96) ( (cid:107) x t (cid:107) ) I o , (72)where S t denotes the shadowing coefficient of the link underconsideration.Let c ( j ) be the cell index that x j resides in, i.e., c ( j ) = k if x j ∈ S k . We consider the extreme cases of (fully) correlatedand independent intra-cell shadowing. In independent shadow-ing, the S c ( j ) are all independent and distributed with CDF F c ( j ) . In correlated shadowing, S c ( j ) and S c ( i ) are independentonly if c ( j ) (cid:54) = c ( i ) , and S c ( j ) = T k for all x j ∈ S k . Bychoosing F k , different shadowing properties can be assignedto the individual cells. B. Performance Analysis
We first characterize the Laplace transform of the interfer-ence. Then, based on it, we derive the mean and variance ofthe interference. Furthermore, we obtain the moments of theCSP Φ under both correlated and independent shadowing.
1) Laplace Transform of the Interference:
To analyze theinterference distribution, we start by characterizing the Laplacetransform of the aggregated interference at the target receiver atthe origin with correlated and independent shadowing, denotedby I Cor o and I Ind o , respectively. Similar to the derivation stepsof the moments of the CSP given a Mat´ern cluster field of in-terferers presented in Appendix A , we first express the Laplacetransform as the expectation of the product of a function of thelocations of the interferers given the distribution of the shadow-ing coefficients. Then, by the PGFLs of MCP given in (34), weconvert the expectation into an integral expression as a functionof the shadowing coefficients. Subsequently, the randomness ofthe shadowing can be averaged out by conditioning on that thedaughter points in each cluster are associated with the same and TABLE VI:
MISR and SIR gain of location-specific users
User Location MISR SIR GainTypical general user / ( α − ρ α / ( α − ρ − α Typical cell-boundary user − ρ α +2 ) / ( α − − ρ ) (1 − ρ ) / (1 − ρ α +2 ) Typical edge user ( α + 2) / ( α −
2) 2 / ( α + 2) Typical vertex user α/ ( α −
2) 1 /α independent shadowing coefficients with correlated shadowingand independent shadowing, respectively.Based on this methodology, the Laplace transforms of I Cor o and I Ind o are given in the following theorem. Theorem 5.
In a Rayleigh fading environment with a Poissonfield of interferers, the conditional Laplace transform of theinterference at the target receiver is L I Cor o ( s ) = (cid:89) k ∈ N E T k (cid:34) exp (cid:18) − λ (cid:90) S k (cid:18) − s(cid:96) ( (cid:107) x (cid:107) ) T k (cid:19) d x (cid:19)(cid:35) , (73)with correlated shadowing, and L I Ind o ( s ) = (cid:89) k ∈ N exp (cid:32) − λ (cid:90) S k (cid:18) − E T k (cid:20) s(cid:96) ( (cid:107) x (cid:107) ) T k (cid:21)(cid:19) d x (cid:33) , (74)with independent shadowing. Proof.
See
Appendix D .According to Jensen’s inequality, i.e., the convex transforma-tion of an expectation equals or exceeds the expectation overthe convex transformation, we have the inequality in (75). Fur-thermore, since exp( − x ) is a completely monotone function,we readily obtain the following observation by comparing (73)and (74). Remark 8 : For all s > , L I Cor o ( s ) > L I Ind o ( s ) .
2) Mean and Variance of Interference:
Due to the singularityof the path-loss function (cid:96) ( x ) , we adopt (cid:96) (cid:15) ( x ) = (cid:15) + (cid:107) x (cid:107) α , α > , (cid:15) > , to evaluate the mean and variance of interference,similar to Sec. III-B1. As the mean of a random variable canbe derived by taking the first derivative of its Laplace transformw.r.t. s and setting s = 0 , i.e., E [ X ] = − d L X ( s )d s | s =0 , we caneasily obtain the mean interference as E (cid:104) I Ind o (cid:105) = E (cid:104) I Cor o (cid:105) = λ (cid:88) k ∈ N E [ T k ] (cid:90) S k (cid:15) + (cid:107) x (cid:107) α d x, which shows that the mean interference at the typical receiverwith independent shadowing and correlated shadowing areidentical, i.e., I Ind o = I Cor o .The same result can also be obtained from Campbell’sTheorem [3, Thm. 4.1], which shows that it holds for allstationary point process and arbitrary fading.As the second moment of a random variable can be derivedby evaluating the second derivative of its Laplace transform w.r.t. s at s = 0 , the variance of I Ind o can be obtained based onthe Laplace transform as V [ I Ind o ] = E (cid:104)(cid:0) I Ind o (cid:1) (cid:105) − E (cid:104) I Ind o (cid:105) = d L I Ind o ( s )d s (cid:12)(cid:12) s =0 − (cid:18) d L I Ind o ( s )d s (cid:12)(cid:12) s =0 (cid:19) = 2 λ (cid:88) k ∈ N E (cid:2) T k (cid:3) (cid:90) S k (cid:0) (cid:15) + (cid:107) x (cid:107) α (cid:1) d x + λ (cid:88) k ∈ N E [ T k ] (cid:18) (cid:90) S k (cid:15) + (cid:107) x (cid:107) α d x (cid:19) . (76)In the same manner, the variance of I Cor o can be derived as V [ I Cor o ] = 2 λ (cid:88) k ∈ N (cid:90) S k E (cid:2) T k (cid:3)(cid:0) (cid:15) + (cid:107) x (cid:107) α (cid:1) d x + λ (cid:88) k ∈ N E [ T k ] (cid:18) (cid:90) S k (cid:15) + (cid:107) x (cid:107) α d x (cid:19) + λ (cid:88) k ∈ N V [ T k ] (cid:18) (cid:90) S k (cid:15) + (cid:107) x (cid:107) α d x (cid:19) a ) = V [ I Ind o ]+ λ (cid:88) k ∈ N V [ T k ] (cid:18) (cid:90) S k (cid:15) + (cid:107) x (cid:107) α d x (cid:19) (cid:124) (cid:123)(cid:122) (cid:125) > , (77)where ( a ) follows from (76). Comparing (76) and (77) yieldsthe following observation. Remark 9 : The variance of the interference at the typicalreceiver with correlated shadowing is greater than that withindependent shadowing, i.e., V [ I Cor o ] > V [ I Ind o ] .
3) Moments of the CSP Φ : Given the point process Φ , theCSP is P [ η > θ | Φ] = P (cid:20) h t S t r − α t I o > θ (cid:12)(cid:12)(cid:12) Φ (cid:21) ( a ) = E S t , ( S j ) (cid:34) (cid:89) j ∈ N (cid:18) θr α t r − αj S j S t (cid:19) − (cid:35) , where ( a ) follows the derivations of (108) in Appendix A .Then, the moments of the CSP Φ can be represented as M P s ( b ) = E (cid:34) (cid:89) j ∈ N (cid:18) θr α t r − αj S j S t (cid:19) − b (cid:35) . By following the same methodology to the proof of Theo-rem 5, we have M P s ( b ) derived in the following theorem. E T k (cid:20) exp (cid:18) − ¯ c (cid:90) S k (cid:16) −
11 + s(cid:96) ( (cid:107) x (cid:107) ) T k (cid:17) d x (cid:19)(cid:21) > exp (cid:18) − ¯ c (cid:90) S k (cid:18) − E T k (cid:20)
11 + s(cid:96) ( (cid:107) x (cid:107) ) T k (cid:21) d x (cid:19)(cid:19) . (75) Theorem 6.
In a Rayleigh fading environment, the conditionalmoments of the CSP Φ under correlated shadowing and inde-pendent shadowing are given, respectively, by (78) and (79),shown on the top of the next page. Remark 10 : It follows from Remark 8 that with Rayleighfading, the moments of the CSP Φ under correlated shadowingare greater than that with independent shadowing.Next, we numerically evaluate the network performanceunder correlated and independent shadowing assumptions. Weassume T k = κ N k ( x ) [103], where κ is the attenuation factoraccounting for the signal loss while penetrating a blockage(e.g., a wall) and N k ( x ) represents the number of blockagesbetween x and the receiver of interest. N k ( x ) is assumedto be a Poisson random variable for each link [104], i.e., N k ( x ) ∼ P ( λ b (cid:107) x (cid:107) ) , where λ b denotes the blockage density.With correlated shadowing, the transmitters in the same cell S k are associated with the same shadowing coefficient T k = κ N k ( d k,o ) , N k ( d k,o ) ∼ P ( λ b d k,o ) , where d k,o represents thedistance between the center of S k and the origin. Differently,with independent shadowing, the transmitters at x j ∈ S k areassociated with i.i.d. shadowing coefficients S j = κ N k ( d k,o ) , N k ( d k,o ) ∼ P ( λ b d k,o ) . Besides, we assume that the linkbetween the target receiver and the serving transmitter is subjectto no shadowing, i.e., S t = 1 . For the spatial configuration ofthe shadowing cells, we consider the example in Fig. 22, where S = { S k } k ∈ N is a set of squares with length L . It is worthnoting that the analytical framework can be extended to othertypes of point processes by changing the distribution of pointsas long as | S k | < ∞ , i.e., the point process has a finite numberof points in each shadowing cell almost surely.Figure 23 illustrates the success probability, i.e., M Cor P s (1) and M Ind P s (1) , as a function of the SIR threshold. It can beobserved that the success probability with correlated shadowingis greater than that with independent shadowing, which agreeswith Remark 10 . Moreover, the gap between the success prob-abilities with correlated and independent shadowing decreasesas the cell size (i.e., L ) becomes smaller, as in this casethe signals from the interferers nearly experience independentshadowing even with the correlated shadowing model.Furthermore, to evaluate the temporal effect of shadowing,we evaluate the temporal CSP with both correlated and indepen-dent fading, i.e., M Cor P s (2) M Cor P s (1) and M Ind P s (2) M Ind P s (1) , as shown in Figure 24.We can observe that correlated shadowing results in more tem-poral dependence between two successful transmission events.The gap of CSPs between correlated and independent shadow-ing decreases as the cell size L becomes larger. Moreover,the dependence between two successful temporal transmission events increases as the cell size becomes smaller, especiallywhen the SIR threshold is small. The reasons for the aboveobservations can be ascribed to the adopted shadowing model,which essentially leads to exponential path loss. C. Summary and Discussion
In this section, we have established an analytical frameworkto model the interference distribution and success probabilitiesin networks with spatially-correlated shadowing and spatially-independent shadowing. In particular, the analytical frameworkcharacterizes deterministic shadowing cells where the transmit-ters inside are associated with the same shadowing effects. Thekey findings are as follows. • Spatially-correlated shadowing generates the same inter-ference as but higher variance than spatially-independentshadowing. • The moments of the CSP Φ under spatially-correlatedshadowing is greater than that with spatially-independentshadowing. The performance increase reduces when thecell size shrinks. • The successful transmission events are more temporallycorrelated with spatially-correlated shadowing than withspatially-independent shadowing.
Open Technical Issues : Most of the existing literature, e.g.,[103], [105]–[108], only investigates network performance un-der the assumptions that the shadowing coefficients of theinterferers in the same shadowing cells are either fully spatiallycorrelated or independent. However, in practice, the interferersin the same shadowing cell may only experience partially cor-related shadowing effect due to their location difference. Moreaccurate shadowing models need to be developed by taking intoaccount the properties (e.g., shape, density and mobility) of theobstacles which may exhibit location-dependency (e.g., urbanand rural areas). Moreover, temporal shadowing variation due tomobility of obstacles (e.g., vehicles), access points (e.g., dronehot spots) or users is an important factor to be investigated inpractical systems.Different from the deterministic shadowing cells introducedin this section, reference [109] introduces a correlated shadow-ing model dependent on Poisson Voronoi cell. This model issuitable for studying coverage-oriented cellular networks whereBSs deployment is configured to guarantee the cell-boundaryusers to acquire sufficient signal strength. An intriguing futuredirection is to extend the study of cell-dependent correlatedshadowing to capacity-oriented cellular networks. M Cor P s ( b ) = (cid:89) k ∈ N E T k ,T c (t) (cid:34) exp (cid:18) − λ (cid:90) S k (cid:18) − (cid:18) θr α t (cid:107) x (cid:107) − α T k /T c (t) (cid:19) b (cid:19) d x (cid:19)(cid:35) , (78) M Ind P s ( b ) = (cid:89) k ∈ N E T c (t) (cid:34) exp (cid:32) − λ (cid:90) S k (cid:18) − E T k (cid:20)(cid:18) θr α t (cid:107) x (cid:107) − α T k /T c (t) (cid:19) b (cid:21)(cid:19) d x (cid:33)(cid:35) . (79) Fig. 22:
Configuration of shadowing cells for simulations.
VI. S
PATIO -T EMPORAL I NTERACTIONS B ETWEEN Q UEUES
The majority of the existing literature heavily relies onthe assumption that each transmitter always has packets inthe buffer to send out, which does not characterize randomtraffic flows. While the temporal randomness of the trafficflows complicates the analysis, it is nevertheless essential tounderstanding system-level performance. This is due to thefact that the traffic patterns in the evolving wireless networksare getting increasingly more dynamic and heterogeneous. Themain difficulty of random traffic characterization originatesfrom the correlation among the buffer statuses of differenttransmitters, often referred to as interacting queues . Since thequeues interact spatially and temporally, an exact analysis ofthe mutual interference is quite challenging.
A. System Model
We consider both Poisson downlink networks (as introducedin Section III-A) and
Poisson bipolar networks [3, Def. 5.8].In a Poisson downlink network, the transmitters (i.e., BSs)and receivers (i.e., users) are distributed following independenthomogeneous PPPs, denoted as Φ B = { x j } j ∈ N and Φ u withintensity λ B and λ u , respectively. The points in Φ B are assumedto be ordered from nearest to farthest to the origin, i.e., (cid:107) x j (cid:107) < (cid:107) x j +1 (cid:107) . It is assumed that each user is associated withits nearest BS for downlink transmission. In a Poisson bipolarnetwork, the transmitters are distributed as a homogeneous PPP Φ with intensity λ . Each transmitter is paired with one receiverin a uniformly random direction with a link distance r t . Withoutloss of generality, we study the performance of the typicalreceiver, conditioned to be at the origin, in both models. We consider a discrete-time transmission and queueingmodel. Specifically, the data transmissions are divided intoequal-duration time slots. We consider fixed-length data packetsand assume it takes exactly one time slot to send out onepacket. If a transmitter is scheduled for transmission, it canonly send out the accumulated packet(s) that arrived prior to thetransmission. In light of queueing, we assume that the incomingpackets at each queue are stored in a buffer with infinite sizeand sent out on a first-in-first-out basis. A transmitted packet isremoved from the head of the queue only if it is successfullydecoded at the target receiver.For Poisson downlink networks, each BS maintains anindividual queue for the arrived packets of each associateduser [30]. Hence, each BS has a number of queues equal tothe number of users in its Voronoi cell. The temporal arrivalof traffic at each queue follows an i.i.d. Bernoulli process witharrival rate ξ u representing the probability of a new arrival pertime slot. The users associated with the same BS are servedbased on random scheduling [30], i.e., each BS randomlyselects one user within its Voronoi cell with equal probabilityto serve in each time slot. If the selected user has a non-empty queue at the BS, the BS is scheduled for transmission.Otherwise, the BS is muted.For Poisson bipolar networks, the packet arrival at eachtransmitter x j ∈ Φ follows an i.i.d. Bernoulli process witharrival rate ξ . At each time slot, the transmitters with non-emptybuffer are all scheduled for transmission [110].Given that the typical receiver is receiving data, its SIRin Poisson downlink and Poisson bipolar networks are given,respectively, by η = h (cid:107) x (cid:107) − α (cid:80) ∞ j =2 ι j h j (cid:107) x j (cid:107) − α , x j ∈ Φ B , (80)and η = h t (cid:107) x t (cid:107) − α (cid:80) j ∈ N ι j h j (cid:107) x j (cid:107) − α , x j ∈ Φ , (81)where ι j denote the state indicator of the transmitter locatedat j which equals 1 and 0 when the transmitter is on and off,respectively. B. Performance Analysis
In large random networks, characterizing the exact queueinteraction among the transmitters is quite challenging. Fortu-nately, in a large-scale network, the correlation among the inter-acting queues tends to be “weak” and “global” [111]. Therefore,the impact of interacting queues tends to be negligible. In the SIR Threshold (MH) S u cc e ss P r obab ili t y Correlated ShadowingIndependent Shadowing
Fig. 23:
Success probability versus SIR threshold under correlatedand independent shadowing ( λ = 1 , κ = 0 . , α = 4 , r t = 1 ). C ond i t i ona l S u cc e ss P r obab ili t y Correlated ShadowingIndependent Shadowing
Fig. 24:
CSP versus SIR threshold under correlated and independentshadowing ( λ = 1 , α = 4 , r t = 1 , θ = 1 ). following, we show how to approximate the success probabilityin Poisson downlink networks with “interacting queues” byexploiting the mean-field property.For this, • we first compute the success probability of the typicaluser’s cell, i.e., the cell containing the origin, based onthe assumption that each BS at x j ∈ Φ \{ x } is activeindependently with probability p A = E [ ι j ] . • we then derive the active probability of the serving BS ofthe typical user as a function of packet arrival rate andsuccess probability (i.e., service rate) based on queueingtheory. By inserting the success probability obtained in theprevious step, we can establish a fixed-point equation of p A . • we finally obtain the success probability by plugging in p A obtained by solving the fixed-point equation.Based on the above methodology, for a Poisson downlinknetwork, we have the following result. Theorem 7.
With infinite buffer size and random scheduling,the success probability of a Poisson downlink network underRayleigh fading can be approximated by P s , obtained bysolving the fixed-point equation P s = (cid:18) ξ u ( F (1 , − δ ; 1 − δ ; − θ ) − (cid:80) ∞ n =1 p N u ( n ) P s /n (cid:19) − , (82)where p N u ( n ) = ν ν Γ( n + ν )( λ u /λ B ) n n !Γ( ν )( λ u /λ B + ν ) n + ν , (83)with δ = 2 /α and ν = 3 . . Proof.
See
Appendix E .Fig. 25 shows the success probability under different packetarrival rates ξ u in a Poisson downlink network. We observethat ignoring the temporal and spatial correlation among queue -20 -10 0 10 20 SIR Threshold (dB) S u cc e ss P r obab ili t y Fig. 25:
Success probability versus SIR threshold in Poisson downlinknetworks ( α = 4 , λ u /λ B = 5 ). iterations closely approximates the success probability achievedin the presence of the interactions. This is due to the “meanfield” effect in a large-scale network. We also note that the traf-fic arrival rate ξ plays a pivotal part in the success probability.For example, to achieve a target success probability of ,the disparity of the supported SIR thresholds with ξ u = 0 . and ξ u = 0 . can be over 10 dB. A larger ξ u decreases thesuccess probability due to the increased density of interferers.Next, we discuss the success probability in Poisson bipolarnetworks. By following the same methodology as used forPoisson downlink networks, we obtain the result presented inthe following theorem. Theorem 8.
With infinite buffer size, the success probabilityof the typical link in a Poisson bipolar network under Rayleighfading is given by (84), where δ = α . P s ≈ max (cid:26) exp (cid:18) W (cid:0) − ξλπr θ δ Γ(1+ δ )Γ(1 − δ ) (cid:1)(cid:19) , exp (cid:16) − λπr θ δ Γ(1+ δ )Γ(1 − δ ) (cid:17)(cid:27) , (84) -10 -5 0 5 10 15 20 25 SIR Threshold (dB) S u cc e ss P r obab ili t y =0.5, Analysis=0.5, Simulation=0.85, Analysis=0.85, Simulation=1.0, Analysis=1.0, Simulation Fig. 26:
Success probability as a function of the SIR threshold inPoisson bipolar networks ( α = 4 , r t = 2 , λ = 0 . ). Proof.
See
Appendix F .Next, we explore the asymptotics of the success probabilityby utilizing the expansion of the Lambert- W function as W ( z ) ∼ z when z → , which follows from xe x ∼ x when x → .Fig. 26 depicts the success probability versus the SIRthreshold with different settings of the packet arrival rate.The analytical results closely match the simulation results,which validates the effectiveness of the adopted approximation.Additionally, the success probability with ξ = 1 . overlaps thatwith ξ = 0 . when the SIR threshold is large (e.g., when θ > dB). The reason is that in both cases the service rateof each queue is below the packet arrival rate, and thus thebuffer is always non-empty. As a result, all the transmittersremain active for transmission which renders the same successprobability under different packet arrival rates. This observationalso shows that the success probability is lower-bounded by thecase with a full load.Fig. 27 further illustrates the success probability in a Poissonbipolar network under different densities. It can be observedthat the approximation tends to lose its accuracy with theincrease of the network density. The reason is that, in a Poissonbipolar network, the interferers can be arbitrarily close to thetarget receiver, and thus the queues of the serving transmittersand the interferers are strongly coupled. The spatio-temporalcorrelation of the buffer status cannot be ignored in this regime. -10 -5 0 5 10 15 20 SIR Threshold (dB) S u cc e ss P r obab ili t y =0.001, Analysis=0.001, Simulation=0.002, Analysis=0.002, Simulation=0.003, Analysis=0.003, Simulation Fig. 27:
Success probability in Poisson bipolar networks underdifferent intensities.
C. Summary and Discussion
This section has developed models to analyze wireless sys-tems with unsaturated buffers. Specifically, by integrating re-sults from queueing theory, we have presented the derivations ofsuccess probabilities for Poisson bipolar and Poisson downlinksystems given the packet arrival rate. The key observations areas follows. • The spatial correlation among the queue statues of dif-ferent transmitters and the temporal correlation amongthe queue statue of the same transmitter can be ignoredwhen different queues are weakly coupled, e.g., in Pois-son downlink networks, and cannot be ignored whenthe queues are strongly coupled, e.g., in Poisson bipolarnetworks. • The temporal correlation among the successful transmis-sion events decreases with increasing packet arrival rate.
Open Technical Issues : In Section VI, we have shown thatin large Poisson downlink networks, the interaction amongthe queues becomes weak, and the impact of the temporaland spatial correlation tends to be negligible. However, thiseffect only appears due to the cellular infrastructure, wherethe interfering BSs are further away from the typical userthan the serving BS. In infrastructure-less networks that donot impose any restrictions on the interferers’ locations, theinteraction between the queues tends to be strong, thus, theirspatio-temporal correlation cannot be ignored. Thus, furtherresearch efforts are needed to characterize the dynamics ofstrongly coupled queues in large-scale networks. VII. S
PATIALLY -C ORRELATED I NTERFERENCE AND R ELAYING
Cooperative relaying is a spatial diversity technique forenhancing the reliability and throughput of traditional point-to-point communication [112]. In multihop relaying, the trans-mission performance at different hops can be impacted by somecommon interferers. Therefore, the spatial characteristics ofthe interferers can have a substantial impact on the end-to-end transmission performance [72]. The goal of this section isto quantify the spatially and temporally-correlated interferenceand demonstrate its effect on the performance of multihoprelaying.
A. System Model
We consider a multihop relay network consisting of a sourcenode and a receiver node, which is M hops away from thetransmitter node, in a random field of interferers. The locationsof the source node S and the m -th hop receiver (i.e., ( m + 1) -th hop transmitter) are deterministic and denoted by x S and z m , respectively. The transceiver node in each hop works in ahalf-duplex fashion. Let d m represent the Euclidean distanceof the m -th hop link, i.e., d = (cid:107) z − x S (cid:107) and d m = (cid:107) z m − z m − (cid:107) , for ≤ m ≤ M . The route from the source node tothe M -th hop receiver is fixed. The decode-and-forward (DF)relaying protocol is adopted such that each hop first decodesthe received signal and forwards the re-encoded version to thenext hop. We consider a Poisson field of interferers where thelocations of the interferers Φ ⊂ R are a PPP with intensity λ , as illustrated in Fig. 28(a). Note that the relaying system ofinterest is considered to be independent of the interferers.We assume that each transmitter in the considered systemuses unit transmit power. Let Φ m = { x j,m } j ∈ N denote thepoint process of interferers during the m -th hop transmission.The SIR at the m -th hop can be expressed as η m = h m d − αm (cid:80) j ∈ N h j,m (cid:107) x j,m − z m (cid:107) − α , (85)where h m represents the power gain of small-scale fading forthe m -th transmission hop and h j,m represents the small-scalefading gain between the interferer j and the m -th hop receiver,which are both i.i.d. exponential random variables with unitmean,For the analysis of the success probability of multihoprelaying, we consider both QSI and FVI with which thetransmission of different hops are subject to the interferencefrom the same point process, i.e., Φ = Φ = · · · = Φ M = Φ ,and independent point processes, respectively. With DF relayingprotocol [113], the end-to-end success probability of an M -hoprelaying system with QSI and FVI are given, respectively, by P QSI M = E (cid:34) P (cid:20) M (cid:92) m =1 (cid:8) η m > θ (cid:9) (cid:12)(cid:12) Φ (cid:21)(cid:35) = E (cid:34) M (cid:89) m =1 P (cid:2) η m > θ | Φ (cid:3)(cid:35) (86) Fig. 28:
Multihop relaying in the presence of random interferers. and P FVI M = P (cid:34) M (cid:92) m =1 (cid:8) η m > θ (cid:9)(cid:35) = M (cid:89) m =1 P (cid:2) η m > θ (cid:3) . (87) B. Moments of the End-to-End CSP Φ This subsection characterizes the moments of the end-to-endCSP Φ of the multihop relaying. For this, • we compute the end-to-end JSP that the transmissionsof the M hops all succeed given the point process, i.e., P (cid:2) (cid:84) Mm =1 { η m > θ } | Φ m (cid:3) . • we derive the moments of the CSP Φ based on the PGFLof the PPP.Following the above methodology, we obtain the moments ofthe end-to-end CSP Φ under both QSI and FVI in the followingtheorem. Theorem 9.
The moments of the end-to-end CSP for an M -hop relaying system in a Poisson field of interferers are givenby (88). Proof.
See
Appendix G .Fig. 29 illustrates the success probability for an M -hoplinear-route multihop relaying system where all relays areplaced on the source-destination line and all M links havedistance l . With both fields of interferers, we can observethat correlated QSI provides a higher success probability thanindependent FVI. This can be understood from the perspectiveof the CSP of m -th hop given that the transmissions of theprevious hops all succeed. With FVI, we can see from (88) that M Poi P s ( b, M ) / M Poi P s ( b, M −
1) = M Poi P s ( b, . This is consistentwith the fact that with FVI, the successful transmission eventof the m -th hop is independent of those of the previous hops.Fig. 30 demonstrates the spatial CSP of the m -th hop giventhat the transmissions in the previous m − hops all succeed M Poi P s ( b, M ) = exp (cid:32) − λ (cid:90) R (cid:18) − M (cid:89) m =1 (cid:18)
11 + θd αm (cid:107) x − z m (cid:107) − α (cid:19) b (cid:19) d x (cid:33) , QSI exp (cid:32) − λ M (cid:89) m =1 (cid:90) R (cid:18) − (cid:18) θd αm (cid:107) x − z m (cid:107) − α (cid:19) b (cid:19) d x (cid:33) , FVI (88)
SIR Threshold (MH) S u cc e ss P r obab ili t y Correlated InterferenceIndependent Interference
Fig. 29:
Success probability of a multihop relaying system in a Poissonfield of interferers ( α = 4 , λ = 0 . , d = d = · · · = d M = 1 ). in the scenario with QSI, i.e., (cid:0) M Poi P s (1 , M ) / M Poi P s (1 , M − (cid:1) for M ≥ . It can be seen that, with QSI, the CSP considerablyincreases given the successful transmission of the first hop,especially when the SIR threshold θ is high. Thus, the end-to-end success probability with QSI exceeds that with FVI.It is worth noting that with temporally correlated QSI, theconditional outage probability, given the outage events of theprevious hops, also increases. However, this does not worsenthe end-to-end success probability, since the outage event ofany single hop leads to an end-to-end outage.Moreover, Fig. 31 depicts the temporal CSP of an M -hop relaying system given a previous end-to-end successfultransmission, i.e., M Poi P s (2 , M ) / M Poi P s (1 , M ) . We can observethat the CSP increases with the number of hops. The reasonis that a successful transmission with a larger number of hopsindicates a good channel condition, and thus the next end-to-end transmission is more likely to succeed. C. Summary and Discussion
In this section, we have discussed the impact of spatio-temporal interference on the performance of multihop relaying.In particular, we have presented the derivations of end-to-endsuccess probability of multihop relaying in Poisson fields ofinterferers under both FVI and QSI. The main observations aresummarized as follows. • Correlated spatio-temporal interference (i.e., QSI) imposedby the same interferers at different hops results in highersuccess probability than independent spatio-temporal in-terference (i.e., FVI). • With correlated spatio-temporal interference, as m in-creases, the successful transmission of the m -th hop ismore dependent on the successful transmissions of theprevious hops. • With independent spatio-temporal interference, the CSPof the m -th hop given the successful transmission of theprevious hops is the same as the success probability of the m -th hop without the condition. • With correlated spatio-temporal interference, the end-to-end successful transmission events are more correlatedwhen the number of hops is larger.
Open Technical Issues : This section has presented the character-ization of end-to-end relaying performance under QSI and FVI,which considers a purely static network and an independentnetwork, respectively. The two types of interference repre-sent two extreme cases with fully correlated and independentinterferer locations. In general, the ambient interferers (e.g.,mobile users) may have a certain degree of movement duringdifferent transmission attempts and result in neither a static norindependent network environment. In such an environment, theresulting temporal interference is only partially correlated. Anaccurate characterization of the spatio-temporal correlation atdifferent locations would be required for performance evalua-tion of multihop relaying systems.VIII. T
EMPORALLY -C ORRELATED I NTERFERENCE AND R ETRANSMISSION
The goal of this section is to derive the success probabil-ities of a target transmission link with retransmissions undertemporally correlated and independent interference.
A. System Model
This section considers the Poisson ad hoc network model (asintroduced in Section III-A). Let Φ ( k ) = { x ( k ) j } j ∈ N denote thenode locations in time slot k . The aggregated interference andreceive SIR at the target receiver located at o in time slot k canbe expressed, respectively, as I ( k ) o = (cid:88) j ∈ N h ( k ) j (cid:107) x ( k ) j (cid:107) − α η ( k ) = h ( k )t (cid:107) x ( k )t (cid:107) − α I ( k ) o , (89) SIR Threshold (MH) S u cc e ss P r obab ili t y Success probability of the first hopConditional success probability of M-th hop
Fig. 30:
The spatial CSP of the M -th hop relaying in a Poisson fieldof interferers with QSI. SIR Threshold (MH) C ond i t i ona l S u cc e ss P r obab ili t y Fig. 31:
The temporal CSP of an M hop relaying system in a Poissonfield of interferers with QSI. where h ( k )t and h ( k ) j denote the small-scale fading gains be-tween the transmitters at x ( k )t , x ( k ) j and the target receiver intime slot k , which are i.i.d. exponential random variables withunit mean.We consider both QSI and FVI with which different trans-mission attempts are influenced by the same set and differentsets of interferers, respectively. B. Performance Analysis
Let A k (cid:44) { η ( k ) > θ } denote the successful transmissionevent in time slot k . The JSP of K transmissions in the caseswith QSI and FVI are defined, respectively, as J QSI K (cid:44) E (cid:34) P (cid:20) K (cid:92) k =1 A k (cid:12)(cid:12) Φ ( k ) (cid:21)(cid:35) = E (cid:34) K (cid:89) k =1 P (cid:2) A k | Φ ( k ) (cid:3)(cid:35) (90)and J FVI K (cid:44) P (cid:20) K (cid:92) k =1 A k (cid:21) = K (cid:89) k =1 P (cid:2) A k (cid:3) . (91)
1) Joint Success Probability:
In this subsection, we showhow to derive the JSP of K transmissions based on the PGFLof the PPP. The final results are presented in the followingtheorem [75]. Theorem 10. ( Temporal JSP ) With a Poisson field of interfer-ers, the probability that a link over distance r t has K successfultransmissions with QSI and FVI are given, respectively, as J QSI K = exp (cid:0) − cλθ δ r D K ( δ ) (cid:1) , (92) J FVI K = exp (cid:0) − cλθ δ r K (cid:1) , (93)where c = π Γ(1 + δ )Γ(1 − δ ) and D K ( δ ) = Γ( K + δ )Γ( K )Γ(1+ δ ) . Proof.
See
Appendix H . Remark 11 : J QSI = J FVI as D ( δ ) = 1 , which indicates thatthe success probability of any single time slot is not influencedby the type of interference experienced. Remark 12 : Since δ < (i.e., α > ), it is readily checkedthat D K ( δ ) < K for K ∈ N , and thus J QSI K > J FVI K . Thisreveals that temporally-correlated QSI results in higher JSP thanthe temporally-independent FVI. Fig. 32 illustrates the JSP for K = 2 , , with both QSI and FVI. Remark 13 : When δ → (i.e., α → ∞ ), it can be foundthat J QSI = J QSI = · · · = J QSI K = e − λπr , equivalently P [ A k +1 | A k ] = 1 for k = 1 , . . . , K . This indicates that despitethe effect of small-scale fading, the successful transmissionevents are fully correlated. Remark 14 : When δ → (i.e., α → ), it can be found that a) J QSI K → e − cλθ δ r K , equivalently P [ A k +1 | A k ] = e − cλθ δ r for k = 1 , . . . , K . This means the successful transmission eventsare independent; b) c → ∞ , which indicates that J QSI K → .
2) Conditional Success Probability:
As a consequence ofTheorem 10, the CSPs of succeeding at K + 1 -th transmissiongiven the previous K successful transmissions can be directlyobtained following Bayes rule as C QSI K +1 ,K = P (cid:2) A K +1 | A , . . . , A K (cid:3) = J QSI K +1 J QSI K = exp (cid:16) − cλθ δ r (cid:0) D K +1 ( δ ) − D K ( δ ) (cid:1)(cid:17) , and C FVI K +1 ,K = J FVI K +1 J FVI K = exp (cid:0) − cλθ δ r (cid:1) . Remark 15 : It can be readily checked that ∂ C QSI K +1 ,K /∂K > which indicates that C QSI K +1 ,K monotonically increases with thenumber of transmission attempts K . Fig. 33 shows the temporal CSPs with QSI when K =1 , , , . The numerical results illustrate the above-discussedproperties. Given that the previous transmissions succeed, theCSP considerably increases, especially with high SIR threshold θ .
3) Correlation Coefficient:
Let Q k = A k be the indicatorthat A k occurs. The correlation coefficient between Q k and Q j with k (cid:54) = j is ζ QSI Q k ,Q j = P [ A k ∩ A j ] − P [ A k ] P [ A k ∩ A k ] − P [ A k ] ( a ) = J QSI − (cid:0) J QSI (cid:1) J QSI (1 − J QSI )= exp( − cλθ δ r ( δ + 1)) − exp( − cλr θ δ )exp( − cλr θ δ )(1 − exp( − cλr θ δ ))= exp( cλθ δ r (1 − δ )) − cλθ δ r ) − , (94)where ( a ) holds as { Q k } k ∈{ ,...,K } are identically distributed.Moreover, ζ FVI Q k ,Q j = 0 as J FVI = (cid:0) J FVI (cid:1) . Since ζ QSI Q k ,Q j > and ζ FVI Q k ,Q j = 0 , it is evident that QSI and FVI result intemporally-correlated and independent successful transmissionevents, respectively. Remark 16 : When δ → and δ → , ζ QSI Q k ,Q j → and ζ QSI Q k ,Q j → , indicating that the Q k and Q j become fullycorrelated and fully uncorrelated, respectively. Remark 17 : For given δ and r , ζ QSI Q k ,Q j is a decreasing functionof λ and θ . This can be checked that since < δ < (i.e., α > ), the denominator scales up at a higher rate than thenumerator of (94) with the increase of λ or θ . Fig. 34 shows thecorrelation coefficient of successful transmission events withQSI.
4) Success Probability With Retransmissions:
In case oftransmission failure, multiple transmissions can be carried outto deliver a message. Let us consider a retransmission protocolwhere the receiver requests the associated transmitter to sendthe message again upon a transmission failure until reaching amaximum number of transmission attempts denoted as K . Atthe receiver side, the received signal at each time slot is decodedindependently. The success probability with retransmissions canbe expressed as P K (cid:44) P (cid:20) K (cid:91) k =1 A k (cid:21) . (95)Note that from the above definition, we have P = J , where J k , k ∈ N , is defined in (7).Let A (cid:44) { A , . . . , A K } denote the set of the successfultransmission events and P( A ) represent the power set of A .By applying the inclusion-exclusion principle, we obtain thesuccess probability with retransmissions as P K = (cid:88) A ∈ P( A ) ( − | A | +1 P (cid:2) A (cid:3) ( a ) = K (cid:88) k =1 ( − k +1 (cid:18) Kk (cid:19) J k , (96) where ( a ) follows as { η k } k ∈{ , ,...,K } are identically dis-tributed, and J k has been obtained in Theorem 10.Fig. 35 shows the success probability with retransmissionswith both QSI and FVI. It can be seen that P FVI K exceeds P QSI K . The reason is that the transmission failures are temporallycorrelated and thus are likely to occur in succession with QSI.By contrast, with FVI, transmissions have a better chanceto succeed as previous transmission failure does not inferlower success probability in the current time slot. This can beunderstood by checking the dependency of transmission failure.Let ¯ S (cid:44) { η ( k ) < θ } . By following the inclusion-exclusionprinciple, the joint outage probability of two transmissionevents is given by ¯ J = P (cid:2) ¯ A ∩ ¯ A (cid:3) = 1 − P (cid:2) A ∪ A (cid:3) = 1 − (2 J − J )= 1 − − cλθ δ r ) + exp( − cλθ δ r (1 + δ )) . Subsequently, ¯ J QSI (cid:0) ¯ J QSI (cid:1) = 1 + exp( cλθ δ r (1 − δ )) − cλθ δ r ) − > , and ¯ J FVI (cid:0) ¯ J FVI (cid:1) = 1 .Moreover, it is evident that the conditional outage probabilityincreases given the occurrence of previous transmission failureby checking P (cid:2) ¯ A | ¯ A (cid:3) = P (cid:2) ¯ A ∩ ¯ A (cid:3) P (cid:2) ¯ A (cid:3) = 1 − exp( − cλθ δ r ) 1 − exp( − δcλθ δ r )1 − exp( − cλθ δ r ) ≥ − exp( − cλθ δ r ) = P (cid:2) ¯ A (cid:3) . C. HARQ Retransmission Schemes
For transmission and decoding, similar to [80], we considertwo categories of HARQ, i.e., Type-I HARQ and Type-IIHARQ schemes. • If the SIR at a target receiver for the initial transmissiondoes not exceed the threshold θ , a one-time retransmissionrequest is sent to the serving transmitter. After receivingthe retransmitted signals, the target receiver with Type-I HARQ abandons the received signal from the initialtransmission and decodes only from the received signalfrom the retransmission. Given a maximum number oftransmissions K and a SIR threshold θ , the successprobability is given by P I = P (cid:20) K (cid:91) k =1 (cid:110) η ( k ) > θ (cid:111)(cid:21) , (97)where η ( k ) is the SIR of k -th transmission of the samecontent with Type-I HARQ retransmission scheme. SIR Threshold (MH) J o i n t S u cc e ss P r obab ili t y QSIFVI
Fig. 32:
JSP with a different number of transmission attempts inPoisson ad hoc networks ( α = 4 , λ = 0 . ). SIR Threshold (MH) C ond i t i ona l S u cc e ss P r obab ili t y Fig. 33:
CSP with a different number of transmission attempts inPoisson ad hoc networks ( α = 4 , λ = 0 . ). -2 -1 C o rr e l a t i on C oe ff i c i en t Fig. 34:
Correlation coefficient in Poisson ad hoc networks ( α = 4 ). SIR Threshold (MH)
QSIFVI
Fig. 35:
Success probability with retransmission in Poisson ad hocnetworks ( α = 4 , λ = 0 . ). • With Type-II HARQ with chase combing (CC) codes andmaximal ratio combining (MRC) of signals from boththe initial transmission and retransmission, the successprobability is given by P II = P (cid:20) K (cid:91) k =1 (cid:110) Υ k > θ (cid:111)(cid:21) , (98)where Υ k = (cid:80) ki =1 η ( k ) is the effective SIR after the k -thtransmission of the same content with the Type-II HARQretransmission scheme.In what follows, the success probabilities of Type-I HARQand Type-II HARQ-CC schemes are derived for the case with K = 2 , i.e., each packet is retransmitted once if the initialtransmission is not successful. The cases with K ≥ can beobtained by following the same methodology straightforwardly.
1) Success Probability of Type-I HARQ:
When K = 2 , thesuccess probability with Type-I HARQ given in (97) can be expressed as P I = P (cid:2) η (1) > θ (cid:3) + P (cid:2) η (1) > θ, η (2) ≤ θ (cid:3) (a) = (cid:88) k =1 P (cid:2) η ( k ) > θ (cid:3) − P (cid:2) η (1) > θ, η (2) > θ (cid:3) (b) = (cid:88) k =1 P (cid:2) η ( k ) > θ (cid:3) − P (cid:2) η (1) > θ, η (2) > θ | Φ (cid:3) , QSI (cid:88) k =1 P (cid:2) η ( k ) > θ (cid:3) − (cid:89) k =1 P (cid:2) η ( k ) > θ (cid:3) , FVIwhere ( a ) applies the inclusion-exclusion principle and ( b ) follows as the point processes across different time slots arethe same and i.i.d. for QSI and FVI, respectively. Subsequently,based on the definition of J QSI K and J FVI K in (90) and (91),respectively. we can obtain the following corollary. Corollary 3.
Given that each unsuccessfully transmitted packetis retransmitted once, the success probability under Type-IHARQ retransmission scheme with QSI and FVI are given,respectively, as P QSI I = 2 J QSI − J QSI , (99) P FVI I = 2 J FVI − (cid:16) J FVI (cid:17) , (100)where J QSI K and J FVI K for K ∈ N are given in (92) and (93),respectively.
2) Success Probability of Type-II HARQ-CC:
Next, wediscuss how to obtain the success probability of Type-II HARQ-CC. When K = 2 , the success probability with Type-II HARQgiven in (97) can be expressed as P II = P (cid:2) η (1) > θ (cid:3) + P (cid:2) η (1) + η (2) > θ, η (1) ≤ θ (cid:3) where the first term on the right-hand sideof the equality has been obtained in Section VIII.C.1. To obtainthe second term, we need to compute the distribution of theSIR of any single time slot. In particular, we first calculate theCDF of the SIR and then obtain the corresponding PDF bytaking the derivative of the CDF. With the PDF of the SIR,we then derive the joint probability that events η (1) + η (2) > θ and η (1) < θ both occur. The final results are presented in thefollowing corollary. Corollary 4.
Given that each unsuccessfully transmitted packetis retransmitted once, the success probability under Type-IIHARQ-CC retransmission scheme with QSI and FVI are given,respectively, as P QSI II = exp( − cλθ δ r )+2 πλ (cid:90) θ (cid:90) ∞ r α t r − α J ( r, u )(1 + ur α t r − α ) r d r × exp (cid:18) − πλ (cid:90) ∞ (cid:18) − J ( r, u )1 + ur α t r − α (cid:19) r d r (cid:19) d u, (101)and P FVI II = exp( − cλθ δ r )+2 πλ (cid:90) θ (cid:90) ∞ r α t r − α (1 + ur α t r − α ) r d r × exp (cid:18) − πλ (cid:90) ∞ (cid:18) −
11 + ur α t r − α (cid:19) r d r (cid:19) × exp (cid:18) − πλ (cid:90) ∞ (cid:16) − J ( r, u ) (cid:17) r d r (cid:19) d u, (102)where c = π Γ(1 + δ )Γ(1 − δ ) and J ( r, u ) = θ − u ) r α t r − α . Proof.
See
Appendix I .Fig. 36 depicts the success probabilities of Type-I HARQ andType-II HARQ-CC when K = 2 . As shown, Type-II HARQ-CC provides a higher success probability at all SIR thresholds.Nevertheless, the success probabilities of the two schemes arecomparable when the SIR threshold is small (e.g., θ < . MH). This indicates that, in the high-coverage regime, the SIRgain due to MRC has an imperceptible effect on the successprobability. The reason is that low SIR thresholds render a highsuccess probability of the first transmission, which makes theimpact of retransmission negligible.
SIR Threshold (MH) S u cc e ss P r obab ili t y Type-I HARQ, QSIType-I HARQ, FVIType-II HARQ-CC, QSIType-II HARQ-CC, FVI
Fig. 36:
Success probabilities of HARQ retransmission schemes inPoisson ad hoc networks ( K = 2 ). D. Summary and Discussion
This section has discussed the influence of temporal interfer-ence correlation on the retransmission performance of a trans-mission link in a Poisson field of interferers. In particular, wehave derived the JSP of multiple transmissions, the correlationcoefficient for two transmissions, and the success probabilityfor a given number of transmission attempts. Furthermore,based on the above results, we have shown how to analyze thesuccess probabilities under Type-I HARQ and Type-II HARQretransmission schemes. The lessons learned are as follows. • Temporally-correlated interference results in a higherJSP than temporally-independent interference. The per-formance gap between the two cases increases with thenumber of transmission attempts. • The CSP of K -th transmission given that the previous K − transmissions all succeed increases with K andremains the same with temporally-correlated interferenceand temporally-independent interference, respectively. • Temporally-correlated interference results in higher suc-cess probability with retransmission than temporally-independent interference. The gap between the successprobabilities with retransmission under the two types ofinterference increases with the number of transmissionattempts. • The performance gain of Type-II HARQ over Type-IHARQ is larger with temporally-correlated interferencethan with temporally-independent interference.
Open Technical Issues : This section has presented models tocharacterize the impact of temporal interference correlation ina retransmission-based (or multi-packet transmission) system.We have considered both Type-I HARQ and Type-II HARQsystems and evaluated both CSP and JSP. From a practical pointof view, the impact of “interacting queues” may need to beconsidered in a retransmission scenario. Also, the performance of HARQ systems in the presence of interference correlationin different wireless systems (e.g., non-orthogonal multipleaccess (NOMA) systems and MIMO systems) will be worthinvestigating.In some emerging scenarios, such as device-to-device com-munication [114] and mobile social networks [115], an infor-mation source may rely on mobile users to spread messagesto multiple destinations. To model the process of informationspreading in such systems, a possible direction is to incorporateuser mobility models into the stochastic geometry analysis.IX. S PATIALLY AND T EMPORALLY C ORRELATED I NTERFERENCE IN M OBILE S YSTEMS
A. Introduction
To study the effect of mobility on the spatio-temporal SIRcorrelation, in this section, we analyze the spatio-temporal JSPfor two different mobile network scenarios. In the first scenario,similar to [71], we consider a downlink communication in aPoisson cellular network, and provide the steps to calculatethe spatio-temporal JSP of a mobile user at two different timeinstants. In the second scenario, similar to [116], we considera Poisson bipolar model where the desired transmitter andreceiver are static while other transmitters in the network moveaccording to a random mobility model. For this scenario, wealso explain the steps to calculate the JSP at the static receiver[117].Let us denote the success event indicator at time t at location u by A = { SIR >θ } and at time t at location u by A = { SIR >θ } , where SIR is the SIR at the receiver at location u at time t , and SIR is the SIR at the receiver at location u at time t ; θ is the target SIR. In the following, we present thesteps to calculate E [ A A ] (i.e., the spatio-temporal JSP) for thetwo mentioned network scenarios. Note that when the networkis stationary over time, E [ A A ] does not depend on t and t ; it only depends on ∆ = | t − t | . Generally, the randomvariables A and A are spatially and temporally correlated.When u = u , E [ A A ] captures the temporal correlation,i.e., the correlation at one location in two different time instants.When t = t , E [ A A ] captures the spatial correlation, i.e.,correlation at two different locations at the same time. B. Analysis of Poisson Downlink Networks1) Model I:
Consider a Poisson downlink network Φ withintensity λ where all the users are associated with their nearestBSs. The small-scale fading is i.i.d. across time and space.Consider a mobile user that is located at distance r fromits associated BS at time t . The user moves away from theassociated BS with a constant speed v at an angle φ at time t , where φ is uniformly distributed in [0 , π ] (Fig. 37). Beforefurther discussion, let us introduce some notations that helpus in deriving the spatio-temporal JSP at time t and t with t > t [71]. • P , P : Location of the serving BS at time t and t ,respectively. 𝓁 𝓁 User’s location at time 𝑡 User’s trajectory User’s location at time 𝑡 S erving BS at 𝑡 and 𝑡 𝜃 𝑟 𝐶 (𝓁 ,𝑟 ) 𝒜(𝓁 ,𝑟 )𝑟 (a) No handoff occurs. 𝓁 𝓁 User’s location at time 𝑡 User’s trajectory User’s location at time 𝑡 S erving BS at 𝑡 and 𝑡 𝜃 𝑟 𝐶 (𝓁 ,𝑟 ) 𝒜(𝓁 ,𝑟 )𝑟 (b) Handoff occurs. Fig. 37:
Scenarios with and without handoff. • r , r : Distance between the mobile user and its associatedBS at time t and time t , respectively. • r : Distance between the mobile user at time t and itsassociated BS at time t . • B ( x, r ) : Closed disk centered at x with radius r . Forbrevity, B ( u , r ) , B ( u , r ) , and B ( u , r ) are denotedby B , B , and B , respectively. • H : Event that at least one handoff occurs in the interval [ t , t ] .As the mobile user moves from u to u , the mobile usermay be served by the same BS (no handoff occurs) or it maybe handed off to other BSs (handoff occurs). At time t , whenthere is no BS in B , the mobile user is still connected to thesame BS as time t , i.e., P = P . In this case, according to thedefinitions and from triangle equations, we have r = r = (cid:112) r + v ( t − t ) + 2 r v ( t − t ) cos φ and B = B . Onthe other hand, when there is a BS in B , at time t , themobile user is served by a new BS, i.e., P (cid:54) = P . In this case r < r and B ⊂ B . In the following, we briefly provide the steps for deriving the JSP at time t and t . Note that since u (cid:54) = u and t (cid:54) = t , the JSP captures the spatio-temporal SIRcorrelation.
2) Steps to Derive the JSP:
Step 0 - From the contactdistribution function of the homogeneous PPP, the PDF of thecontact distance at time t follows as (18). Moreover, due to thesymmetry, we can assume φ is uniformly distributed in [0 , π ] . Step (i) - Calculating P ( H | r , φ ) (handoff probability given r and φ ): P ( H | r , φ ) (a) = 1 − P (Φ( B \ B ) = 0 | r , φ ) (b) = 1 − exp (cid:0) − λ |B \ B | (cid:1) , where ( a ) follows the fact that handoff occurs when there is aBS in B . In ( a ) , B is excluded from B since there is noBS inside B . ( b ) follows from the void probability of the PPP. Step (ii) - Deriving the distribution of r :Given r and φ , when there is no handoff, r = (cid:112) r + v ( t − t ) + 2 r v ( t − t ) cos φ . However,when handoff occurs, as discussed earlier, r < r .In this case, we can derive the conditional PDF of r as f r ( z | H, r , φ ) = dd z F r ( z | H, r , φ ) , where F r ( z | H, r , φ ) is the conditional CDF of r and can beobtained by F r ( z | H, r , φ )= P ( r ≤ z, H | r , φ ) P ( H | r , φ )= P (Φ( B ( u , z ) \ B ) > , Φ( B \ B ) > | r , φ ) P ( H | r , φ ) (a) = P (Φ( B ( u , z ) \ B ) > | r , φ ) P ( H | r , φ )= 1 − exp (cid:0) − λ | B ( u , z ) \ B | (cid:1) P ( H | r , φ ) , (103)for z ∈ [max(0 , r − v ) , r ] . ( a ) follows since B ( u , z ) ⊂B in case of handoff. When r > v , the nearest BS is atleast at distance r − v , therefore, F r ( z | H, r , φ ) = 0 for z < max(0 , r − v ) . Moreover, as explained earlier, F r ( z | H, r , φ ) = 1 for r < z .When handoff occurs, we also need the distribution of ˆ φ ,angle between the vectors −−→ u u and −−−→ u P . Given r , r , and v , one of the two following cases holds : 1) | r − r | < v ≤ r + r ( B and B overlap): ˆ φ is uniformly distributed in (cid:104) − π + arccos (cid:16) r + v − r r v (cid:17) , π − arccos (cid:16) r + v − r r v (cid:17)(cid:105) , 2) r + r < v ( B and B are disjoint): ˆ φ is uniformly distributed in [0 , π ] . Step (iii) - Calculating the spatio-temporal JSP P ( SIR > θ, SIR > θ ) as in (104), where ¯ H is the complementof H and denotes the event that handoff does not occur. Term I in (104) can be obtained by (105), where Φ I denotesthe locations of the interferers. In the case of no handoff ( ¯ H ), Φ I is a PPP with intensity λ in R \ ( B ∪ B ) . ( a ) is obtained If v ≤ | r − r | no handoff occurs as the user moves from u to u . using the fact that, given Φ I , ¯ H , r , and φ , SIR and SIR areindependent since small-scale fading is i.i.d. at different timeinstants and spatial locations. The inner expectation w.r.t. Φ I can be calculated by using the PGFL of PPP, and the outerexpectation by using PDF of r and φ provided in Step 0 . P ( ¯ H | r , φ ) , probability that handoff does not occur, is alsoprovided in Step (i) . Term II in (104) can be derived as (106).In the case of handoff ( H ), Φ I is a PPP with intensity λ in R \ ( B ∪ B ) . Note that Φ I ∪ { P } is the set of interferersat time t and Φ I ∪ { P } is the set of interferers at time t . ( a ) is obtained using the fact that, given Φ I , ˆ φ , r , H , r , and φ , SIR and SIR are independent. The expectation w.r.t. Φ I can be obtained by using the PGFL of PPP. The expectationw.r.t. ˆ φ and r can be obtained from Step (ii) . P ( H | r , φ ) isprovided in Step (i) . Outer expectation w.r.t. r and φ can alsobe obtained by using PDFs given in Step 0 . C. Analysis of Poisson Bipolar Networks1) Model II:
In the second scenario, similar to [116], weconsider a Poisson bipolar network, where, at time t , trans-mitters form a PPP Φ with intensity λ . Each transmitter has adedicated receiver at distance r in a random direction. Due tothe stationarity of PPP, we investigate the performance of thetypical receiver and consider its location as the origin. Sincethe homogeneous PPP is isotropic, without loss of generality,we can assume that the transmitter of this receiver is locatedat [ r , T . From Slivnyak’s theorem [3], at time t , the othertransmitters (interferers), denoted by Φ , form a homogeneousPPP with intensity λ . Assume that the selected receiver and itscorresponding transmitter are static while interferers are mobileaccording to a uniform mobility model. Hence, at any time t ,the interferers form a homogeneous PPP with intensity λ . Westudy the JSP at two different time slots t and t . Note thatsince the desired receiver is fixed at the origin ( u = u ),the JSP captures the temporal correlation. Let w i, ∆ denote thedisplacement vector for transmitter i . We consider a randomindividual mobility model (such as random walk), where eachinterferer moves independently of other interferers followingthe same random mobility model. Thus, w i, ∆ is independentlyand identically distributed for each interferer.
2) Derivation of Spatio-temporal JSP:
We can calculate theJSP in the case of random individual mobility model where u = u for spatially and temporally i.i.d. fading as follows: P ( SIR > θ, SIR > θ )= E Φ I [ P ( SIR > θ, SIR > θ | Φ I )] (a) = E Φ I [ P ( SIR > θ | Φ I ) P ( SIR > θ | Φ I )] (b) = E Φ I (cid:104) P ( SIR > θ | Φ I ) × E ( w ∆ ,i ) (cid:2) P ( SIR > θ | { w ∆ ,i } , Φ I ) (cid:3)(cid:105) , (107)where ( a ) is obtained, since given the initial location of theinterferers Φ I , SIR and SIR are independent. In ( b ) , the inner ∆ reflects the fact that statistics of the displacement vector depends on thetime difference ∆ = | t − t | rather than t and t separately. P ( SIR > θ, SIR > θ ) = E [ P ( SIR > θ, SIR > θ | r , φ )]= E (cid:2) P ( SIR > θ, SIR > θ | ¯ H, r , φ ) P ( ¯ H | r , φ ) (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) Term I + E [ P ( SIR > θ, SIR > θ | H, r , φ ) P ( H | r , φ )] (cid:124) (cid:123)(cid:122) (cid:125) Term II . (104) E (cid:2) P ( SIR > θ, SIR > θ | ¯ H, r , φ ) P ( ¯ H | r , φ ) (cid:3) = E (cid:2) E Φ I (cid:2) P ( SIR > θ, SIR > θ | Φ I , ¯ H, r , φ ) (cid:3) P ( ¯ H | r , φ ) (cid:3) (a) = E (cid:2) E Φ I (cid:2) P ( SIR > θ | Φ I , ¯ H, r , φ ) P ( SIR > θ | Φ I , ¯ H, r , φ ) (cid:3) P ( ¯ H | r , φ ) (cid:3) , (105) E [ P ( SIR > θ, SIR > θ | H, r , φ ) P ( H | r , φ )]= E (cid:104) E r (cid:104) E ˆ φ (cid:104) P ( SIR > θ, SIR > θ | ˆ φ, r , H, r , φ ) (cid:105)(cid:105) P ( H | r , φ ) (cid:105) (a) = E (cid:104) E r (cid:104) E ˆ φ (cid:104) E Φ I (cid:104) P ( SIR > θ | Φ I , ˆ φ, r , H, r , φ ) P ( SIR > θ | Φ I , ˆ φ, r , H, r , φ ) (cid:105)(cid:105)(cid:105) P ( H | r , φ ) (cid:105) . (106)expectation is w.r.t. the displacement vectors of all interferers,where ∆ = | t − t | . D. Numerical Results
To understand the effect of mobility, in this section, weconsider the CSP P ( SIR > θ | SIR > θ ) = P ( SIR > θ, SIR > θ ) P ( SIR > θ ) . Since the network performance depends on the nodes’ displace-ments , we set | t − t | = 1 and only consider the effect of thespeed on the spatio-temporal JSP.In Fig. 38, we illustrate the effect of correlation on the CSPfor both scenarios. For the random individual mobility model,we have assumed all the mobile users move with the samespeed v , and each mobile user moves in a random independentdirection ϕ i , uniformly distributed in the interval [0 , π ] . Tobetter understand the impact of correlation, in Fig. 38, we alsocompare the results with the independent case, where the CSP isequal to P ( SIR > θ ) . According to the displacement theorem[3], at any time t , interferers form a PPP with intensity λ .Thus, in Model II, P ( SIR u ( t ) > θ ) does not depend on t (and consequently v ). Using the stationarity of the PPP, we canobtain the same result for Model I.As shown in Fig. 38, at high speeds, we can ignore the effectof correlation, and assume that the two sets of interferers attime instants t and t are independent. As a result, most ofthe existing works in mobile networks only focus on the highly The displacement of a node that moves at a speed v for a time duration | t − t | = ∆ is the same as when it moves with speed v ∆ for duration | t − t | = 1 . mobile scenario where they can ignore the effect of correlation.It is worthwhile to note that the gap between the results withcorrelation (dashed blue curves) and without correlation (solidred lines) is proportional to the correlation coefficient betweenthe success event at t and success event at t . As illustratedin Fig. 38, the correlation between success events at t and t decreases as the speed increases. E. Summary and Discussion
This section has characterized the spatio-temporal JSP of twonetwork scenarios 1) with mobile users and static interferersand 2) with mobile interferers and mobile users. The mainobservations are summarized as follows. • The spatio-temporal correlation between different trans-mission events decreases with speed and can be ignoredwhen the speed is high enough. • The gap between the spatio-temporal CSP is proportionalto the correlation coefficient between successful transmis-sion events.In Poisson networks, mobility introduces spatial randomnesswhich decreases the correlation across time [71]; thus, spatialdiversity provides time diversity when nodes are mobile. Thisis also discussed in the literature by studying the mean localdelay [118]. In a static network, when there is a close interferer,retransmitting an unsuccessfully received signal is usually nothelpful. As a result, in static networks, the mean local delaymay be infinite, i.e., a significant fraction of users sufferfrom large delays. However, in mobile networks, interferencecorrelation decreases across time and there is a higher chance toreceive a retransmitted signal successfully. Therefore, mobilityreduces the mean local delay by providing spatial diversity. C ond i t i ona l S u cc e ss P r obab ili t y With CorrelationWithout Correlation (a) Model I. C ond i t i ona l S u cc e ss P r obab ili t y With CorrelationWithout Correlation (b) Model II.
Fig. 38:
Simulation of the effect of speed on the spatio-temporal JSPfor Model I in ( a ) and Model II in ( b ) for λ = 0 . and θ = − .In Model II, the desired link distance is set to d = 8 . Specifically, in the highly mobile scenario, the mean local delayis always finite.
Open Technical Issues : Analyzing the performance of mobilewireless networks with correlation effects is challenging formost scenarios and usually yields complicated results. Con-sequently, most of the related works consider simple systemmodels such as downlink communication in a Poisson cellularnetwork (as in Model I) or Poisson bipolar networks withmobile interferers (as in Model II), but even for the latter model,there are no analytical results.In the downlink communication in Model I, we have studiedthe performance of a mobile node in a network of statictransmitters (i.e., BSs). In contrast, in the uplink, we havemobile transmitters (desired transmitter and interferers). Due to the random mobility of the nodes and irregular shapes of thecells, it may not be possible to calculate the load distributionwhile considering the correlation effect, which in turn, makesthe analysis of the uplink scenario very challenging. In fact,an exact analysis is impossible, but approximations, bounds, orasymptotic results can almost always be derived.In Model II, as shown in (107), we need the distribution ofthe displacement vector w ∆ . However, it may not be possibleto derive this distribution for most of the mobility models andfor all values of ∆ . Therefore, most of the existing works resortto a simplified mobility model or consider only one movementstep during which the mobile node moves along a straight line.In addition to the spatio-temporal JSP, the SIR meta distri-bution in cellular networks with mobile BSs has been studiedin [119]. The joint SIR meta distribution and product SIR metadistribution in mobile networks are also worth investigating.X. F UTURE D IRECTIONS
In this section, we discuss future applications of stochasticgeometry analysis and related technical challenges.
A. Novel Performance Metrics and Stochastic Geometry Anal-ysis
Although the existing literature has extensively investigateddifferent types of large-scale communication networks, the ma-jority of them focus on the spatial and temporal average perfor-mance, such as mean success probability, delay and throughput.Recently, the SIR meta distribution [77] has been introduced toprovide insights on the distribution of success probability overspace and/or time. The meta distribution has been extendedto evaluate the distributions of rate and energy [120], meandelay [121], transmission rate [122], secrecy rate [123], andsecrecy success probability [73]. This concept can also beevaluate the distribution of other performance metrics suchas covert probability for low probability of detection/interceptcommunications [124], [125]. Moreover, the age-of-information(AoI) [126], [127] that quantifies the time elapsed from thegeneration to observation of the information explicitly measuresthe information freshness that the conventional metrics (e.g.,link delay) do not. Some recent works have shown that networkprotocol designs based on AoI provide better consumer servicequality in real-time status updating, such as Google Scholar[128]. To enhance information freshness, the existing networkprotocols need to be re-examined and stochastic geometryanalysis can be exploited in understanding AoI in large-scalesystems. Characterization of the above-mentioned performancemetrics by considering the spatial and temporal correlationsintroduced in this tutorial opens a broad research area.
B. Stochastic Geometry Analysis for Cross-layer Study
The majority of the existing literature on stochastic geometryanalysis assumes that the transmission demand is uniformlydistributed and regulated by some medium access control (MAC) protocols such as Aloha or CSMA. However, in prac-tice, the MAC-level transmission scheduling can be affectedby application-level service scheduling [129]. For example,in a mobile edge computing system where the mobiles canoffload computation tasks to the edge, different requests canbe scheduled to different edge servers based on the quality-of-service requirements of the tasks as well as the functionalityand the status of the servers. Characterizing the joint effects oflink-level scheduling and application-level scheduling by usingstochastic geometry models is an open research direction tounderstand cross-layer network performance. C. Machine Learning Approaches for Large-Scale Networksand Stochastic Geometry Analysis
Machine learning (ML) techniques have been increasinglyadopted for resource allocation and control in wireless net-works due to their capabilities in learning network environmentvariations (e.g., due to traffic pattern and channel uncertain-ties), classification of the relevant quantities, predicting futureoutcomes [130]. For instance, deep reinforcement learningapproaches are capable of estimating channel state informa-tion (CSI) without explicit feedback/detection and thus canbe exploited to design resource management schemes withlow communication and computation overhead compared toconventional CSI-based schemes [131], [132]. With ML-baseddesigns, network resource allocations will be more intelligentand environment-aware. Stochastic geometry tools can be usedto analyze the correlation effects introduced by environment-aware resource allocation in large-scale systems. For example,studying the impact of the correlation effects on the perfor-mance of ML-enable designs (e.g., stability, converges, andaccuracy) analytically based on stochastic geometry analysisis an appealing research direction.
D. Stochastic Geometry Analysis of Emerging Network Scenar-ios1) Reconfigurable Environment with Reconfigurable In-telligent Surfaces: A Reconfigurable Intelligent Surface (RIS) [133], [134], also known as software-defined hyper-surface and large intelligent surface/antennas , is a digitally-tunable metasurface. A metasurface is a thin and planarstructure composed of sub-wavelength passive scattering parti-cles which alter the electromagnetic waves through a surfaceimpedance boundary condition. The electromagnetic responseof an RIS can be reprogrammed without re-fabrication. Dueto this distinguishing feature, an RIS can be utilized to dy-namically alter the electromagnetic behavior (e.g., amplitude,phase, and frequency [135]) of the incident signals, and thusreshaping the wireless propagation environments. RIS has beenenvisioned as a candidate technology for future generationnetworks [136] and expected to be widely deployed on, e.g.,surfaces of buildings, vehicles and billboards, to facilitatethe ever-increasing traffic demands. In such an environment,the signal distributions are jointly manipulated by multiple RISs. The incident signals at nearby RISs come from commonsources and exhibit spatial correlations. Characterization of theaggregate effect of large-scale RISs by taking into account thespatial correction is an open research problem.
2) Joint Radar and Communication:
Joint radar and commu-nication (JRC) [137] is a novel paradigm that facilitates radardetection and data communication over the same frequencybands to improve the spectrum efficiency. In JRC systems, adual-functional transmitter (e.g., autonomous cars or unmanneddrones) simultaneously transmits data and detects radar targets(e.g., barriers, mobile vehicles and pedestrians) based on ashared and integrated hardware platform. Specifically, a portionof the transmitted signal received at the target receiver isused for communication, while the other portion of the signalreflected from the radar targets at the dual-function transmitteris utilized for detection [138]. Due to the common signal sourceand interferers, the communication and radar performance inJRC systems are correlated. Furthermore, to accommodate bothradar and communication demands, JRC brings a series ofresource allocation problems, including spectrum sharing withwaveform design, spatial beamforming, and power allocation.Analyzing radar and communication performance in large-scalesystems based on stochastic geometry analysis under differentresource allocation schemes is a promising future researchdirection.
3) Internet of Space Things:
The Internet of Space Things(IoST) [139] is an expansion of the ground Internet of Things(IoT) to the aerial and space domains enabled by dronesand miniaturized satellites/CubeSats [140], respectively. Withthe holistic integration of the ground with aerial and spacecommunication systems, IoST is expected to realize ubiquitousconnectivity virtually to support a broad range of applicationsincluding monitoring, tracking, in-space backhauling [141],wireless power transfer [142], [143] and remote healthcaresolutions. Performance characterization of IoST requires effortsof developing three-dimensional modeling incorporating dis-parate propagation characteristics, mobility patterns, resourceallocation schemes of communication systems as well as theirintricate correlations at different domains.
4) Terahertz Communications:
The terahertz band (i.e., 0.1-10 THz) provides ultra-wide spectrum resources and to enablenew applications for beyond 5G [144]. For example, terabits persecond (Tbps) data rate can be realized to support “fiber-like”communication performance which offers seamless transitionbetween optical fibers and THz wireless links with no latency.Moreover, the micro-scale wavelength makes THz band moresuitable for in-vivo nano-network communications than anyother frequency bands because terahertz waves have strong pen-etrating force and can be absorbed by in-vivo substances suchas liquids and organs [145]. The availability of new spectrumbands will necessitate novel resource allocation designs. Anintriguing research direction is to develop stochastic geometrymodels that take into account the correlation effects from: i)physical blockages, ii) direction of arrival/departure of high-gain beam steering, and iii) temporal broadening effect resulted from frequency selectivity in the ultra-wide THz band.
5) Internet of Nanothings (IoNT):
Internet of Nanoth-ings (IoNT) is a connected molecular system empowered bynanomachines, which are microscopic devices capable of per-forming high-precision functions for many real-world applica-tions such as biomedicine, food industry and military opera-tions [146]. As IoNT is expected to work in the environments offluids, gases or particulates, IoNT will foreseeably function ina manner that is drastically different from the IoT due to differ-ences in propagation environments, the scale of deployment, aswell as physical constraints (e.g., in energy and computations)of miniaturized nano devices. Therefore, it is imperative todevelop novel mathematical models to characterize the featuresof IoNT. Moreover, the protocol stack design and analysis forlarge-scale IoNT still remains an open field for exploration.XI. S
UMMARY
Stochastic geometry tools can be used to develop analyticalframeworks for large-scale wireless systems considering theeffects of spatio-temporal interference correlation, which isgenerally ignored in traditional performance analyses. In thistutorial, we have presented a comprehensive spatio-temporalanalysis of large-scale communications systems. In particular,we have formulated models to characterize correlations ininterference (and hence SIR) due to different effects such asdistributions of the interferers, distribution of contact distance,shadowing, transmission buffer status (or network queues),multihop transmissions, retransmissions, and user mobility.The performance of a target link in a large-scale system hasbeen demonstrated both analytically and numerically, consid-ering the effect of spatio-temporal interference correlation indifferent scenarios. In particular, we have derived the jointdistribution of spatially-correlated SIR with multihop relaying,temporally-correlated SIR with retransmissions and spatiallyand temporally-correlated SIR with mobility. For each of theabove scenarios, we have shed light on the technical challengesin stochastic geometry analysis. Finally, we have discussedfuture research directions in stochastic geometry analysis ofemerging wireless communication systems the performance ofwhich will be affected by spatio-temporal interference correla-tions. A
PPENDIX
A. Proof of Theorem 2 (Moments of the CSP Φ in Random Fieldsof Interferers)Proof. Given Φ , the CSP of a link with transmission distance r t can be derived as ¯ F η | Φ ( θ ) = P (cid:34) h t > θr α t (cid:88) j ∈ N h j (cid:107) x j (cid:107) − α (cid:12)(cid:12)(cid:12) Φ (cid:35) = E ( h j ) (cid:20) exp (cid:16) − θr α t (cid:88) j ∈ N h j (cid:107) x j (cid:107) − α (cid:17) (cid:12)(cid:12)(cid:12) Φ (cid:21) = (cid:89) j ∈ N
11 + θr α t r − αj . (108) The moments of the CSP given Mat´ern and Poisson fieldsof interferers can be obtained by applying their PGFLs as in(109), shown on the top of the next page, where ( a ) applies thePGFLs of MCP and PPP given in (34) and (23), respectively.By inserting f M ( y ) given in (31) in (109), the final result witha Mat´ern cluster field of interferers in (49) directly yields.We further extend the expression for the PPP in (109) as M P s ( b ) ( b ) = exp (cid:32) − πλδ (cid:90) ∞ (cid:18) − (cid:16) − θr α t z + θr α t (cid:17) b (cid:19) z δ − d z (cid:33) ( c ) = exp (cid:18) − πλδ ∞ (cid:88) k =1 (cid:18) bk (cid:19) ( − k +1 θ k r αk t × (cid:90) ∞ z δ − ( z + θr α t ) k d z (cid:19) , (110)where ( b ) employs the change of variable z = r θ − δ x andthe substitution δ = α , ( c ) takes the binomial expansion of (cid:0) − θr α t z + θr α t (cid:1) b for b ∈ C .The integral in (110) can be transformed as follows [148,Eqn. 3.196.2] (cid:90) ∞ z δ − ( z + θr α t ) k d z = θ δ − k r − αk t Γ( k − δ )Γ( δ )Γ( k ) ( d ) = θ δ − k r − αk t ( − k +1 π csc( πδ )Γ( δ )Γ( δ − k + 1)Γ( k )= θ δ − k r − αk t ( − k +1 π csc( πδ ) ( δ − . . . ( δ − k + 1)( k − θ δ − k r − αk t ( − k +1 π csc( πδ ) (cid:18) δ − k − (cid:19) , (111)where ( d ) follows as Γ( k − δ ) = π csc( π ( k − δ ))Γ( δ − k + 1) = ( − k +1 π csc( πδ )Γ( δ − k + 1) . Inserting (111) into (110) yields, M P s ( b ) = exp (cid:32) − λθ δ r πδ csc( πδ ) ∞ (cid:88) k =1 (cid:18) bk (cid:19)(cid:18) δ − k − (cid:19)(cid:33) ( e ) = exp (cid:18) − λθ δ r Γ(1 − δ )Γ( b + δ )Γ( b ) (cid:19) , (112)where ( e ) follows as πδ csc( πδ ) = Γ(1 − δ )Γ( δ ) and (cid:80) ∞ k =1 (cid:0) bk (cid:1)(cid:0) δ − k − (cid:1) = Γ( b + δ )Γ( b )Γ(1+ δ ) .With a β -Ginibre field of interferers, the success probabilitycan be obtained by following the distribution of the interferers.From (108), we have M P s ( b ) ( e ) = E Q j (cid:34) (cid:89) j ∈ N (cid:18) β θr α t Q − α/ j + 1 − β (cid:19) b (cid:35) ( f ) = (cid:89) j ∈ N (cid:90) ∞ ( πλ/β ) j Γ( j ) q j − e − πλq/β M P s ( b ) = E (cid:34) (cid:89) j ∈ N (cid:18)
11 + θ (cid:107) x t (cid:107) α (cid:107) r j (cid:107) − α (cid:19) b (cid:35) ( a ) = exp (cid:32) − λ p (cid:90) R (cid:20) − exp (cid:18) − ¯ c (cid:18) − (cid:90) R (cid:18) θr α t (cid:107) x − y (cid:107) − α (cid:19) b f M ( y )d y (cid:19)(cid:19)(cid:21) d x (cid:33) , MCP exp (cid:32) − λ (cid:90) R (cid:18) − (cid:18)
11 + θr α t (cid:107) x (cid:107) − α (cid:19) b (cid:19) d x (cid:33) , PPP , (109) × (cid:18) β θr α t q − α/ + 1 − β (cid:19) b d q, where ( e ) adopts the substitution Q j = r j and ( f ) followsfrom Q j ∼ G ( j, β/πλ ) .Summarizing the above results, we have the final results inTheorem 2. B. Proof of Theorem 3 (Moments of the CSP Φ of the TypicalCell-center User in a Poisson Downlink Network)Proof. Let Φ c (cid:44) { Φ | o ∈ R c } . Given a Poisson point process Φ c , the CSP of the typical cell-center user is ¯ F c η | Φ c ( θ )= P (cid:2) h > θ (cid:107) x (cid:107) α I o | Φ c (cid:3) = E (cid:20) exp (cid:18) − θ (cid:107) x (cid:107) α ∞ (cid:88) j =2 h j (cid:107) x j (cid:107) − α (cid:19) (cid:12)(cid:12)(cid:12) Φ c (cid:21) ( a ) = E (cid:20) exp (cid:18) − θ ∞ (cid:88) j =2 h j (cid:37) αj (cid:19) (cid:12)(cid:12)(cid:12) Φ Rc (cid:21) = ∞ (cid:89) j =2 (cid:37) ≤ ρ
11 + θ(cid:37) α , (113)where ( a ) changes the PPP to its RDP.Then, the b -th moments of the CSP Φ in a Poisson downlinknetwork can be derived as M c P s ( b )= E (cid:34) ∞ (cid:89) j =2 (cid:18)
11 + θ(cid:37) αj (cid:19) b (cid:12)(cid:12) (cid:37) ≤ ρ (cid:35) = E (cid:34) ∞ (cid:89) j =2 (cid:18)
11 + θρ α (cid:0) (cid:37) j ρ (cid:1) α (cid:19) b (cid:12)(cid:12) (cid:37) ≤ ρ (cid:35) ( b ) = E (cid:34) ∞ (cid:89) j =2 (cid:18)
11 + θρ α (cid:37) αj (cid:19) b (cid:35) ( c ) = (cid:18) (cid:90) (cid:18) − (cid:16) − θρ α (cid:37) α θρ α (cid:37) α (cid:17) b (cid:19) (cid:37) − d (cid:37) (cid:19) − d ) = (cid:18) − ∞ (cid:88) k =1 (cid:18) bk (cid:19) ( − θρ α ) k × (cid:90) (cid:16) (cid:37) α θρ α (cid:37) α (cid:17) k (cid:37) − d (cid:37) (cid:19) − e ) = (cid:18) − ∞ (cid:88) k =1 (cid:18) bk (cid:19) ( − θρ α ) k α × (cid:90) (cid:16) x θρ α x (cid:17) k x − α − d x (cid:19) − f ) = (cid:18) − ∞ (cid:88) k =1 (cid:18) bk (cid:19) ( − θρ α ) k δk − δ × F (cid:0) k, k − δ ; k + 1 − δ ; − θρ α (cid:1)(cid:19) − , (114)where ( b ) holds due to the fact that the probability law of (cid:37) conditioned on the probability law of (cid:37) , (cid:37) , . . . conditionedon (cid:37) < ρ is the same as the law of (cid:37) , (cid:37) , . . . withoutconditioning and ( c ) applies the PGFL of the RDP of a PPP, ( d ) takes the binomial expansion of (cid:0) − θρ α (cid:37) α θρ α (cid:37) α (cid:1) b for b ∈ C , ( e ) applies the change of variable x = (cid:37) α , and ( f ) substitutes α with δ .After some mathematical manipulations of (114), we havethe final results in Theorem 3. C. Proof of Theorem 4 (Moments of the CSP Φ for the TypicalVertex User in a Poisson Downlink Network)Proof. Based on the stationarity of a PPP, we have the successprobability of the typical vertex user conditioned on Φ as ¯ F v η | Φ ( θ )= E (cid:2) e − θr α I o | r = r = r , Φ (cid:3) = E (cid:34) exp (cid:32) − θr α (cid:18) h r − α + h r − α + ∞ (cid:88) j =4 h j r − αj (cid:19)(cid:33) (cid:12)(cid:12)(cid:12) Φ (cid:35) = (cid:0) θ (cid:1) − ∞ (cid:89) j =4
11 + θr α r − αj . (115)With ¯ F v η | Φ ( θ ) in (115), the moments of the CSP Φ can bederived as M v P s ( b )= (cid:16) θ (cid:17) − b E (cid:34) ∞ (cid:89) j =4 (cid:18)
11 + θ (cid:107) x (cid:107) α (cid:107) x j (cid:107) − α (cid:19) b (cid:35) ( a ) = (cid:16) θ (cid:17) − b × E r (cid:34) exp (cid:18) − πλ (cid:90) ∞ r (cid:18) − (cid:18) θr α x − α (cid:19) b (cid:19) x d x (cid:19)(cid:35) ( b ) = (cid:16) θ (cid:17) − b π λ (cid:90) ∞ exp( − λπr ) r × exp (cid:18) − πλ (cid:90) ∞ r (cid:18) − (cid:18) θr α x − α (cid:19) b (cid:19) x d x (cid:19) d r, (116)where ( a ) applies [147, Lemma 1] and the PGFL of a PPP, and ( b ) plugs in the PDF f v r ( r ) given in (57).After some mathematical manipulations of (116), we havethe final results in Theorem 4. D. Proof of Theorem 5 (Laplace Transform of Interference withIndependent and Correlated Shadowing)Proof.
We first derive the Laplace transform of the interferencewith correlated shadowing as follows. L I Cor o ( s )= E (cid:2) exp (cid:0) − sI Cor o (cid:1)(cid:3) = E (cid:20) exp (cid:16) − s (cid:88) j ∈ N h j (cid:96) ( (cid:107) x j (cid:107) ) S j (cid:17)(cid:21) ( a ) = E (cid:20) (cid:88) k ∈ N (cid:88) x j ∈ S k
11 + s(cid:96) ( (cid:107) x j (cid:107) ) T k (cid:21) ( b ) = E ( T k ) (cid:34) exp (cid:32) − λ (cid:88) k ∈ N (cid:90) S k (cid:18) − s(cid:96) ( (cid:107) x j (cid:107) ) T k (cid:19) d x (cid:33)(cid:35) , where ( a ) follows as h j ∼ E (1) and the points within the samecell share the same shadowing coefficient, and ( b ) applies thePGFL of an PPP and holds as the accumulative interferenceof each shadowing cell S k is averaged over the same randomvariable T k for x j ∈ S k .Similarly, the Laplace transform of I Ind o is given by L I Ind o ( s )= E (cid:2) exp (cid:0) − sI Ind o (cid:1)(cid:3) = E (cid:20) exp (cid:16) − s (cid:88) j ∈ N h j (cid:96) ( (cid:107) x j (cid:107) ) S j (cid:17)(cid:21) ( c ) = exp (cid:32) − λ (cid:88) k ∈ N (cid:90) S k (cid:18) − E T k (cid:20) s(cid:96) ( (cid:107) x j (cid:107) ) T k (cid:21)(cid:19) d x (cid:33) , where ( c ) follows as S j is an i.i.d mark for all x j ∈ S k withindependent shadowing, with CDF F k , as T k . E. Proof of Theorem 7 (Success Probability in a PoissonDownlink Network with Unsaturated Queues)Proof.
To bypass the difficulty of modeling coupled queuestatuses, we assume the typical receiver observes temporally-independent interference across different time slots. If each in-terfering BS is active with independent probability p A = E [ ι j ] ,the success probability can be derived as P s = P [ η > θ ] ( a ) ≈ E (cid:20) h (cid:107) x (cid:107) − α (cid:80) ∞ j =2 ι j h j (cid:107) x j (cid:107) − α > θ (cid:21) ( b ) = E (cid:34) ∞ (cid:89) j =2 (cid:18) p A θ (cid:107) x (cid:107) α (cid:107) x j (cid:107) − α + 1 − p A (cid:19)(cid:35) ( c ) = E (cid:34) ∞ (cid:89) j =2 (cid:18) p A θ(cid:37) αj + 1 − p A (cid:19)(cid:35) ( d ) = 11 + 2 (cid:82) p A θ(cid:37) α θ(cid:37) α (cid:37) − d (cid:37) ( e ) = 11 + δp A θ (cid:82) x θx x − δ − d x = 11 + δp A θ − δ F (1 , − δ ; 2 − δ ; − θ ) ( f ) = 11 − p A + p A2 F (1 , − δ ; 1 − δ ; − θ ) , (117)where ( a ) applies the independent interference assumption, ( b ) follows as h , h j ∼ E (1) , ( c ) changes the PPP to its RDP, ( d ) applies the PGFL of a RDP given in (24), ( e ) changes thevariable x = (cid:37) α and substitutes α with δ .As can be seen from (117), p A is the only unknown pa-rameter. To obtain p A , we need to analyze the probability thateach BS is active from the perspective of queue dynamics. Asa BS sends out a packet once its selected queue is not empty,the active probability of the BS is equivalent to the utilizationof its queue. Since the service rate of a BS is equivalent tothe success probability, with random scheduling, we have theservice rate of the typical queue at a BS conditioned on Φ B as µ = 1 N u P [ η > θ | Φ B ] , (118)where N u denotes the number of users associated with thetypical BS.According to the law of total probability, the mean servicerate of any user’s queue at its serving BS is calculated as ¯ µ = ∞ (cid:88) n =1 p N u ( n ) µ = ∞ (cid:88) n =1 p N u ( n ) P [ η > θ ] n , (119)where P [ η > θ ] has been obtained in (117), p N u ( n ) denotesthe PDF of N u . In Poisson downlink network, p N u ( n ) isapproximated as [5] p N u ( n ) ≈ ν ν Γ( n + ν )( λ u /λ B ) n n !Γ( ν )( λ u /λ B + ν ) n + ν , where ν = 3 . .From (117) and (119), the mean service rate ¯ µ is a functionof active probability p A . Based on the random scheduling, theactive probability of a BS equals the utilization factor of thequeue for the typical general user. Thus, we can establish afixed-point equation as follows: p A = ξ u ¯ µ = ξ u (cid:80) ∞ n =1 p N u ( n ) P s /n . Subsequently, we have the success probability of a Poissondownlink network with unsaturated queues given in Theorem7.
F. Proof of Theorem 2 (Success Probability in a Poisson BipolarNetwork)Proof.
Using the assumption that each transmitter is activewith independent probability, the success probability in Poissonbipolar networks is given by P s ( a ) ≈ E (cid:20) h t (cid:107) x t (cid:107) − α (cid:80) j ∈ N \{ ς ( x t ) } ι j h j (cid:107) x j (cid:107) − α > θ (cid:21) ( b ) = E (cid:20) (cid:89) j ∈ N (cid:18) p A θ (cid:107) x t (cid:107) α (cid:107) x j (cid:107) − α + 1 − p A (cid:19)(cid:21) ( c ) = exp (cid:18) − πλp A (cid:90) ∞ θr α t x − α +1 θr α t x − α d x (cid:19) ( d ) = exp (cid:18) − πλp A δ (cid:90) ∞ θr α t u α − u + θr α t d u (cid:19) ( e ) = exp (cid:16) − p A λπr θ δ Γ(1 + δ )Γ(1 − δ ) (cid:17) , (120)where ( a ) applies the independent interference assumption, ( b ) holds due to Slivnyak’s theorem, ( c ) applies the PGFL of a PPPand employs the substitution δ = 2 /α , ( d ) applies the changeof varaible u = x α , and ( e ) follows from [148, Eqn. 3.196.2]that (cid:82) ∞ u δ − u + A d u = A δ Γ(1 − δ )Γ( δ ) .Since the service rate of the typical transmitter is equivalentto the success probability of a time slot, p A can be writtenas min { ξP s , } which is a well-known result from Geo/Geo/1queue [149]. Subsequently, we can establish the followingequation: min (cid:8) ξ/P s , (cid:9) = min (cid:26) ξ exp (cid:0) − p A λπr θ δ Γ(1+ δ )Γ(1 − δ ) (cid:1) , (cid:27) = p A . (121)Since P s is a function of P A as shown in (120), solving theequation in (121) yields p A = min (cid:26) − W (cid:0) − ξλπr θ δ Γ(1+ δ )Γ(1 − δ ) (cid:1) λπr θ δ Γ(1 + δ )Γ(1 − δ ) , (cid:27) , (122)where W denotes the Lambert- W function [150].By inserting p A in (122) into the expression of P s = ξp A , wehave the final results presented in Theorem 8. G. Proof of Theorem 9 (Moments of the End-to-End CSP Φ )Proof. From (86), the end-to-end CSP Φ of M -hop relayinggiven a Poisson field of interferers can be expressed as ¯ F η | Φ ( θ )= P [ h > θd α I , h > θd α I , . . ., h M > θd αM I M | Φ m ]= E ( h j,m ) (cid:34) exp (cid:18) − θ M (cid:88) m =1 (cid:88) j ∈ N h j,m d αm (cid:107) x j,m − y m (cid:107) α (cid:19) (cid:12)(cid:12)(cid:12) Φ m (cid:35) , = (cid:89) j ∈ N M (cid:89) m =1
11 + θd αm (cid:107) x j,m − z m (cid:107) − α . (123)With the end-to-end success probability with a Poisson fieldof interferers in (123), we further compute the b -th momentof it as in (124), where ( a ) follows as h j,m ∼ E (1) and ( b ) applies the PGFL of a PPP. H. Proof of Theorem 10 (JSP of Multiple Transmissions)Proof.
The JSP for K transmissions is given as J K = P (cid:20) K (cid:92) k =1 (cid:110) h ( k ) > θr α t I ( k ) o (cid:111)(cid:21) (a) = E (cid:20) K (cid:89) k =1 exp (cid:16) − θr α t I ( k ) o (cid:17)(cid:21) = E (cid:20) K (cid:89) k =1 exp (cid:18) − θr α t (cid:88) j ∈ N h ( k ) j (cid:107) x ( k ) j (cid:107) − α (cid:19)(cid:21) , (125)where ( a ) follows as h ( k ) ∼ E (1) is independent acrossdifferent time slots.In the scenarios with QSI, the JSP can be expressed as J QSI K (b) = E (cid:34) (cid:89) j ∈ N (cid:18)
11 + θr α t (cid:107) x ( k ) j (cid:107) − α (cid:19) K (cid:35) (c) = exp (cid:32) − πλ (cid:90) ∞ (cid:18) − (cid:18)
11 + θr α t x − α (cid:19) K (cid:19) x d x (cid:33) , (126)where ( b ) follows as K transmissions are subject the samepoint process of interferers, and ( c ) applies the PGFL of aPPP. As (126) is equivalent to the moments of the CSP Φ inPoisson ad hoc networks in (109) with b replaced by K , wehave the final results of J QSI K in (92).Moreover, in the scenarios with FVI, as different transmis-sions are affected by independent point processes, the JSP canbe derived as J FVI K = K (cid:89) k =1 E h ( k ) j (cid:34) exp (cid:18) − θr α t (cid:88) j ∈ N h ( k ) j (cid:107) x ( k ) j (cid:107) − α (cid:19)(cid:35) ( d ) = K (cid:89) k =1 E (cid:34) (cid:89) j ∈ N
11 + θr α t (cid:107) x ( k ) j (cid:107) − α (cid:35) ( e ) = exp (cid:32) − πλ (cid:90) ∞ (cid:18) −
11 + θr α t x − α (cid:19) x d x (cid:33) K ( f ) = exp (cid:16) − cλr θ δ K (cid:17) , where ( d ) holds as h ( k ) j ∼ E (1) , ( e ) follows as the point processof interferers across different transmissions are i.i.d. and appliesthe PGFL of a PPP, and ( f ) follows the same steps of thederivations of (120). M Poi P s ( b, M ) ( a ) = E Φ (cid:34) (cid:89) j ∈ N M (cid:89) m =1 (cid:18)
11 + θd αm (cid:107) x j,m − z m (cid:107) − α (cid:19) b (cid:35) , ( b ) = exp (cid:32) − λ (cid:90) R (cid:18) − M (cid:89) m =1 (cid:18)
11 + θd αm (cid:107) x − z m (cid:107) − α (cid:19) b (cid:19) d x (cid:33) , QSIexp (cid:32) − λ M (cid:89) m =1 (cid:90) R (cid:18) − (cid:18)
11 + θd αm (cid:107) x − z m (cid:107) − α (cid:19) b (cid:19) d x (cid:33) , FVI , (124) I. Proof of Corollary 4 (Success Probability of Type-II HARQ)Proof.
When K = 2 , the success probability with Type-IIHARQ-CC given in (98) can be rewritten as P II = P (cid:2) η (1) > θ (cid:3) + P (cid:2) η (1) + η (2) > θ, η (1) < θ (cid:3) = P (cid:2) η (1) > θ (cid:3) + E (cid:104) P (cid:2) η (2) > θ − η (1) | η (1) (cid:3) { η (1) <θ } (cid:105) = J + (cid:90) T P (cid:2) η (2) > θ − u (cid:3) f η (1) ( u )d u (cid:105) , (127)where J is given in Theorem 10 and f η (1) ( t ) represents thePDF of η (1) .Then, we derive the conditional CDF of η (1) given Φ (1) asfollows: P (cid:2) η (1) ≤ t (cid:3) = E (cid:20) − exp (cid:16) − ur α t (cid:88) j ∈ N h (1) j (cid:107) x j (cid:107) − α (cid:17)(cid:21) = 1 − (cid:89) j ∈ N
11 + ur α t r − αj . (128)Subsequently, the PDF of η (1) can be derived by taking thederivative of (128) with respective to u as follows: f η (1) ( u ) = ∂ (cid:16) − (cid:81) j ∈ N ur α t r − αj (cid:17) ∂u = (cid:88) j ∈ N r α t r − αj (1 + ur α t r − αj ) i (cid:54) = j (cid:89) i ∈ N
11 + ur α t r − αi . (129)By plugging (129) into the second term of (127), we have(130), where ( a ) applies the Campbell-Mecke formula givenin (27).Subsequently, by computing the expectations in (130) basedon the PGFL of the PPP, we have the final results in Corol-lary
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