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Unmanned aerial vehicle (UAV) assisted wireless communication has been recently proposed as an effective solution to provide wireless connectivity to ground users which can improve the network coverage area and spectral efficiency. In this paper, we consider the post‐disaster network scenario where users are isolated from the network infrastructure. We investigate the best positioning of the UAV in terms of the optimal trajectory as well as UAV vertical location to cover as many ground users as possible to improve the spectral efficiency. When the trajectory is optimized, the optimal transmit power within the available power budget is then obtained. The simulation results demonstrate that the proposed UAV‐enabled wireless communication technique improves the overall spectral and energy efficiency. [ABSTRACT FROM AUTHOR]

This paper develops a new approach to the modeling and analysis of device-to-device (D2D) underlaying multi-tier cellular network for dense hotspot communications, which consists of macro base stations (MBSs), pico BSs (PBSs), femto BSs (FBSs). A typicl user equipment (UE) can work either in D2D mode or cellular mode. Considering the dense hotspot communications, this work employs Poisson point process (PPP) to model the locations of MBSs and PBSs, and uses Poisson cluster process (PCP) to model the ones of UEs and FBSs. The locations of PBSs are also modeled as the centers of hotspots, referred to as the centers of PCPs. UEs and FBSs cluster around the common parent process PBSs. To guard the cluster-edge UEs, the clustered-UE classification and modified fractional frequency reuse (FFR) are jointly used, by which both the UEs and FBSs are classified two sets, cluster-center UEs and cluster-edge UEs, cluster-center FBSs and cluster-edge FBSs, respectively. The total frequency band is divided into two orthogonal segments, one of which is shared by D2D devices, cluster-edge FBSs, and PBSs, and the other segment of which is shared by cluster-center FBSs and MBSs. For such clustered multi-tier network, by using the methods from PPP, PCP, and PHP, this paper presents a tractable approach for modeling and analyzing the performance of cellular and D2D networks and gives the statistical descriptions of the experienced interferences at a typical D2D or cellular receiver by using the approximated Poisson hole processes (PHP) theory. This yields the derivations of the coverage probabilities of both the D2D receivers and cellular destinations. In additon, during the analysis of cellular UEs, to derive the coverage probabilities, this paper specially constructs one UE association criterion as well as the derivations of both the association probabilities and the statistical descriptions of association distances for cluster-center and cluster-edge UEs. The simulations results exploit the effect of various network parameters on the network performance and give the insights in terms of the proposed schemes as well as the comparison between cluster-center and cluster-edge UEs. [ABSTRACT FROM AUTHOR]

Wireless networks in next generation will provide users ubiquitous computing services with low delay by devices at the network edge, namely mobile edge computing (MEC). The intensive computation tasks can be partially offloaded to the MEC server via the wireless link and then processed through the MEC computation resources to cater for the delay demand. A parallel computation process is formed in the MEC network consists of local computation at MEC users (MUs) and MEC computation at MEC servers. However, the fluctuating wireless channel environment, changeable spatial distribution of MUs and the randomness of MEC servers’ locations make it hard to characterize and guarantee the end-to-end quality of service requirements. In this work, we are devoted to analyze and optimize the overall delay bound for MEC networks under two orthogonal frequency division multiple access (OFDMA) strategies via stochastic network calculus (SNC). Specifically, Poisson cluster process is utilized to capture the randomness of MEC servers’ and users’ spatial locations and to derive the Laplace transform of interference suffered by an MU of interest. The upper bounds for the delay violation probability of two OFDMA strategies are established by exploiting SNC with the Mellin transform of signal-to-interference ratio. Furthermore, we propose an optimal task offloading scheme by minimizing the overall delay, which balances the local computation delay and MEC delay. [ABSTRACT FROM AUTHOR]

This paper provides an analytical framework for the coexistence of uplink cellular network and D2D network in the same frequency band. D2D devices are assumed to be distributed according to a Poisson Cluster Process (PCP), while the locations of the cellular users and macro BSs are modeled as an independent Poisson Point Process (PPP) respectively. We study the performance of the uplink in cellular network with power control and the performance of this D2D network for two content availability cases: (1) Uniform content availability, which means content of interest to a typical device is available on devices selected randomly and uniformly from the same cluster, and (2) Closest content availability, which means content of interest is available on the closest devices to a typical device in the same cluster. Using this model, the distribution of the Signal to Interference Ratios (SIR) of a typical receiving node (tagged BS or D2D Rx) can be obtained. We derive the coverage probability, area spectrum efficiency (ASE) and average rate for both cellular and D2D links to analyze the performance of the whole network, and our analysis shows that an optimal number of D2D links must be simultaneously activated per cluster in order to maximize ASE. [ABSTRACT FROM AUTHOR]

In typical wireless heterogeneous networks (HetNets), users are clustered around known hotspots, e.g., shopping centers or schools, but such a non-uniform distribution of nodes is difficult to analyze. This paper explicitly models this scenario, with macro base stations (MBSs) modeled by a homogeneous Poisson point process (PPP), and millimeter-wave small base stations (mmWave SBSs) and users clustered around the hotspot centers, forming two Poisson cluster processes (PCPs), respectively. Fractional frequency reuse (FFR) and coordinated multi-point transmission (CoMP) are assumed since they help to limit the co-tier interference and enhance the coverage and capacity of the network. We present a distance-based approach for grouping macro user equipments (MUEs) from the cell center (CC) and cell edge (CE) regions for FFR analysis. We first derive some distance distributions, including joint distance distribution from the typical user to the cooperative open-access mmWave SBS and distance distribution from the typical user to the non-cooperative open-access mmWave SBS. We obtain expressions for various performance metrics, including association probability, signal to interference-plus-noise ratio (SINR) coverage probability, and ergodic capacity, under these conditions. Due to the complexity of the exact expressions, we derive novel approximations, using Alzer's lemma, to obtain the lower bounds on coverage and ergodic capacity, which are shown to be accurate through Monte Carlo simulation. Simulation results analyze the effect of different parameters on the network performance to give some guidance for the design of future networks. Numerical optimization of a key parameter, in terms of association probability, coverage probability, and ergodic capacity, is enabled by our analysis. [ABSTRACT FROM AUTHOR]

In order to satisfy the requirement of extremely high data rate in traffic hotspot regions, millimeter wave (mmWave) has attracted significant attention in wireless communication networks. While the coverage performance of mmWave networks based on the distribution of signal-to-interference-plus-noise ratio (SINR) has been widely studied, it provides only very limited information on the link reliability. In this paper, we provide a fine-grained performance analysis of the mmWave networks with hotspots. Specifically, we first establish a general and tractable framework to investigate the performance of mmWave networks using the Poisson cluster process integrated with several features of the mmWave band. Both open and closed association strategies are considered. To show what fraction of users in the networks achieves target reliability when the SINR is given, we derive the tier association probability and the moments of the conditional SINR distribution, based on which the exact meta distributions of SINR are given. Interestingly, in clustered mmWave networks, the widely used standard and generalized beta approximations do not work well when the blockage effect is severe. To resolve this issue, we provide a modified approximation by scaling the standard beta distribution, which is shown to be closer to the exact results. We conduct extensive simulations to study the impact of mmWave and deployment features on the performance of clustered mmWave networks. Numerical results reveal that the optimal scattering variance of mmWave base stations scales with the cluster size to maximize the number of concurrent reliable links, and increasing the antenna directivity results in more reliable communications than elevating the transmit power of mmWave base stations. [ABSTRACT FROM AUTHOR]

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Device-to-device (D2D) communication is a promising solution to meet rapidly growing demands for data services via spectrum reuse. This paper studies the physical layer security in a D2D underlay cellular network from a network-wide perspective, where the locations of D2D and cellular users are modeled as Poisson cluster processes (PCPs) to characterize the clustering feature of D2D users, the locations of eavesdroppers (Eves) and base stations (BSs) are modeled as a PCP and Poisson point process (PPP), respectively. We establish an analytical framework to assess the coverage and security performance of the network. Two scenarios are considered, i.e., one D2D pair scenario and multiple D2D pairs scenario, where in each cell there is one or multiple D2D users (DUs) sharing the frequency spectrum with the cellular users (CUs) in each time slot of the TDMA scheme adopted by BSs. In each considered scenario, we derive exact expressions for the coverage outage probabilities (COPs) and secrecy outage probabilities (SOPs), respectively, for both the CUs and DUs. Furthermore, the exact expression for the network-wide secrecy throughput (ST) is derived. Numerical results are presented to verify our theoretical derivations and reveal some insights into the impact of various parameters on the system performance. [ABSTRACT FROM AUTHOR]

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes. [ABSTRACT FROM AUTHOR]

Device-to-Device (D2D) communication has become a promising candidate in future cellular networks to improve spectrum efficiency and energy efficiency, while reducing the latency. As the capacity of D2D user equipments (DUEs) increases, it makes DUEs caching possible, and it can offload traffic from macro base stations, perform computation-intensive and latency-critical tasks. In this paper, in-band communication is considered, and the Poisson cluster process is utilized to model and analyze the clustered D2D networks under cluster-centric caching strategy. Firstly, we use the Thomas cluster process to model cellular user equipments (CUEs) and DUEs, and give a deployment scheme of clustered D2D caching networks. Secondly, the aggregated interference of the typical D2D receiver is analyzed in the clustered D2D networks. Then the Laplace transform of the aggregated interference is analyzed, and the expressions of coverage probability, average achievable rate and cache hit probability of the typical D2D receiver are deduced. The simulation results show that we can adjust the path loss exponent, densities of DUEs and CUEs, transmitting power of CUEs, mean of simultaneously active transmitters in each cluster and Zipf exponent to improve the performance of clustered D2D caching networks. [ABSTRACT FROM AUTHOR]

This paper develops a stochastic geometry-based approach for the modeling, analysis, and optimization of wireless cloud caching networks comprised of multiple-antenna radio units (RUs) inside clouds with coordinated multi-point transmissions and guard zones. We consider Poisson cluster processes to model RUs and users, and the probabilistic content placement to cache files in RUs. Accordingly, we study the exact hit probability for a user of interest for two strategies; closest selection, where the user is served by the closest RU that has its requested file, and best power selection, where the serving RU having the requested file provides the maximum instantaneous received power at the user. As key steps for the analyses, the Laplace transform of out of cloud interference, the desired link distance distribution in the closest selection, and the desired link received power distribution in the best power selection are derived. Also, we approximate the derived exact hit probabilities for both the closest and the best power selections in such a way that the related objective functions for the content caching design of the network can lead to tractable concave optimization problems. Solving the optimization problems, we propose algorithms to efficiently find their optimal content placements. Finally, we investigate the impact of different parameters on the caching performance. [ABSTRACT FROM AUTHOR]

In order to satisfy the requirement of high throughput in most UAV applications, the potential of integrating millimeter wave (mmWave) communications with UAV networks is explored in this paper. A tractable three-dimensional (3D) spatial model is proposed for evaluating the average downlink performance of UAV networks at mmWave bands, where the locations of UAVs and users are randomly distributed with the aid of a Poisson cluster process. Moreover, an actual 3D antenna model with the uniform planar array is deployed at all UAVs to examine the impact of both azimuth and elevation angles. Based on this framework and two typical user selection schemes, closed-form approximation equations of the evaluated coverage probability and area spectral efficiency (ASE) are derived. In a noise-limited scenario, an exact expression is provided, which theoretically demonstrates that a large scale of antenna elements is able to enhance the coverage performance. Regarding the altitude of UAVs, there exists at least one optimal height for maximizing the coverage probability. Numerical results verify the proposed insight that non-line-of-sight transmission caused by obstacles have negligible effects on the proposed system. Another interesting result is that the ASE can be maximized by optimizing both the targeted data rate and the density of UAVs. [ABSTRACT FROM AUTHOR]

In this paper, downlink non-orthogonal multiple access (NOMA) systems in large-scale cellular networks are considered and an analytical framework to evaluate the outage performance is developed. Different from the previous work where the users’ distribution is assumed to be completely independent of the Base Station (BS) locations, the deployment of the BSs and the location of NOMA-user pair are modeled as Poisson Cluster Process so that the relation between system throughput and the density of BSs could be effectively formulated and clearly revealed. Under the proposed system model, the closed-form expressions of outage probabilities are derived for 2-user downlink NOMA systems. Furthermore, the power allocation strategy is proposed for minimizing the maximal outage probability among NOMA users. And the average area spectral efficiency is also defined and derived to evaluate the throughput of the network. The accuracy of the theoretical results is verified by the simulations. [ABSTRACT FROM AUTHOR]

Deploying Sub-6GHz networks together with millimeter wave (mmWave) is a promising solution to achieve high data rates in traffic hotspots while guaranteeing sufficient coverage, where mmWave small cells are densely deployed to provide high quality of service. In this paper, we propose an analytical framework to investigate the integrated Sub-6GHz-mmWave cellular networks, in which the Sub-6GHz base stations (BSs) are modeled as a Poisson point process, and the mmWave BSs are clustered following a Poisson cluster process in traffic hotspots. We conduct stochastic geometry-based analysis and derive the performance metrics including the association probability, signal-to-interference-plus-noise ratio coverage probability and average achievable rate, which are validated to be accurate by Monte Carlo simulations. We analyze the impact of various deployment parameters on the network performance to give insights on the network design. In particular, it is shown that deploying mmWave small cells in traffic hotspots will outperform both traditional Sub-6GHz heterogeneous network and isolated mmWave system in terms of the coverage probability. It can also be shown that extremely high and extremely small association weight for mmWave BSs will deteriorate the performance for cell edge users and cell interior users, respectively. Moreover, there exists an optimal pre-decided dispersion parameter of mmWave BSs that contributes to the maximum coverage probability. [ABSTRACT FROM AUTHOR]

Owing to its flexibility in modeling real-world spatial configurations of users and base stations (BSs), the Poisson cluster process (PCP) has recently emerged as an appealing way to model and analyze heterogeneous cellular networks (HetNets). Despite its undisputed relevance to HetNets—corroborated by the models used in the industry—the PCP’s use in performance analysis has been limited. This is primarily because of the lack of analytical tools to characterize the performance metrics, such as the coverage probability of a user connected to the strongest BS. In this paper, we develop an analytical framework for the evaluation of the coverage probability, or equivalently the complementary cumulative density function (CCDF) of signal-to-interference-and-noise ratio (SINR), of a typical user in a $K$ -tier HetNet under a $\max $ power-based association strategy, where the BS locations of each tier follow either a Poisson point process (PPP) or a PCP. The key enabling step involves conditioning on the parent PPPs of all the PCPs, which allows us to express the coverage probability as a product of sum-product and probability generating functionals (PGFLs) of the parent PPPs. In addition to several useful insights, our analysis provides a rigorous way to study the impact of the cluster size on the ${\it SINR}$ distribution, which was not possible using the existing PPP-based models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uplink transmission performance of low-power wide-area (LPWA) networks with regards to coexisting radio modules. We adopt the long-range (LoRa) radio technique as an example of the network of focus, even though our analysis can be easily extended to other situations. We exploit a new topology to model the network, where the node locations of LoRa follow a Poisson cluster process while other coexisting radio modules follow a Poisson point process. Unlike most of the performance analysis based on stochastic geometry, we take noise into consideration. More specifically, two models, with a fixed and a random number of active LoRa nodes in each cluster, respectively, are considered. To obtain insights, both the exact and simple approximated expressions for coverage probability are derived. Based on them, area spectral efficiency and energy efficiency are obtained. From our analysis, we show how the performance of LPWA networks can be enhanced by adjusting the density of LoRa nodes around each LoRa receiver. Moreover, the simulation results unveil that the optimal number of active LoRa nodes in each cluster exists to maximize the area spectral efficiency. [ABSTRACT FROM AUTHOR]

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example. [ABSTRACT FROM AUTHOR]

In this paper, an analytical framework is provided to analyze the uplink performance of device-to-device (D2D)-enabled millimeter-wave (mm-wave) cellular networks with clustered D2D user equipments (UEs). The locations of cellular UEs are modeled as a Poisson point process, while the locations of potential D2D UEs are modeled as a Poisson cluster process. Signal-to-interference-plus-noise ratio outage probabilities are derived for both cellular and D2D links using tools from stochastic geometry. The distinguishing features of mm-wave communications such as directional beamforming and having different path loss laws for the line-of-sight and non-line-of-sight links are incorporated into the outage analysis by employing a flexible mode selection scheme and Nakagami fading. Also, the effect of beamforming alignment errors on the outage probability is investigated to get insight into the performance in practical scenarios. Moreover, area spectral efficiency of the cellular and D2D networks is determined for both underlay and overlay types of sharing. Optimal spectrum partition factor is determined for overlay sharing by considering the optimal weighted proportional fair spectrum partition. [ABSTRACT FROM AUTHOR]

Caching popular contents at the base stations (BSs) is a promising way to reduce the backhaul congestion. In this paper, we consider the scenario of BS cooperation for a cache-enabled HetNets where the helpers with cache are distributed around the macro BSs (MBSs). In this paper, we model the locations of the MBSs and the helpers as a Poisson point process and a Poisson hole process (PHP), respectively. Based on the proposed model, we adopt different caching and BS cooperation strategies depending on the requested file. Due to the intractability of the PHP, we adopt a fitted Poisson cluster process (PCP) to approximate the PHP, which is shown to obtain tight approximation. Based on the approximated PCP, we derive explicit expressions of the average spectrum efficiency and the area power consumption using tools from stochastic geometry. Building upon the analytical results, we then solve the problems of the area power consumption minimization and the average spectrum efficiency maximization, in which a closed-form expression for the optimal segment allocation that maximizes the average spectrum efficiency is obtained. Simulation results justify the significant gain of the proposed design compared with previously known caching schemes. [ABSTRACT FROM AUTHOR]

The growing complexity of heterogeneous cellular networks (HetNets) has necessitated a variety of user and base station (BS) configurations to be considered for realistic performance evaluation and system design. This is directly reflected in the HetNet simulation models proposed by standardization bodies, such as the 3rd Generation Partnership Project (3GPP). Complementary to these simulation models, stochastic geometry-based approach, modeling the locations of the users, and the $K$ tiers of BSs as independent and homogeneous Poisson point processes (PPPs), has gained prominence in the past few years. Despite its success in revealing useful insights, this PPP-based K$ -tier HetNet model is not rich enough to capture spatial coupling between user and BS locations that exists in real-world HetNet deployments and is included in 3GPP simulation models. In this paper, we demonstrate that modeling a fraction of users and arbitrary number of BS tiers alternatively with a Poisson cluster process (PCP) captures the aforementioned coupling, thus bridging the gap between the 3GPP simulation models and the PPP-based analytic model for HetNets. We further show that the downlink coverage probability of a typical user under maximum signal-to-interference-ratio ( \mathtt {SIR} ) association can be expressed in terms of the sum-product functionals over PPP, PCP, and its associated offspring point process, which are all characterized as a part of our analysis. We also show that the proposed model converges to the PPP-based HetNet model as the cluster size of the PCPs tends to infinity. Finally, we specialize our analysis based on general PCPs for Thomas and Matérn cluster processes. Special instances of the proposed model closely resemble the different configurations for BS and user locations considered in 3GPP simulations. [ABSTRACT FROM PUBLISHER]

This paper develops a new approach to the modeling and analysis of heterogeneous cellular networks (HetNets) that accurately incorporates coupling across the locations of users and base stations, which exists due to the deployment of small cell base stations (SBSs) at the places of high user density (termed user hotspots in this paper). Modeling the locations of the geographical centers of user hotspots as a homogeneous Poisson point process (PPP), we assume that the users and SBSs are clustered around each user hotspot center independently with two different distributions. The macrocell BS locations are modeled by an independent PPP. This model is consistent with the user and SBS configurations considered by 3GPP. Using this model, we study the performance of a typical user in terms of coverage probability and throughput for two association policies: 1) Policy 1, under which a typical user is served by the open-access BS that provides maximum averaged received power and 2) Policy 2, under which the typical user is served by the small cell tier if the maximum averaged received power from the open-access SBSs is greater than a certain power threshold; and macro tier otherwise. A key intermediate step in our analysis is the derivation of distance distributions from a typical user to the open-access and closed-access interfering SBSs. Our analysis demonstrates that as the number of SBSs reusing the same resource block increases, coverage probability decreases, whereas throughput increases. Therefore, contrary to the usual assumption of orthogonal channelization, it is reasonable to assign the same resource block to multiple SBSs in a given cluster as long as the coverage probability remains acceptable. This approach to HetNet modeling and analysis significantly generalizes the state-of-the-art approaches that are based on modeling the locations of BSs and users by independent PPPs. [ABSTRACT FROM PUBLISHER]

This paper investigates the interference statistics in a clustered device-to-device network, where the cluster centers form a Poisson point process and each active receiver device selects the transmitter randomly following the uniform distribution. The statistics of the number of intra/inter-cluster interferers is first analyzed for both finite and infinite number of receivers per cluster. In addition, for the special cases where the numbers of transmitters and receivers differ significantly, simplified expressions are given. Then, using these derived probability distributions of the number of interferers, the Laplace transforms of intra/inter-cluster interference are established and simple approximations for the special cases are obtained as well. Our results show the following: 1) with the concurrent uniform transmitter selection, the distribution of the number of intra/inter-cluster interfering devices differs greatly from the truncated Poisson distribution assumed in the literature; 2) the model assumed in the literature underestimates the coverage probability specially for a small number of transmitters/receivers; and 3) unlike the model assumed in the literature, the ASE of the proposed model increases with the number of transmitters/receivers when the average number of active receivers is relatively small, while it does not monotonically vary with the number of transmitters/receivers when the average number of active receivers is large. [ABSTRACT FROM PUBLISHER]

This paper investigates the performance of millimeter wave (mmWave) communications in clustered device-to-device (D2D) networks. The locations of D2D transceivers are modeled as a Poisson Cluster Process. In each cluster, devices are equipped with multiple antennas, and the active D2D transmitter (D2D-Tx) utilizes mmWave to serve one of the proximate D2D receivers. Specifically, we introduce three user association strategies: 1) uniformly distributed D2D-Tx model; 2) nearest D2D-Tx model; and 3) closest line-of-site (LOS) D2D-Tx model. To characterize the performance of the considered scenarios, we derive new analytical expressions for the coverage probability and area spectral efficiency (ASE). Additionally, in order to efficiently illustrating the general trends of our system, a closed-form lower bound for the special case interfered by intra-cluster LOS links is derived. We provide Monte Carlo simulations to corroborate the theoretical results and show that: 1) the coverage probability is mainly affected by the intra-cluster interference with LOS links; 2) there exists an optimum number of simultaneously active D2D-Txs in each cluster for maximizing ASE; and 3) the closest LOS model outperforms the other two scenarios but at the cost of extra system overhead. [ABSTRACT FROM PUBLISHER]

In heterogeneous networks, the random deployment of femto access points (FAPs) and macro base stations (MBSs) with uncoordinated channel access imposes huge intertier interferences. In real-life networks, the process of MBSs’ deployment exhibits the homogeneity; however the FAPs have the behavioral characteristic of clusters formation, such as in malls, apartments, or offices. Therefore, the composite modeling of the MBSs and the FAPs using Poisson point process and Poisson cluster process is employed for the evaluation of coverage probability. The scenario of the real-time traffic for macrotier and the best-effort traffic for femto-tier is considered. Cognition is introduced in the clustered FAPs to control the intertier interference. Furthermore, the impact of the macrotier load is analyzed by exploiting the inherent coupling between coverage probability and activity factor of an MBS. Further, we study the effect of co-channel, orthogonal, and partial spectrum sharing modes on the coverage for given parameters, such as load condition and FAPs/MBSs density. We provide simulation validation for the derived expressions of coverage and present a comparative analysis for the mentioned spectrum sharing modes. [ABSTRACT FROM AUTHOR]

One of the principal underlying assumptions of current approaches to the analysis of heterogeneous cellular networks (HetNets) with random spatial models is the uniform distribution of users independent of the base station (BS) locations. This assumption is not quite accurate, especially for user-centric capacity-driven small cell deployments where low-power BSs are deployed in the areas of high user density, thus inducing a natural correlation in the BS and user locations. In order to capture this correlation, we enrich the existing $K$ -tier Poisson point process (PPP) HetNet model by considering user locations as Poisson Cluster Process with the BSs at the cluster centers. In particular, we provide the formal analysis of the downlink coverage probability in terms of a general density function describing the locations of users around the BSs. The derived results are specialized for two cases of interest: 1) Thomas cluster process, where the locations of the users around BSs are Gaussian distributed and 2) Matérn cluster process, where the users are uniformly distributed inside a disc of a given radius. Tight closed-form bounds for the coverage probability in these two cases are also derived. Our results demonstrate that the coverage probability decreases as the size of user clusters around BSs increases, ultimately collapsing to the result obtained under the assumption of PPP distribution of users independent of the BS locations when the cluster size goes to infinity. Using these results, we also handle mixed user distributions consisting of two types of users: 1) uniformly distributed and 2) clustered around certain tiers. [ABSTRACT FROM PUBLISHER]

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic. [ABSTRACT FROM AUTHOR]

Device-to-device (D2D) communication enables direct communication between proximate devices thereby improving the overall spectrum utilization and offloading traffic from cellular networks. This paper develops a new spatial model for D2D networks in which the device locations are modeled as a Poisson cluster process. Using this model, we study the performance of a typical D2D receiver in terms of coverage probability under two realistic content availability setups: 1) content of interest for a typical device is available at a device chosen uniformly at random from the same cluster, which we term uniform content availability, and 2) content of interest is available at the k^th closest device from the typical device inside the same cluster, which we term $k$-closest content availability. Using these coverage probability results, we also characterize the area spectral efficiency (ASE) of the whole network for the two setups. A key intermediate step in this analysis is the derivation of the distributions of distances from a typical device to both the intra- and inter-cluster devices. Our analysis reveals that an optimum number of D2D transmitters must be simultaneously activated per cluster in order to maximize ASE. This can be interpreted as the classical tradeoff between more aggressive frequency reuse and higher interference power. The optimum number of simultaneously transmitting devices and the resulting ASE increase as the content is made available closer to the receivers. Our analysis also quantifies the best and worst case performance of clustered D2D networks both in terms of coverage and ASE. [ABSTRACT FROM PUBLISHER]

This paper develops a comprehensive analytical framework with foundations in stochastic geometry to characterize the performance of cluster-centric content placement in a cache-enabled device-to-device (D2D) network. Different from device-centric content placement, cluster-centric placement focuses on placing content in each cluster, such that the collective performance of all the devices in each cluster is optimized. Modeling the locations of the devices by a Poisson cluster process, we define and analyze the performance for three general cases: 1) k -Tx case: the receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at the k{\mathrm{ th}} closest device to the cluster center; 2) \ell $ -Rx case: the receiver of interest is the \ell {\mathrm{ th}} closest device to the cluster center and its content of interest is available at a device chosen uniformly at random from the same cluster; and 3) baseline case: the receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at a device chosen independently and uniformly at random from the same cluster. Easy-to-use expressions for the key performance metrics, such as coverage probability and area spectral efficiency of the whole network, are derived for all three cases. Our analysis concretely demonstrates significant improvement in the network performance when the device on which content is cached or device requesting content from cache is biased to lie closer to the cluster center compared with the baseline case. Based on this insight, we develop and analyze a new generative model for cluster-centric D2D networks that allows to study the effect of intra-cluster interfering devices that are more likely to lie closer to the cluster center. [ABSTRACT FROM AUTHOR]

Due to its tractability, a multitier model of mutually independent Poisson point processes (PPPs) for heterogeneous cellular networks (HCNs) has recently been attracting much attention. However, in reality, the locations of the BSs, within each tier and across tiers, are not fully independent. Accordingly, in this paper, we propose two HCN models with inter-tier dependence (Case 1) and intra-tier dependence (Case 2), respectively. In Case 1, the macro-base station (MBS) and the pico-base station (PBS) deployments follow a Poisson point process (PPP) and a Poisson hole process (PHP), respectively. Under this setup and conditioning on a fixed serving distance (distance between a user and its nearest serving BS), we derive bounds on the outage probabilities of both macro and pico users. We also use a fitted Poisson cluster process to approximate the PHP, which is shown to provide a good approximation of the interference and outage statistics. In Case 2, the MBSs and the PBSs follow a PPP and an independent Matern cluster process, respectively. Explicit expressions of the interference and the outage probability are derived first for fixed serving distance and second with random distance, and we derive the outage performance, the per-user capacity, and the area spectral efficiency (ASE) for both cases. It turns out that the proposed Case 2 model is a more appropriate and accurate model for a HCN with hotspot regions than the multitier independent PPP model since the latter underestimates some key performance metrics, such as the per-user capacity and the ASE, by a factor of 1.5 to 2. Overall, the two models proposed provide good tradeoffs between the accuracy, tractability, and practicability. [ABSTRACT FROM PUBLISHER]

Let $$(X_n)$$ be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution $$F$$ . An observation $$X_n$$ is a near-record if $$X_n\in (M_{n-1}-a,M_{n-1}]$$ , where $$M_{n}=\max \{X_1,\ldots ,X_{n}\}$$ and $$a>0$$ is a parameter. We analyze the point process $$\eta $$ on $$[0,\infty )$$ of near-record values from $$(X_n)$$ , showing that it is a Poisson cluster process. We derive the probability generating functional of $$\eta $$ and formulas for the expectation, variance and covariance of the counting variables $$\eta (A), A\subset [0,\infty )$$ . We also obtain strong convergence and asymptotic normality of $$\eta (t):=\eta ([0,t])$$ , as $$t\rightarrow \infty $$ , under mild tail-regularity conditions on $$F$$ . For heavy-tailed distributions, with square-integrable hazard function, we show that $$\eta (t)$$ grows to a finite random limit $$\eta (\infty )$$ and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data. [ABSTRACT FROM AUTHOR]

Future mobile networks are visualized as networks that consist of more than one type of base station to cope with rising user demands. Such networks are referred to as heterogeneous networks. There have been various attempts at modeling and optimization of such networks using spatial point processes, some of which are alluded to (later) in this paper. We model a heterogeneous network consisting of two types of base stations by using a particular Poisson cluster process model. The main contributions are two-fold. First, a complete description of the interference in heterogeneous networks is derived in the form of its Laplace functional. Second, using an asymptotic convergence result which was shown in our previous work, we derive the expressions for the mean and variance of the distribution to which the interference converges. The utility of this framework is discussed for both the contributions. [ABSTRACT FROM PUBLISHER]

The concept of extending traditional macrocell cellular structure with small cells (like femtocells) in next-generation mobile networks (e. g., Long Term Evolution Advanced) provides a great opportunity to improve coverage and enhance data rate. Femtocells are cost efficient, indoor base stations. These femtocells can operate in closed mode i. e. only restricted users connection are allowed. Therefore, if the number of deployed femtocells is significant, that can dramatically modify the interference pattern of a macrocell. Thus mobile service providers have to pay attention for the number of simultaneously operating femtocells and encroach, if necessary, to provide appropriate service level to every mobile user. In this paper we provide an analytic framework to characterize the upper bound of service outage probability for a potential macrocell user in a two-tier mobile system, when the radio channels are infected by Nakagami- $$m$$ fading. In our proposal the femtocells are operating in closed mode and deployed into a designated macrocell, hence every femtocell increases the interference level. The spatial location femtocells is modelled with Poisson cluster process. Compared to traditional grid structure or completely spatial random Poisson point process femtocell deployment, cluster based layout may provides more life realistic deployment scenario. To evaluate the upper bound of service outage we use the tools of stochastic geometry. [ABSTRACT FROM AUTHOR]

A considerable challenge in plant ecology is to understand how interactions, such as competition or facilitation, shape the spatial distribution of plants. The 'stress gradient hypothesis' predicts that facilitation and competition will vary inversely across gradients of abiotic stress or consumer pressure. Surprisingly, few previous studies have explored how the balance between facilitation and competition affects spatial patterns along gradients of stress in a plant population based on field experiments. In order to investigate the effects of consumer pressure, facilitation, and competition on the spatial pattern of plant populations, we conducted a restoration succession series field experiment in the Inner Mongolian steppe in which sample sites of graded consumer pressure, specifically grazing stress, were established. We chose to examine the spatial patterns of Leymus chinensis, a dominant species in our experimental site. In order to test the 'stress gradient hypothesis,' we applied the univariate O-ring statistic to analyze local neighborhood density at different spatial scales. We used the pair-correlation function to detect the characteristics of point patterns using complete spatial randomness, the Poisson cluster process, and the nested double-cluster process. We found that the local densities of L. chinensis were higher under high stress than lower stress environments. This demonstrated the 'stress gradient hypothesis' in that facilitation and competition varied inversely across gradients of consumer pressure. However, we found no differences in the spatial patterns of L. chinensis based on complete spatial randomness when interactions shifted from facilitation to competition along gradients of consumer pressure. Furthermore, we detected the characteristics of point patterns using the Poisson cluster and nested double-cluster processes. The results showed the spatial patterns of L. chinensis to fit well with the nested double-cluster model under highly stressful conditions, while in lower stress environments they were best approximated by the Thomas process. Our results illustrate that a shift in interactions from facilitation to competition along gradients of consumer pressure can shape spatial patterns and that a combination of the Poisson cluster process and nested double-cluster process can detect spatial pattern characteristics which cannot be detected by complete spatial randomness. [ABSTRACT FROM AUTHOR]

In this article, the time from the start of chemotherapy randomization until cancer relapse is of primary interest. Here, cancer relapse refers to the appearance of the first observable malignant clone after therapy. A dynamic model for cancer relapse after chemotherapy is developed. The model differs from the traditional cure rate models in that it takes into consideration the growth kinetics of malignant tumors using a two-stage carcinogenesis model. The survival and hazard functions for cancer relapse time are derived, and a simulation study is performed to validate the underlying model. [ABSTRACT FROM AUTHOR]

The article presents information on a study which uses a wide class of spatial point processes, the shot noise Cox processes (SNCP), to model species occupancies at different spatial scales. The study show that species' spatial aggregation is important for predicting population estimates at fine scales starting from coarser ones.

We study stochastic properties of the empty space for stationary germ-grain models in ℝd: in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞. [ABSTRACT FROM AUTHOR]

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A point process rainfall model is further developed that has storm origins occurring in space-time according to a Poisson process, where each storm origin has a random radius so that storms occur as circular regions in two-dimensional space, where the storm radii are taken to be independent exponential random variables. Each storm origin is of random type z, where z follows a continuous probability distribution. Cell origins occur in a further spatial Poisson process and have arrival times that follow a Neyman-Scott point process. Each cell origin has a radius so that cells form discs in two-dimensional space, where the cell radii are independent exponential random variables. Each cell has a random lifetime and an intensity that remains constant over both the cell lifetime and cell disk area. Statistical properties up to third order are given for the model. Using these properties, the model is fitted to 10 min series taken from 23 sites across the Rome region, Italy. Distributional properties of the observed annual maxima are compared to equivalent values sampled from series that are simulated using the fitted model. The results indicate that the model will be of use in urban drainage projects for the Rome region. [ABSTRACT FROM AUTHOR]

We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ2 of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$. σ2 is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n → ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$. The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$. Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n. [ABSTRACT FROM AUTHOR]

Species spatial turnover, or β-diversity, induces a decay of community similarity with geographic distance known as the distance–decay relationship. Although this relationship is central to biodiversity and biogeography, its theoretical underpinnings remain poorly understood. Here, we develop a general framework to describe how the distance–decay relationship is influenced by population aggregation and the landscape-scale species-abundance distribution. We utilize this general framework and data from three tropical forests to show that rare species have a weak influence on distance–decay curves, and that overall similarity and rates of decay are primarily influenced by species abundances and population aggregation respectively. We illustrate the utility of the framework by deriving an exact analytical expression of the distance–decay relationship when population aggregation is characterized by the Poisson Cluster Process. Our study provides a foundation for understanding the distance–decay relationship, and for predicting and testing patterns of beta-diversity under competing theories in ecology. [ABSTRACT FROM AUTHOR]

Hawkes processes are important in point process theory and its applications, and simulation of such processes are often needed for various statistical purposes. This article concerns a simulation algorithm for unmarked and marked Hawkes processes, exploiting that the process can be constructed as a Poisson cluster process. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work Møller and Rasmussen (2004). We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations. Extensions of the algorithm and the results to more general types of marked point processes are also discussed. [ABSTRACT FROM AUTHOR]

Routers have the ability to output statistics about packets and flows of packets that traverse them. Since, however, the generation of detailed traffic statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this paper, we study both theoretically and practically what information about the original traffic can be inferred when sampling, or ‘thinning’, is performed at the packet level. While basic packet level characteristics such as first order statistics can be fairly directly recovered, other aspects require more attention. We focus mainly on the spectral density, a second-order statistic, and the distribution of the number of packets per flow, showing how both can be exactly recovered, in theory. We then show in detail why in practice this cannot be done using the traditional packet based sampling, even for high sampling rate. We introduce an alternative flow-based thinning, where practical inversion is possible even at arbitrarily low sampling rate. We also investigate the theory and practice of fitting the parameters of a Poisson cluster process, modeling the full packet traffic, from sampled data. [ABSTRACT FROM AUTHOR]

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented. [ABSTRACT FROM AUTHOR]

ABSTRACT. Product quality in the paper-making industry can be assessed by opacity of a linear trace through continuous production sheets, summarized in spectral form. We adopt a class of non-Gaussian stochastic models for continuous spatial variation to describe data of this type. The model has flexible covariance structure, physically interpretable parameters and allows several scales of variation for the underlying process. We derive the spectral properties of the model, and develop methods of parameter estimation based on maximum likelihood in the frequency domain. The methods are illustrated using sample data from a UK mill. [ABSTRACT FROM AUTHOR]

Heterogeneous cellular networks (HetNets) consist of different tiers of base stations (BSs) to meet the ever-increasing mobile traffic demand. Due to the random deployment of BSs, Poisson point process (PPP) is often used to model the BS distribution. However, low power small cells are usually clustered around the popular areas, and PPP can not reflect such a feature. To this end, we in this paper consider base station (BS) cooperation and analyze user rate and energy efficiency of HetNets based on a Poisson cluster process (PCP). A calculable formula for the average data rate (or spectral efficiency) and its approximated closed form are derived. Based on this closed form, a power minimization solution with certain spectral efficiency constraint is proposed, and the optimal cooperation radii are derived. Furthermore, we analyze spectral efficiency under a limited number of cooperative BSs in a two-tier network. Finally, we propose a range expansion (RE) scheme and examine the impact of this scheme. The theoretical analyses are verified by simulation results. [ABSTRACT FROM AUTHOR]

Consider a cognitive radio network with two types of users: primary users (PUs) and cognitive users (CUs), whose locations follow two independent Poisson point processes. The cognitive users follow the policy that a cognitive transmitter is active only when it is outside the primary user exclusion regions. We found that under this setup the active cognitive users form a point process called the Poisson hole process. Due to the interaction between the primary users and the cognitive users through exclusion regions, an exact calculation of the interference and the outage probability seems unfeasible. Instead, two different approaches are taken to tackle this problem. First, bounds for the interference (in the form of Laplace transforms) and the outage probability are derived, and second, it is shown how to use a Poisson cluster process to model the interference in this kind of network. Furthermore, the bipolar network model with different exclusion region settings is analyzed. [ABSTRACT FROM PUBLISHER]