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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]

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]

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]

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]

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]

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]

Abstract: The distribution of the excess process describing heights of extreme values can be approximated by the distribution of a Poisson cluster process. An estimate of the accuracy of such an approximation has been derived in in terms of the Wasserstein distance. The paper presents a sharper estimate established in terms of the stronger total variation distance. We derive also a new bound to the accuracy of negative Binomial approximation to the distribution of the number of exceedances. [Copyright &y& Elsevier]

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]

A cluster point process model is considered for the analysis of fine-scale rainfall time series. The model is based on three Poisson processes. The first is a Poisson process of storm origins, where each storm has a random (exponential) lifetime. The second is a Poisson process of cell origins that occur during the storm lifetime, terminating when the storm finishes. Each cell has a random lifetime that follows an exponential distribution (or terminates when the storm terminates, whichever occurs first). During cell lifetimes, a third Poisson process of instantaneous pulses occurs. The model is essentially an extension of the well-known Bartlett-Lewis rectangular pulses model, with the rectangular profiles replaced with a Poisson process of instantaneous pulse depths to ensure more realistic rainfall profiles for fine-scale series. Model equations, derived in Cowpertwait et al. (2007), are used to fit different sets of properties to a 60 year record of 5-min data taken from Kelburn, New Zealand. As in the previous work, two superposed processes are used to account for two main and distinct precipitation types (convective and stratiform). By treating the within-cell pulses as dependent random variables, it is found, by simulation, that improved fits to extreme values and the proportion of dry intervals are obtained. Citation Cowpertwait, P. S. P., Xie, G., Isham, V., Onof, C. & Walsh, D. C. I. (2011) A fine-scale point process model of rainfall with dependent pulse depths within cells. Hydrol. Sci. J. 56(7), 1110–1117. [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]

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]

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]