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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]

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]

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]

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]

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]

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]

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]