Point Process Distribution Research Articles

EE Optimization for Downlink NOMA-Based Multi-Tier CRANs

Non-orthogonal multiple access (NOMA) is increasingly becoming very attractive in cloud radio access networks (CRANs) to further boost the overall spectral efficiency, connectivity and, capacity of such networks. This paper addresses optimizing the energy efficiency (EE) for the downlink of a NOMA-based two tiers CRAN. The stochastic geometry represented by Poisson Point Process (PPP) distribution is used to decide the number and locations of the base stations (BSs) in each tier within the coverage area. A numerical optimal solution is obtained and compared against a proposed subgradient solution, as well as another proposed unoptimized solution based on the false positioning method. For comparison purposes, two other power allocation techniques are presented to allocate different powers to various BS categories; one allocates the power to each BS based on their relative distances to the cloud-based central station and the other is the bisection based scheme. Two simulation scenarios are presented to examine the performance of the two-tier NOMA-CRANs with NOMA is adopted as the multiple access of each tier in both cases. The first scenario considers heterogeneous CRAN (NOMA-HCRAN) case by using two different BS categories in each tier, namely, the macro-BSs and the RRHs. The second scenario considers a homogeneous CRAN (NOMA-CRAN) case by using the RRHs in both tiers but each tier has different frequency layer to prevent cross tier interference. Simulation results show the promising performance gain can be achieved with the proposed techniques relative to the existing approaches. More specifically, it was illustrated that the proposed subgradient based NOMA CRAN offers better performance than the proposed false positioning based NOMA CRAN, which is in turn better than the existing techniques, in particular, the bisection and the distance based NOMA-CRAN.

SCPP

A collection of individuals is represented by point patterns. Each individual is a finite set of geographical locations representing their visiting pattern to places in a region. We present SCPP, an algorithm for clustering these individuals considering the spatial patterns of their visiting locations. We adopted a probabilistic framework based on the theory of point processes that allows us to derive a non-obvious distance metric between each individual point pattern and the underlying, unobserved continuous intensity function. This metric is the Kullback-Leibler divergence between the true data-generating point process distribution and the model-generating distribution. We also introduce a theoretically based framework for the cost function to be minimized, a functional T (P) taking as arguments the probability distributions underlying the unknown clusters. We present an extensive experimental analysis to show SCPP’s effectiveness using several synthetic datasets and spatial mobility patterns from geo-tagged social media.

Distributed Caching in Converged Networks: A Deep Reinforcement Learning Approach

Content caching is an effective technique to alleviate the burden on backhaul links and reduce the traffic in cellular networks. In converged networks, broadcasting networks can push non-real-time popular services to different types of terminals during off-peak hours, while cellular networks are deployed to meet personalized needs. However, the storage capacity of terminals are usually limited. In this context, we study on the converged networks to push and cache the popular services in the router nodes close to the terminators. In this scheme, the most popular services are transmitted by the broadcasting base station and cached in the router nodes in a distributed cache network. Then, users can access the cached services in a more energy efficient manner. Due to the limited storage capacity of the router node, we assume that the user can access the cached services within two hops. Then, we formulate the service scheduling problem as a Markov Decision Process, aiming to maximize equivalent throughput (ET). Due to the large state space involved in the distributed cache network, it is quite challenging to obtain a tractable solution by the classical optimization algorithms. To handle this problem, a deep reinforcement learning based framework is proposed to tackle this problem. The simulation results show that the proposed algorithms are very effective; and they outperform the conventional one in term of the ET, especially when the users in the network subject to Poisson point process distribution.

Secrecy-Aware Altitude Optimization for Quasi-Static UAV Base Station Without Eavesdropper Location Information

In this letter, the altitude of an unmanned aerial vehicle base station (UAV-BS) is optimized to maximize an average worst case secrecy rate (AWSR) without eavesdropper location information. The air-to-ground channels are modeled based on the derived line-of-sight probability under a general urban model with Poisson point process and Rayleigh distribution for the locations and heights of the buildings, respectively. To solve the AWSR maximization problem, which is intractable, we reformulate it by the first-order Taylor series approximation and apply the Davies–Swann–Campey algorithm to effectively find the solution of the modified AWSR maximization problem. Numerical results verify that the altitude of UAV-BS affects the secrecy rate significantly and that the proposed UAV-BS can improve the secrecy rate compared with a static terrestrial BS.

Energy-Efficient Deployment and Adaptive Sleeping in Heterogeneous Cellular Networks

This paper focuses on energy efficiency that is a key performance metric in heterogeneous cellular networks to two key areas. First, based on the Poisson point process distributions of small-cell base stations (SBSs) and macrocell base stations (MBSs), the energy-efficiency model is formulated, and the effect of base stations’ distribution on energy efficiency is analyzed. For maximizing energy efficiency, the joint optimal densities of SBSs and MBSs are deduced under the constraint of quality of service. Second, according to this, we propose a joint sleep strategy of MBSs and that of SBSs. We deduce the optimal threshold of traffic load according to the joint optimal densities. If the traffic load of SBSs (or MBSs) is less than the optimal threshold of traffic load, these SBSs (or MBSs) go to sleep; otherwise, it is activated. This makes the SBSs and MBSs adaptively and distributively sleep according to their own traffic loads. The simulation results verify that the deduced joint optimal densities of the SBS and the MBS are accurate, and energy efficiency is improved when SBSs and MBSs adaptively sleep.

Analysis and Scheduling in a 5G Heterogeneous Content Delivery Network

Caching popular multimedia content constitutes a promising solution to mitigate the congestion in the ultimate potential for 5G networks. Moreover, a three-tier heterogeneous network, where device-to-device pairs, small cells, and macro base stations (BSs) are included, is suggested in 5G wireless networks. In addition, local caching of popular multimedia content for small-cell base stations (SBSs) and user equipment has been considered to decrease the BS transmission cost without requiring high capacity backhaul or access to the core network. Thus, in this paper, we first analyze the numerous relevant performances in this three-tier content delivery network. For the independent Poisson point process and uniform distribution, where there are two regular different SBS assumption modes, we first provide theoretical analysis of the hitting probability in this heterogeneous network in different user request transmission modes based on a stochastic geometry theory. Then, we further propose a novel resource selection and scheduling algorithm satisfying the signal-to-interference-plus-noise ratio and signal-to-interference ratio criteria and the optimal transmission link selections and scheduling strategy for the three-tier heterogeneous network in order to reasonably allocate restricted resources. Finally, the numerical results reveal that the theoretical results are in agreement with the simulation results of the Zipf-distribution caching scheme. Meanwhile, the proposed algorithm performs well compared with the theoretical limit and an increase in the state of the art.

Cooperative Multicast With Location Aware Distributed Mobile Relay Selection: Performance Analysis and Optimized Design

Mobile relay (MR) selection is critical to the performance and realization of two-stage cooperative multicast (CM). Targeting at the energy efficiency and simplified realization, a location-aware distributed (LAD) MR selection method is proposed, where successful mobile stations (MSs) at the first stage that are farther from the base station would activate themselves to be MRs with higher probabilities. Assuming that the number of MRs close to an unsuccessful MS follows Poisson Point Process distribution, based on stochastic geometry, the coverage performance of two-stage CM with LAD MR selection can be numerically evaluated. Moreover, given the analytical results, an optimized LAD MR selection scheme can be designed, aiming at minimizing the total power consumption of two-stage CM. Numerical and simulation results verify that the analysis based on stochastic geometry is accurate. Overall, the optimized LAD MR scheme provides better energy efficiency and coverage performance than existing distributed MR schemes.

Vehicular Communications for 5G Cooperative Small-Cell Networks

Cooperative transmission is an effective approach for vehicular communications to improve wireless transmission capacity and reliability in fifth-generation (5G) small-cell networks. Based on distances between the vehicle and cooperative small-cell base stations (BSs), the cooperative probability and the coverage probability have been derived for 5G cooperative small-cell networks where small-cell BSs follow Poisson point process distributions. Furthermore, the vehicular handoff rate and the vehicular overhead ratio have been proposed to evaluate the vehicular mobility performance in 5G cooperative small-cell networks. To balance the vehicular communication capacity and the vehicular handoff ratio, an optimal vehicular overhead ratio can be achieved by adjusting the cooperative threshold of 5G cooperative small-cell networks.

Hierarchical Species Distribution Models

Determining the distribution pattern of a species is important to increase scientific knowledge, inform management decisions, and conserve biodiversity. To infer spatial and temporal patterns, species distribution models have been developed for use with many sampling designs and types of data. Recently, it has been shown that count, presence-absence, and presence-only data can be conceptualized as arising from a point process distribution. Therefore, it is important to understand properties of the point process distribution. We examine how the hierarchical species distribution modeling framework has been used to incorporate a wide array of regression and theory-based components while accounting for the data collection process and making use of auxiliary information. The hierarchical modeling framework allows us to demonstrate how several commonly used species distribution models can be derived from the point process distribution, highlight areas of potential overlap between different models, and suggest areas where further research is needed.

Tree-based modelling of redundancy and paths in wireless sensor networks

Wireless sensor networks WSNs consist of a large number of autonomous nodes randomly deployed in the monitoring area. Nodes, having a short distance from each other, gather information and transmit it to the base station. They are used to monitor a given field of interest. They are widely used for military, environmental, and scientific applications, etc. The performance of wireless sensor networks is greatly influenced by their network topology. Node deployment is a fundamental issue to be solved in wireless sensor networks. In spite of their random deployment, nodes have to organise themselves to avoid redundancy and transceiver tasks. The network has to guarantee complete coverage and connectivity as long as possible. In this paper, we address the problem of network coverage and connectivity and propose an hierarchical model for the wireless sensor networks deployment which consists of dividing sensors in sets of equivalent nodes in order to maintain the connectivity of the network. We study the effectiveness of the model under different deployment strategies: random, circular and Poisson point process distributions. We investigate the impact of deployment strategies on: 1 coverage; 2 connectivity ratio; 3 the shortest path to sink.

Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^{2}$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.

Protocol Design and Capacity Analysis in Hybrid Network of Visible Light Communication and OFDMA Systems

Visible light communication (VLC) uses a vast unregulated and free light spectrum. It is considered to be a solution for overcoming the crowded radio spectrum for wireless communication systems. However, duplex communication, user mobility, and handover mechanisms are becoming challenging tasks in a VLC system. This paper proposes a hybrid network model of VLC and orthogonal frequency-division multiplexing access (OFDMA) in which the VLC channel is only used for downlink transmission, whereas OFDMA channels are served for uplinks in any situation or for downlinks only without VLC hotspots coverage. A novel protocol is proposed combined with access, horizontal, and vertical handover mechanisms for mobile terminal (MT) to resolve user mobility among different hotspots and OFDMA system. A new VLC network scheme and its frame format are presented to deal with the multiuser access problems in every hotspot. In addition, a new metric ρ is defined to evaluate the capacity of this hybrid network as the spatial density of interarrival time of MT requests in s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> under the assumption of the homogenous Poisson point process (HPPP) distribution of MTs. Analytical and simulation results show improvements in capacity performance of the hybrid, when compared to OFDMA system.

Asymptotic mean stationarity and absolute continuity of point process distributions

This paper relates - for point processes $\Phi$ on $\mathbb{R}$ - two types of asymptotic mean stationarity (AMS) properties and several absolute continuity results for the common probability measures emerging from point process theory. It is proven that $\Phi$ is AMS under the time-shifts if and only if it is AMS under the event-shifts. The consequences for the accompanying two types of ergodic theorem are considered. Furthermore, the AMS properties are equivalent or closely related to several absolute continuity results. Thus, the class of AMS point processes is characterized in several ways. Many results from stationary point process theory are generalized for AMS point processes. To obtain these results, we first use Campbell's equation to rewrite the well-known Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.

A Stochastic Geometry Framework for Analyzing Pairwise-Cooperative Cellular Networks

Cooperation in cellular networks has been recently suggested as a promising scheme to improve system performance, especially for cell-edge users. In this work, we use stochastic geometry to analyze cooperation models where the positions of Base Stations (BSs) follow a Poisson point process distribution and where Voronoi cells define the planar areas associated with them. For the service of each user, either one or two BSs are involved. If two, these cooperate by exchange of user data and channel related information with conferencing over some backhaul link. Our framework generally allows variable levels of channel information at the transmitters. In this paper we investigate the case of limited channel state information for cooperation (channel phase, second neighbour interference), but not the fully adaptive case which would require considerable feedback. The total per-user transmission power is further split between the two transmitters and a common message is encoded. The decision for a user to choose service with or without cooperation is directed by a family of geometric policies depending on its relative position to its two closest base stations. An exact expression of the network coverage probability is derived. Numerical evaluation allows one to analyze significant coverage benefits compared to the non-cooperative case. As a conclusion, cooperation schemes can improve system performance without exploitation of extra network resources.

Joint limit distributions of exceedances point processes and partial sums of gaussian vector sequence

In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.

Convergence in distribution of point processes on Polish spaces to a simple limit

Let ξ , ξ 1 , ξ 2 , … be a sequence of point processes on a complete and separable metric space ( S , d ) with ξ simple. We assume that P { ξ n B = 0 } → P { ξ B = 0 } and lim sup n → ∞ P { ξ n B > 1 } ≤ P { ξ B > 1 } for all B in some suitable class B , and show that this assumption determines if the sequence { ξ n } converges in distribution to ξ . This is an extension to general Polish spaces of the weak convergence theory for point processes on locally compact Polish spaces found in Kallenberg (1996).

On the conditional distributions of spatial point processes

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

On the conditional distributions of spatial point processes

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

The contour of splitting trees is a Lévy process

Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (v, τ) of some individual v (vertex) in a discrete tree where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping φ from the tree into the real line which preserves this order. The inverse of φ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level τ, we prove that a thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

Fractal characterization of the spatial distribution of geological point processes

Geological point processes can be used to model point patterns occurring frequently in a wide variety of geoscience fields, including the study of mineral deposits, oil producing wells, earthquakes, and landslides. Characterization of the spatial distribution of GPP has implications for understanding the properties of the underlying geological processes and events. Three examples of GPP dealing with (1) metallic mineral deposits, (2) oil producing wells, and (3) aftershocks of the Wenchuan earthquake (on 12 May 2008, magnitude 8.0) are presented to illustrate that (1) the spatial distribution of geological point processes generally shows clustering implying rejection of the Poisson model because L(r)>LPois(r); (2) the clustering statistics of the underlying geological processes are fractal; and (3) the size distribution of geological point processes is scale invariant. These results indicate existence of a fundamental law concerning the fractal nature of the point distributions generated by geological point processes.