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The evolution of the fifth-generation (5G)/ beyond 5G (B5G) mobile communication networks greatly depends on the seamless integration of terrestrial and sky networks over advanced multi-tier heterogeneous network (HetNet) infrastructure. By using the huge potential of unmanned aerial vehicles (UAVs) and mm-Wave, we develop a novel and practical multi-UAV assisted multi-tier HetNet model which consists of clustered aerial UAV base stations (U-BSs) and Poisson point process (PPP) based ground BSs (G-BSs) for emergency and crowded hotspot area. In a typical cluster, both ground user equipments (GUEs) and UAVs are modeled as Poisson cluster process (PCP) and scattered around the common cluster center. Specially, the analysis is conducted by adding two additional tiers which consist of multiple intra-cluster U-BSs and single G-BS in representative cluster so that the point sets of U-BSs and G-BSs are partitioned into two sub-sets, respectively, and a 4-tier network model is formed. To find the contribution of the inter-cluster associations on the performance of networks, we construct 2-tier and 4-tier association schemes. The 2-tier association allows a GUE associated only with intra-cluster U-BSs and G-BS, while the 4-tier association allows all inter/intra-cluster U-BSs and G-BSs can be associated. Then, according to stochastic geometry methods, statistical description of the downlink interferences experienced at a typical GUE is derived. This derivation yields the downlink coverage probabilities and total network throughput are obtained. It is achieved that association probabilities greatly depend on the distribution of daughter point process. UAVs' height and the distributions of U-BSs and GUEs impose non-monotonous impact on coverage probabilities. Due to severe inter-cluster interference, the achievable average area throughput is not always increasing with SINR so that there is an optimal SINR threshold. In addition, our analysis finds the proposed 4-tier association scheme outperforms the simple 2-tier one in terms of the average area throughput.

The deployment of low power small base stations (SBSs) in high density user areas enhances the performance of heterogeneous cellular networks (HCNets). Such user-centric SBS deployment (UCSD) is more appropriate as compared with the conventional uniformly deployed user equipment (UE) based HCNets, because it captures the correlation between UEs and SBSs. In the literature, the downlink (DL) performance of UCSD based HCNet (UC-HCNet) has been analyzed under maximum received signal power (MRSP) association strategy. In this paper, we develop and analyze the uplink (UL) performance of UC-HCNets with decoupled UL-DL association (DUDA) using stochastic geometery. We evaluate the total UL interference considering two special cases of Poisson cluster process (PCP), i.e., Matern cluster process (MCP) and Thomas cluster process (TCP), in UC-HCNet. The proposed DUDA scheme enhances the system performance gain by assuming dual connectivity of UE and BSs in the UL and DL directions. Furthermore, fractional power control (FPC) policy is considered to limit the UL power consumption. The system performance is evaluated in terms of outage probability and rate coverage. The numerical results are validated through simulations. Numerical results show that the performance improvement of DUDA in sparse scenario is higher as compared with dense scenario. Furthermore, it is investigated that the performance gain of DUDA is higher for larger UE variance and cluster radius in case of MCP and TCP, respectively.

In this paper, we investigate the uplink transmission performance of low-power wide-area (LPWA) networks with regards to coexisting radio modules. We adopt 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 (PCP) while other coexisting radio modules follow a Poisson point process (PPP). 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 through 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., Comment: This paper has been accepted by IEEE Trans. Vehicular Technology (TVT)

This paper focuses on a comprehensive framework design and performance analysis for a large-scale hotspot communication scenario where there exist large number of clustered pico base stations (P-BSs) and femto BSs (F-BSs) working on millimeter-Wave (mm-Wave). For such realistic hotspots’ deployment, to exploit the coupling between user equipments (UEs) and base stations (BSs), we model the geographical centers of UE hotspots as independent PPP around which UEs, P-BSs and F-BSs are scattered and form independent and non-homogeneous Poisson cluster processes (PCPs). However, with clustered P-BSs and F-BSs, the UEs would experience the severer intra-cluster interference so that the performance is degraded greatly. To overcome this problem in clustered networks, an effective interference management scheme is proposed by using the ratio of the first and second nearest distances of a typical UE from P-BSs so that a typical UE can be classified as either cluster-center UEs (CCUEs) or cluster-edge UEs (CEUEs). Correspondingly, the total available spectrum band is divided as CCUE-band and CEUE-band, which are occupied by P-BSs associated with CCUEs and CEUEs, respectively. Unlike P-BSs, the F-BSs randomly access the CEUE-band and CCUE-band with a given access portion factor. By using the method from stochastic geometry and the distinguishing feature of mm-Wave, such as directional beamforming with sectored antenna model and modified line of sight ball model for blockage, the UE association probability and Laplace transforms of interference are derived. This yields the derivations of downlink transmission rates. The numerical results show that the proposed clustered heterogeneous scheme with the cluster-UE classification outperforms the traditional ones without the UE classification. This approach not only captures the UE-BS coupling and the non-homogeneous nature of multi-tier networks, but also restricts the severe intra-cluster interference due to clustered BSs.

The mathematical modeling and performance analysis of cellular networks have seen a major paradigm shift with the application of stochastic geometry. The main purpose of stochastic geometry is to endow probability distributions on the locations of the base stations (BSs) and users in a network, which, in turn, provides an analytical handle on the performance evaluation of cellular networks. To preserve the tractability of analysis, the common practice is to assume complete spatial randomness} of the network topology. In other words, the locations of users and BSs are modeled as independent homogeneous Poisson point processes (PPPs). Despite its usefulness, the PPP-based network models fail to capture any spatial coupling between the users and BSs which is dominant in a multi-tier cellular network (also known as the heterogeneous cellular networks (HetNets)) consisting of macro and small cells. For instance, the users tend to form hotspots or clusters at certain locations and the small cell BSs (SBSs) are deployed at higher densities at these locations of the hotspots in order to cater to the high data demand. Such user-centric deployments naturally couple the locations of the users and SBSs. On the other hand, these spatial couplings are at the heart of the spatial models used in industry for the system-level simulations and standardization purposes. This dissertation proposes fundamentally new spatial models based on stochastic geometry which closely emulate these spatial couplings and are conductive for a more realistic and fine-tuned performance analysis, optimization, and design of cellular networks. First, this dissertation proposes a new class of spatial models for HetNets where the locations of the BSs and users are assumed to be distributed as Poisson cluster process (PCP). From the modeling perspective, the proposed models can capture different spatial couplings in a network topology such as the user hotspots and user BS coupling occurring due to the user-centric deployment of the SBSs. The PCP-based model is a generalization of the state-of-the-art PPP-based HetNet model. This is because the model reduces to the PPP-based model once all spatial couplings in the network are ignored. From the stochastic geometry perspective, we have made contributions in deriving the fundamental distribution properties of PCP, such as the distance distributions and sum-product functionals, which are instrumental for the performance characterization of the HetNets, such as coverage and rate. The focus on more refined spatial models for small cells and users brings to the second direction of the dissertation, which is modeling and analysis of HetNets with millimeter wave (mm-wave) integrated access and backhaul (IAB), an emerging design concept of the fifth generation (5G) cellular networks. While the concepts of network densification with small cells have emerged in the fourth generation (4G) era, the small cells can be realistically deployed with IAB since it solves the problem of high capacity wired backhaul of SBSs by replacing the last-mile fibers with mm-wave links. We have proposed new stochastic geometry-based models for the performance analysis of IAB-enabled HetNets. Our analysis reveals some interesting system-design insights: (1) the IAB HetNets can support a maximum number of users beyond which the data rate drops below the rate of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no rate gain is observed with the addition of more SBSs. The third and final direction of this dissertation is the combination of machine learning and stochastic geometry to construct a new class of data driven network models which can be used in the performance optimization and design of a network. As a concrete example, we investigate the classical problem of wireless link scheduling where the objective is to choose an optimal subset of simultaneously active transmitters (Tx-s) from a ground set of Tx-s which will maximize the network-wide sum-rate. Since the optimization problem is NP-hard, we replace the computationally expensive heuristic by inferring the point patterns of the active Tx-s in the optimal subset after training a determinantal point process (DPP). Our investigations demonstrate that the DPP is able to learn the spatial interactions of the Tx-s in the optimal subset and gives a reasonably accurate estimate of the optimal subset for any new ground set of Tx-s. Doctor of Philosophy The high speed global cellular communication network is one of the most important technologies, and it continues to evolve rapidly with every new generation. This evolution greatly depends on observing performance-trends of the emerging technologies on the network models through extensive system-level simulations. Since these simulation models are extremely time-consuming and error prone, the complementary analytical models of cellular networks have been an area of active research for a long time. These analytical models are intended to provide crisp insights on the network behavior such as the dependence of network performance metrics (such as coverage or rate) on key system-level parameters (such as transmission powers, base station (BS) density) which serve as the prior knowledge for more fine-tuned simulations. Over the last decade, the analytical modeling of the cellular networks has been driven by stochastic geometry. The main purpose of stochastic geometry is to endow the locations of the base stations (BSs) and users with probability distributions and then leverage the properties of these distributions to average out the spatial randomness. This process of spatial averaging allows us to derive the analytical expressions of the system-level performance metrics despite the presence of a large number of random variables (such as BS and user locations, channel gains) under some reasonable assumptions. The simplest stochastic geometry based model of cellular networks, which is also the most tractable, is the so-called Poisson point process (PPP) based network model. In this model, users and BSs are assumed to be distributed as independent homogeneous PPPs. This is equivalent to saying that the users and BSs independently and uniformly at random over a plane. The PPP-based model turned out to be a reasonably accurate representation of the yesteryear’s cellular networks which consisted of a single tier of macro BSs (MBSs) intended to provide a uniform coverage blanket over the region. However, as the data-hungry devices like smart-phones, tablets, and application like online gaming continue to flood the consumer market, the network configuration is rapidly deviating from this baseline setup with different spatial interactions between BSs and users (also termed spatial coupling) becoming dominant. For instance, the user locations are far from being homogeneous as they are concentrated in specific areas like residential and commercial zones (also known as hotspots). Further, the network, previously consisting of a single tier of macro BSs (MBSs), is becoming increasingly heterogeneous with the deployment of small cell BSs (SBSs) with small coverage footprints and targeted to serve the user hotspots. It is not difficult to see that the network topology with these spatial couplings is quite far from complete spatial randomness which is the basis of the PPP-based models. The key contribution of this dissertation is to enrich the stochastic geometry-based mathematical models so that they can capture the fine-grained spatial couplings between the BSs and users. More specifically, this dissertation contributes in the following three research directions. Direction-I: Modeling Spatial Clustering. We model the locations of users and SBSs forming hotspots as Poisson cluster processes (PCPs). A PCP is a collection of offspring points which are located around the parent points which belong to a PPP. The coupling between the locations of users and SBSs (due to their user-centric deployment) can be introduced by assuming that the user and SBS PCPs share the same parent PPP. The key contribution in this direction is the construction of a general HetNet model with a mixture of PPP and PCP-distributed BSs and user distributions. Note that the baseline PPP-based HetNet model appears as one of the many configurations supported by this general model. For this general model, we derive the analytical expressions of the performance metrics like coverage probability, BS load, and rate as functions of the coupling parameters (e.g. BS and user cluster size). Direction-II: Modeling Coupling in Wireless Backhaul Networks. While the deployment of SBSs clearly enhances the network performance in terms of coverage, one might wonder: how long network densification with tens of thousands of SBSs can meet the everincreasing data demand? It turns out that in the current network setting, where the backhaul links (i.e. the links between the BSs and core network) are still wired, it is not feasible to densify the network beyond some limit. This backhaul bottleneck can be overcome if the backhaul links also become wireless and the backhaul and access links (link between user and BS) are jointly managed by an integrated access and backhaul (IAB) network. In this direction, we develop the analytical models of IAB-enabled HetNets where the key challenge is to tackle new types of couplings which exist between the rates on the wireless access and backhaul links. Such couplings exist due to the spatial correlation of the signal qualities of the two links and the number of users served by different BSs. Two fundamental insights obtained from this work are as follows: (1) the IAB HetNets can support a maximum number of users beyond which the network performance drops below that of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no performance gain is observed with the addition of more SBSs. Direction-III: Modeling Repulsion. In this direction, we focus on modeling another aspect of spatial coupling imposed by the intra-point repulsion. Consider a device-to-device (D2D) communication scenario, where some users are transmitting some on-demand content locally cached in their devices using a common channel. Any reasonable multiple access scheme will ensure that two nearly users are never simultaneously active as they will cause severe mutual interference and thereby reducing the network-wide sum rate. Thus the active users in the network will have some spatial repulsion. The locations of these users can be modeled as determinantal point processes (DPPs). The key property of DPP is that it forms a bridge between stochastic geometry and machine learning, two otherwise non-overlapping paradigms for wireless network modeling and design. The main focus in this direction is to explore the learning framework of DPP and bring together advantages of stochastic geometry and machine learning to construct a new class of data-driven analytical network models.

The use of unmanned aerial vehicles (UAVs) operating as aerial base stations (BSs) has emerged as a promising solution especially in scenarios requiring rapid deployments (e.g., in the cases of crowded hotspots, sporting events, emergencies, and natural disasters) in order to assist the ground BSs. In this paper, an analytical framework is provided to analyze the signal-to-interference-plus-noise ratio (SINR) coverage probability of UAV assisted cellular networks with clustered user equipments (UEs). Locations of UAVs and ground BSs are modeled as Poison point processes, and UEs are assumed to be distributed according to a Poisson cluster process around the projections of UAVs on the ground. Initially, the complementary cumulative distribution function and probability density function of path losses for both UAV and ground BS tiers are derived. Subsequently, association probabilities with each tier are obtained. SINR coverage probability is derived for the entire network using tools from stochastic geometry. Finally, area spectral efficiency (ASE) of the entire network is determined, and SINR coverage probability expression for a more general model is presented by considering that UAVs are located at different heights. Via numerical results, we have shown that UAV height and path-loss exponents play important roles on the coverage performance. Moreover, coverage probability can be improved with smaller number of UAVs, while better ASE is achieved by employing more UAVs and having UEs more compactly clustered around the UAVs.

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.

As wearables provide new health-related functionalities, they can be employed in hospitals to monitor patients and notify the medical personnel regarding their status. However, in order to be approved by the medical community, wearables need to have reliable communication and high lifetime. In such scenarios, it is important to know the probability of correct notification which is affected mainly by the deployment of the wireless wearables and their energy supply. Typically, rooms in hospitals host multiple people and, thus, a clustered communication model should be adopted for more trustworthy results. Moreover, by employing wireless charging, it is possible to provide an uninterrupted operation with high reliability. Therefore, in this paper, we study the aforementioned probability in a clustered network while the wearable devices are wirelessly charged. We provide an analytical model for the wearables' ability to inform quickly the medical personnel and discuss different trade-offs via extensive simulations.

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. NSA grant [H98230-13-1-0220] info:eu-repo/semantics/publishedVersion

We study stochastic properties of the empty space for stationary germ-grain models in R d ; 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 ∞.

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. Ecology Letters (2008) 11: 904–917

International audience; The goal of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a fundamental integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity in proportion, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called Che approximation.

Cilj je ovog diplomskog rada bio proučiti Poissonov proces s klasterima i njegova osnovna svojstva. Nakon prvoga poglavlja, čitatelju su poznate definicija točkovnih procesa, mjera njihovih momenata te primjene tzv. Laplaceove transformacije i Laplaceova funkcionala točkovnoga procesa. Navedeno je i nekoliko primjera točkovnih procesa te najvažniji točkovni proces, općeniti Poissonov točkovni proces ili Poissonova slučajna mjera (PRM). Pokazano je i kako se dodavanjem nezavisne koordinate točkama poznatoga PRM-a može konstruirati novi PRM. Dalje je navedena definicija i jedna karakterizacija općenitih točkovnih procesa s klasterima. Pretpostavljamo da nam je poznat proces centara klastera. Točke tog procesa generiraju klastere koji su također točkovni procesi. Zbrajanje elemenata po klasterima čini opaženi proces. Ako su središta klastera točke Poissonova procesa, riječ je o Poissonovom procesu s klasterima. U zadnjem poglavlju bavimo se primjenom Poissonova procesa s klasterima u matematici neživotnoga osiguranja. Točku Poissonova procesa interpretiramo kao vrijeme dolaska zahtjeva za isplatu štete, a klaster koji ona uzrokuje opisuje vremena i iznose isplate tog zahtjeva. Proučavamo i chain ladder model. Na kraju se bavimo Poissonovim procesima s Poissonovim klasterima. Analiziramo njihove prve i druge momente kako bismo predvidjeli broj i ukupan iznos isplata šteta. Chain ladder model i Poissnov proces s klasterima često se koriste za procjenu pričuva, što je jedan od najvažnijih praktičnih problema u aktuarstvu. The aim of this thesis was to study the Poisson cluster process as well as its basic properties. After the first chapter the reader is familiar with the definition of point processes, their moment measures, and the application of the so called Laplace transform and the Laplace functional of the point processes. There are several examples of point processes including the most important one, the general Poisson point process or Poisson random measure (PRM). It is also shown how one can adhere an independent coordinate to the points of given PRM to construct a new PRM. In addition, there is a definition and a characterization of general point process with independent clusters. We suppose a point process of cluster centers is known. The points of the center process generate the clusters, which are also point processes. The superposition of the elements within clusters constitutes the observed process. If the cluster centers are the points of a Poisson process, we speak of a Poisson cluster process. In the final chapter we consider the applications of the Poisson cluster process in non-life insurance mathematics. A Poisson process point is interpreted as the arrival time of a claim, and the cluster that point triggers describes the times and amounts of the payment for this particular claim. We also study the chain ladder model. Lastly, we concentrate on the Poisson processes with Poisson clusters. We analyze the first and second moments of those processes in order to predict the claim number and total claim amounts. Chain ladder model and Poisson cluster process are often used to estimate reserves, which is one of the most important insurance practice problems.

Future mobile networks are converging toward heterogeneous multitier networks, where macro-, pico-, and femto-cells are randomly deployed based on user demand. A popular approach for analyzing heterogeneous networks (HetNets) is to use stochastic geometry and treat the location of BSs as points distributed according to a homogeneous Poisson point process (PPP). However, a PPP model does not provide an accurate model for the interference when nodes are clustered around highly populated areas. This motivates us to find better ways to characterize the aggregate interference when transmitting nodes are clustered following a Poisson cluster process (PCP) while taking into consideration the fact that BSs belonging to different tiers may differ in terms of transmit power, node densities, and link reliabilities. To this end, we consider K-tier HetNets and investigate the outage probability, the coverage probability, and the average achievable rate for such networks. We compare the performance of HetNets when nodes are clustered and otherwise. By comparing these two types of networks, we conclude that the fundamental difference between a PPP and a PCP is that, for a PPP, the number of simultaneously covered mobiles and the network capacity linearly increase with K. However, for a PCP, the improvements in the coverage and the capacity diminish as K grows larger, where the curves saturate at some point. Based on these observations, we determine the scenarios that jointly maximize the average achievable rate and minimize the outage probability. Qatar National Research Fund Scopus

Future mobile networks are converging towards being heterogeneous, owing to the co-existence of multi-tier networks within the same geographical area, including macro, pico- and femto-cells. The deployment of such networks is generally based on user demand, which is irregular and random, implying that the deployment of base stations (BSs) is random as well. As a result, analyzing the communication protocols over heterogeneous networks (HetNets) is very challenging. A popular approach is to use stochastic geometry and treat the location of the BSs as points distributed according to a spatial Point Process. Most of the related work on the interference modeling normally assumes homogeneous Poisson point process (PPP). This assumption holds when the nodes are uniformly distributed in space, such as sensor networks or ad-hoc networks. Due to geographical factors, it may be the case for mobile users to cluster around highly populated cities and the PPP assumption does not provide an accurate model for the interference in these conditions. This motivates us to find better ways to characterize the aggregate interference when the transmitting nodes are clustered following a Poisson Cluster Process (PCP). Furthermore, the BSs belonging to different tiers may differ in terms of the transmit power, the node densities, and their link reliabilities. To this end, we consider K-tier HetNets, where, by using the Laplace transform approach, we characterize the aggregate interference at a given destination as a heavy-tailed distribution. Using the derived distribution, we investigate the probability of outage and coverage for such networks. Due to some difficulty in obtaining closed-form expressions for these measures, we derive tight bounds and verify that through numerical examples. We also compare the performance of HetNets when the nodes are clustered and otherwise. We observe that using the PPP results in larger success probability, but using the clustered process results in a larger coverage probability. We also observe that there is an optimal intensity, i.e., number of nodes, that achieves the maximum coverage probability for the given SINR (signal-to-interference-plus-noise ratio) threshold. Qatar National Research Fund Scopus

We analyze a cognitive radio network where the primary users (PUs) and cognitive radio (CR) devices are distributed over the two-dimensional plane according to two independent homogeneous Poisson processes. Any CR that lies within the detection region of some PU switches to a different channel in order to prevent causing harmful interference at the PU. Using the concepts of stochastic geometry, we study the characteristics of the interference caused by the PUs and the CRs to a given PU. Further, these results are used to obtain tight upper and lower bounds for the success probability at the PU; defined as the probability that the signal-to-interferenceplus- noise ratio (SINR) is beyond a certain operating threshold.

Recently developed spatial capture-recapture (SCR) models represent a major advance over traditional capture-recapture (CR) models because they yield explicit estimates of animal density instead of population size within an unknown area. Furthermore, unlike nonspatial CR methods, SCR models account for heterogeneity in capture probability arising from the juxtaposition of animal activity centers and sample locations. Although the utility of SCR methods is gaining recognition, the requirement that all individuals can be uniquely identified excludes their use in many contexts. In this paper, we develop models for situations in which individual recognition is not possible, thereby allowing SCR concepts to be applied in studies of unmarked or partially marked populations. The data required for our model are spatially referenced counts made on one or more sample occasions at a collection of closely spaced sample units such that individuals can be encountered at multiple locations. Our approach includes a spatial point process for the animal activity centers and uses the spatial correlation in counts as information about the number and location of the activity centers. Camera-traps, hair snares, track plates, sound recordings, and even point counts can yield spatially correlated count data, and thus our model is widely applicable. A simulation study demonstrated that while the posterior mean exhibits frequentist bias on the order of 5-10% in small samples, the posterior mode is an accurate point estimator as long as adequate spatial correlation is present. Marking a subset of the population substantially increases posterior precision and is recommended whenever possible. We applied our model to avian point count data collected on an unmarked population of the northern parula (Parula americana) and obtained a density estimate (posterior mode) of 0.38 (95% CI: 0.19-1.64) birds/ha. Our paper challenges sampling and analytical conventions in ecology by demonstrating that neither spatial independence nor individual recognition is needed to estimate population density - rather, spatial dependence can be informative about individual distribution and density., Published in at http://dx.doi.org/10.1214/12-AOAS610 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider risk processes with non-stationary Hawkes claims arrivals, and we study the asymptotic behavior of infinite and finite horizon ruin probabilities under light-tailed conditions on the claims. Moreover, we provide asymptotically efficient simulation laws for ruin probabilities and we give numerical illustrations of the theoretical results.

In the project “statistical image analysis” of CWI we have studied some spatial point patterns that originated from biological observations. These observations were the positions of so called EGF‐receptors on the surface of human carcinoma cells. We propose a stochastic model for these point patterns. Since the EGF‐receptors appear in clusters on the cell surface, we have opted for the Poisson‐cluster‐process as the model. We estimated the three parameters in this process by means of a method described by Diggle. We also did some work in assessing the statistical reliability of our estimates. Copyright

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. Udgivelsesdato: SEP 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.

This article concerns a simulation algorithm for unmarked and marked Hawkes processes. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work. 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. This article concerns a simulation algorithm for unmarked and marked Hawkes processes. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work. 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.

Ponencia presentada en: III Congreso de la Asociación Española de Climatología “El agua y el clima”, celebrado en Palma de Mallorca del 16 al 19 de junio de 2002. [ES]El objetivo de este trabajo es caracterizar la máxima sequía que cabe esperar en un determinado periodo de tiempo. Para ello es necesario disponer de un modelo estocástico que describa el proceso de sequias (proponemos un proceso Poisson cluster para describir la ocurrencia y tres series de variables aleatorias, Longitud, Déficit e Intensidad Máxima, para describir la severidad) y desarrollar los resultados teóricos necesarios sobre la distribución del máximo en una muestra de tamaño Poisson. [EN]This work aims to characterize the largest drought event to occur in a given period of time. A Poisson cluster process is used to model drought occurrence and three series of random variables (Length, Deficit and Maximum Intensity) to describe their severity. Some theoretical results on the distribution of the maximum in a random Poisson size sample are developed for describing the largest drought events.

We consider m-dependent random fields of bounded random vectors (generated by independent random fields) and investigate the analyticity of the cumulant generating function of sums of these random vectors. Using the Kirkwood-Salsburg equations we derive upper bounds for the cumulant generating function and prove its analyticity in a neighbourhood of zero, where the normalized bounds and the neighbourhood are independent of the number of terms in the sum, The results are applied to statistics of Poisson cluster processes and Boolean models (which have a representation in terms of an independent field) and yield probabilities of large deviations as well as Berry-Esseen results for these statistics.

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 [4] 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.