Gilbert's disc model with geostatistical marking

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in $\mathbb{R}^2$ with radii determined by an underlying stationary and ergodic random field $\varphi:\mathbb{R}^2\to[0,\infty)$, independent of the Poisson process. When the random field is independent of the point process one often talks about 'geostatistical marking'. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of $\mathbb{R}^2$ does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.Comment: 22 page

Deployment Strategies of Multiple Aerial BSs for User Coverage and Power Efficiency Maximization

Unmanned aerial vehicle (UAV) based aerial base stations (BSs) can provide rapid communication services to ground users and are thus promising for future communication systems. In this paper, we consider a scenario where no functional terrestrial BSs are available and the aim is deploying multiple aerial BSs to cover a maximum number of users within a certain target area. To this end, we first propose a naive successive deployment method, which converts the non-convex constraints in the involved optimization into a combination of linear constraints through geometrical relaxation. Then we investigate a deployment method based on K-means clustering. The method divides the target area into K convex subareas, where within each subarea, a mixed integer non-linear problem (MINLP) is solved. An iterative power efficient technique is further proposed to improve coverage probability with reduced power. Finally, we propose a robust technique for compensating the loss of coverage probability in the existence of inaccurate user location information (ULI). Our simulation results show that, the proposed techniques achieve an up to 30% higher coverage probability when users are not distributed uniformly. In addition, the proposed simultaneous deployment techniques, especially the one using iterative algorithm improve power-efficiency by up to 15% compared to the benchmark circle packing theory

Towards automatic Markov reliability modeling of computer architectures

The analysis and evaluation of reliability measures using time-varying Markov models is required for Processor-Memory-Switch (PMS) structures that have competing processes such as standby redundancy and repair, or renewal processes such as transient or intermittent faults. The task of generating these models is tedious and prone to human error due to the large number of states and transitions involved in any reasonable system. Therefore model formulation is a major analysis bottleneck, and model verification is a major validation problem. The general unfamiliarity of computer architects with Markov modeling techniques further increases the necessity of automating the model formulation. This paper presents an overview of the Automated Reliability Modeling (ARM) program, under development at NASA Langley Research Center. ARM will accept as input a description of the PMS interconnection graph, the behavior of the PMS components, the fault-tolerant strategies, and the operational requirements. The output of ARM will be the reliability of availability Markov model formulated for direct use by evaluation programs. The advantages of such an approach are (a) utility to a large class of users, not necessarily expert in reliability analysis, and (b) a lower probability of human error in the computation

Betti number signatures of homogeneous Poisson point processes

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.Comment: Submitted to PRE. 11 pages, 10 figure

Recursive analysis and estimation for the discrete Boolean random set model

Random sets provide a powerful class of models for images containing randomly placed objects of random shapes and orientation. Those pixels within the foreground are members of a random set realization. The discrete Boolean model is the simplest general random set model in which a Bernoulli point process (called a germ process) is coupled with an independent shape or grain process. A typical realization consists of many overlapping shapes. Estimation in these models is difficult owing to the fact that many outcomes of the process obscure other outcomes. The directional one-dimensional (ID) model, in which random- length line segments emanate to the right from germs on the line, is analyzed via recursive expressions to provide a complete characterization of these discrete models in terms of the distributions of their black and white runlengths. An analytic representation is given for the optimal windowed filter for the signalunion- noise process, where both signal and noise are Boolean models. Several of these results are extended to the nondirectional case where segments can emanate to the left and right. Sufficient conditions are presented for a two-dimensional (2D) discrete Boolean model to induce a one dimensional Boolean model on an intersecting line. When inducement holds, the likelihood of runlength observations of the two-dimensional model is used to provide maximum-likelihood estimation of parameters of the 2D model. The ID directional discrete Boolean model is equivalent to the discrete-time infinite-server queue. Analysis for the Boolean model is extended to provide densities for many random variables of interest in queueing theory

Perfect simulation of spatial processes

This work presents a review of some of the schemes used to perfect sample from spatial processes

Exploiting Randomly-located Blockages for Large-Scale Deployment of Intelligent Surfaces

One of the promising technologies for the next generation wireless networks is the reconfigurable intelligent surfaces (RISs). This technology provides planar surfaces the capability to manipulate the reflected waves of impinging signals, which leads to a more controllable wireless environment. One potential use case of such technology is providing indirect line-of-sight (LoS) links between mobile users and base stations (BSs) which do not have direct LoS channels. Objects that act as blockages for the communication links, such as buildings or trees, can be equipped with RISs to enhance the coverage probability of the cellular network through providing extra indirect LoS-links. In this paper, we use tools from stochastic geometry to study the effect of large-scale deployment of RISs on the performance of cellular networks. In particular, we model the blockages using the line Boolean model. For this setup, we study how equipping a subset of the blockages with RISs will enhance the performance of the cellular network. We first derive the ratio of the blind-spots to the total area. Next, we derive the probability that a typical mobile user associates with a BS using an RIS. Finally, we derive the probability distribution of the path-loss between the typical user and its associated BS. We draw multiple useful system-level insights from the proposed analysis. For instance, we show that deployment of RISs highly improves the coverage regions of the BSs. Furthermore, we show that to ensure that the ratio of blind-spots to the total area is below 10^5, the required density of RISs increases from just 6 RISs/km2 when the density of the blockages is 300 blockage/km^2 to 490 RISs/km^2 when the density of the blockages is 700 blockage/km^2.Comment: Accepted in IEEE Journal on Selected Areas in Communication