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927 results in Applied probability and stochastic networks

Page 16 of 47

Applied Stochastic Differential Equations
  • Simo Särkkä, Arno Solin
  • Published online:
    16 April 2019
    Print publication:
    02 May 2019
  • Stochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic Runge–Kutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book's practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-driven derivations and computational assignments. MATLAB/Octave source code is available for download, promoting hands-on work with the methods.

  • Concluding Remarks

  • Bartłomiej Błaszczyszyn, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Martin Haenggi, University of Notre Dame, Indiana, Paul Keeler, Sayandev Mukherjee
  • Book:
    Stochastic Geometry Analysis of Cellular Networks
    Published online:
    26 March 2018
    Print publication:
    19 April 2018, pp 168-169

  • Summary

    Future wireless deployments will increasingly rely on HetNets to satisfy society's demands for high data rates coupled with high cell capacities. We argue that the design and deployment of such HetNets present many challenges, which, because of the much larger space of system deployment parameters relative to a single-tier network, cannot be feasibly addressed by the traditional approaches of measurement and simulation alone.

    We show that techniques and results from stochastic geometry can relieve system designers from the burden of having to perform exhaustive simulations of all feasible combinations of deployment parameters, because they yield analytical results for key performance metrics such as coverage probability, thereby permitting (a) a uniform comparison across architectures and (b) quick rejection of certain combinations of deployment parameters without needing to simulate their performance. These theoretical results are illustrated with examples of their application to transmission scenarios specified in the LTE standard.

    These analytical results for coverage probability have, until now, mostly been derived for independent Poisson deployments of base stations in tiers, a scenario for which there are tractable derivations of exact expressions. Although it is well known that real-world base station deployments do not behave like Poisson deployments, we show that the Poisson model is nonetheless fundamental to an analytical treatment of all deployments, for the following reasons: (a) the set of propagation losses to the typical location in an arbitrary network deployment converge asymptotically to that from a Poisson deployment of base stations and (b) long before this asymptotic limit is reached, the coverage results for a Poisson deployment can be employed to obtain very accurate approximations to the coverage results for various regular deployments. The present work is the first book-length treatment of these results.

    We also provide the first book-level exposition of exact results on coverage for certain non-Poisson deployment models that are analytically tractable. Such models include the Ginibre point process and determinantal point processes. We also discuss the challenges of understanding and evaluating the complex analytical expressions so obtained.

    For future work, in addition to stochastic geometry and point process theory covered in this book, other tools from probability may also be employed to analyze cellular networks.

  • 2 - The Role of Stochastic Geometry in HetNet Analysis

  • from Part I - Stochastic Geometry
  • Bartłomiej Błaszczyszyn, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Martin Haenggi, University of Notre Dame, Indiana, Paul Keeler, Sayandev Mukherjee
  • Book:
    Stochastic Geometry Analysis of Cellular Networks
    Published online:
    26 March 2018
    Print publication:
    19 April 2018, pp 13-17

  • Summary

    In Chapter 1, Section 1.8, we list several important and general analytical results for single-tier networks and multitier HetNets that were derived using results from stochastic geometry. Of course, the value of analytical results in yielding insights to help network design has long been recognized by the industry. Stochastic geometric approaches are only the latest, and by far the most successful, of a long line of analytical approaches that began almost immediately after the first “cellular” models for wireless communication were proposed. Before commencing a detailed study of stochastic geometry, it is useful to take a step back and examine competing analytical approaches to get a better idea of why stochastic geometric approaches achieved greater success. To do so, let us revisit the origins of the now-canonical hexagonal cell model.

    The Hexagonal Cellular Concept

    The earliest proposals for wireless coverage via hexagonal “cells” date back to 1947, even before the principles of information theory were published by Shannon in 1948. A more up-to-date overview of the cellular concept may be found in MacDonald's 1979 paper (MacDonald 1979), applied to what is now considered the “first-generation” or “1G” cellular standard, advanced mobile phone service. MacDonald's carefully reasoned rationale for the hexagonal cellular model bears reading even today:

  • He acknowledged that a practical cellular deployment would contain cells of different shapes and sizes (owing to topographic obstacles, restricted rights-ofway, etc.), but made the reasonable argument that systematization of the design and layout of a cellular system is greatly aided by modeling all cells of the system as having the same shape.

  • At the time MacDonald wrote his paper, cellular base stations transmitted using omnidirectional antennas. This in turn means that the boundary of the coverage region of a base station is circular. In other words, omnidirectional transmitting antennas at the base stations mean circular disk-shaped cells.

  • Unfortunately, circular disks do not tile (meaning, cover completely, without overlaps or gaps) the two-dimensional plane.

  • Three shapes that do tile the plane are equilateral triangles, squares, and (regular) hexagons.

  • For each of these three shapes, and with the base station located at the center of a cell with that shape, the farthest point within the cell is a vertex of that shape.

  • Preface

  • Bartłomiej Błaszczyszyn, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Martin Haenggi, University of Notre Dame, Indiana, Paul Keeler, Sayandev Mukherjee
  • Book:
    Stochastic Geometry Analysis of Cellular Networks
    Published online:
    26 March 2018
    Print publication:
    19 April 2018, pp ix-x

  • Summary

    Since 2010, our knowledge of coverage and capacity in heterogeneous cellular networks (HetNets) has expanded rapidly, primarily through analytical results using stochastic geometry. Most of these results assume that the locations of the base stations in a given tier of the HetNet are the points of a homogeneous Poisson point process (PPP). This modeling assumption was made for mathematical tractability, and the coverage results for a single tier were determined to be about as pessimistic (relative to the real coverage in real-world deployments) as the regular hexagonal lattice location model was optimistic (Andrews, Baccelli, & Ganti 2011).

    However, since 2013 we have obtained results showing the fundamental importance of the PPP model to the analysis of real-world deployments in two ways: (a) the set of propagation losses to the typical location in an arbitrary network deployment converge asymptotically to that from a PPP deployment of base stations; (b) long before this convergence is reached, the coverage results for a PPP deployment can be employed to obtain very accurate approximations to the coverage results for various regular deployments.

    This book provides the first detailed expository treatment of these results, and includes additional exact analytical results on coverage for certain special non-PPP deployment models.We expect the book to be of interest to researchers in academia and industry, and anyone interested in the application of stochastic geometry to problems in communication. HetNets are an important component of future cellular network standards (LTE Release 12 and later), and the theoretical results in the book are illustrated with examples of their application to transmission scenarios specified in the LTE standard.

    Although we have made every effort to make the book self-contained, the reader will benefit from having had some prior exposure both to the theory and techniques of stochastic geometry, and to their application to the derivation of coverage results for PPP base station deployments. For an introduction to stochastic geometry with emphasis on wireless communications, we recommend Haenggi's Stochastic Geometry for Wireless Networks (Haenggi 2012) at the introductory level (a condensed version of which is available in Chapter 3) and the two volumes of Baccelli and Błaszczyszyn's Stochastic Geometry and Wireless Networks (Baccelli & Błaszczyszyn 2009a, 2009b) for a comprehensive treatment at the advanced level.

  • 8 - Extensions to Non-Poisson Models

  • from Part II - SINR Analysis
  • Bartłomiej Błaszczyszyn, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Martin Haenggi, University of Notre Dame, Indiana, Paul Keeler, Sayandev Mukherjee
  • Book:
    Stochastic Geometry Analysis of Cellular Networks
    Published online:
    26 March 2018
    Print publication:
    19 April 2018, pp 144-167

  • Summary

    Non-Poisson Point Processes

    Motivation

    In Chapters 5–7 we see how Poisson network models often give tractable results, and how random propagation effects such as fading and shadowing can render a network to appear more Poisson. Nevertheless, the Poisson model may still not be appropriate for certain network layouts. For example, although the Poisson process statistically does not exhibit either repulsion or clustering, it is possible for the clusters or voids of points to exist in network layouts.

    Appropriate point processes

    In Chapter 3 we see just a small selection of the possible spatial point processes. To choose appropriate point processes for network models, researchers have statistically analyzed networks located in various cities to demonstrate that other point processes may be more appropriate, although of course such analyses can be only as good as the data. But when developing stochastic geometry models of cellular networks, the number of suitable and tractable point processes quickly becomes limited. In terms of the SINR for a single user, there are two important and necessary features of a potential point process. One is the knowledge of the Palm distribution, as covered in Section 3.2.4, which immediately limits the choice of possible point processes. The other is being able to write down a calculable expression (at least, numerically) of the Laplace functional of the point process.

    In short, although there is a rich range of possible point processes, the bulk of them unfortunately lack the necessary properties that make them tractable in our setting, which partly explains why the clear majority of network models are based on the Poisson point process. Consequently, we restrict ourselves to results for point processes from two general families that exhibit repulsion and clustering, respectively, determinantal processes and a specific shot noise Cox process.

    Choice of the base station and propagation effects

    In Section 5.1.3, we see the two different model assumptions for how a user at the typical location (the origin) chooses the serving base station under the Poisson model.

  • 1 - Introduction

  • from Part I - Stochastic Geometry
  • Bartłomiej Błaszczyszyn, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Martin Haenggi, University of Notre Dame, Indiana, Paul Keeler, Sayandev Mukherjee
  • Book:
    Stochastic Geometry Analysis of Cellular Networks
    Published online:
    26 March 2018
    Print publication:
    19 April 2018, pp 3-12

  • Summary

    This chapter discusses the reasons for the importance of heterogeneous network (HetNet) architectures in future wireless network deployments. We begin with a brief survey of the state of wireless networks and the chief challenges faced by designers and operators of such networks. We then discuss possible solutions to these problems and single out small-cell networks as probably the best engineering solution. The deployment of small-cell networks then leads directly to the design and configuration of the resulting heterogeneous networks, which is the focus of this book. We make the case for new approaches to complement the traditional simulation-based schemes for design and performance evaluation when the wireless networks in question are heterogeneous, and provide an overview of the principal results that are now known (and that continue to be added to at a rapid pace), mostly as a result of research done since 2010.

    The Demand for Ubiquitous Connectivity

    The Information Age and the Computer Age have converged in the form of portable wireless devices that are full-fledged computers and that can, with a connection to the internet, provide instant always-on access to information anywhere. In other words, we are moving toward the “nomadic computing” vision of Kleinrock (Kleinrock 1996), where we can perform information processing “anytime, anywhere,” but instead of doing so in Kleinrock's “disconnected” world, we do so by ensuring that the world is fully “connected.”

    The main source of such ubiquitous connectivity, especially for mobile and outdoor users, is the cellular network as it has evolved over three decades and four generations of air interfaces (see Appendix B for details). Indoors, the cellular network is often complemented or even supplanted by a mix of wired and wireless (WiFi) networks, but since the indoor scenario is the simplest one in which to maintain connectedness, we shall focus on outdoor and mobile scenarios in the following discussion of ubiquitous connectivity. At any rate, it is a fait accompli that the cellular model, where base stations serve as exclusive access points to the internet within defined geographical areas called cells, is the foundation for any network that offers such ubiquitous connectivity.

Page 16 of 47