Heutige Telekommunikationsnetze sind für die rasch wachsende Nachfrage nach mobilen Datentransfers schlecht gerüstet. Mit der fünften Generation mobiler Netze stehen paradigmatische Verschiebungen im Netzdesign auf der Tagesordnung. Entscheidend ist hierbei die Rolle der Infrastruktur. Mehrschichtige zellulare Netzwerke mit möglichem Einbau von Vermittlungsmechanismen werden nicht nur in der Wissenschaft, sondern auch in der Industrie intensiv untersucht. Alle diese neuen Designs teilen eine rasche Zunahme von Freiheitsgraden im System. Die zentrale Rolle von (teuren) Basisstationen wird zugunsten einer zunehmend wichtigen Rolle von (billigen) Relais reduziert. Insbesondere wird auch den Anwendern im System eine Relaisfunktionalität zugeordnet und das Netzwerk wird dadurch dezentralisiert. Erste Implementierungen von Peer-to-Peer (P2P) Kommunikationen für die öffentliche Nutzung sind bereits verfügbar. Die Erforschung der möglichen Vorteile solcher neuen Architekturen ist in der akademischen und der industriellen Forschung in vollem Gange.

Ein vielversprechender Weg, um mit den neuen und komplexeren Strukturen zurechtzukommen, ist die Nutzung von Methoden der Wahrscheinlichkeitstheorie. In der Tat werden grundlegende Ansätze aus der stochastischen Geometrie (zum Beispiel räumliche Poisson-Prozesse, Kontinuum-Perkolationstheorie, ...) weitgehend für die Modellierung der räumlichen Orte der Benutzer, der Relais, der Basisstationen und ihrer grundlegenden Verbindungseigenschaften verwendet. Zur Beschreibung von zeitlichen Entwicklungen werden Methoden aus den stochastischen Prozessen (stochastisch interagierende Partikelprozesse wie Bootstrap-Perkolation oder Kontaktverfahren) verwendet, um die Verbreitung von Informationen über ein Netzwerk zu modellieren.

Beitrag des Instituts

Das WIAS hat im Rahmen der Leibniz-Gruppe "Probabilistische Methoden für mobile Ad-hoc-Netzwerke", zusammen mit dem Leibniz-Institut für innovative Mikroelektronik (IHP) und in anderen Kooperationen, mathematische Untersuchungen zu Konnektivitäts- und Kapazitätsproblemen in mobilen Relais-erweiterten probabilistischen Modellen über einen Zeitraum von vier Jahren durchgeführt. Seine Kompetenz umfasst die dynamische Modellierung der Nachrichtenausbreitung in dichten Netzwerken, Bottleneck-Verhalten in Device-to-Device (D2D)-Systemen, Verbindungszeiten in großen Netzwerken ohne Infrastruktur und Wifi-erweiterte mobile städtische Kommunikationsmodelle.
Metropolitan street system with random users and additional boxes
Fig1. - Urbanes Strassennetz mit zufälligen Nutzern und zusätzlichen Relais.

Höhepunkte

Eines der Highlights der letzten Jahre ist eine industrielle Kooperation mit einem großen europäischen Telekommunikationsunternehmen mit dem Ziel, große D2D-Netze auf realistischen Straßenmodellen besser zu verstehen.
Drainage network
Fig2. - Gerichtetes Durchfluss-Netzwerk in einer Zelle.

Publikationen

  Artikel in Referierten Journalen

  • CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Stochastic Processes and their Applications, published online on 20.11.2018, urlhttps://doi.org/10.1016/j.spa.2018.11.002, DOI 10.1016/j.spa.2018.11.002 .
    Abstract
    We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

  • CH. Hirsch, B. Jahnel, R.I.A. Patterson, Space-time large deviations in capacity-constrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587--615, DOI 10.30757/ALEA.v15-24 .
    Abstract
    We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Queueing Systems. Theory and Applications, 88 (2018), pp. 349--387 (published online on 28.10.2017).
    Abstract
    We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Advances in Applied Probability, 49 (2017), pp. 1091--1115, DOI 10.1017/apr.2017.35 .
    Abstract
    We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large-deviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 1061--1094.
    Abstract
    We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

  • P. Keeler, N. Ross, A. Xia, B. Błaszczyszyn, Stronger wireless signals appear more Poisson, IEEE Wireless Communications Letters, 5 (2016), pp. 572--575.
    Abstract
    Keeler, Ross and Xia [1] recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects. The aim of this note is to apply some of the main results of [1] in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.
    [1] P. Keeler, N. Ross, and A. Xia:“When do wireless network signals appear Poisson?? ”

  • H. Döring, G. Faraud, W. König, Connection times in large ad-hoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016), pp. 2143--2176.
    Abstract
    We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation.

  • P. Keeler, P.G. Taylor, Discussion on ``On the Laplace transform of the aggregate discounted claims with Markovian arrivals'' by Jiandong Ren, Volume 12 (2), North American Actuarial Journal, 19 (2015), pp. 73--77.

  • B. Blaszczyszyn, P. Keeler, Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures, IEEE Transactions on Information Theory, 61 (2015), pp. 6774--6794.

  • B. Blaszczyszyn, M. Karray, P. Keeler, Wireless networks appear Poissonian due to strong shadowing, IEEE Transactions on Wireless Communications, 14 (2015), pp. 4379--4390.

  Beiträge zu Sammelwerken

  • P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in high-density cellular networks, in: Proceedings of ``16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)'', IEEE Xplore digital library, 2018, pp. 1--6, DOI 10.23919/WIOPT.2018.8362867 .
    Abstract
    Stochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signal-to-interference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the near-far effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.

  Preprints, Reports, Technical Reports

  • CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, Preprint no. 2557, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2557 .
    Abstract, PDF (311 kByte)
    The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.

  • CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Preprint no. 2445, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2445 .
    Abstract, PDF (438 kByte)
    We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

  Vorträge, Poster

  • A. Hinsen, Random Malware Propagation, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

  • B. Jahnel, Telecommunication models in random environments, BIMoS Day : The Mathematics of Quantum Information, May 23, 2018, Technische Universität Berlin, Berlin, May 23, 2018.

  • W. König, Probabilistic Methods in Telecommunication, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

  • A. Wapenhans, Data mobility in ad-hoc networks: Vulnerability & security, Telecom Orange Paris, France, November 17, 2017.

  • B. Jahnel, Continuum percolation for Cox processes, Seminar, Ruhr Universität Bochum, Fakultät für Mathematik, October 27, 2017.

  • B. Jahnel, Continuum percolation theory applied to Device to Device, Telecom Orange Paris, France, November 17, 2017.

  • B. Jahnel, Stochastic geometry in telecommunications, Summer School 2017: Probabilistic and Statistical Methods for Networks, August 21 - September 1, 2017, Technische Universität Berlin, Berlin Mathematical School.

  • CH. Hirsch, Large deviations in relay-augmented wireless networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 22, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • P. Keeler, Signal-to-interference ratio in wireless communication networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 24, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • W. König, Connection times in large ad-hoc mobile networks, Workshop on Dynamical Networks and Network Dynamics, January 18 - 21, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • P. Keeler, Large-deviation theory and coverage in mobile phone networks, Seminar ``Applied Probability'', The University of Melbourne, Department of Mathematics and Statistics, Australia, August 17, 2015.

  • P. Keeler, The Poisson--Dirichlet process and coverage in mobile phone networks, Stochastic Processes and Special Functions Workshop, August 13 - 14, 2015, The University of Melbourne, Melbourne, Australia, August 14, 2015.

  • P. Keeler, When do wireless network signals appear Poisson?, Simons Conference on Networks and Stochastic Geometry, May 18 - 21, 2015, University of Texas, Austin, USA, May 20, 2015.

  • G. Faraud, Connection times in large ad-hoc networks, Ecole de Printemps ``Marches Aléatoires, Milieux Aléatoires, Renforcements'' (MEMEMO2), June 10 - 14, 2013, Aussois, France, June 13, 2013.