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 innerhalb der Wissenschaft und mit der Industrie mathematische Untersuchungen zu Konnektivitäts- und Kapazitätsproblemen in mobilen Relais-erweiterten wahrscheinlichkeits-theoretischen Modellen über einen Zeitraum von über fünf Jahren durchgeführt. Seine Kompetenz umfasst die dynamische Modellierung der Nachrichtenausbreitung in dichten Netzwerken, Bottleneck-Verhalten in Device-to-Device (D2D)-Systemen und Verbindungszeiten in großen Netzwerken ohne Infrastruktur. Darüber hinaus gehört zur Expertise des Instituts die Untersuchung von Wifi-erweiterte mobilen städtische Kommunikationsmodellen und die interferenzbasierte räumliche Analyse der Nachrichtentrajektorien als stochastische Prozesse.

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Fig. 1 - Urbanes Straßennetz mit Abdeckungsbereich (grau) von Infrastruktur (grün), erweitert durch zufällige Nutzer (blau) als Relais.


Ein besonderer Fokus liegt auf der Simulation und Extraktion von relevanten Daten, dem Einfluss von Mobilität auf wichtige Netzwerkcharakteristiken sowie der Untersuchung von möglichen Konsequenzen von Verzögerungen und Initialisierungen von Datenübertragungen.

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Fig. 2 - Ausbreitungsstand von Malware (rot) in einem zufälligen ad-hoc Netzwerk mit Immunisierungsmöglichkeit durch sogenannte white knights (grün) in zwei Zeitpunkten.


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Fig. 3 - Veränderung der Netzwerkverbindungen in einem beweglichen System.

Höhepunkte

Highlights der letzten Jahre beinhalten mehrere Industriekooperationen mit einem großen europäischen Telekommunikationsunternehmen mit dem Ziel, große D2D-Netze auf realistischen Straßenmodellen besser zu verstehen. Hierbei konnten wir beispielsweise kritisches Verhalten im Bezug zu langreichweitigen Verbindungen, auch Perkolation genannt, nachweisen.

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Fig. 4 - Zwei Cox-Punktprozesse in singulären und kontinuierlichen zufälligen Umgebungen.


Ein weiteres Highlight war die theoretische und numerische Untersuchung der typischen Kommunikationszelle in zufälligen zellulären Netzwerken basierend auf nicht-rotationsinvarianten Straßensystemen.

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Fig. 5 - Typische Voronoi-Zelle eines Cox-Punktprozesses auf einem iterierten Manhattan Gitter.

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: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789136/1--17789136/8, 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. Tóbiás, Lower large deviations for geometric functionals, Preprint no. 2632, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2632 .
    Abstract, PDF (1268 kByte)
    This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

  • 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.

  • W. König, A. Tóbiás, Routeing properties in a Gibbsian model for highly dense multihop networks, Preprint no. 2466, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2466 .
    Abstract, PDF (683 kByte)
    We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized

  • 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.

  • W. König, A. Tóbiás, A Gibbsian model for message routing in highly dense multi-hop networks, Preprint no. 2392, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2392 .
    Abstract, PDF (468 kByte)
    We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure.

  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.