Present day telecommunication networks are ill equipped for the rapidly growing demand for mobile data transfers. With the fifth generation of mobile networks, paradigmatic shifts in the design of the network are on the agenda. A critical aspect here is the role of infrastructure. Multilayered cellular networks with possible incorporation of relaying mechanisms are under investigation not only in the scientific community, but also in industry. All these new designs have in common a rapid increase of degrees of freedom in the system. The central role of (expensive) base stations is reduced in favour of an increasingly important rôle of (cheap) relays. In particular, also the users of the system will be attached a relay functionality in the system. As a result, the network becomes more and more decentralised. First implementations of peer-to-peer (P2P) communications for public use are already available. Exploring the possible benefits of such new architectures is in full swing in the academic and the industrial research.

One promising way to cope with the new and more complex structures that arise is to exploit probabilistic methods. Indeed, fundamental ansatzes from stochastic geometry (e.g., spatial Poisson processes, continuum percolation theory, ...) are widely used for modelling the spatial locations of the users, the relays and the base stations and their basic connectivity properties. For the description of temporal developments, standard methods from stochastic processes (stochastic interacting particle processes like bootstrap percolation or the contact process) are commonly used to model the spread of information through a network.

Contribution of the Institute

The WIAS has performed mathematical research on connectivity and capacity problems in mobile relay-augmented probabilistic models over a period of four years within the Leibniz Group "Probabilistic methods for mobile ad-hoc networks" together with the Leibniz-Institute "Innovations for High Performance Microelectronics" (IHP) and in other collaborations. Its expertise includes dynamic modelling of message propagation in dense networks, bottleneck behaviour in Device-to-Device (D2D) systems, connection times in large networks without infrastructure and wifi-augmented mobil urban communications models.
Metropolitan street system with random users and additional boxes
Fig1. - Metropolitan street system with random users and additional relay boxes.


One of the highlights of the past years is an industry collaboration with a large European telecommunications company with the goal to better understand large scale D2D networks build on realistic street models.
Drainage network
Fig2. - Directed drainage network in a single cell.


  Articles in Refereed Journals

  • 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 .
    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, pp. published online on 28.10.2017, url, DOI 10.1007/s11134-017-9555-9 .
    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 .
    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.
    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.
    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.
    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, Institute of Electrical and Electronics Engineers. 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.

  Preprints, Reports, Technical Reports

  • E. Cali, T. En-Najjari, N.N. Gafur, Ch. Christian Hirsch, B. Jahnel, R.I.A. Patterson, Percolation for D2D networks on street systems, Preprint no. 2479, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2479 .
    Abstract, PDF (988 kByte)
    We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.

  • 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.

  Talks, Poster

  • B. Jahnel, Telecommunication models in random environments, BIMoS Day : The mathematics of quantum information, May 23, 2018, Technische Universität Berlin,, Berlin, May 23, 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.