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 relayaugmented probabilistic models over a period of four years within the Leibniz Group "Probabilistic methods for mobile adhoc networks" together with the LeibnizInstitute "Innovations for High Performance Microelectronics" (IHP) and in other collaborations. Its expertise includes dynamic modelling of message propagation in dense networks, bottleneck behaviour in DevicetoDevice (D2D) systems, connection times in large networks without infrastructure and wifiaugmented mobil urban communications models.Highlights
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.Publications
Articles in Refereed Journals

CH. Hirsch, B. Jahnel, R.I.A. Patterson, Spacetime large deviations in capacityconstrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587615, DOI 10.30757/ALEA.v1524 .
Abstract
We consider a singlecell 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 spacetime evolution of frustrated transmitters in the highdensity regime. 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relayaugmented wireless networks, Queueing Systems. Theory and Applications, pp. published online on 28.10.2017, urlhttps://doi.org/10.1007/s1113401795559, DOI 10.1007/s1113401795559 .
Abstract
We analyze a model of relayaugmented 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 nonunique 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. 10911115, 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 linkdensity and a subballisticity 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, Largedeviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 10611094.
Abstract
We study largedeviation 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 signaltointerferenceandnoise ratio is larger than a certain connectivity threshold. First, we show a largedeviation 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 largedeviation 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 importancesampling 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. 572575.
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 adhoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016), pp. 21432176.
Abstract
We study connectivity properties in a probabilistic model for a large mobile adhoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a spacedependent 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 wellknown 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. 7377.

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. 67746794.

B. Blaszczyszyn, M. Karray, P. Keeler, Wireless networks appear Poissonian due to strong shadowing, IEEE Transactions on Wireless Communications, 14 (2015), pp. 43794390.
Contributions to Collected Editions

E. Cali, T. EnNajjari, N.N. Gafur, Ch. Christian Hirsch, B. Jahnel, R.I.A. Patterson, Percolation for D2D networks on street systems, in: Proceedings of ``16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)'', IEEE Xplore digital library, 2018, pp. 16, DOI 10.23919/WIOPT.2018.8362866 .
Abstract
We study fundamental characteristics for the connectivity of multihop 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 PoissonVoronoi or PoissonDelaunay 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 deviceintensity 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. 
P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in highdensity 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. 18, 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 signaltointerference 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 nearfar 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 stochasticgeometry 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 Manhattantype 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. 
E. Cali, T. EnNajjari, 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 multihop 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 PoissonVoronoi or PoissonDelaunay 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 deviceintensity 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 nontrivial sub and supercritical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, highdensity 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 adhoc 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 relayaugmented 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, Signaltointerference 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 adhoc 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, Largedeviation 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 PoissonDirichlet 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 adhoc networks, Ecole de Printemps ``Marches Aléatoires, Milieux Aléatoires, Renforcements'' (MEMEMO2), June 10  14, 2013, Aussois, France, June 13, 2013.