Stochastic Geometry and Wireless Networks

Multiclass wireless birth-and-death processes

This work studies a multiclass spatial birth-and-death (SBD) processes on a compact region of the Euclidean plane modeling wireless interactions. In this model, users arrive at a constant rate and leave at a rate function of the interference created by other users in the network. The novelty of this work lies in the addition of service differentiation, inspired by bandwidth partitioning present in 5G networks: users are allocated a fixed number of frequency bands and only interfere with transmissions on these bands. The first result of the paper is the determination of the critical user arrival rate below which the system is stochastically stable, and above which it is unstable. The analysis requires symmetry assumptions which are defined in the paper. The proof for this result uses stochastic monotonicity and fluid limit models. The monotonicity allows one to bound the dynamics from above and below by two adequate discrete-state Markov jump processes, for which we obtain stability and instability results using fluid limits. This leads to a closed form expression for the critical arrival rate. The second contribution consists in two heuristics to estimate the steady-state densities of all classes of users in the network: the first one relies on a Poisson approximation of the steady-state processes. The second one uses a cavity approximation leveraging second-order moment measures, which leads to more accurate estimates of the steady-state user densities. The Poisson heuristic also gives a good estimate for the critical arrival rate.

  • P. Popineau and F. Baccelli, “On multiclass spatial birth-and-death processes with wireless-type interactions”, arXiv, 2023. https://arxiv.org/abs/2205.15799

The results of the last paper rely on a symmetry assumption which is relaxed in

  • P. Popineau, S. Shneer, “An instability condition for queueing systems with state-dependent departure rates”, ArXiv 2023 – https://arxiv.org/abs/2304.08853

Low Earch Orbit Satellite Constellation based Communication Networks

This work, in collaboration with Chang Sik Choi, develops an analytical framework for downlink low earth orbit (LEO) satellite communications, leveraging tools from stochastic geometry. We propose a tractable approach to the analysis of such satellite communication systems accounting for the fact that satellites are located on circular orbits. We accurately characterize this geometric property of such LEO satellite constellations by developing a Cox point process model that jointly produces orbits and satellites on these orbits. Our work differs from existing modeling studies that have assumed satellites’ locations as completely random binomial point processes. For this newly developed Cox model, we derive the outage probability of the proposed network and the distribution of the signal-to-interference-plus-noise ratio (SINR) of an arbitrarily located user in the network. By presenting various key network performance metrics as functions of key network parameters, this work allows one to assess the statistical properties of downlink LEO satellite communications and thus can be used as a system-level design tool.

Spatial Network Calculus

A new tool that combines the stochastic modeling of wireless networks using stationary point processes and the methodology of regulation/control from network calculus was introduced. It is based on spatial regulation properties for stationary spatial point processes and develops the first steps of a calculus for this regulation, which can be seen as an extension to space of the classical network calculus. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which are shown equivalent and leads to upper bounds on the interference power; the other one includes void regulation, which lower constraints the signal power. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter only requires the regulations to hold as observed by a jointly stationary point process. Using this approach, we derive performance guarantees in device-to-device, ad hoc, and cellular networks under proper regulations, respectively. We give universal bounds on the SINR for all links, which gives link service guarantees based on information theoretic achievability. They are combined with classical network calculus to provide end-to-end latency guarantees for all packets in wireless queuing networks. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks.

  • K. Feng and F. Baccelli, ‘Spatial Network Calculus and Performance Guarantees in Wireless Networks’, Arxiv (2023), URL : https://doi.org/10.48550/arXiv.2302.02001, Under revision by Transactions on Wireless Communications.

Stochastic Geometry of Beam Management in 5G Networks

Beam management is central in the operation of dense 5G cellular networks. Focusing the energy radiated to mobile terminals by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency. This benefit, however, comes with unavoidable overheads that increase with the number of beams and the mobile terminal speed. In industry, this crucial problem of beam management is often studied using system-level simulations that are expensive and time-consuming. To accelerate system-level simulations, Sanket Kalamkar and Francois Baccelli from INRIA Paris and Luis G. Uzeda Garcia, Fuad Abinader and Andrea Marcano from Nokia Bell Labs Paris jointly proposed a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antennas, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective area spectral efficiency that the typical user gets in this context. This in turn allows one to find, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz, the number of beams per cell that offers the best global trade-off between these benefits and costs. This work also discusses different systemic trade-offs of mmWave and sub-6 GHz 5G deployments.

  • S. Kalamkar, F. Baccelli, F. Abinader, A. Marcano and L. Uzeda Garcia, ‘Beam Management in 5G: A Stochastic Geometry Analysis’, IEEE Trans. Wirel. Commun., volume 21, number 4, pages 2275-2290, (2022),  https://doi.org/10.1109/TWC.2021.3110785.

Beam management also leads to new natural stochatic geometry models investigated in

  • F. Baccelli, S. S. Kalamkar, “On Point Processes Defined by Angular Conditions on Delaunay Neighbors in the Poisson-Voronoi Tessellation”, Journal of Applied Probability, 58(4), 2021. https://arxiv.org/abs/2010.16116

  • F. Baccelli, B. Liu, L. Decreusefond, and R. Song. ‘A User Centric Blockage Model for Wireless Networks’. In: IEEE Transactions on Wireless Communications (2022). https://hal.archives-ouvertes.fr/hal-03646854.

It also leads to new user association algorithms studied in

  • P. Popineau, S. S. Kalamkar, F. Baccelli, “On Velocity-based Association Policies for Multi-tier 5G Wireless Networks”, Proceedings of IEEE Globecom 2021,  December 2021.

Stochastic Geometry for Bandwidth Part in 5G Networks

Inspired by a new feature in 5G called bandwidth part (BWP), Francois Baccelli and Sanket S. Kalamkar proposed a bandwidth allocation (BA) model that allows one to adapt the bandwidth allocated to users depending on their data rate needs. Specifically, in adaptive BA, a wide bandwidth is divided into chunks of smaller bandwidths and the number of bandwidth chunks allocated to a user depends on its needs or type. Although BWP in 5G mandates allocation of a set of contiguous bandwidth chunks, the proposed BA model also allows other assumptions on chunk allocation such as the allocation of any set of bandwidth chunks, as in, e.g., LTE resource allocation, where chunks are selected uniformly at random. The proposed BA model is probabilistic in that the user locations are assumed to form a realization of a Poisson point process and each user decides independently to be of a certain type with some probability. This model allows one to quantify spectrum sharing and service differentiation in this context, namely to predict what performance a user gets depending on its type as well as the overall performance. This is based on exact representations of key performance metrics for each user type, namely its success probability, the meta distribution of its signal-to-interference ratio, and its Shannon throughput. Authors show that, surprisingly, the higher traffic variability stemming from adaptive BA is beneficial: when comparing two networks using adaptive BA and having the same mean signal and the same mean interference powers, the network with higher traffic variability performs better for all these performance metrics. With respect to Shannon throughput, the proposed BA model is roughly egalitarian per Hertz and leads to a linear service differentiation in aggregated throughput value.

  • F. Baccelli and S.S. Kalamkar, “Bandwidth Allocation and Service Differentiation in D2D Wireless Networks,” Proceedings of the IEEE International Conference on Computer Communications IEEE INFOCOM, Toronto, Canada, July 2020. https://arxiv.org/abs/2108.11472

Sanket S. Kalamkar studied the effect of bandwidth partitioning on the reliability and delay performance in infrastructureless wireless networks in this 2019 Globecom paper. The reliability performance is characterized by the density of concurrent transmissions that satisfy a certain reliability (outage) constraint and the delay performance by so-called local delay, defined as the average number of time slots required to successfully transmit a packet. The focus is on the ultrareliable regime where the target outage probability is close to 0. The bandwidth partitioning has two conflicting effects: while the interference is reduced as the concurrent transmissions are divided over multiple frequency bands, the signal-to-interference ratio (SIR) requirement is increased due to smaller allocated bandwidth if the data rate is to be kept constant. Instead, if the SIR requirement is to be kept the same, the bandwidth partitioning reduces the data rate and in turn increases the local delay. For these two approaches with adaptive and fixed SIR requirements, the author has obtained closed-form expressions of the local delay and the maximum density of reliable transmissions in the ultrareliable regime. The analysis shows that, in the ultrareliable regime, the bandwidth partitioning leads to the reliability-delay tradeoff.

  • S. S. Kalamkar, ” Reliability and Local Delay in Wireless Networks: Does Bandwidth Partitioning Help? ” December 2019, Proceedings of GLOBECOM 2019 – 2019 IEEE Global Communications Conference. https://arxiv.org/abs/1909.00913

Recent and Past Papers on the Stochastic Geometry Analysis of Wireless Networks Arranged by Topics

Beam Forming

Cellular Networks

Cognitive Radio Networks

Connectivity and Coverage

Cooperation

CSMA

Device-to-Device

Dynamics

General

Initial Access

Internet of Things

Shadowing

Spectral efficiency

Vehicular Networks

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