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Spatial Network Calculus and Performance Guarantees in Wireless Networks
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Abstract
This work develops a novel approach towards performance guarantees for all links in arbitrarily large wireless networks. It introduces spatial regulation properties for stationary spatial point processes, which model transmitter and receiver locations, and develops the first steps of a calculus for this regulation. This spatial network calculus can be seen as an extension to space of the initial network calculus which is available with respect to time. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which upper constraint the total power of interference generated by other links; the other one includes void regulation, which lower constraints the signal power. Notable examples satisfying the first class of regulation are hardcore processes, and a notable counter-example is the Poisson point process. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter, which relies on Palm calculus, only requires the regulations to hold at the atoms of a jointly stationary observer point process. Using this approach, we show how to derive performance guarantees for various types of device-to-device and cellular networks. We show that, under appropriate spatial regulation, universal bounds hold on the SINR for all links. The bounds are deterministic in the absence of fading and stochastic in the case with fading, respectively. This leads to service guarantees for all links based on information theoretic achievability when treating interference as noise. This can in turn be combined with classical network calculus to provide end-to-end latency guarantees for all packets in queuing processes taking place in all links of a large wireless network. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks.
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This paper is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here.
The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues of the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils).
An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper,
allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of non-necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant.
Eternal Galton-Watson Trees and Eternal Multitype Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes.
The calculation of the SIR distribution at the typical receiver (or,
equivalently, the success probability of transmissions over the typical link)
in Poisson bipolar and cellular networks with Rayleigh fading is relatively
straightforward, but it only provides limited information on the success
probabilities of the individual links. This paper introduces the notion of the
meta distribution of the SIR, which is the distribution of the conditional
success probability P given the point process, and provides bounds, an exact
analytical expression, and a simple approximation for it. The meta distribution
provides fine-grained information on the SIR and answers questions such as
"What fraction of users in a Poisson cellular network achieve 90% link
reliability if the required SIR is 5 dB?". Interestingly, in the bipolar model,
if the transmit probability p is reduced while increasing the network density
such that the density of concurrent transmitters stays
constant as , P degenerates to a constant, i.e., all links have
exactly the same success probability in the limit, which is the one of the
typical link. In contrast, in the cellular case, if the interfering base
stations are active independently with probability p, the variance of P
approaches a non-zero constant when p is reduced to 0 while keeping the
mean success probability constant.
We study the impact of interferences on the connectivity of large-scale ad hoc networks, using percolation theory. We assume that a bi-directional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contribution of interferences from all other nodes, weighted by a coefficient γ, and of a background noise. We find that there is a critical value of γ above which the network is made of disconnected clusters of nodes. We also prove that if γ is nonzero but small enough, there exist node spatial densities for which the network contains a large (theoretically infinite) cluster of nodes, enabling distant nodes to communicate in multiple hops. Since small values of γ cannot be achieved without efficient CDMA codes, we investigate the use of a very simple TDMA scheme, where nodes can emit only every nth time slot. We show that it achieves connectivity similar to the previous system with a parameter γ/n.
The signal-to-interference-ratio (SIR) meta distribution (MD) characterizes the link performance in interference-limited wireless networks: it evaluates the fraction of links that achieve an SIR threshold
with a reliability above
x
. In this work, we show that in Poisson networks, for any independent fading and power-law path loss with exponent
, the SIR MD can be expressed as the product of
and a function of
x
when
is in the so-called “separable region”. We show by simulation that the separable form serves as a good approximation of the SIR MD in Ginibre and triangular lattice networks when
is chosen large enough. Given the quest for ultra-reliable transmission, we study the asymptotics of the SIR MD as
for general cellular networks with Rayleigh fading. Finally, we apply our results to characterize the distribution of the link rate, where each link transmits with a rate satisfying a given reliability
x
, and the asymptotic distribution of the local delay, defined as the number of transmissions needed for a message to be received successfully.
Many industrial automation systems rely on time-synchronized (and timely) communication among sensing, computing, and actuating devices. Advances in Ethernet enabled by time-sensitive networking (TSN) standards, being developed by the IEEE 802.1 TSN Task Group, are significantly improving time synchronization as well as worst case latencies. Next-generation industrial systems are expected to leverage advances in distributed time coordinated computing and wireless communications to enable greater levels of automation, efficiency, and flexibility. Significant progress has been made in extending accurate time synchronization over the air (e.g., 802.1AS profile for IEEE 802.11/Wi-Fi). Given the inherently unreliable, varying capacity and latency prone characteristics associated with wireless communications, proving the feasibility of worst case latency performance over the wireless medium is a major research challenge. More specifically, understanding what levels of capacity, reliability, and latency could be guaranteed over wireless links with high reliability are important research questions to guide the development of new radios, protocols, and time coordinated applications. This paper provides an overview of the potential applications, requirements, and unique research challenges to extend TSN capabilities over wireless. The paper also describes advances in wireless technologies (e.g., next-generation 802.11 and 5G standards) toward achieving reliable and accurate time distribution and timeliness capabilities. It also provides a classification of wireless applications and a reference architecture for enabling the integration of wired and wireless TSN capabilities in future industrial automation systems.
A fundamental problem for the delay and backlog analysis across multi-hop paths in wireless networks is how to account for the random properties of the wireless channel. Since the usual statistical models for radio signals in a propagation environment do not lend themselves easily to a description of the available service rate, the performance analysis of wireless networks has resorted to higher-layer abstractions, e.g., using Markov chain models. In this work, we propose a network calculus that can incorporate common statistical models of fading channels and obtain statistical bounds on delay and backlog across multiple nodes. We conduct the analysis in a transfer domain, which we refer to as the SNR domain, where the service process at a link is characterized by the instantaneous signal-to-noise ratio at the receiver. We discover that, in the transfer domain, the network model is governed by a dioid algebra, which we refer to as (min, ×) algebra. Using this algebra we derive the desired delay and backlog bounds. An application of the analysis is demonstrated for a simple multi-hop network with Rayleigh fading channels.
An extensive update to a classic text
Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis.
The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital applications to spatial statistics and as a very interesting field of mathematics in its own right.
This edition: presents a wealth of models for spatial patterns and related statistical methods;
provides a great survey of the modern theory of random tessellations, including many new models that became tractable only in the last few years;
includes new sections on random networks and random graphs to review the recent ever growing interest in these areas;
provides an excellent introduction to theory and modelling of point processes, which covers some very latest developments;
illustrate the forefront theory of random sets, with many applications;
adds new results to the discussion of fibre and surface processes;
offers an updated collection of useful stereological methods;
includes 700 new references;
is written in an accessible style enabling non-mathematicians to benefit from this book;
provides a companion website hosting information on recent developments in the field www.wiley.com/go/cskm
Stochastic Geometry and its Applications is ideally suited for researchers in physics, materials science, biology and ecological sciences as well as mathematicians and statisticians. It should also serve as a valuable introduction to the subject for students of mathematics and statistics.
In this paper, a very general discrete-time queueing model with one single server and an infinite waiting room is studied. The number of arrivals during any discrete time-unit (henceforth called a “slot”) as well as the service time of each customer (expressed in slots) have general probability distributions. Both single and bulk arrival processes are considered. The system is analysed by using a two-dimensional state description. Results of the analysis include: the probability generating function of the number of customers in the system at various sets of observation epochs, and the probability generating functions of the unfinished work, the system time of the customers, and the busy period (expressed in slots). Also a number of fundamental relationships between the main performance measures of discrete-time queueing systems in general are derived. The results of the paper are applicable in the study of queueing phenomena in the area of digital communications systems.
The asymptotic distribution of orbits for discrete subgroups of motions in Euclidean and non-Euclidean spaces are found; our principal tool is the wave equation. The results are new for the crystallographic groups in Euclidean space and for those groups in non-Euclidean spaces which have fundamental domains of infinite volume. In the latter case we show that the only point spectrum of the Laplace-Beltrami operator lies in the interval (]; furthermore we show that when the subgroup is nonelementary and the fundamental domain has a cusp, then there is at least one eigenvalue in this interval.
This monograph surveys recent results on the use of stochastic geometry for the performance analysis of large wireless networks. It is structured in two volumes. Volume I - Theory (see http://hal.inria.fr/inria-00403039) focuses on stochastic geometry and on the evaluation of spatial averages within this context. It contains two main parts, one on classical stochastic geometry (point processes, Boolean models, percolation, random tessellations, shot noise fields, etc.) and one on a new branch of stochastic geometry which is based on information theoretic notions, such as signal to interference ratios, and which is motivated by the modeling of wireless networks. This second part revisits several basic questions of classical stochastic geometry such as coverage or connectivity within this new framework. Volume II bears on more practical wireless network modeling and performance analysis. It leverages the tools developed in Volume I to build the time-space framework needed for analyzing the phenomena which arise in these networks. The first part of Volume II focuses on medium access control protocols used in mobile ad hoc networks and in cellular networks. The second part bears on the analysis of routing algorithms used in mobile ad hoc networks. For readers with a main interest in wireless network design, the monograph is expected to offer a new and comprehensive methodology for the performance evaluation of large scale wireless networks. This methodology consists in the computation of both time and space averages within a unified setting which inherently addresses the scalability issue in that it poses the problems in an infinite domain/population case. For readers with a background in applied probability, this monograph is expected to provide a direct access to an emerging and fast growing branch of spatial stochastic modeling.
This monograph surveys recent results on the use of stochastic geometry for the performance analysis of large wireless networks. It is structured in two volumes. Volume I focuses on stochastic geometry and on the evaluation of spatial averages within this context. It contains two main parts, one on classical stochastic geometry (point processes, Boolean models, percolation, random tessellations, shot noise fields, etc.) and one on a new branch of stochastic geometry which is based on information theoretic notions, such as signal to interference ratios, and which is motivated by the modeling of wireless networks. This second part revisits several basic questions of classical stochastic geometry such as coverage or connectivity within this new framework. Volume II - Applications (see http://hal.inria.fr/inria-00403040) bears on more practical wireless network modeling and performance analysis. It leverages the tools developed in Volume I to build the time-space framework needed for analyzing the phenomena which arise in these networks. The first part of Volume II focuses on medium access control protocols used in mobile ad hoc networks and in cellular networks. The second part bears on the analysis of routing algorithms used in mobile ad hoc networks. For readers with a main interest in wireless network design, the monograph is expected to offer a new and comprehensive methodology for the performance evaluation of large scale wireless networks. This methodology consists in the computation of both time and space averages within a unified setting which inherently addresses the scalability issue in that it poses the problems in an infinite domain/population case. For readers with a background in applied probability, this monograph is expected to provide a direct access to an emerging and fast growing branch of spatial stochastic modeling.