##
###
Stochastic Geometry and Wireless Networks: Volume I Theory

François Baccelli

2009
*
Foundations and Trends® in Networking
*

Preface A wireless communication network can be viewed as a collection of nodes, located in some domain, which can in turn be transmitters or receivers (depending on the considered network, nodes will be mobile users, base stations in a cellular network, access points of a WiFi mesh etc.). At a given time, some nodes simultaneously transmit, each toward its own receiver. Each transmitter-receiver pair requires its own wireless link. The signal received from the link transmitter is jammed by the
## more »

... signals received from the other transmitters. Even in the simplest model where the signal power radiated from a point decays in an isotropic way with Euclidean distance, the geometry of the node location plays a key role since it determines the signal to interference and noise ratio (SINR) at each receiver and hence the possibility of establishing simultaneously this collection of links at a given bit rate. The interference seen by a receiver is the sum of the signal powers received from all transmitters, except its own transmitter. Stochastic geometry provides a natural way of defining and computing macroscopic properties of such networks, by averaging over all potential geometrical patterns for the nodes, in the same way as queuing theory provides response times or congestion, averaged over all potential arrival patterns within a given parametric class. Modeling wireless communication networks in terms of stochastic geometry seems particularly relevant for large scale networks. In the simplest case, it consists in treating such a network as a snapshot of a stationary random model in the whole Euclidean plane or space and analyzing it in a probabilistic way. In particular the locations of the network elements are seen as the realizations of some point processes. When the underlying random model is ergodic, the probabilistic analysis also provides a way of estimating spatial averages which often capture the key dependencies of the network performance characteristics (connectivity, stability, capacity, etc.) in function of a relatively small number of parameters, which are typically the densities of the underlying point processes and the parameters of the involved protocols. By spatial average, we mean an empirical average made over a large collection of 'locations' in the considered domain; depending on the cases, these locations will simply be certain points of the domain, or nodes located in the domain, or even nodes on a certain route defined on this domain. These various kinds of spatial averages will be defined in precise terms in the monograph. This is a very natural approach e.g. for iii ad hoc networks, or more generally to describe user positions, when these are best described by random processes. But it can also be applied to represent both irregular and regular network architectures as observed in cellular wireless networks. In all these cases, such a space average is performed on a large collection of nodes of the network executing some common protocol and considered at some common time when one takes a snapshot of the network. Simple instances of such averages could be the fraction of nodes which transmit, the fraction of space which is covered or connected, the fraction of nodes which transmit their packet successfully, or the average geographic progress obtained by a node forwarding a packet towards some destination. This is rather new to classical performance evaluation, compared to time averages. Stochastic geometry, which we use as a tool for the evaluation of such spatial averages, is a rich branch of applied probability particularly adapted to the study of random phenomena on the plane or in higher dimension. It is intrinsically related to the theory of point processes. Initially its development was stimulated by applications to biology, astronomy and material sciences. Nowadays, it is also used in image analysis and in the context of communication networks. In this latter case, its role is similar to this played by the theory of point processes on the real line in classical queuing theory. The use of stochastic geometry for modeling communication networks is relatively new. The first papers appeared in the engineering literature shortly before 2000. One can consider Gilbert's paper of 1961 (Gilbert 1961) both as the first paper on continuum and Boolean percolation and as the first paper on the analysis of the connectivity of large wireless networks by means of stochastic geometry. Similar observations can be made on (Gilbert 1962) concerning Poisson-Voronoi tessellations. The number of papers using some form of stochastic geometry is increasing fast. One of the most important observed trends is to better take into account specific mechanisms of wireless communications in these models. Time averages have been classical objects of performance evaluation since the work of Erlang (1917). Typical exmples are the random delay to transmit a packet from a given node, the number of time steps required for a packet to be transported from source to destination on some multihop route, the frequency with which a transmission is not granted access due to some capacity limitations, etc. A classical reference on the matter is (Kleinrock 1975) . These time averages will be studied here either on their own or in conjunction with space averages. The combination of the two types of averages unveils interesting new phenomena and leads to challenging mathematical questions. As we will see, the order in which the time and the space averages are performed matters and each order has a different physical meaning. This monograph surveys recent results of this approach and is structured in two volumes. Volume I focuses on the theory of spatial averages and contains three parts. Part I in Volume I provides a compact survey on classical stochastic geometry models. Part II in Volume I focuses on SINR stochastic geometry. Part III in Volume I is an appendix which contains mathematical tools used throughout the monograph. Volume II bears on more practical wireless network modeling and performance analysis. It is in this volume that the interplay between wireless communications and stochastic geometry is the deepest and that the time-space framework alluded to above is the most important. The aim is to show how stochastic geometry can be used in a more or less systematic way to analyze the phenomena that arise in this context. Part IV in Volume II is focused on medium access control (MAC). We study MAC protocols used in mobile ad hoc networks (MANETs) and in cellular networks. Part V in Volume II discusses the use of stochastic geometry for the quantitative analysis of routing algorithms in MANETs. Part VI in Volume II gives a concise summary of wireless communication principles and of the network architectures considered in the iv monograph. This part is self contained and readers not familiar with wireless networking might either read it before reading the monograph itself, or refer to it when needed. Here are some comments on what the reader will obtain from studying the material contained in this monograph and on the possible ways of reading it. 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 (see e.g. the proceedings of conferences such as IEEE Infocom, ACM Sigmetrics, ACM Mobicom, etc. or the special issue (Haenggi, Andrews, Baccelli, Dousse, and Franceschetti 2009)). By mastering the basic principles of wireless links and of the organization of communications in a wireless network, as summarized in Volume II and already alluded to in Volume I, these readers will be granted access to a rich field of new questions with high practical interest. SINR stochastic geometry opens new and interesting mathematical questions. The two categories of objects studied in Volume II, namely medium access and routing protocols, have a large number of variants and of implications. Each of these could give birth to a new stochastic model to be understood and analyzed. Even for classical models of stochastic geometry, the new questions stemming from wireless networking often provide an original viewpoint. A typical example is that of route averages associated with a Poisson point process as discussed in Part V in Volume II. Reader alreadly knowledgeable in basic stochastic geometry might skip Part I in Volume I and follow the path: Part II in Volume I ⇒ Part IV in Volume II ⇒ Part V in Volume II, using Part VI in Volume II for understanding the physical meaning of the examples pertaining to wireless 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 from the very beginning. We show that this methodology has the potential of providing both qualitative and quantitative results. • Some of the most important qualitative results pertaining to these infinite population models are in terms of phase transitions. A typical example bears on the conditions under which the network is spatially connected. Another type of phase transition bears on the conditions under which the network delivers packets in a finite mean time for a given medium access and a given routing protocol. As we shall see, these phase transitions allow one to understand how to tune the protocol parameters to ensure that the network is in the desirable "phase" (i.e. well connected and with small mean delays). Other qualitative results are in terms of scaling laws: for instance, how do the overhead or the end-to-end delay on a route scale with the distance between the source and the destination, or with the density of nodes? • Quantitative results are often in terms of closed form expressions for both time and space averages, and this for each variant of the involved protocols. The reader will hence be in a position to discuss and compare various protocols and more generally various wireless network organizations. Here are typical questions addressed and answered in Volume II: is it better to improve on Aloha by using a collision avoidance scheme of the CSMA type or by using a channel-aware extension of Aloha? Is Rayleigh fading beneficial or detrimental when using a given MAC scheme? v How does geographic routing compare to shortest path routing in a mobile ad hoc network? Is is better to separate the medium access and the routing decisions or to perform some cross layer joint optimization? The reader with a wireless communication could either read the monograph from the beginning to the end, or start with Volume II i.e. follow the path Part IV in Volume II ⇒ Part V in Volume II ⇒ Part II in Volume I and use Volume I when needed to find the mathematical results which are needed to progress through Volume I. We conclude with some comments on what the reader will not find in this monograph: • We will not discuss statistical questions and give no measurement based validitation of certain stochastic assumptions used in the monograph: e.g. when are Poisson-based models justified? When should one rather use point processes with some repulsion or attraction? When is the stationarity/ergodicity assumption valid? Our only aim will be to show what can be done with stochastic geometry when assumptions of this kind can be made. • We will not go beyond SINR models either. It is well known that considering interference as noise is not the only possible option in a wireless network. Other options (collaborative schemes, successive cancellation techniques) can offer better rates, though at the expense of more algorithmic overhead and the exhange of more information between nodes. We believe that the methodology discussed in this monograph has the potential of analyzing such techniques but we decided not to do this here.

doi:10.1561/1300000006
fatcat:3f3rgmgmtncvbeief3wpdo2vd4