Marios Kountouris Marios Kountouris

Department of Telecommunications

SUPELEC, France

marios.kountouris@supelec.fr

COMELEC General Seminar

TELECOM ParisTech

Paris

October 2, 2014

Wireless Networks

A collection of communication nodes (TXs or RXs, base stations, access points)

located in some domain

� Nodes access the medium and transmit simultaneously, often using the same spectrum resources at the same geographical area

� The signal received suffers from:� Interference: sum of signal powers received from all other

located in some domain

� Interference: sum of signal powers received from all other unintended TXs

� Signal power attenuation: due to distance-dependent pathloss, fading, shadowing

� Performance depends strongly on the locations of the users or nodes. These locations are subject to considerable uncertainty ï modeled as a stochastic process of points

Wireless Ecosystem Overview� Mobile Data Tsunami

� Over 100%/year growth in data traffic

� At least a 1000x every decade 18 fold

increase

� In 2016, global traffic will reach 1/100 of a zettabyte (= 1 trillion GBs)

� Capacity Crunch & Spectrum “El Dorado”

� Explosion of Connected Devices (50 billion Internet-capable devices by 2020)

� Diverged traffic and revenue growth

18 fold

increase

over 2011

Wireless Traffic Trends:

� Huge fraction of traffic in mobile video

� Most traffic is generated indoors

� Bandwidth-greedy applications are usedvery frequently

Unprecedented demands on capacity and coverage

� Technological progress has always occurred at exponential pace (“Kurzweil argument of accelerating returns”)

Wireless “Moore’s Law”

� Engineers have been exponentially increasing the achievable wireless rate for a very long time

� Cooper’s “Law”: “Wireless capacity has doubled every 30 months over the last 104 years’’ months over the last 104 years’’

� Mostly driven by smaller cells� more base stations has been the key

� Enhanced per link comm. engineering and new standards have had little impact: 4G, 5G, xG may not be the answer

Reaching the PHY limits

Number of antennas

More Spectrum

AMC, interference management, etc.

� Bit rate is divided among the number of users per cell

bit/s/Hz/m2Exploit space, time, frequency dimensions

Topology

Extreme densification: Re-use Shannon’s Law Everywhere!

Heterogeneous Cellular Networks

� Broadly speaking, Heterogeneous Cellular Networks orHeterogeneous Networks (HetNet) refers to the a multi-Heterogeneous Networks (HetNet) refers to the a multi-tier RAN architecture, in which macrocells, microcells,picocells, femtocells, relays, and remote radio headscoexist under the same umbrella

Heterogeneous Cellular Networks

Source: Qualcomm

� Network Densification: Bring the Network closer to the User

� Unprecedented Capacity, Coverage and Traffic Offloading Gains

� Emerging Networks: dense, heterogeneous and complex!

Source: Qualcomm

QQQQ: How to model and analyze dense, heterogeneous networks

and their uncoordinated/dynamic interactions?

Stochastic Geometry and Wireless Networks

Performance of Wireless Networks

� Network Geometry plays a key role:determines the Signal to Interference plus Noise Ratio (SINR) at each receiverand hence the possibility of communicating at a given rate.

� Rationale: Calculate spatial averages that capture the key dependencies ofthe network performance as functions of a relatively small number of systemparameters

� Why using Stochastic Geometry?� locations of network nodes are seen as the realizations of some point

process in space (and time)� network is a snapshot of a stationary spatial random model and is analyzed

in a probabilistic wayin a probabilistic way� relevant for large-scale, dense networks

� Stochastic Geometry provides a natural way of defining and computingmacroscopic properties of large networks by averaging over all potentialgeometrical patterns for the nodesï permits statements about entire classes of wireless networks, instead of justabout one specific configuration of the network

Stochastic Geometry

� Complicated geometrical patterns occur in many scientific andtechnological areas and often require statistical analysis

� Examples: geological structures, sections of porous media, solidbodies, biological tissues, and patterns formed by the distinctionbetween wood and field in a landscape.

Stochastic Geometrythe area of mathematical research that seeks to provide

suitable mathematical models and appropriate statistical

� A rich branch of applied probability with various applications:material science, image analysis and stereology, astronomy,biology, forestry, geology, communications, …

suitable mathematical models and appropriate statistical methods to study and analyze random spatial patterns

Stochastic Geometry� Term first coined in 1969 by Klaus Krickeberg and David

G. Kendall

� Also used in 1963 by H. L. Frisch and J. M. Hammersley("Percolation processes and related topics". SIAM Journal

� Related to Geometric Probability: deals with random geometric objects andconfigurations of simple nature (points, lines, planes, convex bodies, andintersections).

� Geometric Probability considers problems concerning a finite number ofgeometrical objects of fixed form, whose positions are completely random and (insome sense) uniformly distributed.

("Percolation processes and related topics". SIAM Journalof Applied Mathematics 11 (4): 894–918)

some sense) uniformly distributed.

� Stochastic Geometry employs more sophisticated models (such as random sets andparticle processes) and random geometrical patterns (which may be infinite in extent)of more complicated distribution.

� Stereology: branch of stochastic geometry which studies the problem of recoveringinformation on 3-dimensional structures when the only information available is 2- or1-dimensional, obtained by planar or linear section.

Geometric Probability

� In 1733, Georges-Louis Leclerc, Comte de Buffonpresented to the Académie Royale des Sciences his« Mémoire sur le Jeu de Franc-Carreau »« Mémoire sur le Jeu de Franc-Carreau »

Voici un problème qui m’a occupé ces jours passés, et qui sera peut-être du goût de Mr de Moivre.Vous ne savez peut-être pas ce que nous appelons en français le jeu du franc-carreau.Dans une chambre pavée de carreaux, on jette en l’air un écu.S’il retombe sur un seul carreau, on dit qu’il tombe franc, et celui qui l’a jeté gagne.S’il tombe sur deux ou plusieurs carreaux, celui qui l’a jeté perd.C’est un problème à résoudre et qui n’a point de difficulté : trouver la probabilité de gagner ou de perdre,les carreaux et l’écu étant donnés.

� Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor.

What is the probability that the needle will lie across a line between two strips?

les carreaux et l’écu étant donnés.

Buffon’s Needle Problem� Clean tile problem: players bet on the number of different tiles a thrown coin will

partially cover on a floor that is regularly tiled.

� Special cases of this game give the Buffon-Laplace needle problem (for a square� Special cases of this game give the Buffon-Laplace needle problem (for a squaregrid) and Buffon's needle problem (for infinite equally spaced parallel lines).

� Given a needle of length ℓ dropped on a plane ruled with parallel lines t unitsapart, what is the probability that the needle will cross a line?

� The solution depends on whether we have a short or a long needle.

Point Process

� Basic ingredients of geometry are points.

� Similarly, random point-patterns (or point processes) play a fundamental role in stochastic geometrya fundamental role in stochastic geometry

� Visually and loosely speaking… A point process in E (general locally compact space) is that of a random collection of isolated points in E (simple point process).

Spatial pattern in tropical forests (lowland hill dipterocarp forest in Sinharaja, Sri Lanka)

A Basic Example of a Point Process

� A trivial point process

� Contains only one point

� The random point x is uniformly distributed in a bounded set A

Examples of Point Processes

Thomas Cluster ProcessLongleaf Pine Point Process

Locations and diameters of 584 Longleaf pine (Pinus palustris) trees in a 200 x 200 meter region in

southern Georgia (USA).

Examples of Point Processes

Ginibre Point Process Stars in Night Sky Point Process

Examples of Point Processes

Crème Brûlée Point Process Cappuccino Point Process

Poisson Point Process

� The most widely used model for spatial node locations

� Analytical tractable

� “Gaussian of point processes”

(spatially white and maximum entropy point process)

� No dependence between node locations

� Random number of nodes

� Can be defined on the entire plane

� Limiting distribution of a Binomial point process (BPP)

Poisson Point Process

A homogeneous PPP with density λis a random point set such that

� The number of points N(A) for any bounded set A ⊂ Rd follows a Poisson distribution with

E(N(A)) = λ×area of (A)

� Independence� Independence

the numbers of isolated pointsfalling within two regions A andB are independent randomvariables if A and B do notintersect each other

Poisson Point Process

Modeling Emerging Cellular Networks

The need for novel models

Cellular Models

� Grid-based and Honeycomb models� Useful in studying and simulating

macrocells� Not easily scalable to multiple tiers� Not easily scalable to multiple tiers

� Linear (Wyner) Model� Not very useful for realistic cellular

analysis� No notion of outage since SINR is fixed

and deterministic

� Random Spatial Models� BS locations are assumed to form a

realization of some random point process

� Scalable and surprisingly tractable

How Current Wireless Networks Look Like?

1

2Frequency bands: 4

−2 −1 0 1 2−2

−1

0

Actual 4G macrocells 3-tier Hetnet Real data 3-tier Hetnet PPPSource: http://arxiv.org/abs/1103.2177

� Challenges:

� How to model and analyze multi-tier wireless networks?

� How to characterize and quantify the aggregate interference?

� How to derive key performance measures (coverage, area spectral efficiency, average rate)?

Random Spatial Models

� Statistical Model based on Stochastic Geometry

� K tiers of base stations (HetNets)

� BS locations taken from independent Poisson point process� Base Station Density: λi BS/area� Transmit Power: Pi Watts� SINR Target βi

� Per-tier path loss exponent αi

� Cell Association (closest BS, strongest signal, max SINR)

� Key features:� Key features:� Permits analysis of system performance, e.g. coverage

probability, area spectral efficiency, average rate� Averages performance of different cell sizes and

deployment

� May include other quantities (traffic factor, bias, load balancing, range extension, …)

� Clustering, repulsion, irregularity can be incorporated maxSINR association: macros (red), picos (green), and femtos (black).

Baseline Example

Network Model

� PPP of intensity λ� PPP of intensity λ

� ALOHA with probability p

o: typical receiver

(assumed at the origin)

x: desired transmitter x: desired transmitter

(not part of the process)

� Objective:

� Interference at the origin o ?

� Success/Coverage probability

Interference

Mean Interference

Laplace Transform

Laplace Transform

Interference Distribution

� Interference is not Gaussian evenfor λ ö ∞ (no CLT)

� Closed-form expressions for thedistribution only exist for α = 4(Lévy distribution).

� It has a heavy tail (expected fromthe fact that it does not have amean)

� The type of fading is irrelevant, only the fractional momentmatters

� The distribution of the interference without considering theclosest interferer has a different tail of smaller order

Downlink Cellular Model

� Cells are Poisson-Voronoi tessellations

What is the SINR distribution of a typical mobile user?

J. G. Andrews, F. Baccelli, and R. K. Ganti. A tractable approach to coverage and rate in cellular networks. IEEE Trans. on Communications, 59(11):3122-3134, November 2011

Poisson – Voronoi Tessellation

� Given a collection of points Φ = {Xi} and a given point x, the Voronoi cell C (ϕ) of this point is defined as the Cx(ϕ) of this point is defined as the subset of the plane of all locations that are closer to x than to any point of Φ

� When Φ = {Xi} is a Poisson point process,

we call the (random) collection of cells {CXi(Φ)} the Poisson - Voronoi Tessellation (PVT).

SINR Analysis

Coverage Probability

Key Result

Laplace Transform of Interference

Special Cases

Interference is Rayleigh fading, No noise

Model Validation

HetNet Model Accuracy

� Nearly as accurate as a grid model (for macrocell tier)tier)� Grid provides upper

bound on coverage probability

� PPP provides lower bound on coverage probability

� Result accurate down to � Result accurate down to an SIR target of about –4dB

� (in practice, rarely have a target below – 3 or – 5 dB • Coverage probability comparison in

a 2-tier network• Second tier (femtocells) assumed to be PPP in all cases

Source: H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, “Modeling and

Analysis of K-Tier Downlink Heterogeneous Cellular Networks", IEEE JSAC, Apr. 2012

The Poisson Model for BSs

� Clearly, BSs are not placed independently

� Alternative random distributions introducing correlation, repulsion, inhibition, etc have to be used

� No evidence that the Poisson model is any less accurate than the ubiquitous grid modelmodel� PPP is actually pessimistic

� Poisson model is tractable, allowing for a large class of powerful results and analytical tools

Coverage probability vs. SIR threshold (experimental data from urban region)

Parting Comments

� Cellular networks are undergoing a fundamental change

Shifting the Cellular ParadigmShifting the Cellular Paradigm

Pretty much all cellular knowledge and conventional wisdom from last 2-3 decades must be re-thought in next 3-5 years

� The rapid trend towards heterogeneity in wireless networks requires many longstanding models and the associated conventional wisdom to be reevaluated.

� Powerful results and analytical tools are available for HetNets

Stochastic Geometry can be used to analyze tradeoffs and provide guidelines for optimizing dense networks with spatial randomness

� Models and tools can be applied to other settings (sensor and ad hoc networks, vehicular networks, social networks, …)

Merci !