1 stochastic geometry as a tool for the modeling of telecommunication networks prof. daniel kofman,...

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Stochastic Geometry as a tool for the modeling of telecommunication networks

Prof. Daniel Kofman,ENST - Telecom Paris

Dr. Anthony Busson

IEF – University of Orsay-Paris 11

TAU – 25/11/2004

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S.G. and Network Modeling

When modeling a network, two main types of characteristics need to be captured: the dynamics imposed by the traffic evolution at

different time scales time properties

the spatial distribution and movement of network elements (terminals, antennas, routers, etc.) geometric properties

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Examples of Geometric Properties

Modeling of UMTS/WiFi antennas location

optimal cost under coverage constraints Sensor networks

optimal cost under coverage, connectivity and lifetime constraints Ad-Hoc Networks CDN servers location for optimal content distribution Multicast capable routers of a CBT architecture Reliable Multicast Servers for optimal retransmission of missed

information Networks Interconnection points Optimal placement of fix access networks concentrators Others

Daniel Kofman

On ne connait pas le terrain a priori mais on represente aleatoirement les zones d couverture pour evaluer le cout de la couverture

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Why Stochastic Geometry

The efficiency of a protocol/mechanism/ dimensioning rule, etc. depends on its adaptability to different network topologies and users distribution The performance metrics of interest have usually to be

obtained as an average over A large set of possible network topologies A large set of possible users location distribution

Members of the various multicast groups Clients of the different available content

A large set of users behaviors Mobility Content popularity

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Content

Introduction Application domains in the telecommunication world Why Stochastic Geometry (S.G.)?

A Simple example to illustrate what S.G. is Network infrastructure optimization

Theoretical framework, part 1 – Tessellation processes Other application examples (CDNs, Multicast routing) Theoretical framework, part 2 – Coverage processes More application examples (CDMA, Ad-hoc and sensor

networks) Summary: Main mathematical objects, Main known

results Conclusions and Perspectives

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Content






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A simple example: Network infrastructure optimization

Network topology to be modeled: Users are connected to the closest

Service Provider Point of Presence (PoP) PoP are hierarchically connected to the

closest concentrator Higher layer concentrators are connected

to the closest core equipment Core equipment are “meshed”

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Architecture

Access Network

PoP

Conc.

PoP

Core

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Clients are represented by a Point Process on the plane

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PoPs and their Voronoï cells

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Concentrators and their Voronoi cells

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Access Hierarchy

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Access Hierarchy

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Delaunay Graph

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Meshed Core and Delaunay graph

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Questions we can answer

For a given distribution of users and for a given cost function, under Poisson hypothesis, we can compute the Optimal number of hierarchical levels Optimal intensity of the various point processes Average number of users per PoP Average cost of the network Routing cost in number of hops when connection two

clients as a function of their distance For the detailed analysis of this model see

F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev. Stochastic geometry and architecture of communication networks. J. Telecommunication Systems, 7:209-227, 1997.

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Point Processes and Voronoï Tessellations

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Stationary Poisson point process in d

Definition The number of points in a set B of d follows a

discrete Poisson law of parameter ||B||, where is the intensity of the process

Let B1…Bn be disjoint sets of d, the number of points in B1 … B2 are independent.

Consequence Given n the number of points in B, the points are

independently and uniformly distributed in B.

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Poisson Voronoï tessellation The point process

generating the Voronoï tessellation is a stationary Poisson point process.

The mathematical theory is studied by Møller See [Møller 89,94]

Main characteristics λ : pp intensity λ0 =2 λ (vertices intensity)

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Poisson Voronoï Tessellation The point process

generating the Voronoï tessellation is a stationary Poisson point process.

The mathematical theory is studied by Møller [Møller]

Main characteristics : pp intensity 0 =2 1 =3 (sides intensity)

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Characteristic of the « typical cell »

Number of sides (6 in average)

Area (1/ in average)

Average perimeter length :

Probability that the cell has n sides

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

3 4 5 6 7 8 9 10

number of sides

pro

bab

ilit

ies

Area of the typical cell

0

0,2

0,4

0,6

0,8

1

Area

De

ns

ity

fu

nc

tio

n

/4

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Cost function

dxNxfENxfE xNx

i

i

)],([),( 0

Nx

i

i

NxfE ),(

A point at x add a cost f(x,N). In this case, the mean of the cost function is:

By the refined Campbell formula, we have:

Expectation under Palm measure

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Palm measure: intuitive introduction

D(1)/D(0,8) 1

time

Number ofpackets

1

0

D0,8 D

U(1)Arrival

Departure

Prob (Queue empty)=0,2Prob (Queue empty at arrival times)=1Prob0(Queue empty)=1

PASTA: Poisson Arrivals See Time Averages

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Feller’s Paradox for a Poisson Process

Bus inter-arrival process: Poisson of parameter Bus inter-arrival times sequence: i.i.d., exp() Waiting time for a passenger arriving at time t:

exp() Time since last bus arrival before time t: exp() Probability distribution of the inter-arrival containing

time t: Erlan-2 of parameter Average inter-arrival time 1/ Average length of the inter-arrival containing time t:

2

time

t

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Feller’s paradox and Palm theory

Since we look at stationary processes, time t could be whatever.

We will concentrate without loss of generality in the case t=0.

By definition of Palm probability (at time 0), we have Prob0(T0=0) = 1 The inter-arrival time sequence is i.i.d., exp()

Since the intervals generated by each point of the process are equivalent, we can concentrate in any of them, like the one starting at 0, when analyzing the performances of the system.

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Plane case

E(C0()) = / with =1.280E0(C0()) = 1/

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Back to Campbell Formula

dxNxfENxfE xNx

i

i

)],([),( 0

Nx

i

i

NxfE ),(

A point at x add a cost f(x,N). In this case, the mean of the cost function is:

By the refined Campbell formula, we have:

Expectation under Palm measure

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Summary

The location of the various elements is modeled by point processes

Voronoï Tessellations are used to partitioning the plane and deducing the elements connectivity

Delaunay graph/tessellations can be used for the same purposes

A cost function is defined as a functional of the previous processes

Palm theory is used to evaluate this cost function we want to optimize

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Content






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Example 2: Content Distribution

Internet

Content Provider Server

User

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Content Delivery Network

Problems : The provided QoS depends on the network

performances Thus, the content provider cannot control this quality

The content on the cash servers cannot be controlled

Solution : To deploy a set of servers

Expensive To share the resources of a CDN between

various Content Providers

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What is the optimal location of the CDN servers ?

Internet

Content Providers

Users

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The role of Stochastic Geometry

Dimensioning difficulty: several parameters are not known a priori Clients evolution – Content Providers

location and content Number and location of users Popularity of content Network topology Network distribution cost

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A Simplified Stochastic Model

A point process will represent the various possible server locations (ISPs, etc.) A non Euclidian distance can be used, like the

transmission cost Two marks are associated with each point

The fist one indicates the number of users associated with the corresponding point (ISP, etc.)

The second one indicates whether a server is deployed in the corresponding point or not

A function of the distance between each client and the nearest server describes the QoS perceived by the users A non Euclidian distance can be used, like the

transmission cost

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Marked Point Process

(x,mx)

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Servers locations and corresponding Voronoï cells

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Cost Function

From the QoS point of view, the best solution is to deploy servers in each available location

This approach leads to a high CAPEX and OPEX The cost function we optimize will consider

The cost of the servers, denoted by α (we denote the number of servers by S) The number of users at point j, denoted by mj (we denote by L the set of

possible locations) A measure of the QoS degradation, denoted by f(xj), where xj is the distance

between the users that are related with location j and their nearest server.

Cost

Cost

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A more general model

Several server classes can be considered Servers of different classes have different cost E.g. Many small servers for a reduced number of very

popular content and a reduced number of big servers for the less popular content

Each object is located in a server of a given class Different location policies can be implemented

Based on objects popularity Random Others

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Main Results

Optimal intensity of the point processes representing the different classes of servers

Analysis of the impact of the various parameters on the performances of the system

Evaluation of the cost of the CDN For a detailed analysis of this model see

A. Busson, D. Kofman and Jean-Louis Rougier Optimization of Content Delivery Networks server placement, International Teletraffic Congress,ITC-18, 2003

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Example 3, Hierarchical CBT Multicast Trees

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Point Process on the place representing routers location

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Stochastic Geometry Model

Routers are represented by a Point Process in the plane

The routers participating to the tree are obtained by thinning the previous point process

« Rendez-vous » (RP) points are modeled by independent point process of lower intensity RP are active if they have an active router (RV

point of the lower level) in their Voronoi cell

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CBT distribution and corresponding Voronoi cells

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Stochastic Geometry Model

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Hierarchical CBT optimization

Model

Palm

P.P.

1

0 1

*

2

H

k k

kk p

ExplicitFormulae

Mathematical

Tools

..., ,kk p

Parameters

Optimal dimensioning of the various parameters (H, intensities, thinning probabilities)

Dimensioning

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Reference

For a detailed analysis of this model see: F.Baccelli, D.Kofman, J.L.Rougier, « Self-

Organizing Hierarchical Multicast Trees and their Optimization », IEEE Infocom'99, New-York (E.U.), March 1999

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Exemple 4 : Optical access network

Optical backboneEgress Node

Access Node

Passive splitter

Base station

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Evaluation of optical access network

Estimate the cost P of a ring N ring access networks may be

evaluated as NP If the ring intensity is λ, the cost of a

network covering A is λ||A||P The problem is reduced to the

estimation of the cost of a typical ring architecture.

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Rings modeling

Poisson point process of intensity λ.

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PONs Modeling

The Access nodes are the node of the Voronoï cell.

A Poisson point process represents the passive splitters

Another PPP represents the base stations.

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PONs Modelling

Every splitter is connected to the closest node of the Voronoï cell it lies in.

Every base station is connected to the closest splitter.

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The cost function (1)

Cost of the ring and access nodes

)))((())(( 00 VcardcVdlP accfeed

accfeed cd

PE 64

][0

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Cost of the splitters

))(())((,())((,( 01)(

20201

01

VcVHxdaVHxdaP splVx

iisplitters

i

1

010 ))](([ splspl cVEc

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Cost of the base stations

0

21

00 12

][ bsl

l

lbs c

bPE

l

)( )()( )( 01 1001 10 Vy Vx

bsVy Vx l

jilbs

i iyj

l

i iyj

cxybP

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Conclusions for the example 4

Economical studied of the access network Evaluation of the costs with regard to the number of

equipment access nodes splitters base stations

Evaluation of the optimal intensities describing the different equipments

For a detailed analysis of this model see: C.Farinetto, S. Zuyev, “Stochastic geometry modelling of

hybrid optical networks”, Performance Evaluation 57, 441-452, 2004.

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Dual problem

Tessellation: the process define the geometry properties of a way to partition the plane from which the topology of the network is deduced The connectivity between neighbors equipment is

deduced from the geometric properties of the processes Coverage: the processes defined the topology of

the network from which the geometry of the coverage of the plane is deduced The geometric properties we are interested on are

deduced from the connectivity properties between neighbors equipment (like those deduced from the radio channel model)

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Coverage processes

Motivation Historical applications

Structure of the paper Distribution of the heather in a forest Modeling the crystallization in metals Etc

Modeling of communication systems Modeling node and connectivity of an ad-hoc network Modeling the coverage of a CDMA network Modeling coverage and connectivity in sensor networks Routing in ad-hoc networks Others

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Boolean Model-Definition

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Boolean model - example

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Boolean model example The compact

sets here are circles, centered in 0, of random radius uniformly distributed in [0,1]

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Capacity functional

Probability that the intersection between the Boolean model Ξ and a finite closed set K is not empty

The capacity functional determines uniquely the distribution of the Boolean model.

Where is the Lebesgue measure in the plane Remark: the probability of K being covered is not known

in general Of course it is when K is a singleton set

)()emptynot is ()( KPKPKT

K)(exp1)( 02 KGEKT

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Capacity functional – our example

K={0} In this case, the capacity functional

is the probability that 0 belong to Ξ

exp1)0( 02 GET

0

3

102 GE

3

1exp1)0(

T

Capacity functional

0

0,2

0,4

0,6

0,8

1

1,2

0,1

0,3

0,5

0,7

0,9

1,1

1,3

1,5

1,7

1,9

2,1

2,3

2,5

2,7

2,9

Process intensity

cap

acit

y fu

nct

ion

al f

or

K=

{0}

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Contact distribution function

If a point is not covering by Ξ, how far is the boolean model?

Let’s take B(R)=B(0,R) a test set covering 0 We define

)0)((1)0)(()( RBPRBPH RB

)0(

)(1)(

P

RBPH RB

)0(1

))((11)(

T

RBTH RB

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Contact distribution function – our example

)(exp1 2)( RRH RB

R

0

Contact distribution function

00,10,20,30,40,50,60,70,80,9

1

0,01

0,04

0,07 0,1

0,13

0,16

0,19

0,22

0,25

0,28

0,31

0,34

0,37 0,4

0,43

0,46

0,49

Distance between 0 and the boolean model

con

tact

dis

trib

uti

on

fu

nct

ion

67

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CDMA Coverage - Boolean Model

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Example

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Known results

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CDMA coverage Model

Remark: Not a Boolean model since the compact sets are not independent

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CDMA coverage Model

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What can we calculate

Coverage probability Distribution of the number of cells

covering a given location

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Conclusions on the CDMA coverage model

Tool for estimating the network cost How many antennas (on average) for a given coverage ?

Tool for predicting the impact of network evolution What about coverage when increasing the number of antennas

The model can be extended to include random attenuation, correlation between marks, etc.

The movement of terminals can be modelled by line processes Evaluation of number of hand-overs Evaluation of traffic and required capacity

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References

For a detailed analysis of these models see: F. Baccelli and B. Blaszczyszyn. On a coverage process

ranging from the boolean model to the poisson voronoi tessellation, with applications to wireless communications. Adv. Appl. Prob., 33(2), 2001.

F. Baccelli, B. Blaszczyszyn, and F. Tournois. Spatial averages of coverage characteristics in large CDMA networks. Technical Report 4196, INRIA, June 2001.

F. Baccelli and S. Zuyev. Stochastic geometry models of mobile communication networks. In Frontiers in queueing, pages 227-243. CRC Press, Boca Raton, FL, 1997.

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Modeling ad-hoc and sensor networks

Let N be a random variable representing the number of devices

For a given realization of N, N points are independently and uniformly distributed in the square of size LxL

Two points x and y are said to be connected if d(x,y)<R.

Application: connectivity in ad-hoc and sensor networks

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Random geometric graph

The N points

L

L

80


Radio range of the points

81


Connectivity

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Random geometric graph obtained by simulation

100 nodes 3000 nodes

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Percolation – Finite domain

A network is said to be fully connected when it exist a path between any pair of nodes

What is the probability of the network being fully connected based on the random geometric graph model? Depends only on the mean number of

direct neighbors (mean size of the 1-neighborhoud)

84

Percolation – Finite domain – results

Let G(n,r(n)) be the random geometric graphs with n points and with radius r(n).

Let be Pc(n,r(n)) the probability that all the nodes are connected.

85

Percolation – Finite domain – results

Determine r(n) such that Pc(n,r(n)) goes to one as n →+∞.

Theorem:

)( iff 1)(P

then,)(log

)( if

c

2

ncnn

ncnnπr

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Percolation – Infinite domain – the line

Let’s consider a Boolean model with fixed radius. Question: What is the size of the clusters (clumps of

ball)? Answer: In one dimension, the network is almost

surely disconnected. There are an infinite number of bounded clusters.

87

Percolation – Infinite domain – the plane

Let be a Poisson Boolean model in the plane with balls of fixed radius.

Theorem [Meester99]: There exists a critical density λc>0 such that If λ<λc, all clusters are bounded almost

surely (sub-critical case) If λ>λc, there exists a unique unbounded

cluster almost surely (supercritical case)

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Percolation in a more realistic model

STIRG : Signal to Interference Ration Graph A node j can receive data from node i iff

Two nodes are neighbors if they can exchange data in both directions

jik

jkk

jii

xxlPN

xxlP

,0

89

Percolation in a more realistic model : results

When γ=0,it is a boolean model and the previous theorem holds.

When γ>0, The number of neighbors is bound. A node can

have at most 1+1/ γβ neighbors. Under certains assumptions on the attenuation

function l(.), there exists λc<∞ s.t. for all λ> λc

there exists 0<γc(λ) s.t. for γ <γc(λ) the probability that a node belongs to an inifinite cluster is strictly greater than zero.

Dousse, Baccelli, Thiran, « Impact of interfernces on connectivity in Ad Hoc Networks », Infocom 2003.

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Other interesting problems

Optimizing a sensor network composed of heterogeneous devices

Taking into account layer 3 routing mechanisms when evaluating an ad-hoc or sensor network connectivity

Taking into account the MAC layer and radio channel properties when modeling sensor networks

Link with graph theory (e.g. small worlds), percolation theory, etc.

Others

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Modeling Heterogeneous Wireless Sensor Networks

Application-specific nature of sensor networks Two main classes (based on applications)

Data gathering sensor networks: e.g. environment monitoring, temperature monitoring and control

Event detection sensor networks: e.g. forest re detection Data gathering sensor networks

Periodic data gathering cycles, correlated measurements, data aggregation

Clustering for aggregation and protocol scalability Hierarchical clustering

Guarantee system lifetime

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Modeling Heterogeneous Wireless Sensor Networks

Random deployment of nodes, 2-D homogeneous Poisson process

Each cluster is a Voronoi cell Use simple tools from stochastic geometry

to determine the relaying load on critical nodes, P0

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Others

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Routing in dense ad-hoc or sensor networks

High number of nodes and high connectivity requires: New addressing

paradigms New routing approaches New algorithms for

multicast and broadcast Etc.

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Self-organization of the network

Each node elect the node in its neighborhood with the highest metric.

Metric examples : Degree of a node : number

of neighbors for this node Density of a node : number

of edges between neighbors of the node

96


If a node has the highest metric in its neighborhood, it elects himself has a cluster head. Example : the degree as metric

97


1000 nodes – radius 0.1 3000 nodes – radius 0.1

Simulation in a random geometric graph (in a square of size 1x1)

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Self-organization of the network : results

Geometry sotchastic gives: Bound on the number of clusters, Bound on the probability that a node is a cluster head, Mean and variance of the metrics.

Other results are obtained by simulation: Degree of the nodes in the cluster tree Behavior of the cluster when the node are moving (mobile ad-

hoc netwkorks) Number of broadcast messages received by the nodes.

Mitton, Busson, Fleury, “Self Organization in Large Scale Ad Hoc Networks”, MedHoc-Net 2004.

Mitton, Fleury, “Self-Organization in Ad Hoc Networks”, reserah report INRIA, RR-5042.

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Others

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References related with the last cited topics

Vivek Mahtre, Catherine Rosenberg, Daniel Kofman, Ravi Mazumdar, Ness Shroff, A Minimum Cost Surveillance Sensor Network with a Lifetime Constraint, to appear in IEEE Transactions of Mobile Computing (TMC).

Sunil Kulkarni, Aravind Iyer, Catherine Rosenberg, Daniel Kofman, Routing Dependent Node Density Requirements for Connectivity in Multi-hop Wireless Networks, accepted, Globecom 2004



O. Douse, F. Baccelli, P. Thiran, Impact of Interferences on Connectivity in Ad-Hoc Networks, in Proc. IEEE Infocom 2003

O. Douse, P. Thiran and M. Hasler, Connectivity in ad-hoc and hybrid networks”, in Proc. IEEE Infocom, 2002

M. Grossglauser and D. TSe, Mobility increases the capacity of ad-hoc woireless networks, in Proc. Infocom 2001

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Content






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Targeted results of S.G. modeling

Modeling complex systems through a reduced number of parameters

Capturing Spatial/Geometric Properties A priori evaluation of the cost of a network/system

to be deployed, E.g. Mobile network: before knowing the exact position of

each antenna, an estimation of the future cost of the network can be obtained

Optimization of main parameters Estimation of the amount of equipment that has to be

deployed Not applicable to find the optimal location of system

equipment over a deterministic known infrastructure

103

Main tools

Point Processes on the space E.g. to represent the elements of the network

and their variability on time and space Stochastic Geometry

To represent how these elements are structured (service zones represented by tessellations, coverage zones, etc.)

Palm theory To calculate the required performance metrics

expressed as functionals of the previous stochastic objects.

104

Main used processes and objects

Processes Poisson Processes Clustering Processes Boolean Processes Coverage Processes Line Processes

Objects Voronoi Tesselations Delaunay Graph Markovian routing

Moller Theorem

105

Conclusion

Stochastic Geometry is a powerful and useful tool to Model spatial properties of big size systems With a reduced number of parameters To evaluate average performance measures and costs And to optimize main parameters

The number of applications in the telecommunication world has exploded during the past 3 years

The approach has been used by the telecom operators; for example, to estimate the cost of access networks

There is an important ongoing work, both on theoretic and applied problems To consider more sophisticated models

Hybrid models capturing both time and geometric properties To model the non-homogeneous distribution of equipment To obtain formulae for measures other than « averages » To analyze new type of systems like peer-to-peer architectures, WiFi

deployments, sensor networks, etc.

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Short Bibliography (1)

See http://www.di.ens.fr/~mistral/sg/

Books Stoyan, Kendall and Mecke. « Stochastic geometry and its applications. » Ed : Wiley.

(main results on point process, palm calculus, boolean model and other models). Okabe, Boots, Sugihara, and Chiu « Spatial tesselations ». Concepts and

applications of Voronoï diagrams. Ed : Wiley. Penrose. « Random Geometric graphs ». Ed : Oxford University Press.

Poisson Voronoï tesselations [MØLLER] MØLLER. “Random tesselation in d ». Adv. Appl. Prob. 24. 37-73. MØLLER. “Lectures on random Voronoï Tesselations.” Lectures notes in statistics

87. Springer Verlag, New York, Berlin, Heidelberg.

Percolation Gupta & Kumar, « Critical power for asymptotic connectivity in wireless networks »,

1998. [Meester1996] Continuum percolation. Ed : Cambridge University Press. Dousse, Baccelli, Thiran, « Impact of interfernces on connectivity in Ad Hoc

Networks », Infocom 2003.

107


SG applied to Network performance evaluation F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev. Stochastic geometry and

architecture of communication networks. J. Telecommunication Systems, 7:209-227, 1997.

Stochastic geometry modelling of hybrid optical networks. (with C.Farinetto) Performance Evaluation 57, 441-452, 2004.

Baccelli, Blaszczyszyn, « On a coverage process ranging from the boolean model to the Poisson voronoï tesselation, with applications to wireless communications », Adv. Appl. Prob., vol. 33(2), 2001.

Busson, Rougier, Kofman, « Impact of Tree Structure on Retransmission Efficiency for TRACK”. NGC 2001.

Busson, Kofman, Rougier, « Optimization of Content Delivery Networks Server Placement”, ITC 18, Berlin.

Baccelli, Kofman, Rougier. “Self organizing hierarchical multicast trees and their optimization”. IEEE INFOCOM'99, New York (USA), March 1999.

Baccelli,Tchoumatchenko, Zuyev. “Markov paths on the Poisson-Delaunay graph with applications to routing in mobile networks.” Adv. Appl. Probab., 32(1):1-18, 2000.

Baccelli, Gloaguen, Zuyev. “Superposition of planar voronoi tessellations”. Comm. Statist. Stoch. Models, 16(1):69-98, 2000.


108


SG applied to Network performance evaluation F. Baccelli, B. Blaszczyszyn, and F. Tournois. Spatial averages of coverage

characteristics in large CDMA networks. Technical Report 4196, INRIA, June 2001. F. Baccelli and S. Zuyev. Stochastic geometry models of mobile communication

networks. In Frontiers in queueing, pages 227-243. CRC Press, Boca Raton, FL, 1997.

Vivek Mahtre, Catherine Rosenberg, Daniel Kofman, Ravi Mazumdar, Ness Shroff, A Minimum Cost Surveillance Sensor Network with a Lifetime Constraint, to appear in IEEE Transactions of Mobile Computing (TMC).

Sunil Kulkarni, Aravind Iyer, Catherine Rosenberg, Daniel Kofman, Routing Dependent Node Density Requirements for Connectivity in Multi-hop Wireless Networks, accepted, Globecom 2004

O. Douse, F. Baccelli, P. Thiran, Impact of Interferences on Connectivity in Ad-Hoc Networks, in Proc. IEEE Infocom 2003

O. Douse, P. Thiran and M. Hasler, Connectivity in ad-hoc and hybrid networks”, in Proc. IEEE Infocom, 2002

M. Grossglauser and D. TSe, Mobility increases the capacity of ad-hoc woireless networks, in Proc. Infocom 2001



1 stochastic geometry as a tool for the modeling of telecommunication networks prof. daniel kofman,...

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meshed slide

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different network topologies

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