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Stochastic Geometry as a tool for the modeling of

telecommunication networks

- Daniel Kofman,
- Telecom Paris ENST
- Anthony Busson
- IEF University of Orsay-Paris 11

Euro-NGI Summer School Bamberg-September 2004

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

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

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

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

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

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

Architecture

PoP

Core

Conc.

PoP

Access Network

Clients are represented by a Point Process on the

plane

PoPs and their Voronoï cells

PoPs are represented by a Point Process on the

plane

PoPs and their Voronoi cells

Concentrators and their Voronoi cells

Access Hierarchy

Access Hierarchy

Meshed Core and Delaunay graph

Delaunay Graph

Delaunay Graph

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, 7209-227, 1997.

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

Point Processes and Voronoï Tessellations

Stationary Poisson point process in ?d

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

discrete Poisson law of parameter l.B, where

l 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.

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
- ? pp intensity
- ?0 2 ? (vertices intensity)

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
- l pp intensity
- l0 2 l
- l1 3 l (sides intensity)

Characteristic of the typical cell

- Number of sides (6 in average)
- Area (1/ l in average)
- Average perimeter length

Delaunay Graph

Cost function

- 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

Palm measure intuitive introduction

D(1)/D(0,8)

1

Number of packets

0,8 D

D

1

Arrival

U(1)

Departure

0

time

Prob (Queue empty)0,2 Prob (Queue empty at

arrival times)1 Prob0(Queue empty)1

PASTA Poisson Arrivals See Time Averages

Fellers Paradox for a Poisson Process

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

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

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

time t 2/ l

t

time

Fellers 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 t0. - By definition of Palm probability (at time 0), we

have - Prob0(T00) 1
- The inter-arrival time sequence is i.i.d., exp(l)
- 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.

Plane case

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

Back to Campbell Formula

- 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

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

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

Example 2 Content Distribution

User

Content Provider Server

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

What is the optimal location of the CDN servers ?

Users

Content Providers

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

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

Marked Point Process

(x,mx)

Servers locations and corresponding Voronoï cells

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 a (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

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

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

Example 3, Hierarchical CBT Multicast Trees

Point Process on the place representing routers

location

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

CBT distribution and corresponding Voronoi cells

Stochastic Geometry Model

Hierarchical CBT optimization

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

Exemple 4 Optical access network

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 ?, the cost of a network

covering A is ?AP - The problem is reduced to the estimation of the

cost of a typical ring architecture.

Rings modeling

Poisson point process of intensity ?.

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.

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.

The cost function (1)

- Cost of the ring and access nodes

The cost function (2)

- Cost of the splitters

The cost function (3)

- Cost of the base stations

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.

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

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)

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

Boolean Model-Definition

Boolean model - example

Boolean model example

- The compact sets here are circles, centered in 0,

of random radius uniformly distributed in 0,1

Capacity functional

- Probability that the intersection between the

Boolean model ? and a finite closed set K is not

empty - The capacity functional determines uniquely the

distribution of the Boolean model. - Where n2 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

Capacity functional our example

- K0
- In this case, the capacity functional is the

probability that 0 belong to ?

Contact distribution function

- If a point is not covering by ?, how far is the

boolean model? - Lets take B(R)B(0,R) a test set covering 0
- We define

Contact distribution function our example

R

0

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

CDMA Coverage - Boolean Model

Example

Known results

CDMA coverage Model

Remark Not a Boolean model since the compact

sets are not independent

CDMA coverage Model

What can we calculate

What can we calculate

- Coverage probability
- Distribution of the number of cells covering a

given location

Conclusions on the CDMA coverage model

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

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.

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)ltR. - Application connectivity in ad-hoc and sensor

networks

Random geometric graph

L

The N points

L

Random geometric graph

Radio range of the points

Random geometric graph

Connectivity

Random geometric graph obtained by simulation

100 nodes

3000 nodes

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)

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.

Percolation Finite domain results

- Determine r(n) such that Pc(n,r(n)) goes to one

as n ?8. - Theorem

Percolation Infinite domain the line

- Lets 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.

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 ?cgt0 such that - If ?lt?c, all clusters are bounded almost surely

(sub-critical case) - If ?gt?c, there exists a unique unbounded cluster

almost surely (supercritical case)

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

Percolation in a more realistic model results

- When ?0,it is a boolean model and the previous

theorem holds. - When ?gt0,
- The number of neighbors is bound. A node can have

at most 11/ ?ß neighbors. - Under certains assumptions on the attenuation

function l(.), there exists ?clt8 s.t. for all ?gt

?c there exists 0lt?c(?) s.t. for ? lt?c(?) 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.

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

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

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

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

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.

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

Self-organization of the network

- If a node has the highest metric in its

neighborhood, it elects himself has a cluster

head. - Example the degree as metric

Self-organization of the network

Simulation in a random geometric graph (in a

square of size 1x1)

1000 nodes radius 0.1

3000 nodes radius 0.1

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.

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

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 - 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. - 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

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

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

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.

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

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.

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.

Short Bibliography (2)

- 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, 7209-227, 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. - Mitton, Busson, Fleury, Self Organization in

Large Scale Ad Hoc Networks, MedHoc-Net 2004.

Short Bibliography (3)

- 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 - 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.