Title: Stochastic Geometry as a tool for the modeling of telecommunication networks
1Stochastic 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
2S.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
3Examples 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
4Why 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
5Content
- 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
6Content
- 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
7A 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
8Architecture
PoP
Core
Conc.
PoP
Access Network
9Clients are represented by a Point Process on the
plane
10PoPs and their Voronoï cells
11Concentrators and their Voronoi cells
12Access Hierarchy
13Access Hierarchy
14Delaunay Graph
15Meshed Core and Delaunay graph
16Questions 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.
17Content
- 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
18Point Processes and Voronoï Tessellations
19Stationary 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 B1Bn 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.
20Poisson 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)
21Poisson 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)
22Characteristic of the typical cell
- Number of sides (6 in average)
- Area (1/ l in average)
- Average perimeter length
23Cost 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
24Palm 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
25Fellers 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
26Fellers 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.
27Plane case
E(C0(?)) ?/? with ?1.280 E0(C0(?)) 1/?
28Back 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
29Summary
- 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
30Content
- 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
31Example 2 Content Distribution
User
Content Provider Server
32Content 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
33What is the optimal location of the CDN servers ?
Users
Content Providers
34The 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
35A 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
36Marked Point Process
(x,mx)
37Servers locations and corresponding Voronoï cells
38Cost 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
39A 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
40Main 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
41Example 3, Hierarchical CBT Multicast Trees
42Point Process on the place representing routers
location
43Stochastic 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
44CBT distribution and corresponding Voronoi cells
45Stochastic Geometry Model
46Hierarchical CBT optimization
47Reference
- 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
48Exemple 4 Optical access network
49Evaluation 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.
50Rings modeling
Poisson point process of intensity ?.
51PONs 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.
52PONs 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.
53The cost function (1)
- Cost of the ring and access nodes
54The cost function (2)
55The cost function (3)
- Cost of the base stations
56Conclusions 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.
57Content
- 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
58Dual 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)
59Coverage 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
60Boolean Model-Definition
61Boolean model - example
62Boolean model example
- The compact sets here are circles, centered in 0,
of random radius uniformly distributed in 0,1
63Capacity 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
64Capacity functional our example
- K0
- In this case, the capacity functional is the
probability that 0 belong to ?
65Contact 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
66Contact distribution function our example
R
0
67Content
- 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
68CDMA Coverage - Boolean Model
69Example
70Known results
71CDMA coverage Model
Remark Not a Boolean model since the compact
sets are not independent
72CDMA coverage Model
73What can we calculate
74What can we calculate
- Coverage probability
- Distribution of the number of cells covering a
given location
75Conclusions on the CDMA coverage model
76Conclusions 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
77References
- 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.
78Modeling 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
79Random geometric graph
L
The N points
L
80Random geometric graph
Radio range of the points
81Random geometric graph
Connectivity
82Random geometric graph obtained by simulation
100 nodes
3000 nodes
83Percolation 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)
84Percolation 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.
85Percolation Finite domain results
- Determine r(n) such that Pc(n,r(n)) goes to one
as n ?8. - Theorem
86Percolation 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.
87Percolation 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)
88Percolation 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
89Percolation 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.
90Other 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
91Modeling 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
92Modeling 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
93Other 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
94Routing 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.
95Self-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
96Self-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
97Self-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
98Self-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.
99Other 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
100References 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
101Content
- 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
102Targeted 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
103Main 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.
104Main used processes and objects
- Processes
- Poisson Processes
- Clustering Processes
- Boolean Processes
- Coverage Processes
- Line Processes
- Objects
- Voronoi Tesselations
- Delaunay Graph
- Markovian routing
- Moller Theorem
105Conclusion
- 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.
106Short 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.
107Short 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.
108Short 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.