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Stochastic Geometry as a tool for the modeling of telecommunication networks

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Title: Stochastic Geometry as a tool for the modeling of telecommunication networks


1
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
2
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

3
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

4
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

5
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

6
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

7
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

8
Architecture
PoP

Core
Conc.
PoP
Access Network
9
Clients are represented by a Point Process on the
plane
10
PoPs and their Voronoï cells
11
Concentrators and their Voronoi cells
12
Access Hierarchy
13
Access Hierarchy
14
Delaunay Graph
15
Meshed Core and Delaunay graph
16
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.

17
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

18
Point Processes and Voronoï Tessellations
19
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.

20
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)

21
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)

22
Characteristic of the typical cell
  • Number of sides (6 in average)
  • Area (1/ l in average)
  • Average perimeter length

23
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

24
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
25
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
26
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.

27
Plane case
E(C0(?)) ?/? with ?1.280 E0(C0(?)) 1/?
28
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

29
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

30
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

31
Example 2 Content Distribution
User
Content Provider Server
32
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

33
What is the optimal location of the CDN servers ?
Users
Content Providers
34
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

35
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

36
Marked Point Process
(x,mx)
37
Servers locations and corresponding Voronoï cells
38
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
39
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

40
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

41
Example 3, Hierarchical CBT Multicast Trees
42
Point Process on the place representing routers
location
43
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

44
CBT distribution and corresponding Voronoi cells
45
Stochastic Geometry Model
46
Hierarchical CBT optimization
47
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

48
Exemple 4 Optical access network
49
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.

50
Rings modeling
Poisson point process of intensity ?.
51
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.

52
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.

53
The cost function (1)
  • Cost of the ring and access nodes

54
The cost function (2)
  • Cost of the splitters

55
The cost function (3)
  • Cost of the base stations

56
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.

57
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

58
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)

59
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

60
Boolean Model-Definition
61
Boolean model - example
62
Boolean model example
  • The compact sets here are circles, centered in 0,
    of random radius uniformly distributed in 0,1

63
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

64
Capacity functional our example
  • K0
  • In this case, the capacity functional is the
    probability that 0 belong to ?

65
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

66
Contact distribution function our example
R
0
67
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

68
CDMA Coverage - Boolean Model
69
Example
70
Known results
71
CDMA coverage Model
Remark Not a Boolean model since the compact
sets are not independent
72
CDMA coverage Model
73
What can we calculate
74
What can we calculate
  • Coverage probability
  • Distribution of the number of cells covering a
    given location

75
Conclusions on the CDMA coverage model
76
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

77
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.

78
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

79
Random geometric graph
L
The N points
L
80
Random geometric graph
Radio range of the points
81
Random geometric graph
Connectivity
82
Random geometric graph obtained by simulation
100 nodes
3000 nodes
83
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 ?8.
  • Theorem

86
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.

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 ?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)

88
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

89
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.

90
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

91
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

92
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

93
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

94
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.

95
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
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

97
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
98
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.

99
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

100
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

101
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

102
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.

106
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
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.

108
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.
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