Mod

1
Modélisation macroscopique géométrique des
réseaux d'accès en télécommunication

• Catherine Gloaguen
• Orange Labs
• catherine.gloaguen_at_orange-ftgroup.com

Journée inaugurale SMAI-MAIRCI, Issy, 19 Mars 2010
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Summary

1. Introduction
2. Network Topology Synthesis (NTS) principle
3. Models for road systems
4. Computation of shortest path length between nodes
5. Validation on real network data (Paris, cities,
   non denses zones)
6. Potential applications and optimization problems
7. Conclusion

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1

• Introduction

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The access network merges in civil engineering
Closest to the customer
Side street
Path of Distribution cables
Main road
Path of Transport cables
Approximate scale 200m x 200m
5
Road systems are complex
The morphology of the road system depends on the
scale and the type of town
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• France Telecom needs reliable tools with the
  ability to
• analyze complex large scale networks in a short
  time
• compensate for too voluminous or incomplete real
  data sets
• address rupture situations in technology or
  network architecture
• Our approach proposes
• an explicit separation of the topologies of the
  territory and the network
• analytical models for road systems and access
  networks
• Joint work with Volker Schmidt and Florian Voss
• Institute of Stochastics, Ulm University, Germany
• Volker.Schmidt, Florian.Voss_at_uni-unlm.de
• NETWORK TOPOLOGY SYNTHESIS
• (NTS)

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2

• NTS principle
• illustrated on fixed acces problematic

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NTS is a macroscopic model
A small part of the access network
Length distribution of connections
Dis_DistLH(PLT, 26, 0.043, x)
Analytical formula
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Reality
Objects
Model
Road system
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Action area
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Principle Voronoï cell of center H
Action area
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Principle Voronoï cell of center H
Action area
"Low" node
Number of "Low" nodes sites
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Principle Voronoï cell of center H
Action area
"Low" node
Number of "Low" nodes sites
Connection
Principle shortest path on roads from L to H
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Principle Voronoï cell of center H
Action area
"Low" node
Number of "Low" nodes sites
Connection
Principle shortest path on roads from L to H
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Reality
Objects
Model
Road system
"High" node
Number of "High" nodes sites
Principle Voronoï cell of center H
Action area
"Low" node
Number of "Low" nodes sites
Connection
Principle shortest path on roads from L to H
Analytical formula
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L'analyse repose sur une vision globale

• Quelques règles simples et logiques pour décrire
  un réseau d'accès fixe
• Les noeuds colocalisés (sites) sont situés le
  long de la voirie
• La zone d'action d'un noeud H est représentée
  comme l'ensemble des points les plus proches de H
• L'ensemble du territoire est couvert par au moins
  un des sous réseaux
• La connexion se fait au plus court chemin sur la
  voirie
• Simplifier la realité
• en conservant les caractères structurants
• utiliser la variabilité observée
• les ensembles de sous réseaux sont considérés
  comme échantillons statistiques d'un sous réseau
  virtuel aléatoire
• on décrit les lois de ces sous réseaux
• "La science remplace le visible compliqué par de
  l'invisible simple" (J. Perrin)

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3

• Models for road system

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Mathematical models for road system

• Just throw objects in the plane in a random way
  to generate a "tessellation" that can be used as
  a road system.
• Several models are available built on stationary
  Poisson processes
• Simple tessellations

PLT
PDT
PVT
Poisson Voronoï throw points, construct Voronoï
cells erase the points
Poisson Delaunay throw points relate each points
to its neighbors
Poisson Line throw lines
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"Best" model choice

• A constant g defines a stationary simple
  tessellation
• The meaning of g depends on the tessellation type
• Theoretical vector of intensities specific for
  each model

Mean values model ? per unit area ? Mean values model ? per unit area ? PLT g L-1 PDT g L-2 PVT g L-2
Number of nodes (crossings) l1 g 2/ p g 2 g
Number of edges (street segments) l2 2 g 2/ p 3g 3g
Number of cells (quarters) l3 g 2/ p 2 g g
Total edge length (length streets) l4 g 32 vg /(3 p) 2 vg
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Fitting procedure
Raw data
Preprocessed data
133 dead ends
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Fitting procedure
Raw data
Preprocessed data
133 dead ends
Unbiased estimators for intensities
634 crossings 1502 street segments 418
quarters 112 km length streets
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Fitting procedure
Raw data
Preprocessed data
133 dead ends
Unbiased estimators for intensities
Theoretical vector for potential models
634 crossings 1502 street segments 418
quarters 112 km length streets
Minimization of distance
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Fitting procedure
Raw data
Preprocessed data
133 dead ends
Unbiased estimators for intensities
Theoretical vector for potential models
634 crossings 1502 street segments 418
quarters 112 km length streets
Minimization of distance
Best simple tessellation PVT g 45.3 km-2
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Fitting procedure
Raw data
Preprocessed data
133 dead ends
712 crossings 1068 street segments 356
quarters 106 km length streets 133
dead ends
Unbiased estimators for intensities
Theoretical vector for potential models
634 crossings 1502 street segments 418
quarters 112 km length streets
Minimization of distance
Best simple tessellation PVT g 45.3 km-2
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More realistic iterated tessellations
27
Data basis for urban road system in one Excel
sheet
Parametric representation of the road system
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Modélisation de la voirie urbaine (ex Lyon)

• PhD thesis (T. Courtat) on town segmentation and
  morphogenesis
• New road models and tools

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Why should we bother to construct /use models ?

• A model captures the structurant features of the
  real data set
• a "good" choice takes into account the history
  that created the observed data
• ex PDT roads system between towns
• Statistical characteristics of random models only
  depend on a few parameters
• the real location of roads, crossings, parks is
  not reproduced but
• the relevant (for our purpose) geometrical
  features of the road system are reproduced in a
  global way.
• Models allow to proceed with a mathematical
  analysis (of shortest paths)
• final results take into account all possible
  realizations of the model
• no simulation is required

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4

• Computation of shortest path length between nodes

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Recall on the access network problem
Geographical support
Network nodes location
Topology of connection
HLC
LLCs

• Random equivalent network model
• Road system an homogeneous random model
• 2-level network nodes (LLC and HLC) randomly
  located on the roads
• Connection rules logical physical
• What about the distance LLC?HLC?
• The aim is to provide approximate reliable
  analytical formulas for mean values and
  distributions

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Serving zones

• The action area of HLC is a Voronoï cell
• Every LLC is connected to the nearest HLC,
  measured in straight line
• The serving zones define a Cox-Voronoï
  tessallation
• random HLC are located on random tesellations
  (PLT, PVT, PDT) and not in the plane

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Typical serving zone

• It is representative for all the serving zones
  that can be observed
• same probability distribution as the set of cells
  in the plane
• or conditional distribution of the cell with a
  HLC in the origin
• Simulation algorithms for the typical zone are
  specific to the model

Point process of HLC
1 realization of the typical cell by the
simulation algorithm
infinite number of realizations
all the cells
Distribution of cell perimeter
PLCVTPoisson-Line-Cox-Voronoï-Tessellation
Typical PLCVT cell
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Typical shortest path length C

• Same probability distribution as the set of the
  paths in the plane

• Marked point process
• the length of the shortest path to its HLC is
  associated to every LLC
• "Natural" computation
• Simulate the network in a sequence of increasing
  sampling windows Wn
• compute all the paths and their lengths
• the average of some function of the length is

Point process of HLC
Point process of LLC
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Alternative computation of C

• Equivalent writings for the typical shortest path
  length LLC-gtHLC
• distribution of the path length from a LLC
  conditionned in O
• Neveu exchange formula for marked point processes
  in the plane applied to XC (LLC marked by the
  length) and XH
• distribution of the path length to a HLC
  conditionned in O

• Computation in the typical serving zone

length of the path from a point y to O
Linear intensity of HLC
HLC in O
Typical segment system in the typical serving zone
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Probability density of C

• Simulate only the typical serving zone and its
  content
• Density estimation
• the segment system is divided into M line
  segments Si Ai ,Bi
• probability density
• estimated by a step function on n simulations

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Scaling properties

• no absolute length -gt 1/ g is chosen as unit
  length
• up to a scale factor, same model for fixed k g
  / l (roads / HLC)
• k measures the density of roads in the typical
  cell

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Parametric density fitting

• Choice of a parametric family
• theoretical convergence results to known
  distributions limit values
• limited number of parameters, but applies to all
  cases and k values,
• Truncated Weibull distributions

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Library of parametric formulae C

• From extensive simulations made once.
• density estimation n50000, PVT, PDT, PLT
• find parameters a and b for 1lt k lt2000
• approximate functions a ( k ) and b ( k ) for
  each type
• . to instantaneous results explicit morphology
  of the road system

Road PLT intensity 26 km-1 Network HLC
intensity 0.043 km-1
with prob. 85 , the length lt 827 m
Mean length 536 m
Dis_DistLH(PLT, 26, 0.043, x)
Maj_DistLH (PLT, 26, 0.043, q)
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5

• Validation on real network data

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Geometrical analysis of the network in Paris

• Synthetic spatial view
• identification of 2-level subnetworks
• partition of the area in serving zones for every
  subnetwork

• Architecture
• nodes logical links
• Copper technology

wire center station
WCS
Large scale
Transport (primary)
Distribution
WCS
service area interface
SAI
ND
Middle scale
Transport (secondary)
SAI
secondary service area interface
Low scale
SAIs
ND
SAIs
Distribution
ND
network interface device
ND
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C for larger scale subnetwork

• Subnetwork WCS-SAI
• Mean area of a typical serving zone total area
  /(mean number of WCS)
• k 1000 (total length of road /area) x (total
  lenght of road / numbre HLC)
• on average 50 km road in a serving zone

WCS
SAI
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C for middle scale subnetwork

• Subnetwork SAI-SAIs or SAI-ND
• Mean area of a typical serving zone total area
  /(mean number of SAI)
• k 35, on average 2 km roads in a serving zone

SAI
ND
SAIs
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C for lower scale subnetwork

• Subnetwork SAIs-ND
• Mean area of a typical serving zone total area
  /(mean number of SAIs)
• k 5, on average 300 m road in a serving zone

SAIs
ND
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Straigthforward application to other cities

• Same formulae
• Use the fitted road system(s) on the town under
  consideraion
• Right choice of parameters for the network nodes
• Ex. of end to end connexions ND-WCS in a middle
  size French town

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6

• Potential applications and optimization problems

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Stochastic geometry is a powerful toolbox

• Most network problems can be described by
• juxtaposition and/or superposition of 2 level
  subnetworks
• suitable choice of random processes for nodes
  location versus road system
• nodes may also ly in the plane
• logical connexion rules -gt Voronoï cells
• aggregated cells, connexion to the 2nd, 3rd
  closest H node
• "physical" connexion rules
• Euclidian distance or shortest path on roads
• The result is obtained by analysing ad hoc
  functionals of the typical cell

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Shortest path lengths for fixed acces networks

• Both L and H nodes on roads
• Connexion shortest path on roads

Look at the shortest path distance for all points
of the typical segment system in the typical
serving zone
Density of L- H distances on roads
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Realistic cell description

• H nodes on roads

Look at the geometrical charateristic of the
typical cell area, perimeter, number of sides
(neighbouring HLC)
Example of density of cell perimeter
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Euclidian distances

• H nodes on roads
• L nodes in the plane
• Connexion Euclidian distance

Value of the distribution function in x look at
the area of the intersection of the ball centered
in H with the typical serving zone
Density of L-H Euclidian distance
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Euclidian distances

• H and L nodes on roads
• Connexion Euclidian distance

Value of the distribution function in x look at
the area of the intersection of the ball centered
in H with the typical segment system in the
typical serving zone
Density of L-H Euclidian distance
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Cell analysis for mobile networks purpose
Analysis on a typical cell and its neigbouring.
Propagation parameters and conditions are
included in the functional, the road model and k

• H nodes on roads
• L nodes in the plane
• Connexion "propagation" distance
• Current work J.M. Kelif

Distribution of SINR ratio for point x
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Optimization and planning

• NTS performance is not sensitive to the number of
  elements
• Best to describe huge and complex networks
• NTS provides fast and global answers
• Determination of optimal choices by variyng
  parametres
• only in a macroscopic way
• Entry point for further fine optimization
  processes

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An "old" example hierarchical network

• Core network without road dependency
• What is known
• number of levels
• Mean number of lowest and highest nodes
• Cost functions (fixed and distance dependant )
• Question
• find the number of middle level nodes that
  minimizes the cost

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Impact of new technologies on QoS

• Several technologies are available for optical
  fibre networks
• Choice of nodes to be equipped under constraint
  of eligibility threshold

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Impact of new technologies on QoS
Upper bound at 95
Given technology, coupling devices, losses
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7

• Conclusion

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• This validates NTS approach
• NTS allows to address a variety of networks
  situations
• Modular
• Explicits underlying geometry and technology
• Road system MUST be taken into account in specifc
  problems
• Cabling trees cannot be obtained without street
  system
• Correction by a coefficient is not sufficient

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• F. Baccelli, M. Klein, M. Lebourges, S. Zuyev,
  "Géométrie aléatoire et architecture de réseaux",
  Ann. Téléc. 51 n3-4, 1996.
• C. Gloaguen, H. Schmidt, R. Thiedmann, J.-P.
  Lanquetin and V. Schmidt, " Comparison of Network
  Trees in Deterministic and Random Settings using
  Different Connection Rules" Proceedings of
  SpasWin07, 16 Avril 2007, Limassol, Cyprus
• C. Gloaguen, F. Fleischer, H. Schmidt and V.
  Schmidt "Fitting of stochastic telecommunication
  network models via distance measures and
  Monte-Carlo tests" Telecommunications Systems 31,
  pp.353-377 (2006).
• F. Fleischer, C. Gloaguen, H. Schmidt, V. Schmidt
  and F. Voss. "Simulation of typical
  Poisson-Voronoi-Cox-Voronoi cells " Journal of
  Statistical Computation and Simulation, 79, pp.
  939-957 (2009)
• F. Voss, C. Gloaguen and V. Schmidt, "Palm
  Calculus for stationary Cox processes on iterated
  random tessellations", SpaSWIN09, 26 Juin 2009,
  Séoul, South Korea.
• http//www.uni-ulm.de/en/mawi/institute-of-stocha
  stics/research/projekte/telecommunication-networks
  .html