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Modélisation macroscopique géométrique des réseaux d'accès en télécommunication
Catherine Gloaguen Orange Labscatherine.gloaguen@orange-ftgroup.com
Journée inaugurale SMAI-MAIRCI, Issy, 19 Mars 2010
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p2
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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p3 Orange Labs
1Introduction
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p4 Orange Labs
The access network merges in civil engineering
Path of Distribution cables
Side street
Main
road
Approximate scale 200m x 200m
Path of Transport cables
Closest to the customer
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p5 Orange Labs
Road systems are complex
The morphology of the road system depends on the scale and the type of town
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p6 Orange Labs
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 VossInstitute of Stochastics, Ulm University, Germany{Volker.Schmidt, Florian.Voss}@uni-unlm.de
NETWORK TOPOLOGY SYNTHESIS (NTS)
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p7
2NTS principle illustrated on fixed acces problematic
8Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p8
Dis_DistLH(PLT, 26, 0.043, x)
Analytical formula
Length distribution of connectionsA small part of the access network
NTS is a macroscopic model
9Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p9
Road system
ObjectsReality
Model
Mathematical model for roads
10Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p10
"High" node
Road system
ObjectsReality
Model
Number of "High" nodes sites
Mathematical model for roads
11Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p11
"High" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Mathematical model for roads
12Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p12
"High" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Principle Voronoï cell of center H
Mathematical model for roads
13Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p13
"High" node
"Low" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Principle Voronoï cell of center H
Number of "Low" nodes sites
Mathematical model for roads
14Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p14
"High" node
"Low" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Principle Voronoï cell of center H
Number of "Low" nodes sites
Connection Principle shortest path on roads from L to H
Mathematical model for roads
15Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p15
"High" node
"Low" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Principle Voronoï cell of center H
Number of "Low" nodes sites
Connection Principle shortest path on roads from L to H
Mathematical model for roads
Analytical formula
16Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p16
"High" node
"Low" node
Action area
Road system
ObjectsReality
Model
Number of "High" nodes sites
Principle Voronoï cell of center H
Number of "Low" nodes sites
Connection Principle shortest path on roads from L to H
Mathematical model for roads
Analytical formula
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p17
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)
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p18
3Models for road system
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p19 Orange Labs
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
Poisson Line
throw lines
PLT PDT PVT
Poisson Delaunay
throw points
relate each points to its neighbors
Poisson Voronoï throw points,
construct Voronoï cells
erase the points
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p20 Orange Labs
"Best" model choice
A constant defines a stationary simple tessellation The meaning of depends on the tessellation type Theoretical vector of intensities specific for each model
Mean values model → per unit area ↓
PLT [L]-1
PDT [L]-2
PVT [L]-2
Number of nodes (crossings) 2/ 2
Number of edges (street segments) 2 2/ 3 3
Number of cells (quarters) 2/ 2
Total edge length (length streets) 32 √ /(3 ) 2 √
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p21 Orange Labs
Fitting procedureRaw data Preprocessed
data
+ 133 dead ends
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p22 Orange Labs
Fitting procedureRaw data Preprocessed
data
+ 133 dead ends
634 crossings1502 street segments 418 quarters 112 km length streets
Unbiased estimators for intensities
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p23 Orange Labs
Fitting procedureRaw data Preprocessed
data
+ 133 dead ends
634 crossings1502 street segments 418 quarters 112 km length streets
Theoretical vector for potential models
Minimization of distance
Unbiased estimators for intensities
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p24 Orange Labs
Fitting procedureRaw data Preprocessed
data
+ 133 dead ends
634 crossings1502 street segments 418 quarters 112 km length streetsBest simple tessellation
PVT = 45.3 km-2
Theoretical vector for potential models
Minimization of distance
Unbiased estimators for intensities
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p25 Orange Labs
Fitting procedureRaw data Preprocessed
data
+ 133 dead ends
634 crossings1502 street segments 418 quarters 112 km length streets
Theoretical vector for potential models
Minimization of distance
712 crossings1068 street segments 356 quarters 106 km length streets
+ 133 dead ends
Unbiased estimators for intensities
Best simple tessellationPVT
= 45.3 km-2
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p26 Orange Labs
More realistic iterated tessellations
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p27 Orange Labs
Data basis for urban road system in one Excel sheet
Parametric representati
on of the road system
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p28
Modélisation de la voirie urbaine (ex Lyon)
PhD thesis (T. Courtat) on town segmentation and morphogenesis
New road models and tools
PLT
21,6 km-1
PVT
45,4 km-2
PVT
20,7 km-2
PVT
20,5 km-2
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p29 Orange Labs
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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p30 Orange Labs
4Computation of shortest path length between nodes
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p31 Orange Labs
Recall on the access network problem
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
Geographical support Network nodes location Topology of connection
LLCs
HLC
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p32 Orange Labs
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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p33 Orange Labs
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
Typical serving zone
1 realization of the typical cell by the simulation algorithm
PLCVTPoisson-Line-Cox-Voronoï-
Tessellation
Typical PLCVT cell
infinite number of realizations
all the cells
Distribution of cell perimeter
Point process of HLC
HLC in O
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p34 Orange Labs
Same probability distribution as the set of the paths in the plane
Typical shortest path length C*
Marked point processthe 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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p35 Orange Labs
Equivalent writings for the typical shortest path length LLC->HLC 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
Alternative computation of C*
Computation in the typical serving zone
HLC in O
length of the path
from a point y to OLinear intensity
of HLC
Typical segment system in
the typical serving zone
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p36 Orange Labs
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
Probability density of C*
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p37 Orange Labs
Scaling properties no absolute length -> 1/ is chosen as unit length
up to a scale factor, same model for fixed = /(roads/HLC) measures the density of roads in the typical cell
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p38 Orange Labs
Choice of a parametric family theoretical convergence results to known distributions & limit
values limited number of parameters, but applies to all cases and
values, Truncated Weibull distributions
Parametric density fitting
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p39 Orange Labs
From extensive simulations made once…. density estimation n=50000, PVT, PDT, PLT: find parameters and for 1< <2000 approximate functions ( ) and ( ) for each type
…. to instantaneous results & explicit morphology of the road system
Library of parametric formulae C*
Maj_DistLH (PLT, 26, 0.043, q)
Road : PLT intensity 26 km-1
Network : HLC intensity 0.043 km-1
Mean length 536 m
with prob. 85% , the length < 827 m
Dis_DistLH(PLT, 26, 0.043, x)
LLC
HLC
Orange LabsGloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p40
5Validation on real network data
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p41 Orange Labs
Transport (primary)
WCS
ND
ND
SAI
SAIs
Transport (secondary)
SAIs ND
ND
SAI
WCS
Geometrical analysis of the network in Paris
secondary service area interface
Distribution
network interface device
wire center station
service area interface
Distribution
Architecture nodes & logical links
Copper technology
Synthetic spatial view identification of 2-level subnetworks partition of the area in serving zones
for every subnetwork
Large scale
Middle scale
Low scale
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p42 Orange Labs
C* for larger scale subnetwork Subnetwork WCS-SAI
Mean area of a typical serving zone = total area /(mean number of WCS)
~1000 = (total length of road /area) x (total lenght of road / numbre HLC)
on average 50 km road in a serving zone
WCS
SAI
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p43 Orange Labs
C* for middle scale subnetwork Subnetwork SAI-SAIs or SAI-ND
Mean area of a typical serving zone = total area /(mean number of SAI)
~ 35, on average 2 km roads in a serving zone ND
SAI
SAIs
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p44 Orange Labs
C* for lower scale subnetwork Subnetwork SAIs-ND
Mean area of a typical serving zone = total area /(mean number of SAIs)
~ 5, on average 300 m road in a serving zone
ND
SAIs
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p45 Orange Labs
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
PVT 107 km-2
PVT 17 km-2
PVT 40 km-2
PVT 50 km-2
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p46 Orange Labs
6Potential applications and optimization problems
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p47 Orange Labs
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 -> 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
Stochastic geometry is a powerful toolbox
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p48 Orange Labs
Shortest path lengths for fixed acces networks
Both L and H nodes on roads Connexion : shortest path on roads
Density of L- H distances on roads
Look at the shortest path distance for all points of the typical segment system in the typical
serving zone
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p49 Orange Labs
Realistic cell description
H nodes on roads
HLC in O
Example of density of cell perimeter
Look at the geometrical charateristic of the typical cell : area, perimeter, number of sides
(neighbouring HLC)…
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p50 Orange Labs
Euclidian distances
Density of L-H Euclidian distance
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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p51 Orange Labs
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
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p52 Orange Labs
Cell analysis for mobile networks purpose
Distribution of SINR ratio for point x
H nodes on roads L nodes in the plane Connexion : "propagation"
distance
Current work J.M. Kelif
Analysis on a typical cell and its neigbouring. Propagation parameters and conditions are
included in the functional, the road model and
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p53 Orange Labs
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
Optimization and planning
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p54 Orange Labs
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
An "old" example : hierarchical network
32
21C421B2
10B10A03121
/
,
),,(/
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p55 Orange Labs
Several technologies are available for optical fibre networks Choice of nodes to be equipped under constraint of eligibility
threshold
Impact of new technologies on QoS
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p56 Orange Labs
Impact of new technologies on QoS
Upper bound at 95%
Given technology, coupling devices, losses…
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p57 Orange Labs
7Conclusion
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p58 Orange Labs
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
incoming capacity
Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p59 Orange Labs
F. Baccelli, M. Klein, M. Lebourges, S. Zuyev, "Géométrie aléatoire et architecture de réseaux", Ann. Téléc. 51 n°3-4, 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-stochastics/research/projekte/telecommunication-networks.html
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