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Modélisation macroscopique géométrique des réseaux d'accès en télécommunication Catherine Gloaguen Orange Labs Journée inaugurale SMAI-MAIRCI, Issy, 19 Mars 2010

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p2 Summary Introduction Network Topology Synthesis (NTS) principle Models for road systems Computation of shortest path length between nodes Validation on real network data (Paris, cities, non denses zones) Potential applications and optimization problems Conclusion

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p3 1 Introduction

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p4 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p5 Road systems are complex The morphology of the road system depends on the scale and the type of town

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p6 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, NETWORK TOPOLOGY SYNTHESIS (NTS)

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p7 2 NTS principle illustrated on fixed acces problematic

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8 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p8 Dis_DistLH(PLT, 26, 0.043, x) Analytical formula Length distribution of connections A small part of the access network NTS is a macroscopic model

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9 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p9 Road system Objects Reality Model Mathematical model for roads

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10 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p10 "High" node Road system Objects Reality Model Number of "High" nodes sites Mathematical model for roads

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11 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p11 "High" node Action area Road system Objects Reality Model Number of "High" nodes sites Mathematical model for roads

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12 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p12 "High" node Action area Road system Objects Reality Model Number of "High" nodes sites Principle Voronoï cell of center H Mathematical model for roads

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13 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p13 "High" node "Low" node Action area Road system Objects Reality Model Number of "High" nodes sites Principle Voronoï cell of center H Number of "Low" nodes sites Mathematical model for roads

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14 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p14 "High" node "Low" node Action area Road system Objects Reality 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

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15 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p15 "High" node "Low" node Action area Road system Objects Reality 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

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16 Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p16 "High" node "Low" node Action area Road system Objects Reality 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

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Orange Labs Gloaguen, 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)

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p18 3 Models for road system

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p19 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 PLTPDTPVT Poisson Delaunay throw points relate each points to its neighbors Poisson Voronoï throw points, construct Voronoï cells erase the points

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p20 "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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p21 Fitting procedure Raw data Preprocessed data dead ends

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p22 Fitting procedure Raw data Preprocessed data dead ends 634crossings 1502 street segments 418 quarters 112 km length streets Unbiased estimators for intensities

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p23 Fitting procedure Raw data Preprocessed data dead ends 634crossings 1502 street segments 418 quarters 112 km length streets Theoretical vector for potential models Minimization of distance Unbiased estimators for intensities

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p24 Fitting procedure Raw data Preprocessed data dead ends 634crossings 1502 street segments 418 quarters 112 km length streets Best simple tessellation PVT = 45.3 km -2 Theoretical vector for potential models Minimization of distance Unbiased estimators for intensities

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p25 Fitting procedure Raw data Preprocessed data dead ends 634crossings 1502 street segments 418 quarters 112 km length streets Theoretical vector for potential models Minimization of distance 712crossings 1068 street segments 356 quarters 106 km length streets dead ends Unbiased estimators for intensities Best simple tessellation PVT = 45.3 km -2

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p26 More realistic iterated tessellations

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p27 Data basis for urban road system in one Excel sheet Parametric representation of the road system

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Orange Labs Gloaguen, 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p29 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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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p30 4 Computation of shortest path length between nodes

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p31 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 LLCHLC? The aim is to provide approximate & reliable analytical formulas for mean values and distributions Geographical supportNetwork nodes locationTopology of connection LLCs HLC

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p32 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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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p33 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 PLCVT Poisson-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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p34 Same probability distribution as the set of the paths in the plane Typical shortest path length C* 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 W n 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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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p35 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 X C (LLC marked by the length) and X H 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 O Linear intensity of HLC Typical segment system in the typical serving zone

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p36 Simulate only the typical serving zone and its content Density estimation the segment system is divided into M line segments S i = [A i,B i ] probability density estimated by a step function on n simulations Probability density of C*

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p37 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p38 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p39 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 km -1 Mean length 536 m with prob. 85%, the length < 827 m Dis_DistLH(PLT, 26, 0.043, x) LLC HLC

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p40 5 Validation on real network data

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p41 Transport (primary) WCS ND SAI SAIs Transport (secondary) SAIs 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p42 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p43 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p44 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p45 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p46 6 Potential applications and optimization problems

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p47 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p48 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p49 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)…

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p50 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p51 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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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p52 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p53 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p54 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p55 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p56 Impact of new technologies on QoS Upper bound at 95% Given technology, coupling devices, losses…

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p57 7 Conclusion

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p58 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

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Orange Labs Gloaguen, Journée SMAI-MAIRCI 19 Mars 2010 – p59 F. Baccelli, M. Klein, M. Lebourges, S. Zuyev, "Géométrie aléatoire et architecture de réseaux", Ann. Téléc. 51 n°3-4, 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 (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 (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. networks.html

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