Stochastic geometry & access telecommunication networks Catherine GLOAGUEN – Orange Labs joint work with V. Schmidt and F. Voss – Institute of Stochastics,

stochastic geometry & access telecommunication networksCatherine GLOAGUEN – Orange Labs joint work with V. Schmidt and F. Voss – Institute of Stochastics, Ulm University, germany

7 Septembre 2010, Journées MAS, Bordeaux

2 C. Gloaguen Stochastic geometry and networks MAS2010

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summary

partie 1 the complexity of telecommunication networks

partie 2 the interest to "think stochastic geometry"

partie 3 random models for roads

partie 4 typical cell and estimation of shortest path length

partie 5 network modeling and validation on real data

partie 6 conclusion


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the complexity of telecommunication networks


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the access telecommunication network

What is a network? A collection of equipements and links that aims to enable the customer to reach any possible service she subscribes.

This is realized by means of a suitable architecture defines how to aggregate links and to organize nodes in order to reduce costs while providing a good quality of service.

The fixed network is very important with new technologies like optical fiber; the existing Copper network remains a major cost point

The access network is the part closest to the customer It is very sensitive to the demography & geography and exhibits two major levels of complexity


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complexity in cable pathes

the acces network merges in civil engineering

equipements are inside or in front of buidings

cables ly under the pavement or follow the road system

huge number and a variety of equipments

Approximate scale

100 m x 200 m


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The morphology of the road system depends on the scale of analysis since it is designed for various purposes

complexity of the underlying road system

major citieswidth 12km

inner city and suburbs

Lyon

townswidth 9 km

Amiens and transition to rural areas

nationwidewidth 950 km

motorways, national and some secondary roads


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some challenges for the network operator

for cost reduction or global planning purposes in adequacy with the topography and population density.

to analyze large scale networks in a short time full reconstruction of realistic optimized networks is impossible, partial reconstruction is limited in size.

to use external public data as inputto compensate for too voluminous databasis, that are not always complete nor reliable and often need dedicated software

to address rupture situations in technology and architecture by definition no databasis are available and extrapolation from actual situation may be dubious


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first positive point

even such complex systems as access networks can be described in a global way by simple and logical principles due to the underlying careful building.

they can be decomposed in 2 levels sub-networks connecting L(ow) nodes to H(igh) nodes

a serving zone is associated to each H node with respect to L nodes

the physical connexion L -> H is achieved according to a "shortest path" rule, which meaning depends on the technology


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the interest to build a global vision

it is questionable to work on detailed analysis with the aim to deduce for the purpose of detailed reconstructions when possible are sometimes used to estimate global behaviour

allows to simplify the reality only keeping strcturing features

allows to turn the observed variability and complexity as an advantage

– considering the network areas as a statistical set of realizations of a random network

second positive point


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the interest to "think stochastic geometry"


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stochastic geometry

spatial variabilityrandom spatial processes

node locationchoice of point process

avarage number as global parameter

geometrical characteristicsestimated via the right functionnals

connexion rulesgeometrical considerations

serving zoneapply logical connexion rule to process for node

global visionrelationship between the process parameters

contains all structuring geometrical features

In fine instantaneous results

the "translation" of the problem is easy


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simplest networkFully described by 2 intensities

the simplest 2 levels network as an example

L and H nodes location as independant Poisson point process in R2 , 2 intensities

logical connexion rule from L the nearest H euclidian distance defines the serving zone a Voronoï cell

the physical connexion follows the straight line

analytical global results for distributions of geometrical features

– distances L -> H as Exp (intensity H)

– action area characteristics : area, perimeter..

"Géométrie aléatoire et architecture de réseaux", F. Baccelli, M. Klein, M. Lebourges, S. Zuyev, Ann. Téléc. 51 n°3-4, 1996


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The typical serving zone is representative for all the serving zones that can be observed (ergodicity). Efficient simulation algorithms are derived.

a key object : the typical serving zone

Poisson Voronoï tessellation

Point process of H nodes

probability distribution

typical cellConditioned with a H node in the origin

Empirical

distribution of all

cells

Distribution of the

typical cell

perimeter


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simulation algorithm for PVT typical cell

"Spatial stochactic network models" F. Voss Doctoral dissertation, Dec. 2009, Ulm


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a real network involves the road system

as a support for nodes location

as a support for physical connections following a shortest path principle

Road system L node

connection

H node

Serving zone

"Comparison of network trees in deterministic and random settings using different connection rules. " Gloaguen C, Schmidt H, Thiedmann R, Lanquetin JP, Schmidt V SpaSWiN, Limassol, 2007


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stochastic modelling in realistic settings

with the following methodology

stochastic models for road systems

typical cell for nodes located on the road systems

dedicated simulation algorithm for typical cells

geometric characteristics are expressed as functionals of the processes and estimated from the content of the typical cell

We focus on the estimation of the distribution length of the shortest path connexions as an example


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Random models for roads


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throw points or line in the plane in a random way to generate a "tessellation" that can be used as a road system. More sophisticated models (iterated, aggetagted) are available

simple Poissonian models for road systems

LineThrow lines

DelaunayThrow points and relate them to their neighbours

VoronoïThrow points, draw Voronoi tesselation, erase the points


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models are discriminated by mean values

A stationary model is fully described by its intensity

"Stationary iterated random tessellations" Maier R, Schmidt V ,Adv Appl Prob (SGSA) 35:337-353, 2003


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partition of urban area

fitting algorithm to find the "best" model to represent real data

automatized segmentation

morphogeneis of urban street systems --> new stationary models

PVT 37 km-2

PVT 18 km-2

PVT 163 km-2

PVT 52 km-2

Bordeaux built up area

"Mathematics and morphogenesis of the city" T. Courtat,Workshop Transportation networks in nature and technology, 24 juin 2010 Paris

"Fitting of stochastic telecommunication network models, via distance measures and Monte-Carlo tests" Gloaguen C, Fleischer F, Schmidt H, Schmidt V, Telecommun Syst 31:353—377, 2006


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databasis for road systems in a single Excel sheet


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why road 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

– final results take into account all possible realizations of the model

– no simulation is required


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typical cell and estimation of shortest path length


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the serving zone revisited to incorporate streets

H nodes are randomly located on random tessellations (PVT, PDT, or PLT) and not in the plane

the serving zone has the same formal definition as a Voronoï cell

the serving zones define a Cox-Voronoï tessellation (PLCVT, PDCVT or PVCVT)

Road system (PLT)

H node

Serving zone

PLCVT Poisson-Line-Cox-Voronoï-tessellation


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simulation algorithm for PLCVT typical cell

Initial line l1 through 0, orientation angle ~ U[0,2p) Add one point at the origin d0

Nearest points to 0 P1 and P2. Radial simulation of line l2 and P3 and P4

Construction of first initial cell and radius =2 max (|Opi|)

Further simulated points on l2 and radial simulation of other lines

Further simulated points on l2 and radial simulation of other lines

Distance are Exp distributed


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shortest path on streets

H nodes are located on a random tessellation (ex PLT)

L nodes are located on the same system independantely from H-nodes

L node belongs to one serving zone and is connected to its nucleus

the connexion is the shortest path on the road system : edge set of the tessellation

road system

serving zone

H node

L node

Shortest path with PLT model for streets

Euclidian

along the edge set


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shortest path length C*

the length of the shortest path to its H node is associated to every L as a marked point process

"natural" computation simulate the network in a sequence of increasing sampling windows Wn and compute some function of the length of all paths and average

process for H nodes

process for H nodes

marked process with path length


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typical PLCVT celland its line segment content L*H

representation of the distribution of length C*

consider the distribution of the path length from a L node conditionned in O

use Neveu exchange formula for marked point processes in the plane applied to XC and XH

write the distribution in terms of a H node conditionned in O

the result

– depends on the inside line system– does not depend on L nodes process

Length from y to 0H nodes


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density estimation the distribution of length C*

simulatethe typical cell and the (Palm) line segment system it contains

explicitthe line segments

computethe estimator of the density as a step function

simulates exact distributions, no runtime or memory problems, unbiased and consistent estimator, convergence theorems for maximal error, but needs to develop the simulation algorithm for the correponding serving zone

0

S1

S2

Si


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Nodes locationon iterated tessellations or as thinned vertex set

available algorithms

indirect simulation algorithms

– simulate random cells and weigth it

– PVCVT and PDCVT other processes for nodes location

– Cox on iterated tessellations

– thinned vertex sets

"Simulation of typical Poisson-Voronoi-Cox-Voronoi cells, F. Fleischer, C. Gloaguen, H. Schmidt, V. Schmidt and F. Voss. " Journal of Statistical Computation and Simulation, 79, pp. 939-957 ,2009


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for simple tesselations, the statistical properties of functionals of the typical cell only depend on a scale factor

scaling invariance

PLCVT cell = 1000

PLCVT cell = 1


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library of fitted formulas for densities

empirical densities are computed from n simulations

large range of values

all available road models

PDCVT PLCVT PVCVT

= 50, = 1

n = 50 000


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selection of parametric families to fit empirical densities

ensuring theoretical convergence to known distributions & limit values

not too many parameters

best if one family for all models

truncated Weibull distribution

= 250

= 750

= 2000

PDCVT

fittedempirical


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Area to be equipped

parameters for road model

number of H nodes ->

Length distribution (road model, )

bloc de texte

2 level subnetwork case is solved

instantaneous results for 2 level networks

analytical parametric formulas for the repartition function, majoration of the length, averages and moments

explicit dependancy on the morphology of the road system


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network modeling and validation on real data


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real networks

A synthetic spatial view of real networks is obtained from the identification of 2-level subnetworks and the partitionning of the area in serving zones for every subnetwork. It maps the architecture on the territory (here on Paris).

SAIs ND

ND

SAI

WCS

Large scale

Middle scale

"Parametric Distance Distributions for Fixed Access Network Analysis and Planning". Gloaguen C, Voss F, Schmidt V, ITC 21, Paris, 2009


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the mean area of a typical serving zone = total area /(mean number of WCS); containing an average of 50 km road.

~1000 = (total length of road /area) x (total length of road / number H nodes)

the family of parametric densities at work

large scale subnetwork WCS-SAI


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the mean area of a typical serving zone = total area /(mean number of SAI); containing an average of 2 km road.


medium scale subnetwork SAI-SAIs or SAI-ND


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the mean area of a typical serving zone = total area /(mean number of SAI); containing an average of 300 m road.


small scale subnetwork SAIs-ND


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no computational time : the time investment comes form the mapping of the architecture on the area, i.e. describing the interweaving of 2 level networks. The models and parameters for the road systems (Excel sheet) are determined once and do not vary in time.

global analysis of a network

middle size French town Partitioned in homogeneous road models

customer-WCS connexionobtained by convolutions and ponderated average of 2 level subnetworks


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obvious application to optical networks. Given the architecture, the technology (coupling devices, optical losses) and the number of nodes, the probability distribution of the optical gain of the end to end connexion is easily deduced.

impact of new technologies on QoS

middle size French town optical gain of the end to end connexions for optical network

the optical fiber gaindepends on the number of nodes and technology


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conclusion


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key points

global analysis of fixed access networks explicitely accounting for regional specificities, without runtime problems

analytical formulas for network geometric characteristics

analytical models for road systems

– with potential use in mobility problems

– can't be ignored to model cabling systems open methodology : choice of functionals

mathematical results for convergence, limit theorems, fitting & simulation tools

merci

Stochastic geometry & access telecommunication networks Catherine GLOAGUEN – Orange Labs joint work with V. Schmidt and F. Voss – Institute of Stochastics,

Documents

access networks

gloaguen stochastic

stochastic geometry

large scale networks

levels subnetworks

network modeling

network operator

fixed network