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stochastic geometry & access telecommunication networks

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### stochastic geometry & access telecommunication networks

Catherine GLOAGUEN – Orange Labs joint work with V. Schmidt and F. Voss – Institute of Stochastics, Ulm University, germany

7 Septembre 2010, Journées MAS, Bordeaux

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 5network modeling and validation on real data

partie 6 conclusion

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

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

complexity of the underlying road system

major cities

width 12km

inner city and suburbs

Lyon

towns

width 9 km

Amiens and transition to rural areas

nationwide

width 950 km

motorways, national and some secondary roads

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

some challenges for the network operator

- for cost reduction or global planning purposesin adequacy with the topography and population density.
- to analyze large scale networks in a short timefull 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 architectureby definition no databasis are available and extrapolation from actual situation may be dubious

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

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

the "translation" of the problem is easy

global vision

relationship between the process parameters

contains all structuring geometrical features

In fine instantaneous results

spatial variability

random spatial processes

node location

choice of point process

avarage number as global parameter

stochastic geometry

geometrical characteristics

estimated via the right functionnals

connexion rules

geometrical considerations

serving zone

apply logical connexion rule to process for node

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..

simplest network

Fully described by 2 intensities

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

a key object : the typical serving zone

Poisson Voronoï tessellation

Point process of H nodes

probability distribution

typical cell

Conditioned with a H node in the origin

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

Empirical distribution of all cells

Distribution of the typical cell

perimeter

simulation algorithm for PVT typical cell

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

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

H node

Serving zone

L node

connection

"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

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

simple Poissonian models for road systems

Line

Throw lines

Delaunay

Throw points and relate them to their neighbours

Voronoï

Throw points, draw Voronoi tesselation, erase the points

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

models are discriminated by mean values

A stationary model is fully described by its intensity g

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

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

Bordeaux built up area

PVT 163 km-2

PVT 52 km-2

PVT 37 km-2

PVT 18 km-2

"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

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

the serving zone revisited to incorporate streets

PLCVT Poisson-Line-Cox-Voronoï-tessellation

- 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

simulation algorithm for PLCVT typical cell

Distance are Exp distributed

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

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

Further simulated points on l2 and radial simulation of other lines

Further simulated points on l2 and radial simulation of other lines

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

shortest path on streets

Shortest path with PLT model for 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

Euclidian

along the edge set

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

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 0

H nodes

typical PLCVT cell

and its line segment content L*H

S1

0

Si

density estimation the distribution of length C*simulate

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

explicit

the line segments

compute

the 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

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

Nodes location

on iterated tessellations or as thinned vertex set

"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

scaling invariance

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

PLCVT cell

k = 1

PLCVT cell

k = 1000

PLCVT

PVCVT

library of fitted formulas for densitiesk = 50, g = 1

n = 50 000

- empirical densities are computed from n simulations
- large range of k values
- all available road models

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

PDCVT

k = 250

k = 750

k = 2000

empirical

fitted

2 level subnetwork case is solved

Area to be equipped

parameters for road model

- 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

number of H nodes -> k

Length distribution (road model, k)

bloc de texte

WCS

Middle scale

SAI

ND

SAIs

ND

real networksA 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).

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

the family of parametric densities at work

large scale subnetwork WCS-SAI

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

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

medium scale subnetwork SAI-SAIs or SAI-ND

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

k ~35 = (total length of road /area) x (total length of road / number H nodes)

small scale subnetwork SAIs-ND

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

k ~5 = (total length of road /area) x (total length of road / number H nodes)

global analysis of a network

middle size French town

Partitioned in homogeneous road models

customer-WCS connexion

obtained by convolutions and ponderated average of 2 level subnetworks

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.

impact of new technologies on QoS

middle size French town

optical gain of the end to end connexions for optical network

the optical fiber gain

depends on the number of nodes and technology

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.

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

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